service life modelling of r.c. highway structures...

SERVICE LIFE MODELLING OF R.C.HIGHWAY STRUCTURES EXPOSED TO CHLORIDES

by

Beatriz Martın-Perez

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Civil EngineeringUniversity of Toronto

c© Copyright by Beatriz Martın-Perez 1999

Abstract

The detrimental effect of chloride-induced corrosion on the service life of reinforced con-

crete highway structures highlights the need for the development of mathematical models

that describe the mechanisms of corrosion and damage build-up in reinforced concrete. The

objective of this study was to develop a generalized computer model for the prediction of

service life of reinforced concrete highway structures exposed to de-icing salts. The math-

ematical formulation idealized the corrosion sequence as a two-stage process: an initiation

stage, during which chloride ions penetrate to the reinforcing steel layer and depassivate it,

and a propagation stage, in which active corrosion takes place until cracking of the concrete

cover has occurred.

In modelling chloride transport to the reinforcing steel (initiation stage), a modified

version of Fick’s 2nd law was used, in which the processes of diffusion, chloride binding,

and convection due to water movement were taken into account. The migration of chloride

ions through concrete was coupled with moisture diffusion and heat transfer within concrete.

Corrosion was assumed to initiate when the chloride concentration at the steel layer reaches a

specified threshold value. To model the stage corresponding to active corrosion (propagation

stage), the rate of corrosion was linked to oxygen availability at the cathodic areas. This

was done by establishing the mass conservation equation corresponding to oxygen diffusion

into concrete.

The governing partial differential equations that idealize the problem were solved nu-

merically in space as a boundary-value problem and in time as an initial-value problem by

means of a two-dimensional finite element formulation, in which appropriate boundary con-

ditions were enforced to simulate the seasonal variations in exposure conditions. A time-step

integration procedure was applied to determine the variation in time of the various species

concentrations at the level of the reinforcement. To assess the mechanical damage resulting

from the expansion of the corrosion products, an equivalent uniform internal pressure was ap-

plied around the steel/concrete interface, and the resulting state of stress in the surrounding

concrete was evaluated by means of an elastic analysis.

The numerical performance of the model was examined in few example cases to assess the

relative importance of the different mechanisms considered on the service life of reinforced

ii

concrete structures exposed to chloride environments. The aim of this research was to reach

a better understanding of the physical mechanisms underlying the deterioration process

of reinforced concrete associated with chloride-induced corrosion and to propose a reliable

method for estimating these processes and their effects.

iii

To Soufiane

husband, best friend, mentor

iv

Acknowledgments

I would like to express my sincere gratitude to my co-supervisors Professor Pantazopoulou

and Professor Thomas for their continuous interest and guidance during the course of this

work. I am grateful to them for being part of this project, which has proven to be enormously

interesting and challenging at times. Their different expertise on the subject has been

extremely helpful during the entire project. It has been a pleasure to be their student.

I would also like to thank the examining committee members Professor Bonacci, Professor

Collins, and Professor Hooton from the University of Toronto and Professor Marchand from

the University of Laval for their helpful and valuable comments of the research in general

and of this manuscript in particular.

I would like to express my appreciation for the financial support provided by the Ministry

of Transportation of Ontario, the Natural Sciences and Engineering Research Council of

Canada (NSERC), the Ontario Graduate Scholarship program (OGS), and the University of

Toronto.

I would also like to thank my colleagues Dr. Amr El-Dieb and Dr. Surali Shashiprakash

for having been such a friendly source of help during the course of my stay at the University

of Toronto.

Finally, I would like to acknowledge the unconditional love and support so generously

given to me by Soufiane, Gregorio, Pilar, David, and Daniel.

v

Contents

Abstract ii

Acknowledgments vi

List of Tables ix

List of Figures xiii

Notation xvii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Principles of reinforcing steel corrosion . . . . . . . . . . . . . . . . . . . . . 3

1.3 Objectives of the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Literature Review 8

2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Modelling of chloride ingress . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Chloride diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Chloride diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Chloride diffusion and binding . . . . . . . . . . . . . . . . . . . . . . 22

2.2.4 Chloride diffusion and convection . . . . . . . . . . . . . . . . . . . . 26

2.2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Chloride threshold concentration . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Modelling of active corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vi

Contents vii

2.4.1 Electrical resistivity of concrete . . . . . . . . . . . . . . . . . . . . . 40

2.5 Modelling of concrete damage due to corrosion . . . . . . . . . . . . . . . . . 41

3 Service Life Model 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Chloride transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Chloride diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.2 Chloride diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.3 Chloride diffusion and binding . . . . . . . . . . . . . . . . . . . . . . 56

3.2.4 Chloride diffusion and convection . . . . . . . . . . . . . . . . . . . . 60

3.2.5 Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . 62

3.3 Moisture transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.1 Humidity diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . 67

3.3.2 Adsorption isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


3.4 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


3.5 Kinetics of corrosion reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.6 Oxygen transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.6.1 Oxygen diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . 82


3.7 Rate of rust production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Mechanical Effects of Corrosion 87

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Estimate of expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Concrete ring model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.1 Uncracked elastic stage . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.2 Partly-cracked elastic stage . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 End of service life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.5 Effect of corrosion on bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Contents viii

5 Finite Element Formulation 98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Elements description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.1 Linear triangular element . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3.2 Bilinear rectangular element . . . . . . . . . . . . . . . . . . . . . . . 105

5.3.3 Mapped infinite elements . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Evaluation of element integrals . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.1 Linear triangular element . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.2 Bilinear rectangular element . . . . . . . . . . . . . . . . . . . . . . . 113

5.4.3 Mapped infinite elements . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.5 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5.1 Numerical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.5.2 Solution algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6 Analytical Results 126

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.3 Chloride penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3.2 Diffusion and convection . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3.3 Effect of environmental conditions . . . . . . . . . . . . . . . . . . . . 144

6.4 Active corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Onset of cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7 Closure 156

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . 159

Bibliography 161

List of Tables

2.1 Recommended upper limits for Cl− coming from mixing ingredients. . . . . . 33

3.1 Typical values for the effective chloride diffusion coefficient Dc. . . . . . . . . 53

5.1 Correspondence between Eq. 5.1 and the governing differential equations. . . 99

5.2 Correspondence between L and M and the imposed boundary conditions. . . 103

5.3 Mapping and shape functions for infinite elements. . . . . . . . . . . . . . . . 109

6.1 Material parameters used in the numerical simulations. . . . . . . . . . . . . 131

ix

List of Figures

1.1 Corrosion cell in reinforced concrete. . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Relative volume for various corrosion products. . . . . . . . . . . . . . . . . 5

2.1 Corrosion process of reinforcing steel according to Tuutti (1982). . . . . . . . 9

2.2 Design nomogram for service life predictions. . . . . . . . . . . . . . . . . . . 15

2.3 Relation between the effective diffusion coefficient Dc and time. . . . . . . . 18

2.4 Correlation between the ratio of achieved to potential chloride diffusivity and

the time of exposure for some marine reinforced concrete structures. . . . . . 19

2.5 Decrease of chloride diffusivity due to increase of equivalent maturation time. 20

2.6 Total chloride profiles with depth when no binding and binding are taken into

account. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Chloride binding isotherm for concrete. . . . . . . . . . . . . . . . . . . . . . 24

2.8 Effect of temperature on chloride binding. . . . . . . . . . . . . . . . . . . . 26

2.9 Moisture and chloride variation in the concrete cover of structures subjected

to cycles of wetting and drying. . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Comparison between chloride profiles obtained from a convection-diffusion

model and a pure diffusion model. . . . . . . . . . . . . . . . . . . . . . . . . 29

2.11 Influence of the degree of mortar pore saturation on the corrosion rate of the

reinforcement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.12 Model for corrosion of reinforcing steel. . . . . . . . . . . . . . . . . . . . . . 37

2.13 Influence of concrete quality, concrete cover, and degree of pore saturation on

the corrosion current density. . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.14 The effect of water saturation on the resistivity of concrete. . . . . . . . . . . 41

x

List of figures xi

2.15 Evolution of the corrosion rate and the electrical resistance with time for

mortar specimens subjected to moisture cycles. . . . . . . . . . . . . . . . . 42

2.16 Dependence of concrete electrical resistivity on concrete moisture content. . . 43

2.17 Consequences of rebar corrosion. . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.18 Quantification of the propagation period for different corrosion rates. . . . . 45

2.19 Progress of chloride-induced deterioration. . . . . . . . . . . . . . . . . . . . 47

3.1 Conceptual service life model. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Dependence of Dc on temperature, age, and pore relative humidity. . . . . . 55

3.3 Relationship between coefficient m and w/c for various mixes. . . . . . . . . 56

3.4 Chloride binding isotherms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 The effect of chloride binding on the apparent diffusion coefficient. . . . . . . 60

3.6 Advancement of water and chloride fronts into concrete. . . . . . . . . . . . 63

3.7 Environmental values for the concentration of applied de-icing salts, the at-

mospheric relative humidity, and the daily average temperature. . . . . . . . 65

3.8 Dependence of Dh on pore relative humidity. . . . . . . . . . . . . . . . . . . 69

3.9 Water vapour adsorption isotherms. . . . . . . . . . . . . . . . . . . . . . . . 72

3.10 Moisture capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.11 Evans diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.12 Dependence of Do on the concrete moisture content. . . . . . . . . . . . . . . 83

4.1 Increase in volume due to accumulation of corrosion products. . . . . . . . . 88

4.2 Concrete ring model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Concrete ring with internal cracks. . . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Load-carrying capacity of concrete ring versus concrete cover thickness. . . . 94

4.5 Schematic representation of radial bond forces. . . . . . . . . . . . . . . . . . 95

4.6 Bond stresses along the reinforcement. . . . . . . . . . . . . . . . . . . . . . 96

5.1 Linear triangular and bilinear rectangular elements. . . . . . . . . . . . . . . 101

5.2 Element boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 Bilinear rectangular element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

List of figures xii

5.4 Bilinear rectangular element in the natural coordinate system ξ − η with its

corresponding shape functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5 Mapped infinite elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.6 Mesh for a corner of a reinforced concrete member. . . . . . . . . . . . . . . 119

5.7 Influence of time step on the calculated pore relative humidity profile. . . . . 121

5.8 Dependence of the number of iterations needed for convergence on the time

step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.9 Dependence of the number of iterations needed for convergence on the relax-

ation factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.10 Solution algorithm for proposed computational service life model. . . . . . . 125

6.1 Finite element mesh simulating a linear semi-infinite concrete strip. . . . . . 127

6.2 Calculated free chloride profiles for meshes with 1× 1 mm, 2.5× 2.5 mm, and

5× 5 mm rectangular elements. . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3 Free chloride profiles after 25 years of exposure given by the 5× 5 mm finite

element mesh solution and Eq. 2.6. . . . . . . . . . . . . . . . . . . . . . . . 129

6.4 Free chloride build-up at x = 50 mm over an exposure period of 100 years. . 129

6.5 Influence of Bc on the calculated free chloride profiles. . . . . . . . . . . . . . 132

6.6 Effect of temperature on chloride penetration due to diffusion. . . . . . . . . 134

6.7 Effect of time of exposure on chloride penetration due to diffusion. . . . . . . 136

6.8 Idealized binding isotherms for a concrete with 40% slag replacement level

and w/cm = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.9 Effect of chloride binding on free chloride profiles after (a) 1 year, (b) 5 years,

and (c) 25 years of exposure, respectively. . . . . . . . . . . . . . . . . . . . 138

6.10 Effect of chloride binding on total chloride profiles after (a) 1 year, (b) 5 years,

and (c) 25 years of exposure, respectively. . . . . . . . . . . . . . . . . . . . 139

6.11 Comparison between free chloride profiles calculated by the model and mea-

sured by Sergi et al. (1992). . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.12 Comparison between free chloride profiles calculated by the model and re-

ported by Sandberg (1998). . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.13 Calculated humidity profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

List of figures xiii

6.14 Effect of convection on chloride penetration. . . . . . . . . . . . . . . . . . . 145

6.15 Effect of the winter and spring seasons on chloride penetration. . . . . . . . . 147

6.16 Effect of the summer and fall seasons on chloride penetration. . . . . . . . . 148

6.17 Dissolved oxygen concentration profiles for a cathode-to-anode ratio equal to

10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.18 Time to first cracking of the concrete cover versus corrosion rate. . . . . . . . 151

6.19 Splitting failure pattern of the concrete cover. . . . . . . . . . . . . . . . . . 153

6.20 Load-carrying capacity of concrete cover versus cover-to-diameter ratio. . . . 154

6.21 Time to spalling of the concrete cover versus corrosion rate. . . . . . . . . . 155

Notation

The following symbols are used in this thesis:

Bc = surface chloride transfer coefficient (m/s);

Bh = surface moisture transfer coefficient (m/s);

BT = convection heat transfer coefficient (W/m2·C);

Cbc = concentration of bound chlorides (kg/m3 of concrete);

Cen = concentration of applied de-icing salt (kg/m3 of solution);

Cfc = concentration of free chlorides (kg/m3 of pore solution);

Co = concentration of oxygen dissolved in the pore solution (kg/m3 of pore solution);

Cso = oxygen concentration at the external concrete surface (kg/m3 of solution);

Cs = chloride surface concentration (kg/m3);

Ctc = concentration of total chlorides in concrete (kg/m3 of concrete);

c = concrete cover (m);

cq = concrete specific heat capacity (J/kg·C);

Dc = effective chloride diffusion coefficient (m2/s);

Dc,ref = effective chloride diffusion coefficient determined at some specified reference con-

ditions (m2/s);

D∗c = apparent chloride diffusion coefficient (m2/s);

Dh = humidity diffusion coefficient (m2/s);

Do = oxygen diffusion coefficient (m2/s);

Dw = moisture diffusion coefficient (m2/s);

d = reinforcing steel diameter (m);

∆d = increase in rebar diameter due to accumulation of rust (m);

xiv

Notation xv

E = electrical potential (V);

E0a = equilibrium potential of the anodic reaction (V);

Ec = elastic modulus of concrete (MPa);

E0c = equilibrium potential of the cathodic reaction (V);

e = inner radius of uncracked ring in concrete cylinder with internal cracks (m);

F = Faraday’s constant (= 96,486.7 C/mol of electrons);

f ′c = cylinder compressive strength of concrete (MPa);

f ′sp = tensile splitting strength of concrete (MPa);

GBFS = granulated blast-furnace slag;

h = pore relative humidity;

hen = atmospheric relative humidity;

hmax = maximum daily average atmospheric relative humidity in a year;

hmin = minimum daily average atmospheric relative humidity in a year;

i = current density vector (A/m2);

ia = current density of the anodic reaction (A/m2);

ic = current density of the cathodic reaction (A/m2);

icorr = corrosion rate (A/m2);

ioa = exchange current density for the anodic reaction (A/m2);

ioc = exchange current density for the cathodic reaction (A/m2);

Jc = flux of chloride ions due to diffusion (kg/m2·s);

Jsc = flux of chloride ions normal to the concrete surface (kg/m2·s);

J ′c = flux of chloride ions due to convection (kg/m2·s);

Jm = moisture flux through concrete due to diffusion (m/s);

Jo = flux of oxygen due to diffusion (kg/m2·s);

Jao = rate of O2 consumption at the anode (kg/m2·s);

J co = rate of O2 consumption at the cathode (kg/m2·s);

Jfh = rate of Fe(OH)2 production at the anode (kg/m2·s);

Jr = rate of Fe(OH)3 production at the anode (kg/m2·s);

K = hygrothermic coefficient (1/C);

Notation xvi

m = age reduction factor;

mr = mass of rust per unit length of rebar (kg/m);

ms = mass of steel consumed to produce rust per unit length of rebar (kg/m);

OPC = ordinary Portland cement;

PFA = pulverized-fuel ash;

pi = pressure resulting from rust accumulation at the steel/concrete interface (MPa);

pe = pressure acting at the inner surface of uncracked ring in concrete cylinder with

internal cracks (MPa);

q = heat flux (W/m2);

R = gas constant (8.314×10−3 kJ/K·mol);

r = radius of concrete ring where stresses are calculated (m);

SF = silica fume;

T = temperature in the concrete (C), K = C + 273.15;

Ten = temperature in the surrounding environment (C);

Tmax = maximum daily average temperature of the surrounding environment (C);

Tmin = minimum daily average temperature of the surrounding environment (C);

t = time (s);

tcr = time at which concrete cover cracks due to the expansion of corrosion products

(s);

te = equivalent hydration time (s);

ti = time to reinforcing steel depassivation (s);

tp = time corresponding to active corrosion (s);

∆t = time step (s);

U = activation energy (kJ/mol);

w/c = water-to-cement ratio;

w/cm = water-to-cementitious ratio;

x = x-coordinate (m);

y = y-coordinate (m);

z = ionic valence of the ion going into solution;

Notation xvii

α = chloride binding constant;

β = chloride binding constant;

βa = activation tafel slope for the anodic reaction (V);

βc = activation tafel slope for the cathodic reaction (V);

γ = angle of the outward normal with respect to the x-direction;

εv = volumetric strain;

θ = finite-difference weighting factor;

λ = thermal conductivity in concrete (W/m·C);

ν = Poisson’s ratio of concrete;

ρ = electrical resistivity of concrete (Ω·m);

%c = density of concrete (kg/m3);

%r = density of rust (kg/m3);

%s = density of steel (kg/m3);

σr = radial stress in concrete (MPa);

σt = tangential stress in concrete (MPa);

σmaxt = maximum tangential stress in concrete (MPa);

ω = relaxation factor;

ωe = evaporable water content in concrete (m3 of water/m3 of concrete);

ωsat = evaporable water content in saturated concrete (m3 of water/m3 of concrete).

Chapter 1

Introduction

1.1 Background

Chloride-induced corrosion of steel bars in reinforced concrete is one of the major causes

of deterioration of reinforced concrete structures in North America. Highway bridges and

parking garages exposed to de-icing salts are among the structures most affected by corrosion-

induced damage. This damage is usually manifested in the form of cracking and spalling

of the concrete cover due to the expansion of corrosion products accumulating around the

reinforcement. Over time it can also cause structural distress due to either loss of bond along

the steel/concrete interface resulting from splitting of the concrete cover, or reduction of the

cross-sectional area of the reinforcing bars.

In general concrete provides a high degree of protection against corrosion of embedded

reinforcing steel. The highly alkaline medium (pH> 13.0) within the pores of the hardened

cement matrix provided by the hydration of cement maintains the reinforcement in a passive

state where the corrosion rate is insignificantly low due to the formation of a layer of iron

oxide (Fe2O3) on the steel. Physical protection is also provided by the concrete cover, which

acts as a barrier against access of aggressive species, such as chloride ions, and reduces the

flow of electrical currents if the electrical resistivity is high. Depassivation of reinforcing

steel can occur because of two reasons: (1) neutralization of the Portland cement paste

by atmospheric CO2, reducing the pH to about 9 (carbonation-induced corrosion); and,

(2) localized breakdown of the passive layer (pitting corrosion) when there is a sufficient

1

Introduction 2

amount of chloride ions dissolved in the pore solution in contact with the reinforcing steel

(chloride-induced corrosion).

It is general practice nowadays in concrete design against corrosion of reinforcing steel to

use low-permeability concrete, attained by low water-to-cementitious ratios and the use of

supplementary cementing materials, and to provide a concrete cover for the reinforcement

that satisfies certain minimum requirements. But besides the implementation of these design

rules, there is no widely accepted approach to corrosion estimation available yet, in which the

corrosion process and its effect on the long-term durability of reinforced concrete structures

may be established. The reason for this is that corrosion of reinforcing steel involves many

factors that are not currently very well understood, such as uncertainties with the material

itself and the corrosivity of a specific environment of exposure. Furthermore, no general

criteria exist to define and quantify the level of corrosion.

Given that the costs associated with the maintenance and repair of reinforced concrete

structures are very high, it is important from the point of view of design and construction to

estimate the amount of deterioration that a structure may suffer during its service life, i.e.,

the time until the deterioration reaches an unacceptable level and repairs or replacement have

to be considered. Reliable service life models can help engineers to make decisions on the

type of materials used and concrete covers required to ensure the design life of the structure

is met. Besides durability design, service life estimations are also useful for rehabilitation

and maintenance planning of in-service reinforced concrete structures. Life predictions are

also used with life-cycle cost analyses which consider both long-term durability and costs

associated with various construction or repair alternatives (Clifton, 1993).

There are two major methodologies for making service life predictions (ACI Committee

365, 1997):

1. service life design of new structures, which involves proper mix design and structural

details so that the structures meet their required design life.

2. estimation of residual life of in-service structures, where decisions are being made

about the need for repair or replacement of the deteriorating structure, and appropriate

maintenance is planned. Information regarding the present condition of the structure

(e.g., the current state and rate of deterioration) is needed for this type of evaluation.

Introduction 3

In both of the two approaches mentioned above a criterion for the end of service life needs

to be defined.

The detrimental effect of corrosion on the service life of reinforced concrete structures

highlights the need for the development of mathematical models of the process of corrosion

and the mechanics of damage buildup. The advantage of developing a mathematical service

life model is in the breadth of possible applications of such a model to a range of different

materials and conditions. Even though a lot of research on reinforcement corrosion has been

done in the last two decades, reliable prediction models of the expected or remaining service

life of reinforced concrete structures do not exist yet. An understanding of the mechanisms

and kinetics of the corrosion process as well as the rates of deterioration is the basis for

making quantitative predictions of the service life of reinforced concrete structures exposed

to chloride environments. It is also necessary to define the end of service life condition,

i.e., the stage when a structure no longer performs its intended use (different criteria for

different situations). For example, alternative definitions which however would correspond

to vastly different damage levels are: (a) the aesthetics of a structure due to rust-staining

becomes unacceptable; (b) the structure does not meet current building standards; or, (c) the

structural safety due to material degradation is compromised (ACI Committee 365, 1997).

Service life models will permit a systematic approach to be utilized in optimizing the

design and rehabilitation of reinforced concrete structures in terms of cost and performance

criteria under the influence of a given environmental condition (ACI Committee 365, 1997).

In addition to the current strength-based design philosophy, the availability of a service life

prediction tool will also contribute to the rational development of durability-based design

standards.

1.2 Principles of reinforcing steel corrosion

Corrosion of reinforcing steel in concrete is an electrochemical process that requires the

presence of an anode and a cathode. For corrosion to occur, the anode and the cathode

must be connected by a metallic conductor (reinforcing bar) for electrons to move and

by an electrolyte conductor (concrete pore solution) for ions to move (see Fig. 1.1). The

corrosion process initiates when a difference in potential due to any non-uniformity within

Introduction 4

CONCRETE

STEEL

O H O2 2,

e−

Fe++OH−

Fe+++2OH- Fe(OH)2

ferrous hydroxide

ANODE

Fe Fe+++2e-

CATHODE

1/2O2+H2O+2e- 2OH-

Fig. 1.1: Corrosion cell in reinforced concrete.

the corroding system is established. This electrochemical potential may be generated when

two dissimilar metals are embedded in concrete, when there are significant variations along

the reinforcing steel surface, or when there are differences in the concentration of ions present

in the pore solution along the steel/concrete interface. Due to this differential of potential,

iron is oxidized at the anode according to:

Fe → Fe++ + 2e− (1.1)

The released electrons move through the rebar towards the cathode, while ferrous ions are

dissolved in the concrete pore solution. In normal concrete structures in which the pore

solution is alkaline and there is access to air, reduction of dissolved oxygen is the main

cathodic reaction (Rosenberg et al., 1989; Gulikers, 1996), i.e.,

O2 + 2H2O + 4e− → 4OH− (1.2)

It can be observed from Eq. 1.2 that there is a need for sufficient moisture and oxygen to be

present at the steel surface for the cathode half-cell reaction to take place.

The hydroxyl ions released at the cathode migrate under the influence of an electric

field towards the anode where they combine with the dissolved ferrous ions to yield ferrous

hydroxide Fe(OH)2, i.e.,

Fe++ + 2OH− → Fe (OH)2 (1.3)

Given sufficient oxygen at the anodic sites, ferrous hydroxide can be further oxidized into

other corrosion products. The transformation of ferrous ions into higher oxidation states is

accompanied by an increase in volume. Depending on the level of oxidation, iron can expand

Introduction 5

0 2 4 6 8Volume

Fe

FeO

Fe3O4

Fe2O

Fe(OH)2

Fe(OH)3

Fe(OH)3 3H2O

Fig. 1.2: Relative volume for various corrosion products (adapted from Rosenberg et al. (1989)).

as much as six times its original volume (Fig. 1.2). This increase in volume exerts tensile

stresses in the surrounding concrete which result in cracking and spalling of the concrete

cover if the concrete tensile strength is exceeded.

Pitting corrosion occurs when a differential of potential is established between large

cathodic areas (passive steel) and small anodic sites with a high concentration of chloride

ions (active steel). The high alkali content of the cement paste in contact with the cathodic

sites hinders the lateral growth of the pits thus favouring their deepening. In contrast, a

state of general corrosion occurs as a result of general depassivation of the entire reinforcing

bar. This is the case when steel depassivation takes place due to carbonation of the concrete

cover. However, general corrosion can also be due to high chloride concentrations along

the reinforcing bar where a large number of closely spaced pits are formed and eventually

coalesce.

When local corrosion occurs, the localized nature of the attack can result in an extreme

loss of reinforcing bar cross-sectional area before any other form of deterioration is detected,

affecting significantly the tensile capacity of the corroded member. This is particularly crit-

ical for prestressed concrete structures. However, damage associated with general corrosion

is usually manifested in the form of cracking and spalling of the concrete cover before a

significant reduction of the reinforcing bar cross-sectional area has taken place.

The exact role of chloride ions in the process of depassivating the steel is not thoroughly

understood yet. It is believed that ferrous chloride FeCl2 forms when the chlorides penetrate

the passivating film and react with iron (Fe++ + 2Cl− → FeCl2). Ferrous chloride then

Introduction 6

combines with oxygen and water according to (Sandberg, 1998):

4FeCl2 + O2 + 6H2O → 4FeOOH + 8HCl (1.4)

The released chloride ions continue to react with ferrous ions, thus reducing the protective

oxide film. The lowering of the concentration of ferrous ions and production of hydrogen

ions (acidification) at the anode lead to a change in the anodic half-cell potential forcing

further oxidation of the iron. Chloride ions therefore act as catalysts of iron dissolution.

1.3 Objectives of the research

The main objective of the present study was to develop a generalized computer model for

the prediction of service life of reinforced concrete highway structures exposed to de-icing

salts. The aim in developing the model was to integrate all of the aspects related to chloride-

induced corrosion of reinforcing steel, namely the time to steel depassivation, the process of

active corrosion, and the mechanics of damage build-up in the surrounding concrete.

It was an objective to formulate the model upon general physical laws so it would not

be limited to a specific type of concrete or environmental conditions. Several phenomena

taking place during the occurrence of the physical processes being described were included in

the governing equations on the basis of experimental observations reported in the literature.

Although the model was intended to be as general as possible, the main focus of this thesis

research was on reinforced concrete highway structures exposed to de-icing salts. For the

sake of simplicity, it was further assumed that concrete was uncracked.

Upon completion of the model and its implementation in a two-dimensional finite-element

computer program, the numerical performance of the mathematical model was examined by a

sensitivity study of the different input parameters as well as different mechanisms considered.

The aim of this thesis was to reach a better understanding of the physical mechanisms

underlying the deterioration process of reinforced concrete associated with chloride-induced

corrosion and to develop a numerical tool that in the future could serve as a complement to

engineering decisions related to durability-based design and assessment of reinforced concrete

structures.

Introduction 7

1.4 Outline of the thesis

Chapter 2 presents a literature survey of research related to service life modelling of reinforced

concrete structures subjected to chloride-induced corrosion. The review focuses on published

experimental studies and mathematical models dealing with the different stages of reinforcing

steel corrosion, namely chloride ingress into concrete, active corrosion of steel reinforcement,

and corrosion-induced damage in reinforced concrete.

Formulation of the proposed mathematical model of service life of reinforced concrete

highway structures exposed to de-icing salts is outlined in Chapter 3. This chapter describes

the governing differential equations that idealize the physical problem as well as the initial

and boundary conditions imposed to simulate seasonal variations in exposure conditions.

Chapter 4 presents a concrete ring model to assess the damage caused in concrete by

corroding reinforcing bars. Criteria used to define the end of service life are introduced in

this chapter as well.

The mathematical problem is discretized and expressed by a two-dimensional finite el-

ement formulation. Solution of the system of nonlinear partial differential equations is de-

scribed in Chapter 5.

Chapter 6 examines the numerical performance of the finite element formulation by a

sensitivity study of the different model parameters and different mechanisms considered.

Closed-form solutions and calculated results are compared to reported values wherever such

are available in the literature.

Finally, conclusions drawn from this research as well as recommendations for future work

are discussed in Chapter 7.

Chapter 2

Literature Review

2.1 General

Most of the published service life models associated with corrosion of reinforcing steel in

concrete have followed a simplified model that was first introduced by Tuutti (1982), wherein

the mechanism of corrosion is considered as a two-stage process as illustrated in Fig. 2.1:

1. Initiation period, during which the steel remains in a passive state. The onset of

corrosion corresponds to reinforcement depassivation due to either carbonation of the

concrete cover or accumulation of chloride ions at the reinforcing steel layer.

2. Propagation period, during which the structure deteriorates as a result of loss of re-

inforcing steel cross-sectional area and accumulation of corrosion products around the

bar surface. This phase lasts until an unacceptable degree of corrosion damage has

occurred.

For chloride-induced corrosion, the length of the initiation period depends on the rate of

penetration of chloride ions in concrete, the depth of the concrete cover, as well as the

chloride concentration required to start the corrosion process, also known as the threshold

value (Tuutti, 1993). Once the reinforcing steel has been depassivated, the corrosion rate

is the rate-determining parameter of the progress of corrosion-induced damage (Andrade

and Alonso, 1996a). Critical factors in determining the rate of corrosion include the rate

of oxygen diffusion to the reinforcing steel, the resistivity of the concrete pore solution, and

8

Literature review 9

Fig. 2.1: Corrosion process of reinforcing steel according to Tuutti (1982).

the temperature in the concrete (Tuutti, 1982; Lopez and Gonzalez, 1993). The lifetime

of a reinforced concrete structure affected by corrosion is assumed to terminate when an

unacceptable degree of damage has been reached, which Tuutti (1982) defined in his original

model as the amount of corrosion that causes visible cracks with a width of approximately

0.1 to 0.2 mm. Research on chloride transport in concrete and corrosion of reinforcing

steel has indicated which variables influence each stage of the model illustrated in Fig. 2.1;

however, because of the complexity of the physico-chemical phenomena taking place and

their interaction, there is not currently a widely accepted mathematical model that makes

Tuutti’s model a workable design procedure for service life estimations of reinforced concrete

structures.

Most of the mathematical models found in the literature dealing with chloride-induced

reinforcement corrosion have only taken into account the initiation stage, identifying the end

of service life with the onset of reinforcing steel depassivation (Browne, 1982; Funahashi,

1990; Maage et al., 1995; Tang and Nilsson, 1996a). Reasons for this approach include

(1) the fact that a relatively short period compared to the initiation time is required to

crack normal concrete as a result of accumulation of corrosion products around the rebars,

often 5 to 10 years (Collins and Grace, 1997), and (2) the difficulty in quantifying the

propagation stage, due to its dependence on too many random factors such as the onset of

corrosion, the rate and geometry of corrosion build-up, and the type of corrosion products

Literature review 10

formed. Nevertheless, some researchers argue that the structural strength and safety of

structures subjected to reinforcement corrosion are only reduced when significant loss of the

reinforcing steel cross-sectional area (reduction in tensile capacity) or loss of bond along

the steel/concrete interface has taken place, and that these structures can remain safe and

serviceable beyond the onset of corrosion if suitable repairs and corrosion protection measures

are carried out (Rasheeduzzafar et al., 1992). Thus, if certain amount of steel and concrete

deterioration is considered to be part of the design service life of new structures or of the

residual life of existing structures, then the corrosion rate that governs the propagation period

should be included in service or residual life estimations (Andrade and Alonso, 1996a). The

general approach in service life modelling associated with reinforcement corrosion has been to

focus on the initiation stage when considering the design of new structures, in order to ensure

the longest possible service period before steel depassivation, and on the propagation stage

when assessing in-service structures whose reinforcing steel may already be depassivated, in

order to evaluate different strategies for maintenance, repair or even replacement.

The following sections present a review of mathematical models found in the literature

dealing with the different stages illustrated in Fig. 2.1, namely chloride ingress into concrete

(initiation period), active corrosion of reinforcing steel (propagation period), and corrosion-

induced damage in reinforced concrete.

2.2 Modelling of chloride ingress

Chloride ions may be incorporated into the concrete from the mix ingredients at the time

of the manufacture (aggregates, mixing water, accelerating admixtures) or from external

sources (road de-icing salts, seawater, groundwater). Current standards of concrete technol-

ogy provide guidelines on the allowable amounts of chloride ions coming from the former

source, and compliance with these specifications should preclude chloride-induced corrosion

unless chlorides from external sources penetrate the concrete.

Chloride ions coming from external sources can enter the concrete by a combination of

several transport mechanisms:

• by ionic diffusion due to the existing chloride concentration gradient between the ex-


posed surface and the pore solution of the cement matrix, provided the concrete is

saturated.

• by absorption of water containing chlorides into concrete that is not saturated (capillary

sorption). This is a typical situation in bridge decks and parking slabs exposed to de-

icing salts where the concrete cover is likely to be partially-saturated, or in the tidal

and splash zones of marine structures.

• by water flow in concrete due to a pressure gradient (permeation). This type of chlo-

ride ingress is typical of offshore or subsurface concrete structures where hydrostatic

pressure increases with depth.

• by dispersion of diffusing chloride ions due to the effects of capillarity, moisture gradi-

ents within the concrete, or a hydrostatic pressure head.

• by the movement of water through concrete which is exposed to water in one surface

and is dry on the other (wick action). This case is typical of tunnel liners.

A large amount of the work done on service life modelling associated with chloride-

induced corrosion has taken diffusion as the main transport mechanism of chloride ions within

concrete, under the assumption that the concrete cover is fully saturated. Other aspects such

as capillary sorption, chloride binding, dispersion of chloride ions, or the effect of time of

exposure on chloride diffusivity have sometimes been included in these types of models by

modifying the boundary conditions and material properties of the problem. Diffusion models

are applicable to concrete structures that are permanently wet, but they underestimate

the amount of chlorides penetrating the concrete of structures subjected to wetting and

drying cycles, as it is the case for the splash and tidal zones of structures exposed to marine

environments or for highway structures exposed to de-icing salts. Chloride profiles in these

structures depend strongly on the moisture distribution in the concrete cover. Diffusion is

a very slow process compared to the convective movement of chloride ions due to moisture

flow in concrete. It is for this reason that the amount of chlorides being accumulated in a

partially-saturated concrete cover is largely determined by this moisture flow. It is therefore

important to define the type of exposure conditions, since these are going to determine to

a great extent the degree of saturation of the concrete cover and thus the mechanisms of


chloride ingress into concrete. A presentation of different modelling approaches previously

employed to consider these phenomena follows.

2.2.1 Chloride diffusion

Provided that concrete is in a saturated state, chloride ions enter the concrete by ionic

diffusion due to the existing concentration gradient between the exposed surface and the

pore solution of the cement matrix (diffusion driving force). This process is mathematically

described by Fick’s 1st law of diffusion, according to which the rate of transfer Jc of diffusing

chlorides through a plane perpendicular to the direction of diffusion is proportional to the

concentration gradient, which for one-dimensional flow is given by:

Jc = −Dc ·∂Cfc

∂x

[kg/(m2 · s)

](2.1)

where Jc is the flux of chlorides due to diffusion in the x-direction (kg/m2·s), Dc is the effective

chloride diffusion coefficient (m2/s), and Cfc is the concentration of chlorides dissolved in the

pore solution (kg/m3 of pore solution). The negative sign in Eq. 2.1 indicates that diffusion

occurs in the opposite direction to that of increasing concentration. Note that the value of

the diffusion coefficient Dc is different if Cfc is expressed in different units (Nilsson, 1993).

If the diffusion coefficient is denoted as Dc when the concentration gradient of free chlorides

is given as kg/m3 of concrete, then the relation between Dc and Dc is:

Dc = Dc · ωe

[m2/s

](2.2)

where ωe is the evaporable water content expressed per unit volume of concrete, given the

assumption that this is the water in which diffusion occurs (Sergi et al., 1992; Nilsson et al.,

1994). For a given concrete, the two diffusion coefficients, Dc and Dc, may differ by more

than one order of magnitude (Nilsson, 1993), which makes difficult to interpret reported

values when concentration units are not being considered properly. The difference between

both coefficients is that Dc refers to a chloride flux crossing an area of solution, whereas Dc

refers to a chloride flux crossing an area of concrete. The diffusion coefficient Dc is usually

measured from a steady-state diffusion cell test given the assumption that the concrete pore

solution is at equilibrium with the external chloride solution (Nilsson, 1993).

Chloride ions in concrete can be present in three different forms (Neville, 1995):


• chemically bound to the hydration products of the cement.

• physically sorbed on the surfaces of the gel pores.

• dissolved in the pore solution (free chlorides).

The total amount of chlorides in concrete is therefore the sum of bound and free chlorides,

i.e.,

Ctc = Cbc + ωeCfc

[kg/m3 of concrete

](2.3)

where Ctc is the total chloride concentration (kg/m3 of concrete) and Cbc is the concentration

of bound chlorides (kg/m3 of concrete). Conservation of chloride mass in concrete requires

that a change in the chloride diffusive flow be balanced by the change in the amount of total

chlorides present in the concrete, i.e.,

∂Ctc

∂t=∂Cbc

∂t+ ωe

∂Cfc

∂t= −∂Jc

∂x

[kg/(m3 · s)

](2.4)

where Jc is given by Eq. 2.1 for one-dimensional flow. For the simple case where chloride

binding is neglected (i.e., Cbc = 0), substituting Eq. 2.1 into Eq. 2.4 yields Fick’s 2nd law of

diffusion, which in one-dimension is given by:

∂Cfc

∂t=

∂

∂x

(Dc ·

∂Cfc

∂x

) [kg/(m3 · s)

](2.5)

From a theoretical point of view it is possible to use the total chloride concentration as the

driving force for diffusion, even though the physical mechanism of diffusion involves only the

chlorides dissolved in the pore solution. A similar expression to Eq. 2.5 results if chlorides

are expressed in terms of total chlorides, since for no binding Ctc = ωeCfc.

For the initial condition Cfc (x > 0, t = 0) = 0 and boundary condition Cfc (x = 0, t >

0) = Cs, a closed-form solution of Eq. 2.5 for a semi-infinite medium (i.e., from x = 0 to

x = ∞) can be obtained as follows (Crank, 1975):

Cfc (x, t) = Cs

[1− erf

(x

2√Dc t

)] [kg/m3] (2.6)

where Cfc (x, t) is the concentration of free chlorides at depth x after time t (kg/m3 of pore

solution), Cs is the chloride concentration at the surface x = 0 (kg/m3 of solution), and

erf is the error function. The solution implemented in Eq. 2.6 is only valid when both


the diffusion coefficient Dc and the surface concentration Cs are assumed to be constant in

space and time. Given the heterogeneous nature of concrete and the evolution of its material

properties with time due to cement hydration, the range of application of Eq. 2.6 is very

limited. According to Nilsson et al. (1996) the assumption of a constant diffusion coefficient

Dc can be valid for structures that have been in service for a long time. The assumption of

a constant surface chloride concentration Cs means that equilibrium between the external

solution and the concrete pore solution has been reached, and that the supply of chloride

ions coming from the external solution remains constant over time. This type of scenario is

applicable to reinforced concrete structures permanently submerged in a salty solution such

as seawater (Amey et al., 1998). The use of the expression given by Eq. 2.6 then allows for

a quick estimation of chloride profiles at a given time for these types of situations. However,

this is a simplification of the problem since it is known that Dc varies with temperature,

maturity of concrete and moisture content, and Cs depends on the concrete quality and

exposure conditions.

Using the closed-form solution given by Eq. 2.6, Browne (1982) formulated a method

for designing reinforced concrete structures against chloride-induced corrosion. His model

was based on the conceptual model proposed by Tuutti (1982), where the design life of re-

inforced concrete structures was assumed to be the time for the concrete chloride content at

the reinforcing steel to exceed a critical value (0.4% by weight of cement). Browne (1982)

ignored the stage corresponding to active corrosion in his durability-based design methodol-

ogy, although he recognised that for significant corrosion-induced damage to take place a low

concrete electrical resistivity (5–10,000 Ω·cm) and enough oxygen at cathodic areas would

be required. The author evaluated chloride diffusion coefficients Dc and surface chloride

contents Cs from Eq. 2.6 by determining the best fit diffusion curve through experimental

data obtained from chloride profiles at different exposure times. From the results, a design

nomogram was constructed in which a family of curves representing the variation in chloride

content with depth and time for given values of Dc and Cs is plotted (see Fig. 2.2). In

creating such a nomogram the intention was to allow the design engineer to select concretes

and cover thicknesses that would ensure reinforced concrete structures maintenance-free life

against reinforcement corrosion.

A lot of work on chloride penetration into concrete has been done since then, and much of


Fig. 2.2: Nomogram illustrating life prediction estimations for structures according to Eq. 2.6(reproduced from Browne (1982)).


it has followed the same approach adopted by Browne (1982). Oftentimes, apparent values

for Dc and Cs are calculated by fitting the solution given by Eq. 2.6 to chloride profiles

obtained from either core samples of real structures or specimens tested in the laboratory.

Some researchers (Funahashi, 1990; Sergi et al., 1992) have implemented different solutions

to Fick’s 2nd law of diffusion than the one given by Eq. 2.6 (e.g., finite-difference schemes);

however, least-squares methods have still been employed in these studies to find the apparent

chloride diffusivity Dc for which calculated theoretical profiles best fitted experimental data.

The problem with this approach is that the calculated apparent chloride diffusivity includes

the average impact of all the chloride ingress mechanisms (not only diffusion) that have

taken place in the concrete over the exposure period (Maage et al., 1995), and thus it does

not reflect an intrinsic material property.

2.2.2 Chloride diffusion coefficient

Experimental evidence has shown that the rate of chloride diffusion in concrete, usually

defined by the chloride diffusion coefficient Dc, depends on internal material parameters

such as temperature, porosity, cement type, cation type associated with Cl− ions, moisture

content, and curing conditions. The assumption implied in Eq. 2.6 that Dc is constant is thus

not correct. To take into account some of these observations, Saetta et al. (1993) expressed

the coefficient Dc as a function of the concrete temperature T , degree of cement hydration

te, and pore relative humidity h by correcting a reference value Dc,ref evaluated at standard

conditions (T = 23C, te = 28 days, h = 100%) according to:

Dc = Dc,ref · f1(T ) · f2(te) · f3(h)[m2/s

](2.7)

where the function f1(T ) considers the dependence of Dc on temperature, f2(te) expresses

the decrease of chloride diffusivity with increasing degree of hydration, and f3(h) relates the

diffusion coefficient Dc to the concrete pore relative humidity h. Tang and Nilsson (1996a,

1996b) followed a similar approach by defining Dc as a function of temperature, age, and

concrete depth (f4(x)).


Effect of temperature

The dependence of the chloride diffusivity on temperature has been estimated by several

researchers (Saetta et al., 1993; Berke and Hicks, 1994; Tang and Nilsson, 1996a, 1996b;

Amey et al., 1998) using an Arrhenius-type equation of the form:

f1(T ) = exp

[U

R·(

1

Tref

− 1

T

)](2.8)

where U is the activation energy of the chloride diffusion process (kJ/mol), R is the gas con-

stant (kJ/K·mol), Tref is the reference absolute temperature at which the chloride diffusivity

Dc,ref has been measured (K), and T is the actual absolute temperature in the concrete (K).

Page et al. (1981) reported activation energies for the chloride diffusion process in Portland

cement pastes of 41.8, 44.6, and 32.0 kJ/mol for water-to-cementitious ratios of 0.4, 0.5, and

0.6, respectively.

Unless the transfer of heat within concrete is also modelled, the assumption of a constant

temperature in concrete over time (T in Eq. 2.8) is not very realistic for structures exposed to

climates where temperatures drop below zero in winter and rise to much warmer values in the

summer. Amey et al. (1998) have suggested to take a weighted average temperature based

on the number of months and average seasonal temperatures to correct for the temperature-

dependence of Dc according to Eq. 2.8. Based on temperature records, others have adopted

a sinusoidal variability of the temperature over the period of one year to reflect the seasonal

variations in exposure conditions (Tang and Nilsson, 1996a, 1996b). Saetta et al. (1993)

approached this problem by coupling heat transfer within concrete to chloride ingress and

also considering sinusoidal variations of the environmental temperature.

Effect of time of exposure

Reported experimental research has shown that the chloride diffusion coefficient decreases

with time due to the increase in maturity of the exposed concrete (see Fig. 2.3). The

time dependency is considered to be caused by a combination of the physical and chemical

phenomena taking place in concrete over time and the variation of intensity of the exposure

environment (Nilsson et al., 1996). From reviewing experimental data mainly on marine

structures, various researchers have proposed a similar mathematical description of the decay


Fig. 2.3: Relation between the effective diffusion coefficient Dc and time for mixes A (OPC,w/cm = 0.4) and B (26% PFA, w/cm = 0.4) exposed to marine spray cycles (MC) and tidal

cycles (TC) (reproduced from Mangat and Molloy (1994)).

of Dc with time (Mangat and Molloy, 1994; Maage et al., 1995, 1996; Bamforth, 1998):

Dc = Dc,ref

(tref

t

)m [m2/s

](2.9)

where Dc,ref is the effective diffusion coefficient (m2/s) determined at time tref (s), t is the

time of exposure (s), and m is the age reduction factor.

In the relationship proposed by Mangat and Molloy (1994), Dc,ref was assumed to be

the effective diffusion coefficient at time equal to one second. The authors reported that

parameter m depends on the concrete mix proportions and the type of curing applied to the

concrete, and they observed that the most significant decreases in Dc occurred with mixes

containing fly ash and slag. Similar observations with respect to fly ash were reported by

Thomas and Jones (1996). Mangat and Molloy (1994) obtained values for m from various

mixes (OPC, 26% PFA, 60% GBFS, 15% SF) by a linear regression analysis of experimental

data coming from concretes exposed during a period of 5 years, and, in spite of their findings

with respect to the beneficial influence of supplementary cementing materials, they expressed

m as a function of water-to-cement ratio w/c only:

m = 2.5w/c− 0.6 (2.10)


Fig. 2.4: Correlation between the ratio of achieved to potential chloride diffusivity, δ = Da/Dp =(to/t)γ , and the time of exposure for some marine reinforced concrete structures (reproduced from

Maage et al. (1996)).

However, other authors have found that the water-to-cementitious ratio w/cm has a bigger

impact on parameter m than the water-to-cement ratio w/c alone. According to Maage

et al. (1996) and Bamforth (1998), the water-to-cementitious ratio w/cm mainly influences

the initial chloride diffusivity of the concrete, while the content and type of supplementary

cementing materials influence the value taken by m. The following design values for m were

proposed by Bamforth (1998) after reviewing a large amount of published data coming from

a range of concrete mix types:

Mix m

OPC 0.264

PFA 0.699

GBFS 0.621

where OPC stands for ordinary Portland cement, PFA stands for pulverized-fuel ash, and

GBFS stands for granulated blast-furnace slag.

A similar expression to the one given by Eq. 2.9 was used by Maage et al. (1995, 1996),

who made a distinction between a time reduction in the diffusivity due to changes in the


material properties (continuing hydration) and that due to environmental conditions. The

authors defined two types of chloride diffusivities: a potential chloride diffusivity Dp (t),

which is a time-dependent material parameter, and an achieved or in situ chloride diffusivity

Da (t), which is both a time-dependent material and environment parameter (see Fig. 2.4).

Service life estimations taking into account the time-dependency of Dc for a concrete cover

of 52 mm with a threshold and chloride surface concentrations of 0.4% and 2.35% by binder,

respectively, resulted in an increase from 23 years without considering the time dependence

of Dc to 79 years with a time-dependent Dc (Maage et al., 1995).

Saetta et al. (1993) also used an exponential decay to describe the decrease ofDc with age,

wherein the diffusivity converges asymptotically to a final value once hydration is completed

(Fig. 2.5). The expression they used is given by:

f2(te) = ζ + (1− ζ)

(28

te

)1/2

(2.11)

where te denotes the “equivalent maturation time”, which the authors calculated according

to the expression proposed by Bazant and Najjar (1972), and ζ, a value that ranges from 0

to 1, measures the rate of decay of the chloride diffusivity with time.

Based on their own experimental work,

Fig. 2.5: Decrease of chloride diffusivity due to in-crease of equivalent maturation time te (reproduced

from Saetta et al. (1993)).

Tang and Nilsson (1996a, 1996b) mod-

elled the decrease of Dc with time by as-

suming it reaches a constant value after

a specified period of time tc according

to:

f2(t) =

(

tct

)βtif t < tc,

1 if t ≥ tc.

(2.12)

where tc is the age when the diffusivity

becomes constant, taken as 180 days by

the authors, and βt is a constant, esti-

mated to range from 0.3 to 0.5 (Tang

and Nilsson, 1996a, 1996b). These values are within the range of values for m reported by

Bamforth (1998).


A closed-form solution of Eq. 2.5 that incorporates the time dependence of Dc can be

obtained by means of the following substitution (Poulsen, 1993):

∂t′ = Dc(t) · ∂t[m2

](2.13)

Equation 2.5 then becomes:

∂Cfc

∂t′=∂2Cfc

∂x2

[kg/m5] (2.14)

whose solution for a semi-infinite medium is similar to the one given by Eq. 2.6, i.e.,

Cfc(x, t) = Cs

[1− erf

(x

2√t′

)] [kg/m3] (2.15)

where variable t′ is given by:

t′ =

∫ t

0

Dc(t) dt[m2

](2.16)

Note that Eq. 2.15 still assumes the surface chloride concentration Cs to remain constant

over time.

Effect of relative humidity

Chloride diffusion is a process which can only occur if water is present in the capillary pores

of the concrete. Thus a decrease in the moisture content of the concrete cover hinders the

process of chloride diffusion towards the reinforcement. This phenomenon was simulated in

the chloride diffusion model proposed by Saetta et al. (1993) by introducing an empirical

relationship wherein chloride diffusivity decreases with a decrease in the concrete pore relative

humidity:

f3(h) =

[1 +

(1− h)4

(1− hc)4

]−1

(2.17)

where h is the concrete pore relative humidity and hc is the humidity at which Dc drops

halfway between its maximum and minimum values, taken to be 0.75 from the work done

by Bazant and Najjar (1971) on drying of concrete.

Effect of concrete depth

Because of the different composition of concrete at the external surface (more binder) from

concrete at the inner zone, it has been observed that the chloride diffusivity is usually lower


in the outer parts of the concrete cover (Tang and Nilsson, 1996b). By analysing test results,

Tang and Nilsson (1996a, 1996b) expressed the dependence of Dc on the concrete depth x

as:

f4(x) =

ϕ+ (1− ϕ)(

xxs

)βx

if x < xs,

1 if x ≥ xs.

(2.18)

where ϕ is the ratio of the diffusivity at the surface to that at the inner zone (Dsc/Dc), xs

is the thickness of the surface zone, and βx is a constant. In their numerical simulations ϕ

and xs ranged from 0.21 to 0.53 and 20 to 40 mm, respectively.

This “skin effect” was also modelled by Andrade et al. (1995) by assuming that the

concrete chloride diffusivity takes different values in the outer and inner parts of the concrete

cover. The authors obtained closed-form solutions to Fick’s 2nd law by assuming a two-

layered interface (skin and bulk), and their resulting expressions were also dependent on the

Dsc/Dc ratio and xs.

2.2.3 Chloride diffusion and binding

The main form of chloride binding in concrete is by reaction with the C3A in Portland

cement to form calcium chloro-aluminate hydrate (3CaO·Al2O3·CaCl2·10H2O), commonly

known as Friedel’s salt. Cements containing high levels of C3A can bind a larger amount

of chloride ions. Also, larger quantities of chlorides are bound when these come from the

mix ingredients. However, the amount of bound chlorides is not permanent. There is a

state of equilibrium between free and bound chlorides, so there is always free chlorides in

the liquid phase (Papadakis et al., 1996). The presence of sulphate ions may also result in

the liberation of chloride ions due to the preferential formation of calcium sulpho-aluminate

hydrates (Neville, 1995). Likewise, carbonation of the concrete cover has a similar effect in

releasing bound chloride ions into the pore solution (Neville, 1995; Andersen, 1997).

Only the chlorides dissolved in the pore solution (free chlorides) are responsible for initiat-

ing the process of corrosion (Tuutti, 1982), since only in this state they are able to penetrate

through the concrete cover. It is for this reason that the ability of the cement hydrates to

bind chloride ions has an important effect on the chloride ionic mobility. The diffusion of

chloride ions into concrete is accompanied by both chemical and physical binding, reducing


Fig. 2.6: Total chloride profiles with depth when no binding (diffusion without reaction) andbinding (diffusion with reaction) are taken into account (reproduced from ACI Committee 222

(1985)).

the concentration of free chlorides at any particular depth, which in turn decreases the ten-

dency for further inward diffusion (Fig. 2.6). The effect of chloride binding in concrete on

the initiation of corrosion is two-fold: (1) the rate of ionic transport in concrete is reduced,

since the amount of available mobile ions is also reduced by the binding mechanisms; and,

(2) the reduction of free chlorides in concrete results in lower amounts of chlorides being

accumulated at the reinforcing steel layer (Nilsson et al., 1996).

The binding properties of a specific cementitious system are usually defined in the form

of a binding isotherm (see Fig. 2.7), with the amount of bound chlorides Cbc expressed

as a function of the chlorides present in the pore solution Cfc at a given temperature.

Binding isotherms describe the relation between bound and free chlorides at equilibrium,

i.e., they assume that equilibrium conditions are obtained instantaneously in the concrete

pores when conditions change (Nilsson, 1993). Chloride binding isotherms are unique to each

cementitious system since they are influenced by the components making up that system,

such as C3A content, supplementary cementing materials, and pH of the pore solution.


Fig. 2.7: Chloride binding isotherm for a concrete with w/cm = 0.5, 300 kg/m3 of cementitiousmaterial content, and ωe = 7.6% (reproduced from Nilsson (1993)).

Nilsson et al. (1994) have stated that they may also be a function of the concrete age and

degree of hydration.

Several researchers have considered chloride binding mechanisms in chloride ingress mod-

els by modifying existing diffusion models based on Fick’s 2nd law (Sergi et al., 1992; Tang

and Nilsson, 1993; Nilsson et al., 1994; Papadakis et al., 1996). To take into account chloride

binding in the solution of Eq. 2.5, binding isotherms have to be determined experimentally

for each cementitious system under consideration. By substituting Eq. 2.1 into Eq. 2.4, the

following modified Fick’s 2nd law equation in one-dimension results:

∂Cfc

∂t=

∂

∂x

(D∗

c

∂Cfc

∂x

) [kg/(m3 · s)

](2.19)

with

D∗c =

Dc

1 + 1ωe

∂Cbc

∂Cfc

[m2/s

](2.20)

where D∗c is the apparent diffusion coefficient (m2/s) and ∂Cb/∂Cf is the “binding capacity”

of the concrete binder (m3 of pore solution/m3 of concrete). The binding capacity is the

ability of a concrete cementitious system to bind chlorides when the chloride concentration


changes (Nilsson et al., 1994). The binding capacity, which is given by the slope of the

corresponding binding isotherm, is not constant especially at low levels of free chlorides

(Nilsson et al., 1996). The effect of chloride binding on Eq. 2.20 is to reduce the diffusivity

of chloride ions in concrete Dc by an amount which is representative of the binding capacity

of the specific cementitious system.

The diffusion coefficient Dc is usually determined by fitting the solution of Fick’s 2nd

law to measured chloride profiles, generally expressed as total chloride concentrations. Since

chloride binding is not taken into account in the solution of Fick’s 2nd law in many cases, what

is in fact determined by following the above procedure is the apparent diffusion coefficient

D∗c , not the effective diffusion coefficient Dc nor the diffusion coefficient Dc obtained from a

steady-state test.

In a recent study carried out by Tang and Nilsson (1996a), the influence of the pH and

temperature of the pore solution were also integrated in the numerical modelling of chloride

binding. Their expression to take into account all these effects was given as:

Cbc = exp

[αOH

(1− [OH−1]

[OH−1]i

)]· exp

[Ub

R

(1

T− 1

Tref

)]·W (gel) · f(Cfc) (2.21)

where Cbc is the amount of bound chlorides (kg/m3 of concrete), αOH is a constant estimated

to be 0.56 from data reported in the literature, [OH−1] and [OH−1]i are the actual and

initial hydroxide concentrations, respectively, Ub is the activation energy for chloride binding

(J/mol), T is the actual absolute temperature (K), Tref is the reference absolute temperature

at which the binding isotherm was obtained (K), W (gel) is the amount of the hydrate gel

(kg/m3 of concrete), and f(Cfc) is the chloride binding isotherm function (per unit weight

of gel). Note that according to Eq. 2.21 the amount of the hydrate gel W (gel) controls the

chloride binding mechanism as opposed to the aluminate content of the cement. The effect

of temperature on Eq. 2.21 is to reduce the concentration of bound chlorides with increasing

temperature as illustrated in Fig. 2.8, where calculated free chloride profiles are plotted for

different seasons of the year (Tang and Nilsson, 1998). To implement Eq. 2.21 in their chloride

diffusion model, Tang and Nilsson (1996a) also evaluated the counter diffusion of hydroxyl

ions in a similar way to chloride penetration by applying Fick’s 2nd law. Sergi et al. (1992)

also coupled outward hydroxyl diffusion to chloride ingress in concrete with the intention of

evaluating the ratio of chloride-to-hydroxyl ions as a critical parameter for corrosion risk.


Fig. 2.8: Effect of temperature on chloride binding (reproduced from Tang and Nilsson (1998)).

However, the latter encountered the difficulty in quantifying hydroxyl desorption as a result

of chloride binding with reasonable accuracy. They concluded that hydroxyl ion profiles

could be obtained from charge balance equations after diffusion of all other species had been

considered.

2.2.4 Chloride diffusion and convection

Diffusion models are valid tools for estimating chloride profiles provided the concrete is under

saturated conditions. However, in environments where the concrete is subjected to wetting

and drying of the surface, such as the case of reinforced concrete highway structures, the

transport of chlorides due to water ingress into concrete also needs to be taken into account.

When a partially-saturated concrete surface is being wet, the water is quickly absorbed

through capillary suction bringing with itself dissolved chloride ions. During dry periods,

the water at the surface evaporates leaving the chloride ions behind in the concrete pore

solution. Due to these two phenomena, the concentration of chlorides will be fluctuating near

the concrete surface and will reach a maximum value some millimetres inside the concrete.

Because of these observations, Tuutti (1993) made a distinction between a convection zone

and a diffusion zone in the concrete cover as illustrated in Fig. 2.9. Whereas a diffusion

model is applicable to concrete with a constant moisture level (diffusion zone), the transport


of chloride ions due to water flow has to be considered for the outer part where moisture

conditions vary with time (convection zone). The depth of the convection zone depends

on the concrete quality and exposure conditions and varies from 10 to 20 mm in ordinary

concretes (Andrade and Alonso, 1996b).

Chloride convection is known as the

maximum Cl-

concentration

constantmoisture content

effectivecover

surface subjected to wetting and drying

convectionzone

diffusionzone

rebar

Cl-

Fig. 2.9: Moisture and chloride variation in the con-crete cover of structures subjected to cycles of wetting

and drying (adapted from Tuutti (1993)).

flow of chloride ions due to the move-

ment of water in which they are dissolved.

This transport mechanism is mathemat-

ically described by the product of the

moisture flux and the chloride concen-

tration (Nilsson and Tang, 1996), i.e.,

J ′c = Cfc · Jm

[kg/(m2 · s)

](2.22)

where J ′c is the convective flux of chlo-

ride ions (kg/m2·s) and Jm is the flux of

moisture in concrete (m/s). Saetta et al.

(1993) and Akita and Fujiwara (1995)

used the expression given in Eq. 2.22 in

their chloride conservation statement by adding J ′c to the diffusive flux Jc given by Eq. 2.1.

The implementation of their model showed the effectiveness of drying-wetting cycles in in-

troducing chloride ions into concrete.

Collins and Grace (1997) proposed a modified diffusion model for estimating the time

to corrosion in marine concrete structures based on both diffusion and convection chloride

penetration. The model was solved by a finite-difference method since Dc and Cs were

allowed to vary with time. The transport processes in the splash and atmospheric zones

were described by:

∂C

∂t= (Dc + k · v) ∂

2C

∂x2︸︷︷︸diffusion

− v · ∂C∂x︸︷︷︸

convection

[kg/(m3 · s)

](2.23)

where C is the chloride concentration at depth x and time t (kg/m3), k is the dispersion

constant (m), and v is the flow velocity of the capillary pore water (m/s). Parameter k in


Eq. 2.23 accounts for the dispersion of chloride ions that takes place when flow of water in

concrete occurs, and it has been determined experimentally as being in the order of 5 mm

for typical concretes (Grace, 1991). When there is no moisture flow in concrete (i.e., v = 0),

Eq. 2.23 reduces to the pure diffusion case. By implementing Eq. 2.23 into a finite-difference

model, Grace (1991) concluded that service life estimations based on chloride diffusion alone

could significantly overestimate the time to reinforcing steel depassivation as illustrated in

Fig. 2.10, which plots chloride profiles (solid lines) at 5 and 25 years of exposure for a concrete

with a w/cm of 0.46 (the dotted lines represent the calculated moisture distribution along

the concrete cover).

Collins and Grace (1997) argued that the application of Eq. 2.23 to actual structures is

difficult since there is a need to determine moisture profiles in concrete and to define wet/dry

cycles at the surface in order to adequately model the convective movement of chloride ions.

They solved Fick’s 2nd law of diffusion instead, where Dc was defined as a “penetration”

coefficient that combines both the mechanisms of dispersion and diffusion of chloride ions,

i.e.,

Dc = Dultc

[(KD − 1) ·

(1− t

t+ TDc

)+ 1

] [m2/s

](2.24)

where KD is a constant governing the influence of dispersion of ions and TDc is a parameter

that defines the rate of reduction of Dc to Dultc over time (s).

Saetta et al. (1993) incorporated the contribution of moisture flux in transporting chloride

ions into partially-saturated concrete in a one-dimensional finite element model by modifying

Fick’s 2nd law of diffusion and considering suitable boundary conditions. Their expression

for the variation in time of the total chloride concentration Ctc is given by:

∂Ctc

∂t=

∂

∂x

(Dc

∂Ctc

∂x

)︸︷︷︸

diffusion

+Ctc

α· ∂ωe

∂t︸︷︷︸convection

[kg/(m2 · s)

](2.25)

where Dc is the apparent diffusion coefficient (m2/s), which was modified to consider the

effects of temperature, hydration, and humidity on diffusion, and α is a binding constant

(linear binding where Cbc = αCfc was assumed in their model). The authors also took into

account in their formulation the simultaneous diffusion of humidity and heat in concrete by

coupling the equations of moisture and heat flow with the solution of Eq. 2.25.


Fig. 2.10: Comparison between chloride profiles obtained from a convection-diffusion model (topfigure) and a pure diffusion model (bottom figure) after 5 and 25 years of exposure (reproduced

from Grace (1991)).


A similar approach was adopted by Akita and Fujiwara (1995) who also coupled moisture

diffusion to a similar expression as the one given by Eq. 2.25. The authors implemented their

model by means of the method of control volume. However, whereas Saetta et al. (1993)

assumed that chloride binding was defined by a linear relationship, Akita and Fujiwara (1995)

used non-linear chloride binding in their formulation. Their analyses showed that chloride

ingress rates with water flow are about five times higher than chloride ingress rates due to

diffusion alone after 12 weeks of exposure.

2.2.5 Boundary conditions

The solution of the equations describing the mechanisms of chloride ingress into concrete

requires that the conditions existing at the boundaries be defined. However, one of the prob-

lems faced on service life modelling associated with chloride-induced reinforcement corrosion

is the lack of information regarding the influence that different types of exposure conditions

exert on different concrete structures (Nilsson et al., 1996). A general approach in the diffu-

sion models has been to determine an “apparent” surface chloride concentration from best

fits of measured chloride profiles to Eq. 2.6 (Browne, 1982).

Some empirical data relating to surface chloride concentrations from curve-fitted chloride

profiles have been reported for some structures exposed to different environments. It has

been observed from field data that the surface chloride content Cs tends to increase the

longer a structure is in service. Uji et al. (1990) proposed an expression for the boundary

condition of Eq. 2.5, in which Cs is assumed to be a linear function of the square root of the

concrete age, i.e.,

Cs = S√t [% of concrete weight] (2.26)

where S is a surface chloride coefficient (1/√

s) dependent on the type of structure and zone

of exposure and t is the exposure time (s). In order to get an estimate of the values S

takes, the authors evaluated Cs by fitting Eq. 2.6 to chloride profiles obtained from offshore

structures which were in service from 7 to 58 years, and then calculated S from Eq. 2.26. The

range of values obtained for S in the splash zone (5.31–16.6 ×10−6 1/√

s) was found to be

higher than that for the tidal zone (18.2–23.5 ×10−6 1/√

s) or the atmospheric zone (1.56–

5.57 ×10−6 1/√

s). By performing a dimensional analysis, they found that S depends on


the chloride flux across the surface, which in turn varies with the environmental conditions.

Thus, wetting and drying cycles in the splash zone explain the higher fluctuation in the

amount of chloride ions being deposited at the concrete surface.

Measurements of surface chloride concentrations in marine structures by Collins and

Grace (1997) have indicated, however, that after an initial increase of Cs over the first few

years of exposure, the concentration of chlorides at the concrete surface eventually becomes

constant. According to the authors, the magnitude of this ultimate level is related to the

cementitious system of the concrete and the porosity of the surface layer. Based on their

observations, Collins and Grace (1997) developed a new relationship for Cs, which is given

by:

Cs = Cults · t

t+ TCs

[% of concrete weight] (2.27)

where Cults is the ultimate surface concentration (found to be 0.6% for an OPC concrete with

w/cm = 0.4 and cement content equal to 450 kg/m3), t is the time of exposure in days, and

TCs is a parameter governing the rate of buildup of chlorides at the concrete surface (days).

As parameter S in Eq.2.26, the value of TCs depends on the type of concrete and type of

exposure; a value of TCs approaching zero means a faster buildup of chlorides at the surface.

Instead of specifying a value for Cs, Grace (1991) and Saetta et al. (1993) took a different

approach by defining the flux of chlorides crossing the concrete surface. When simulating

concretes exposed to chloride ions by immersion, Saetta et al. (1993) evaluated the chloride

flux across the surface Jsc from:

Jsc = Bc (Cfc − Cen)︸︷︷︸

diffusion

+Cen · Jsm︸︷︷︸

convection

[kg/(m2 · s)

](2.28)

where Jsc is the chloride flux normal to the concrete surface (kg/m2·s), Bc is the surface

chloride transfer coefficient (m/s), Cen is the concentration of the external solution (kg/m3 of

solution), and Jsm is the influx of moisture across the concrete surface (m/s). The convective

term reflects the effect of moisture flow in transporting chloride ions into concrete. The

transfer coefficient Bc was determined from experimental data and was found to range from 1

to 6 m/s for concretes immersed in a salt solution. Saetta et al. (1993) also defined a second

type of boundary condition simulating reinforced concrete structures exposed to marine

atmospheres, in which the amount of chlorides on the concrete surface layer is controlled by


the deposition process of the chlorides present in the air and the opposing washing effect

of the rain. The amount of chloride ions deposited or washed in a unitary time and for a

unitary surface area was calculated from:

dQc

dt= kdep · Catm︸︷︷︸

deposition

− kdil · Ctc︸︷︷︸washing away

[kg/(m2 · s)

](2.29)

where kdep · Catm is the chloride ion deposition rate (kg/m2· s), Catm is the chloride ion

concentration in the atmospheric air (kg/m3), and kdil·Ctc is the washing away rate (kg/m2·s).

The deposition and washing-away coefficients, kdep and kdil, respectively, were determined

by fitting of experimental data.

2.3 Chloride threshold concentration

As seen in Section 1.1, depassivation of reinforcing steel in concrete can occur because of

carbonation of the concrete cover or localized breakdown of the passive layer when a sufficient

amount of chloride ions dissolved in the pore solution reach the reinforcing steel. It is

this amount of chloride ions that is known as the chloride threshold concentration. The

attainment of this amount of chlorides at the depth of the reinforcement corresponds to the

onset of corrosion and, therefore, initiation of the propagation stage as illustrated in Fig. 2.1.

Since many service life models assume the end of service life to coincide with the removal of

reinforcing steel passivity, the specified chloride threshold is therefore a critical parameter

in resulting service life estimations.

There is a difference in the critical amount of chlorides required to initiate corrosion

depending on whether chloride ions are present in the original concrete mix or penetrate

the concrete from the external environment. A higher amount of chlorides added to the

concrete from the mix ingredients will chemically combine with the hydrating cement paste,

the amount of resulting bound chlorides being a function of the C3A content of the cement.

In general, higher chloride contents can be tolerated when these are added to the concrete

mix (Glass and Buenfeld, 1995).

In order to prevent corrosion from chlorides already present in the concrete mix, concrete

standards have provided guidelines on the allowable amount of chloride ions coming from

the original mix ingredients. Some of these values, specified as % of the cement content, are


Table 2.1: Recommended upper limits for Cl− coming from mixing ingredients (% by mass ofcementing material).

CSA A23.1–94 ACI 318–89 ACI 222 BS 8110–85 ENV 206–92

(water-soluble) (water-soluble) (acid-soluble) (acid-soluble) (acid-soluble)

Prestressed concrete 0.06 0.06 0.08 0.1 0.1

R.C. exposed to Cl− 0.15 0.15 0.2 0.4 0.4

shown in Table 2.1. The conservative approach of ACI Committee 222 as compared to that

of BS 8110–85 and ENV 206–92 reflects the difficulty in defining the service environment

to which reinforced concrete structures are likely to be exposed. ACI 222 recommends

to maintain the lowest possible chloride content in the original mixture of the concrete in

order to maximize the structure’s service life before the critical chloride content is reached

and a high risk of corrosion develops. The reason for recommending more strict values

for prestressed concrete structures is because they are more vulnerable to pitting corrosion

effects such as a localized reduction in the cross-sectional area of the strands.

It is difficult to establish a precise level of chloride content below which corrosion does

not occur, since the chloride threshold concentration depends on several parameters. Factors

that affect the chloride threshold level are:

• the condition of the steel/concrete interface, which influences the level of active in-

hibitor through its effective buffering capacity and the availability of Cl− ions by re-

stricting their mobility (Glass and Buenfeld, 1995; Sandberg et al., 1995).

• the properties of the concrete, such as its binding capacity (Hussain et al., 1996), the

pH level of the pore solution (Funahashi, 1990; Tuutti, 1993), and its barrier properties

(Sandberg et al., 1995).

• the exposure conditions (Glass and Buenfeld, 1995; Sandberg et al., 1995), such as

the source and type of Cl− contamination (Papadakis et al., 1996), temperature, and

moisture content (Funahashi, 1990; Tuutti, 1993).

Chloride threshold values reported in the literature have been usually expressed as:

• a percentage of total chlorides by mass of concrete.


• a percentage of total chlorides by mass of cement or cementing material content. Re-

ported values range from 0.17 to 2.5% (Glass and Buenfeld, 1995).

• a free chloride concentration in the concrete pore solution. Published values from

mortar and concrete specimens vary between 0.14 and 1.8 mol/l (Glass and Buenfeld,

1995).

• a concentration ratio of free chloride ions to hydroxyl ions in the concrete pore solution.

Laboratory studies have reported values ranging from 0.26 to 40 (Glass and Buenfeld,

1995).

The chloride threshold level given in terms of a total chloride content by mass of dry con-

crete shows a strong relation to the concrete cement or cementing material content; however,

this dependency is decreased if the threshold value is given as a percentage of the cement

or cementing material content (Nilsson et al., 1996). Work undertaken to directly assess the

threshold level in terms of free chlorides is limited due to the difficulty in determining the

concentration of chloride ions dissolved in the concrete pore solution (Glass and Buenfeld,

1995; Nilsson et al., 1996). Furthermore, since the amount of chloride ions dissolved in the

pore solution depend on the ability of the cement matrix to bind chlorides, expressing the

threshold content in terms of free chlorides may overestimate the onset of corrosion, since

weakly bound chlorides can present a significant corrosion risk (Glass and Buenfeld, 1995).

It is known that hydroxyl ions act as an inhibitor in concrete, continuously opposing

the corrosive action of the chloride ions by the film-repairing action (passivating layer).

This is why earlier work suggested that the Cl−/OH− should be used to represent corrosion

risk. From laboratory studies of steel immersed in alkaline solutions, Hausmann (1967)

suggested that a ratio of Cl−/OH− greater than 0.6 would no longer maintain the steel

in a passive state. However, later studies of reinforcing steel embedded in concrete have

reported higher values (Hussain et al., 1996). The reason why Hausmann (1967) obtained a

more conservative value is that steel in simulated pore solution environments tends to exhibit

higher corrosion rates than steel embedded in concrete (Sandberg et al., 1995). The interface

between the reinforcement and the concrete may have an extra protective effect resulting in

higher threshold values (Tang and Nilsson, 1996a). Yet, expressing the threshold value in


terms of the Cl−/OH− ratio still poses the problem of determining the amount of chlorides

dissolved in the pore solution.

Thus the most practical way of presenting threshold levels is by expressing them as a

percentage of total chlorides by mass of cementing material content. According to a litera-

ture survey on critical chloride values done by Glass and Buenfeld (1995), this representation

results in a reduction in the range of reported values when compared to critical free chloride

contents or chloride-to-hydroxyl concentration ratios. Moreover, the total chloride content

expressed relative to the mass of cementing material content reflects the ratio of total poten-

tial aggressive ion content to total potential inhibitor content provided by the cement (Glass

and Buenfeld, 1995; Nilsson et al., 1996).

A chloride content in excess of the threshold amount is not the only decisive parameter

in determining the service life of reinforced concrete structures exposed to chloride environ-

ments, as assumed by many models. The subsequent rate of corrosion depends on other

variables such as temperature, oxygen availability at the steel surface, and moisture content.

In fact, the moisture distribution in concrete is a significant factor for corrosion to proceed.

From a study on corrosion of reinforcing steel embedded in mortar specimens without Cl−

and with 2% Cl− by cement weight, Lopez and Gonzalez (1993) observed a strong relation

between the reinforcement corrosion rate and the degree of pore saturation of the mortars

(see Fig. 2.11). For saturation levels lower than 60%, the corrosion rate markedly decreases

due to an increase in the electrical resistivity of the cement matrix, whereas for saturated

concretes the kinetics of the corrosion reactions will be limited by oxygen diffusion to the re-

inforcing steel. Because of the interaction of these two factors, optimum corrosion conditions

will then occur when the concrete cover is partially-saturated (Browne, 1982; Tuutti, 1993).

Lopez and Gonzalez (1993) found these optimum conditions to correspond to saturation

levels between 60% and 70%.

2.4 Modelling of active corrosion

As seen in Section 2.2, many service life models associated with chloride-induced corrosion

of steel reinforcement assume the service life to end with the onset of corrosion and thus

ignore the modelling aspects related to the propagation stage or active corrosion. However,


Fig. 2.11: Influence of the degree of mortar pore saturation on the corrosion rate of the reinforce-ment icorr (reproduced from Lopez and Gonzalez (1993)).

there have been few attempts in quantifying the propagation stage based on the diffusivity

of oxygen in concrete and the kinetics of the electrochemical reactions taking place at the

anodic and cathodic sites (Bazant, 1979a; Harker et al., 1987; Walton and Sagar, 1987;

Walton et al., 1990; Kranc and Sagues, 1994; Balabanic et al., 1996; Matsushima et al.,

1996; Yokozeki et al., 1997).

The simplest approach in estimating the corrosion rate of reinforcing steel after it has

been depassivated is to assume that it is limited by oxygen availability at the cathode (Walton

et al., 1990). This has usually been done by assuming that all the oxygen diffusing toward

the reinforcement is being consumed in the cathodic reaction (Kranc and Sagues, 1994;

Balabanic et al., 1996; Matsushima et al., 1996; Yokozeki et al., 1997). Assuming that the

main corrosion product formed is ferric hydroxide, i.e., Fe(OH)3, the resulting corrosion rate

is obtained from:

rcorr =4

3JomFe

%Fe

[(kg ·m)/(mol · s)] (2.30)

where rcorr is the corrosion rate (kg·m/mol·s), Jo is the oxygen diffusion flux (kg/m2·s),

mFe is the molecular weight of iron (kg/mol), and %Fe is the density of the reinforcing bars

(kg/m3). Equation 2.30 assumes that 3/4 moles of O2 are being consumed to produce 1 mol


of Fe(OH)3. The diffusive oxygen flux Jo can be obtained by solving the one-dimensional

oxygen diffusion equation given by:

Jo = −Do∂Co

∂x

[kg/(m2 · s)

](2.31)

where Do is the oxygen diffusion coefficient (m2/s) and Co is the dissolved oxygen concen-

tration at depth x (kg/m3 of solution).

This approach has been used by Matsushima

Fig. 2.12: Model for corrosion of reinforcingsteel (reproduced from Yokozeki et al. (1997)).

et al. (1996), who obtained the dissolved oxygen

profile and the corresponding corrosion rate by

establishing mass conservation in concrete and

developing a similar expression to Eq. 2.30. In

their equation for the corrosion rate, the au-

thors introduced the area ratio of the cathode

to anode as a factor, which they assumed to

decrease exponentially over time. As a result

of this assumption the calculated corrosion rate

showed a gradual decrease with time regard-

less of the temperature level of the concrete or

the relative humidity of the exposure environ-

ment (Andrade and Alonso, 1996a; Newhouse

and Weyers, 1996). Following a similar procedure, Yokozeki et al. (1997) assumed the cor-

rosion rate to be constant and proportional to the steady-state diffusion flow of oxygen

(Fig. 2.12). However, the conversion of Jo to the rate of corrosion was made by considering

different percentages for the different corrosion products formed: 4% for Fe(OH)2 and 96%

for Fe(OH)3.

More complex models for active corrosion have also considered the kinetics of the corro-

sion reactions to be rate limiting. In modelling uniform corrosion of radioactive waste steel

canisters surrounded by concrete, Harker et al. (1987) and Walton and Sagar (1987) used

Butler-Volmer kinetics to approximate the kinetics of the anodic and cathodic reactions.

Kranc and Sagues (1994) used the same approach to define the polarization characteristics

of corroding reinforcing steel in reinforced concrete structures exposed to marine environ-


ments. According to Butler-Volmer kinetics, the kinetics of iron dissolution at anodic sites

can be estimated from:

ia = ioa exp

(2.3

E − Eoa

βa

) [A/m2] (2.32)

where ia is the current density of the anodic reaction (A/m2), ioa is the exchange current

density for iron dissolution (A/m2), E is the potential at the concrete pore solution immedi-

ately next to the steel surface (V), Eoa is the equilibrium potential of the anodic reaction (V),

and βa is the activation tafel slope for the anodic reaction (V). Equation 2.32 assumes that

the kinetics of iron dissolution are only subject to activation polarization, i.e., the reaction

rate is limited by transfer of electrochemical charge across the steel/concrete pore solution

interface. Similarly, the kinetics of oxygen reduction at cathodic sites, assuming this is the

main cathodic reaction, can be approximated by:

ic = iocCo

Cso

exp

(2.3

Eoc − E

βc

) [A/m2] (2.33)

where ic is the current density of the cathodic reaction (A/m2), ioc is the exchange current

density for oxygen reduction (A/m2), Co is the dissolved oxygen concentration at the steel

surface (kg/m3 of solution), Cso is the oxygen concentration at the external concrete surface

(kg/m3 of solution), Eoc is the equilibrium potential of the cathodic half-cell reaction (V),

and βc is the activation tafel slope for the cathodic reaction (V). Equation 2.33 assumes

that the kinetics of the cathodic reaction are subject to both activation and concentration

polarizations, i.e., oxygen diffusion in concrete can also be a rate-limiting process. This

is especially the case of submerged reinforced concrete structures whose concrete cover is

completely saturated.

Kranc and Sagues (1994) used Eqs. 2.32 and 2.33 to define the conditions at the steel/concrete

interface in their finite-difference model of reinforcing steel corrosion which was based on the

continuity equation, given by:

∇ i = 0[A/m3] (2.34)

where i is the current density vector (A/m2). Assuming steady-state conditions, the authors

applied the model to compute the distribution of electrical potential in concrete and coupled

it to oxygen diffusion towards the reinforcement. The current density vector i was related


to the electrical potential E through Ohm’s law, i.e.,

ρ i = −∇E [V/m] (2.35)

where ρ denotes the electrical resistivity of concrete (Ω·m). The negative sign in Eq. 2.35

indicates that the direction of the current occurs along a voltage drop.

Balabanic et al. (1996) also developed

Fig. 2.13: Influence of concrete quality (w/c), con-crete cover (dcon), and degree of pore saturation onthe corrosion current density ir (reproduced from Bal-

abanic et al. (1996)).

a numerical procedure based on the fi-

nite element and finite difference meth-

ods to solve the set of equations corre-

sponding to electrical potential distribu-

tion and oxygen transport through con-

crete. However, they assumed anodic

sites to be unpolarisable whereas cathodic

sites were subjected to concentration po-

larization only. Based on previous ex-

perimental observations by Lopez and

Gonzalez (1993), the authors introduced

in their model the dependence of Do and ρ on the degree of water saturation of concrete.

Their numerical results indicated that the influence of the degree of pore saturation on the

calculated corrosion current is more evident when changing the quality of the concrete, which

was expressed by reducing the w/cm from 0.7 to 0.4 (see Fig. 2.13).

Bazant (1979a) formulated a very comprehensive model of the process of corrosion in

marine reinforced concrete structures where both the initiation and propagation stages were

considered. The main aspects of the model included:

• the transport of Cl−, H2O and O2 through the concrete cover and production of

Fe(OH)2 at the steel surface by establishing conservation of mass equations.

• the mass sinks and sources of O2 and Fe(OH)2 at the steel surface due to the cathodic

and anodic reactions, respectively, by means of Faraday’s law.

• the flow of electric current through the concrete pore solution as given by Eq. 2.34.


• the cathodic and anodic electrical potential equations determined by concentration

polarization of the electrodes.

• the rust production rate based on reaction kinetics.

Subramanian and Wheat (1989) solved the four mass conservation equations corresponding

to the transport of moisture, chlorides, oxygen, and ferrous hydroxide in concrete proposed

by Bazant (1979a) by means of a one-dimensional formulation based on the method of

lines with cubic Hermite polynomials. Their analysis was meant to estimate the time to

reinforcing steel depassivation rather than characterize the kinetics of the corrosion process.

The authors concluded that the variability in oxygen diffusivity was negligible on the time

to corrosion initiation. Reasons for this were the facts that they based their criterion for

depassivation solely on chloride concentrations, and the equations used to describe chloride

and oxygen transport were independent of each other.

2.4.1 Electrical resistivity of concrete

Electrical currents resulting from the corrosion of reinforcing steel are passed predominantly

by the movement of ions in the pore water (Elkey and Sellevold, 1995). Thus the pore

structure characteristics of the concrete, the degree of water saturation, and the total ionic

concentration of the pore solution are relevant factors affecting the electrical resistivity of

concrete. Since the concrete pore solution acts as an electrolyte with lower resistivity than

the cement matrix, the moisture content plays in fact a significant role in determining the

electrical properties of concrete (Lopez and Gonzalez, 1993). Fully saturated OPC paste has

an electrical resistivity in the order of 14 Ω·m; however, there is approximately an increase of

three orders of magnitude from that value by progressively drying initially water-saturated

concrete, as illustrated in Fig. 2.14 (Rosenberg et al., 1989).

The electrical resistivity of concrete is a major factor in controlling the corrosion rate of

the reinforcement (Eq. 2.35). However, according to Andrade and Alonso (1996a), it is the

relative changes in resistivity due its dependence on moisture content that causes significant

changes in the resulting corrosion rate (see Fig. 2.15). If this phenomenon were taken into

account, the concrete resistivity ρ should be formulated as a function of the degree of water

saturation (see Fig. 2.16).


Fig. 2.14: The effect of water saturation on the resistivity of concrete (reproduced from Rosenberget al. (1989)).

The concrete electrical resistivity is also affected by the level of temperature in the con-

crete, being reduced with increasing temperature. At higher temperatures, more ions will

dissolve into the concrete pore solution facilitating the flow of electrical current (Elkey and

Sellevold, 1995). The relationship between concrete resistivity and temperature can be ob-

tained from the Hinrichson-Rasch law (Hope and Ip, 1987; Elkey and Sellevold, 1995), which

is given by:

ρ2 = ρ1 exp

[U

R

(1

T2

− 1

T1

)][Ω ·m] (2.36)

where ρ1 and ρ2 (Ω·m) are the concrete resistivities at temperatures T1 and T2 (K), respec-

tively, U is the activation energy (kJ/mol), and R is the gas constant (kJ/K·mol). Hope and

Ip (1987) used 2900 K for the value of U/R when using Eq. 2.36.

2.5 Modelling of concrete damage due to corrosion

The main corrosion-induced damage mechanisms in reinforced concrete as illustrated in

Fig. 2.17 are (Andrade and Alonso, 1996a):


Fig. 2.15: Evolution of the corrosion rate (Icorr) and the corresponding electrical resistance(Rohm) with time for mortar specimens subjected to moisture cycles (reproduced from Andrade

and Alonso (1996a)).


0

2

4

6

8

0.0 0.2 0.4 0.6 0.8 1.0

Degree of pore saturation, ω e /ω sat

log

( ρ )

Bazant (1979a)

Balabanic et al. (1996)

Fig. 2.16: Dependence of concrete electrical resistivity ρ on concrete moisture content.

• a decrease in the rebar cross-sectional area.

• a possible loss of steel ductility.

• cracking and spalling of the concrete cover.

• loss of bond along the steel/concrete interface.

Although most of the research on chloride-induced corrosion has been on chloride ingress into

concrete, there has been an increasing effort in recent years to quantify the damage progress

in reinforced concrete due to the advance of steel reinforcement corrosion (Andrade et al.,

1993; Andrade and Alonso, 1996a, 1996b; Rodriguez et al., 1996; Newhouse and Weyers,

1996). This aspect is of great relevance in the structural assessment of deteriorated reinforced

concrete structures with corroded reinforcement, and it must be included in estimations of

residual service life.

From an experimental study on the amount of current needed to induce cracking of the

concrete cover, Andrade et al. (1993) established a relationship between the corrosion rate

of reinforcing steel and the loss of steel cross-sectional area, given by:

d′ = d− 0.023 · icorr · (t− ti) [mm] (2.37)


where d′ is the rebar diameter (mm) at time t, d is the initial rebar diameter (mm), icorr is the

corrosion rate (µA/cm2), and t− ti is the time elapsed since the onset of steel depassivation

(years), and 0.023 is a factor that converts µA/cm2 into mm/yr. The expression given by

Eq. 2.37 assumes that icorr is constant, i.e., it does not vary with environmental conditions

such as temperature or humidity levels. In order to obtain a representative value from

Eq. 2.37, an average of icorr over the time of interest has to be established by periodical

measurements of corrosion rates. Ongoing research by the same authors is now focused on

establishing mathematical links between the rate of corrosion of reinforcing steel and the rate

of decay of the remaining damage mechanisms listed above (Andrade and Alonso, 1996a).

Such an approach is applicable for residual life calculations of already existing structures,

where on-site measurements can facilitate an estimation of actual corrosion rates; however,

in order to include these types of expressions in service life models aimed at the design of

new structures, a prediction model for corrosion rates given some material parameters and

environmental conditions needs to be developed.

Andrade and Alonso (1996b) quanti-

Fig. 2.17: Consequences of rebar corrosion (repro-duced from Andrade and Alonso (1996a)).

fied the propagation period in Tuutti’s

corrosion model using as a damage in-

dex the loss of steel cross-sectional area

expressed in Eq. 2.37 (Fig. 2.18). The

authors classified corrosion risk into four

levels depending on the value of the cor-

rosion rate icorr: (1) negligible, where

icorr < 0.1 µA/cm2; (2) low, where 0.1 <

icorr < 0.5 µA/cm2; (3) moderate, where

0.5 < icorr < 1 µA/cm2; and, (4) high, where icorr > 1 µA/cm2.

Based on a comprehensive mathematical formulation of the corrosion process in reinforced

concrete, Bazant (1979b) developed the following theoretical equation for evaluating the time

to cracking of the concrete cover:

tcr = %corrd∆d

s jr[s] (2.38)


Fig. 2.18: Quantification of the propagation period for different corrosion rates for a rebar of 20mm in diameter (reproduced from Andrade and Alonso (1996b)).

where tcr is the time to concrete cracking (s), %corr is a function of the densities of steel and

rust (kg/m3), d is the reinforcing bar diameter (m), ∆d is the increase in rebar diameter due

to corrosion products accumulation (m), s is the spacing between rebars (m), and jr is the

rate of rust production (kg/m2·s), assumed to be constant over time. The increase in rebar

diameter ∆d was estimated by applying a uniform pressure around the reinforcing bar and

assuming concrete to be a homogeneous elastic material. From a study on the influence of

corrosion rates on time to first cracking, Newhouse and Weyers (1996) concluded that the

expression developed by Bazant (1979b) significantly underestimated the time to concrete

cracking when using uniform corrosion rates based on measured metal loss. Reasons given

for the difference between calculated and measured values were based on the observations

that not all the corrosion products exert an internal pressure around the reinforcing bar

perimeter (corrosion products were found as far as 5 cm away from the reinforcement), nor

do they result in the same volume increase.

The time needed for first cracking of the concrete cover as a result of the expansive forces

exerted by the corrosion products mainly depends on the tensile strength of the concrete

f ′t and the concrete cover-to-rebar diameter ratio c/d. Based on experimental studies, (Ro-

driguez et al., 1996) evaluated the attack penetration xo corresponding to crack initiation


from:

xo = 83.8 + 7.4c

d− 22.6f ′t [µm] (2.39)

where xo is the attack penetration or the decrease in the reinforcing bar radius (µm), c/d is

the cover/diameter ratio and f ′t is the concrete tensile strength (MPa). The same authors

observed that crack width evolution, however, depends on the position of the rebar in the

reinforced concrete element (top versus bottom) and to a lesser extent on the corrosion rate

icorr. As a result of their experimental observations, they developed the following empirical

equation to assess damage in corroded reinforced concrete structures in terms of crack widths:

w = 0.05 + β [x− xo] [w ≤ 1.0 mm] (2.40)

where w is the estimated crack width (mm), x is the corrosion attack penetration (µm), xo

is the attack penetration corresponding to crack initiation (µm), and β is a coefficient which

depends on the position of the reinforcing bar (0.01 and 0.0125 for top and bottom placement,

respectively). Both of the above expression were obtained from experiments where humidity

was kept constant; the authors commented on the possibility of higher values when concrete

close to the reinforcing bar is subject to variable pore relative humidity conditions.

Dagher and Kulendran (1992) developed a two-dimensional finite element model to es-

timate the damage in concrete resulting from the volume expansion of corrosion products.

Their corrosion-damage model was based on the smeared crack approach with a number

of options for modelling crack formation and propagation (local-versus-nonlocal continuum

approach and strength-versus-fracture mechanics criteria). The model assumed as input

parameters the onset and rate of corrosion as well as the geometry of corrosion buildup. Vol-

ume expansion due to corrosion products accumulation was modelled by imposing predefined

displacement fields on the nodes around the perimeter of the rebars. For each increment of

bar expansion, the model evaluated stresses and strains in the surrounding concrete as well

as crack locations and directions. The authors applied the model to study the damage in

concrete bridge decks subjected to uniform corrosion. The analyses showed that just only

0.008 mm of uniform radial bar expansion can cause a plane of delamination in a deck

with No. 20 reinforcing bars spaced 152 mm apart. It was also observed that as the bar

spacing increased (from 152 mm to 254 mm), the likelihood of delamination decreased and


Fig. 2.19: Progress of chloride-induced deterioration (reproduced from Yokozeki et al. (1997)).

that of the formation of pot holes increased. Given estimates of actual corrosion rates, this

type of analysis proves to be useful in assessing the condition of existing reinforced concrete

structures.

Molina et al. (1993) also developed a numerical model based on the finite element method

to simulate cracking in concrete due to corrosion of the reinforcement. However, in this model

both concrete and steel elements were included in the formulation. Corroding reinforcing

steel was modelled by superposition of a decrease in its stiffness, which varied linearly from

iron to rust, and an increase in its specific volume, which was achieved by imposing an initial

strain on the element being corroded. In fact, this initial strain was imposed as an equivalent

thermal load on the structure. The ratio between the specific volumes of rust and iron was

taken as 2. Concrete cracking was modelled by a smeared-fixed-crack approach with linear

softening. The process of active corrosion was simulated by applying three cumulative load

combinations, each representing an advance of 20 µm in the attack penetration, which was

assumed to be uniform around the rebar perimeter. The authors concluded that the value

given to the specific volume of rust was a determinant factor on crack growth.

Following a similar approach, Yokozeki et al. (1997) formulated a non-linear elasto-plastic

finite element model for predicting first cracking of concrete after the onset of corrosion,


and they integrated it in a complete service life prediction model. The authors considered

the accelerated damage period following the onset of cracking, as illustrated in Fig. 2.19,

to be much faster than the time for corrosion initiation and time for first cracking of the

concrete cover, and they neglected it in service life estimations. In their corrosion-damage

model, the mechanical properties of concrete, steel and corrosion products were defined as

input parameters. Concrete was modelled by means of a fracture mechanics approach with a

smeared-crack analysis. The stiffness of the steel elements were reduced according to the rust

concentration around the rebars. Analyses were performed assuming a state of plane-strain,

in which the corrosive expansion force was simulated by assigning strains to the reinforcing

bar elements according to:dε

dΦ= 3√α (2.41)

where ε denotes the applied strain, Φ is the rust concentration, and α is the volumetric

expansion ratio, which was given a value of 3.20.

Chapter 3

Service Life Model

3.1 Introduction

The mathematical model proposed for the prediction of the service life of reinforced con-

crete highway structures exposed to de-icing salts is outlined in the present chapter. The

formulation is based on the physical model for steel corrosion of reinforced concrete sea

structures proposed by Bazant (1979a). It idealizes the corrosion sequence as a two stage

process (Tuutti, 1982): an initiation stage, during which chloride ions reach the reinforcing

steel in sufficient quantities to depassivate it, and a propagation stage, in which active cor-

rosion takes place until an unacceptable level of cracking of the surrounding concrete cover

has occurred (see Fig. 3.1). Hence, the total service life is defined as:

tcr = ti + tp (3.1)

where tcr is the time at which the concrete cover cracks as a result of the expansion of the

corrosion products, ti is the time to reinforcing steel depassivation (initiation period), and

tp is the time corresponding to active corrosion subsequent to breakup of the passive layer

(propagation period). The end of service life tcr is defined here as the time at which internal

cracks resulting from the expansion of the corrosion products propagate through the concrete

cover and appear at the external surface.

In modelling chloride transport to the reinforcing steel (initiation stage), a modified

version of Fick’s 2nd law is used, in which the processes of ionic diffusion, convection due

to water movement, and chloride binding are taken into account. The effect of moisture

49

Service life model 50

Time

Cor

rosi

on le

vel

Cl-

Diffusion + convection

Chloride thresholdreached at steel level

Cracking of concrete

O2

2Fe+O2+2H2O 2Fe(OH)2

Initiation, ti Propagation, tp

Service life, tcr

Fig. 3.1: Conceptual service life model (adapted from Tuutti (1982)).

flux in transporting dissolved chloride ions (convection) is considered by coupling moisture

diffusion to chloride ion migration through concrete. Determining the moisture distribution

in concrete also enables the evaluation of material properties that strongly depend on the

actual moisture content. Heat transfer is also coupled to moisture and chloride transport

since many material parameters, such as diffusivity, depend on the temperature level of the

concrete. Corrosion is assumed to initiate when the chloride concentration at the steel layer

reaches a specified threshold value. The length of the initiation period therefore depends

on the chloride penetration rate and the chloride concentration required to depassivate the

reinforcing steel and initiate the corrosion process.

To model the stage corresponding to active corrosion (propagation stage), the rate of

corrosion is linked to the polarization characteristics of the anode and cathode as well as

the oxygen availability at cathodic areas. This is done by establishing the mass conservation

equation corresponding to oxygen diffusion into concrete.

The different physical and chemical phenomena leading to corrosion of reinforcing steel

can therefore be described by the mechanisms of heat and moisture diffusion through con-

crete, chloride and oxygen transport to the reinforcement, and polarization effects at anodic

and cathodic sites. The mathematical formulation is based directly on basic physical laws,

such as conservation of mass and energy. This renders the model applicable to a wide range of

reinforced concrete structures including those made of high-performance concrete, on which


few data on long-term performance have been established at this time.

The governing differential equations that idealize the problem are reduced to a set of

conservation equations with two space variables, x and y, and one time variable, t. Because

the third dimension (oriented along the length of the reinforcing bar) is assumed to be

infinitely long and subject to uniform conditions, it is ignored and the problem reduces from

three dimensions to two. For the sake of simplicity, it is further assumed that concrete is

not cracked.

The governing equations describing the physical phenomena are presented in the follow-

ing sections. The model describing the damage induced by the expansion of the corrosion

products as well as the criterion used for end of service life are presented in Chapter 4.

3.2 Chloride transport

Transport mechanisms relevant to chloride ingress into reinforced concrete highway struc-

tures exposed to de-icing salts include water surface absorption in partially-saturated con-

crete (capillary sorption or convection) and ionic diffusion in saturated concrete. These

two transport mechanisms have been considered here by modifying Fick’s 2nd law of diffu-

sion (mass conservation statement). The applicable equations describing each process are

presented in the following.

3.2.1 Chloride diffusion

The flow of chloride ions in saturated concrete due to diffusion is governed by Fick’s 1st law

of diffusion, according to which the flux of chloride ions Jc is expressed as:

Jc = −Dc

(∂Cfc

∂x+∂Cfc

∂y

) [kg/(m2 · s)

](3.2)

where

Jc = flux of chloride ions due to diffusion (kg/m2·s)

Dc = effective chloride diffusion coefficient (m2/s)

Cfc = concentration of chlorides dissolved in the pore solution (free chlorides) at

depths x and y (kg/m3 of concrete)


Equation 3.2 assumes concrete to be isotropic, i.e., the effective diffusivity Dc is the same in

the x and y directions. Note that the effective chloride diffusion coefficient Dc is given for a

concentration gradient expressed in kg/m3 of concrete. If the concentration of free chlorides

is expressed as kg/m3 of pore solution instead, Eq. 3.2 becomes:

Jc = − Dc · ωe︸︷︷︸Dc

(∂Cfc

∂x+∂Cfc

∂y

) [kg/(m2 · s)

](3.3)

where ωe is the evaporable water content (expressed per unit volume of concrete), used as

the fraction of capillary porosity that contains a liquid acting as a solvent (Nilsson, 1993;

Nilsson et al., 1994). It is assumed here that the water in which diffusion occurs is equal to

ωe (Sergi et al., 1992). In the remaining of the report, Cfc will be given as kg/m3 of pore

solution.

Establishing chloride mass conservation in saturated concrete results in:

∂Ctc

∂t= −∂Jcx

∂x− ∂Jcy

∂y

[kg/(m3 · s)

](3.4)

where

Ctc = total chloride concentration (kg/m3 of concrete)

Jcx = flux of chloride ions due to diffusion in the x-direction (kg/m2·s)

Jcy = flux of chloride ions due to diffusion in the y-direction (kg/m2·s)

If Eq. 3.3 is substituted into Eq. 3.4, the equation governing the mechanism of chloride

diffusion in saturated concrete (Fick’s 2nd law) is given by:

∂Ctc

∂t=

∂

∂x

(Dc · ωe

∂Cfc

∂x

)+

∂

∂y

(Dc · ωe

∂Cfc

∂y

) [kg/(m3 · s)

](3.5)

3.2.2 Chloride diffusion coefficient

Because of experimental evidence showing the change of concrete chloride diffusivity with

temperature, time and relative humidity, the effective chloride diffusion coefficient of con-

crete, determined at some defined reference conditions, Dc,ref , can be modified according to

(Saetta et al., 1993):

Dc = Dc,ref · f1(T ) · f2(t) · f3(h)[m2/s

](3.6)


Table 3.1: Typical values for the effective chloride diffusion coefficient Dc.

Source Material w/cm Dc (m2/s)

Page et al. (1981) OPCa paste 0.5 4.5×10−12

OPC paste + 30% PFAb 0.5 1.5×10−12

OPC paste + 65% GBFSc 0.5 0.4×10−12

McGrath and Hooton (1997) OPC concrete 0.4 6.30×10−12

OPC + 8% GBFS concrete 0.4 2.25×10−12

OPC + 8% PFA concrete 0.4 3.35×10−12

OPC + 8% SFd concrete 0.4 0.88×10−12

OPC + 40% GBFS + 8% SF concrete 0.31 0.21×10−12

OPC + 30% PFA + 8% SF concrete 0.31 0.3×10−12

aordinary Portland cementbpulverized-fuel ashcgranulated blast-furnace slagdsilica fume

where f1(T ), f2(t), and f3(h) account for the dependence of Dc on temperature T , time of

exposure t, and pore relative humidity h, respectively. It is assumed here that T , t, and h

act as independent multipliers, i.e., functions f1(T ), f2(t), and f3(h) are independent of each

other. Typical values for the effective chloride diffusion coefficient Dc found in the literature

are given in Table 3.1. It should be noted here that values of Dc reported in the literature

are often determined by different methods, and the resulting values are influenced by the

methodology employed. However, reported values of Dc appear to be sensitive to the same

parameters. A review of experimental techniques to determine Dc is beyond the scope of

this thesis but the reader is referred to Delagrave et al. (1996).

Effect of temperature

The dependence of the chloride diffusivity on temperature as expressed by function f1(T ) is

estimated here using Arrhenius’ law, which is given by:

f1(T ) = exp

[U

R·(

1

Tref

− 1

T

)](3.7)


where

U = activation energy of the chloride diffusion process (kJ/mol)

R = gas constant (8.314×10−3 kJ/K·mol)

Tref = reference temperature at which the chloride diffusivity Dc,ref has been evalu-

ated (K)

T = temperature in the concrete (K)

The dependence of Dc on the temperature level of the concrete according to Eq. 3.7 is

graphically illustrated in Fig. 3.2(a).

Effect of time of exposure

In order to take into account the time dependence of Dc in the chloride diffusion model, the

following mathematical expression has been used (see Fig. 3.2(b)):

f2(t) =

(tref

t

)m

(3.8)

where

tref = time of exposure at which Dc,ref has been measured (s)

t = actual time of exposure (s)

m = age reduction factor

The values for m used in Eq. 3.8 were the ones obtained by Mangat and Molloy (1994) for

various mixes by means of a linear regression analysis of experimental data obtained from

concretes exposed in the field during a period of 5 years. These values are plotted against

the water-to-cement ratio of the various mixes in Fig. 3.3. Although Eq. 2.10 ignores the

effect of cementing material type on m, an important criterion as seen in Chapter 2, it is

adopted here for simplicity. It is also recognized that the values reported by Mangat and

Molloy (1994) take into account the decrease of Dc over time due to both hydration and

chloride binding, whereas the time dependency of Eq. 3.8 is only linked to the on-going

process of hydration. However, the effect of parameter m on the calculated chloride profiles

is investigated in Chapter 6.


0.0

0.5

1.0

1.5

2.0

2.5

3.0

-10 0 10 20 30 40

Temperature, T (oC)

f 1 ( T

)

U = 50

U = 40

U = 30

T Crefo= 23

( )f TU

R T Tref

1

1 1= ⋅ −

exp

(a)

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5

Time, t (years)

f 2 ( t

)

m = 0.2m = 0.4

m = 0.6

t ref = 120 days(b)

( )f tt

tref

m

2 =

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Relative humidity, h

f 3 ( h

)

h c = 0.75

(c)

( ) ( )( )

f hh

hc

3

4

4

1

11

1= +

−−

−

Fig. 3.2: Dependence of Dc on (a) temperature for different values of U (kJ/mol), (b) age, and(c) pore relative humidity.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2 0.4 0.6 0.8 1.0

w/c

m

m w c= −2 5 0 6. / .

Fig. 3.3: Relationship between coefficient m and w/c for various mixes: OPC, 26% PFA, 60%GBFS, 15% SF (Mangat and Molloy, 1994).

Effect of relative humidity

Function f3(h) in Eq. 3.6 takes into account the fact that chloride diffusion in concrete only

occurs if water is present in the capillary pores. This phenomenon is considered in the model

by reducing the chloride diffusivity Dc with the level of pore relative humidity h according

to:

f3(h) =

[1 +

(1− h)4

(1− hc)4

]−1

(3.9)

where hc is the humidity at which Dc drops halfway between its maximum and minimum

values (Fig. 3.2(c)). As seen in Section 2.2.2, Saetta et al. (1993) used Eq. 3.9 by establishing

an analogy between chloride and moisture diffusion in partially-saturated concrete. The

latter phenomenon was described by Bazant and Najjar (1972) in a similar fashion, who

found hc to be 0.75 by best fitting of experimental data on drying of concrete.

3.2.3 Chloride diffusion and binding

Equation 3.5 expresses the change of total chloride concentration with time as a function

of the spatial gradient of free chlorides. If chloride binding is disregarded, the solution of

Eq. 3.5 could be obtained without modification since Ctc = ωeCfc. However, if binding of


Free chlorides, C fc

Bo

un

d c

hlo

rid

es, C

bc

Linear isotherm,

Langmuir isotherm,

Freundlich isotherm,

C Cbc fc

= α

CC

Cbc

fc

fc

=+α

β1

C Cbc fc= α β

Fig. 3.4: Chloride binding isotherms.

chloride ions in concrete is to be accounted for, Eq. 3.5 needs to be re-written in terms of the

same chloride concentration. The total, bound, and free chloride concentrations in concrete

are related by:

Ctc = Cbc + ωeCfc

[kg/m3of concrete

](3.10)

where Cbc is the concentration of bound chlorides (kg/m3 of concrete). The binding proper-

ties of a specific cement composition are usually defined in the form of a binding isotherm

(Fig. 3.4), with the amount of bound chlorides Cbc expressed as a function of the chlorides

present in the pore solution Cfc. To take into account chloride binding in the solution

of Eq. 3.5, binding isotherms have to be determined experimentally for each cementitious

system under consideration.

By applying mass conservation to Eq. 3.10 and substituting into Eq. 3.5, the following

modified Fick’s 2nd law equation results:

∂Cfc

∂t=

∂

∂x

(D∗

c

∂Cfc

∂x

)+

∂

∂y

(D∗

c

∂Cfc

∂y

) [kg/(m3 · s)

](3.11)

with

D∗c =

Dc

1 + 1ωe

∂Cbc

∂Cfc

[m2/s

](3.12)


where D∗c is the apparent diffusion coefficient (m2/s) and ∂Cbc/∂Cfc is the “binding capacity”

of the concrete cementitious system (m3 of pore solution/m3 of concrete) as defined by Nilsson

et al. (1994). The binding capacity of a specific cementitious system is given by the slope of

the corresponding binding isotherm.

The following is a review of some of the theoretical curves used in the literature to describe

chloride binding. The corresponding binding capacities and the resulting apparent diffusion

coefficients are listed as well.

No binding

Cbc = 0 ,∂Cbc

∂Cfc

= 0 , D∗c = Dc (3.13)

Linear isotherm

Cbc = αCfc ,∂Cbc

∂Cfc

= α , D∗c =

Dc

1 + αωe

(3.14)

where α is the slope of the line (m3 of pore solution/m3 of concrete). Note that the value of

α depends on the units used for Cfc and Cbc. Although several experimental works have re-

ported that the relationship between bound and free chlorides is non-linear (Tritthart, 1989;

Sergi et al., 1992; Tang and Nilsson, 1993; Nilsson et al., 1994), chloride binding is often

assumed to be linear for modelling purposes (Saetta et al., 1993). However, as pointed out

by Nilsson et al. (1996), a linear binding relationship oversimplifies the binding mechanism

process, since it underestimates the amount of bound chlorides at low free chloride concen-

trations and overestimates the amount of bound chlorides for high chloride concentrations in

the pore solution. Tuutti (1982) obtained linear relationships that fit very well his chloride

binding data; however, he only reported values of free chloride concentrations lower than 20

kg/m3. Saetta et al. (1993) used linear binding relationships in chloride-penetration profiles

calculations of OPC concretes with w/cm ranging from 0.4 to 0.75 exposed to different en-

vironmental conditions (α was taken as 0.7 m3 of pore solution/m3 of concrete, and Cfc and

Cbc were expressed as kg/m3 of pore solution and kg/m3 of concrete, respectively).

Langmuir isotherm

Cbc =αCfc

1 + β Cfc

,∂Cbc

∂Cfc

=α

(1 + β Cfc)2, D∗

c =Dc

1 + αωe (1+β Cfc)2

(3.15)


where the binding constants α (m3 of pore solution/m3 of concrete) and β (m3 of pore

solution/kg) vary according to the concrete binder composition. These are also dependent

on the units used for Cfc and Cbc. Sergi et al. (1992) obtained the values of α and β as

1.67 ml/g of cement and 4.08 l/mol, respectively, by linear regression analysis of data from

OPC paste samples with w/cm = 0.5 (Cfc and Cbc were expressed as mol/l and mmol/g of

cement, respectively). In spite of the fact that the authors recognized the non-linear nature

of the relationship, they concluded that a linear approximation, i.e., β = 0, gives a good

description of the phenomenon. Tang and Nilsson (1993) have stated that the relationship

between bound and free chlorides is best described by the Langmuir isotherm when the

concentration level of chlorides in the pore solution is less than 1.773 kg/m3. The slope of

the Langmuir isotherm (binding capacity) tends to approach zero as the concentration of

free chlorides increases, suggesting that there is an upper-bound in the amount of chlorides

being bound (Fig. 3.4).

Freundlich isotherm

Cbc = αCβfc ,

∂Cbc

∂Cfc

= αβ Cβ−1fc , D∗

c =Dc

1 + 1ωeαβ Cβ−1

fc

(3.16)

where α (m3 of pore solution/m3 of concrete) and β are binding constants dependent on the

cementitious system used and on the units employed for Cfc and Cbc. Tang and Nilsson (1993)

found that this relationship works best for free chloride concentrations greater than 0.355

kg/m3 of pore solution. Their experimental results showed that concrete binders still exhibit

a binding capacity at high concentrations of free chlorides, as opposed to what is suggested

by the Langmuir relation. According to the authors, monolayer adsorption occurs when the

free chloride concentration is very low (described by the Langmuir isotherm); however, a

more complex adsorption phenomenon takes place at high levels of chloride concentration,

described mathematically by the expression given in Eq. 3.16, wherein the amount of bound

chlorides increases with the concentration of free chlorides.

The effect of chloride binding in Eq. 3.12 is to reduce the diffusivity of chloride ions

in concrete by an amount which is representative of the binding capacity of the specific

cementitious system (see Fig. 3.5). In the case of ignoring binding or considering a linear


Free chlorides, C fc

Dc*/

Dc

No binding

Langmuir

Freundlich

Linear

D

D C

C

c

c

e

bc

fc

*

=+

1

11

ω∂∂

Fig. 3.5: The effect of chloride binding on the apparent diffusion coefficient.

binding isotherm, the apparent diffusion coefficient D∗c is constant as chlorides penetrate the

concrete; however, when non-linearities are taken into account in the bound-free chloride

relation (Langmuir or Freundlich isotherms), D∗c is allowed to vary through the concrete

depth, thus reflecting the dependence of the chloride binding capacity of the concrete binder

on the levels of chloride ions dissolved in the pore solution (Nilsson et al., 1994).

3.2.4 Chloride diffusion and convection

Chloride transport in partially-saturated concrete occurs both by diffusion and capillary

sorption (convection). The flux of chlorides due to capillary sorption J ′c can be mathemati-

cally expressed as the product of the free chloride concentration Cfc and the flux of moisture

through concrete Jm (Saetta et al., 1993; Nilsson and Tang, 1996), i.e.,

J ′c = ωe · Cfc · Jm

[kg/(m2 · s)

](3.17)

The flux of moisture through concrete Jm (m/s) will be defined in Section 3.3. Equation 3.17

implies that chloride ions enter the concrete at the same rate as water, so that the concen-

tration of the absorbing solution remains constant.

The total flow of chlorides into partially-saturated concrete can be obtained by combining


the flow of chlorides due to diffusion (Eq. 3.3) and the flow of chlorides due to convection

(Eq. 3.17). Establishing chloride mass conservation results in:

∂Ctc

∂t= − ∂Jcx

∂x− ∂Jcy

∂y︸︷︷︸diffusion

− ∂J ′cx

∂x−∂J ′

cy

∂y︸︷︷︸convection

[kg/(m3 · s)

](3.18)

where

J ′cx = flux of chloride ions due to convection in the x-direction (kg/m2·s)

J ′cy = flux of chloride ions due to convection in the y-direction (kg/m2·s)

By considering chloride binding in the diffusion process, the following modified Fick’s 2nd

law equation applicable to partially-saturated concrete results:

∂Cfc

∂t=

∂

∂x

(D∗

c

∂Cfc

∂x

)+

∂

∂y

(D∗

c

∂Cfc

∂y

)︸︷︷︸

diffusion

− ∂

∂x

(ωe · Cfc · Jmx

)− ∂

∂y

(ωe · Cfc · Jmy

)︸︷︷︸

convection

[kg/(m3 · s)

](3.19)

where Jmx and Jmy are the moisture fluxes along the x and y directions, respectively. Equa-

tion 3.19 is the governing equation of chloride ingress into partially-saturated concrete. Note

that it is written in terms of the free chloride concentration Cfc (kg/m3 of pore solution).

Once the binding isotherm that characterizes the specific concrete cementitious system is

known and the concentration of free chlorides is determined by solving Eq. 3.19, the total

amount of chlorides in concrete is calculated from Eq. 3.10.

A note on chloride binding and convection

The mechanism of chloride binding in the chloride ingress model presented in this thesis has

only been taken into account in the diffusion process (Eq. 3.19). However, this is a simpli-

fication of the problem, since it has been seen that chloride binding reduces the amount of

free chlorides available in the concrete pore solution, and, therefore, it lowers the amount

of dissolved chloride ions being transported by moisture flow. This phenomenon has been

observed by McCarter et al. (1992) in experimental studies on absorption of chloride con-

taminated water by concrete, wherein the authors reported that the chloride front moved


into the concrete at a slightly lower rate than the water in which chlorides were dissolved

(Fig. 3.6). One could consider the effect of chloride binding on the convective movement of

chloride ions by applying mass conservation to Eq. 3.10 and substituting into Eq. 3.18. By

simplifying the problem to one dimension, the following expression would result:(∂Cbc

∂Cfc

+ ωe

)∂Cfc

∂t=

∂

∂x

(Dc · ωe

∂Cfc

∂x

)︸︷︷︸

diffusion

− ∂

∂x

(ωe · Cfc · Jmx

)︸︷︷︸

convection

[kg/(m3 · s)

](3.20)

By defining the moisture flux Jmx in terms of the concrete pore relative humidity h as

−Dh (∂h/∂x), where Dh represents the concrete humidity diffusion coefficient, Eq. 3.20 is

reduced to:

∂Cfc

∂t=

∂

∂x

(D∗

c

∂Cfc

∂x

)︸︷︷︸

diffusion

+∂

∂x

(Cfc ·D∗

h

∂h

∂x

)︸︷︷︸

convection

[kg/(m3 · s)

](3.21)

where D∗c is the apparent chloride diffusion coefficient defined in Eq. 3.12 and D∗

h is an

apparent humidity diffusion coefficient in relation to the transport of dissolved chloride ions,

and it is given by:

D∗h =

Dh

1 + 1ωe

∂Cbc

∂Cfc

[m2/s

](3.22)

Thus the effect of chloride binding in the convective movement of chlorides in concrete would

be to reduce the diffusivity of moisture transporting dissolved chloride ions by an amount

which is representative of the binding capacity of the specific cementitious system (similar

to the reduction of Dc shown in Fig. 3.5).

3.2.5 Initial and boundary conditions

To obtain a solution for Eq. 3.19, the initial conditions of the reinforced concrete member as

well as the conditions existing at its boundaries need to be defined. It is common practice

in chloride penetration problems to assume that the concrete does not contain an initial

amount of chlorides, unless chlorides have been incorporated in the concrete from the mix

ingredients at the time of manufacture. However, if the background level of the concrete

under consideration is known, the value of Cfc at t = 0 can be specified.

The boundary conditions simulating chloride ingress into concrete are enforced by spec-

ifying the chloride flux crossing the concrete surface (Neumann boundary condition), which


Fig. 3.6: Advancement of water and chloride fronts into concrete: (a) 1 hour; (b) 4 hours; (c) 9hours; (d) 25 hours (reproduced from McCarter et al. (1992)).

depends on the concentration of applied de-icing salt, the concentration of chlorides at the

surface, and the influx of moisture in the concrete. Chloride transfer across the concrete

surface is calculated from (Saetta et al., 1993):

Jsc = Bc (Cfc − Cen)︸︷︷︸

diffusion

+Cen · Jsm︸︷︷︸

convection

[kg/(m2 · s)

](3.23)

where

Jsc = chloride flux normal to the concrete surface (kg/m2·s)

Bc = surface chloride transfer coefficient (m/s)

Cfc = concentration of free chlorides at the surface (kg/m3 of pore solution)

Cen = concentration of applied de-icing salt (kg/m3 of solution)

Jsm = entering moisture flux representing wetting of the concrete surface (m/s)

The convective term, which represents the contribution of water in transporting chloride ions

into the concrete, is set to zero when the surface is under drying conditions, since chlorides

will remain in the concrete surface as water evaporates (Saetta et al., 1993).


For the case of saturated concrete immersed in a chloride aqueous solution, Saetta et al.

(1993) reported the surface chloride transfer coefficient Bc to range between 1 to 6 m/s by

fitting of experimental data. A large value of Bc in this type of exposure conditions means

that the concrete pore solution and the external solution will reach equilibrium rather quickly,

i.e., Cfc = Cen at the surface. The authors carried out a sensitivity analysis of their diffusion

model in permanently saturated concrete to study the effect of Bc on the resulting chloride

profiles and concluded that variability in the values given to Bc was negligible to the final

results. However, no values were reported for the case of partially-saturated concrete.

The exposure of reinforced concrete highway structures to de-icing salts only occurs

during the winter time. Thus the application of chlorides to concrete is discontinuous. The

value of parameter Cen, i.e., the amount of applied chlorides coming from de-icing salts, is

evaluated in the proposed model by assuming a step function, wherein chlorides are applied

to the concrete for only a specified time during the year, as illustrated in Fig. 3.7(a). During

the months when Cen is zero, the chloride flux crossing the concrete surface Jsc given by

Eq. 3.23 is set to zero since chlorides will remain in the concrete unless washing-away effects

such as falling rain are taken into account.

Chloride concentrations coming from de-icing salts will depend on the salt usage and

exposure conditions. Saturated salts applied to the road environment will quickly get diluted

by melting of snow and ice and by splashing of the traffic. There are two types of conditions

for reinforced concrete structural members exposed to de-icing salts: (1) horizontal concrete

surfaces (e.g., parking slabs and bridge decks), and (2) traffic splash zones (e.g., bridge

columns). The different exposure conditions just mentioned can be reflected in the model

by values given to Cen, with higher values assigned for the first case. Reported chloride

concentrations near the concrete surface on bridge decks range from 1 to 18 kg/m3, suggesting

chloride surface build-up rates of approximately 0.9 to 1.3 kg/m3·yr (Berke and Hicks, 1994;

Amey et al., 1998). Such a wide range of values may be an indication that the chloride surface

concentration Cs depends not only on exposure conditions but also on material properties

of the concrete exposed, such as binder content and chloride binding characteristics.


Oct

.

Dec

.

Feb

.

Apr

il

June

Aug

.

Oct

.

Month of the year

Ap

plie

d c

hlo

rid

e sa

lts

C en

(a)

60

70

80

90

Oct

.

Dec

.

Feb

.

Apr

il

June

Aug

.

Oct

.Month of the year

Rel

ativ

e h

um

idit

y, h

(%)

h max

h min

(b)

-25

-15

-5

5

15

25

35

Oct

.

Dec

.

Feb

.

Apr

il

June

Aug

.

Oct

.

Month of the year

Tem

per

atu

re, T

(oC

)

T max

T min

(c)

Fig. 3.7: Environmental values for (a) the concentration of applied de-icing salts, (b) the atmo-spheric relative humidity, and (c) the daily average temperature.


3.3 Moisture transport

As seen in Section 3.2, chloride transport in partially-saturated concrete is coupled with

moisture diffusion in concrete (Eq. 3.19). Furthermore, several concrete properties, such as

the chloride diffusivity (Eq. 3.6), depend strongly on the concrete moisture content. Deter-

mining the moisture distribution within concrete is therefore of significant relevance if these

phenomena are to be considered in a chloride-ingress model.

Moisture flow in concrete can be expressed in two different ways (Xi et al., 1994). It can

be defined in terms of the free water content gradient, i.e.,

Jm = −Dw

(∂ωe

∂x+∂ωe

∂y

)[m/s] (3.24)

or in terms of the pore relative humidity gradient, i.e.,

Jm = −Dh

(∂h

∂x+∂h

∂y

)[m/s] (3.25)

where

Jm = moisture flux (m/s)

Dw = moisture diffusion coefficient (m2/s)

ωe = evaporable water content at depths x and y (m3 of water/m3 of concrete)

Dh = humidity diffusion coefficient (m2/s)

h = pore relative humidity at depths x and y

Coefficients Dw and Dh have different physical meanings and therefore take different values

for a given moisture flux Jm (Xi et al., 1994). The mass conservation equations corresponding

to Eqs. 3.24 and 3.25 are given by:

∂ωe

∂t= −∂Jmx

∂x− ∂Jmy

∂y=

∂

∂x

(Dw

∂ωe

∂x

)+

∂

∂y

(Dw

∂ωe

∂y

)[1/s] (3.26)

∂ωe

∂t=

∂ωe

∂h

∂h

∂t= −∂Jmx

∂x− ∂Jmy

∂y=

∂

∂x

(Dh

∂h

∂x

)+

∂

∂y

(Dh

∂h

∂y

)[1/s] (3.27)

where ∂ωe/∂h is the moisture capacity as defined by Xi et al. (1994).

According to Bazant and Najjar (1971, 1972), the use of Fick’s diffusion laws in terms

of ωe involves some error when hydration proceeds, since the distribution of pore volume


available to evaporable water (capillary porosity) becomes non-uniform with time. It is for

this reason that the pore relative humidity h instead of the free water content ωe will be

used here as the field variable describing moisture movement in concrete.

When both the progress of cement hydration and heat diffusion are taken into account

in characterizing moisture transport (Bazant and Najjar, 1972), Eq. 3.27 can be modified

into:∂ωe

∂h

∂h

∂t=

∂

∂x

(Dh

∂h

∂x

)+

∂

∂y

(Dh

∂h

∂y

)+∂hs

∂t+K

∂T

∂t[1/s] (3.28)

where

∂hs/∂t= pore relative humidity variation due to self-desiccation

K = hygrothermic coefficient (1/C), i.e., the change in humidity due to one degree

change in temperature T at a constant water content ωe

∂T/∂t = variation in temperature over time

For normal concretes, the drop in humidity h due to self-desiccation is known to be quite

small (hs ≥ 0.97) and can be neglected as an approximation even if hydration has not yet

terminated (Bazant and Najjar, 1972; Xi et al., 1994). If drying of concrete were formulated

in terms of evaporable water content rather than pore relative humidity, the term accounting

for the rate of loss in evaporable water content due to hydration would have to be included,

since this term is not negligible unless hydration has ceased. It is acknowledged that self-

desiccation is of importance in high-performance concrete (Neville, 1981); however, there is

not sufficient information that will enable to quantify the variation of hs with time. Because

of the aforementioned reasons, the term ∂hs/∂t in Eq. 3.28 is set to zero. Furthermore, the

effect of heat in moisture transport can also be ignored, since the contribution of this term has

appeared to be rather small for a normal range of temperatures (Bazant and Thonguthai,

1978; Akita et al., 1997). Thus the equation governing moisture transport in concrete is

reduced to:∂ωe

∂h

∂h

∂t=

∂

∂x

(Dh

∂h

∂x

)+

∂

∂y

(Dh

∂h

∂y

)[1/s] (3.29)

3.3.1 Humidity diffusion coefficient

In order to take into account the dependence of Dh on the level of pore relative humidity

h, temperature T , and degree of hydration of the concrete, characterized by an equivalent


hydration period te, the humidity diffusion coefficient is modified according to (Bazant and

Thonguthai, 1978):

Dh = Dh,ref · F1(h) · F2(T ) · F3(te)[m2/s

](3.30)

where Dh,ref is the humidity diffusion coefficient (m2/s) determined at some specified refer-

ence conditions (input parameters). Bazant and Najjar (1971, 1972) reported typical values

of Dh to range from 1.157×10−10 to 4.630×10−10 m2/s (w/cm ranging from 0.45 to 0.8).

Function F1(h) in Eq. 3.30 accounts for the dependence of Dh on the pore relative hu-

midity of the concrete and is evaluated from (Bazant and Najjar, 1971, 1972; Bazant and

Thonguthai, 1978):

F1(h) = αo +1− αo

1 +(

1−h1−hc

)n (3.31)

where

αo = parameter that represents the ratio of Dh,min/Dh,max. Its values have been

found to be quite close for different concretes and equal to 0.05 (Bazant and

Najjar, 1971, 1972).

hc = humidity at which Dh drops halfway between its maximum and minimum

values, found to be 0.75 for different concretes and cement pastes (Bazant and

Najjar, 1971, 1972).

n = parameter characterizing the spread of the drop in Dh as illustrated in Fig. 3.8.

Its value was found to range from 6 to 16 by fitting of experimental data

(Bazant and Najjar, 1971, 1972).

Bazant and Najjar (1971) found the function expressed by Eq. 3.31 to give the best fit over

the whole range of their experimental data. Its graphic representation is shown in Fig. 3.8.

Function F2(T ), which considers the dependence of Dh on the temperature level of the

concrete, is obtained from the expression given by Eq. 3.7 for the chloride diffusion coefficient,

U being in this case the activation energy of the moisture diffusion process. Typical values

of U/R found in the literature range from 2,700 K (Bazant and Thonguthai, 1978; Saetta

et al., 1993) to 4,700 K (Saetta et al., 1993).

The humidity diffusion coefficient Dh also depends on the degree of hydration attained in

the concrete in question. The hydration degree of the concrete is represented by an equivalent


0.00

0.25

0.50

0.75

1.00

0 0.2 0.4 0.6 0.8 1

Pore relative humidity, h

F1(

h)

n =6

αo h c

n =16

Fig. 3.8: Dependence of Dh on pore relative humidity according to Eq. 3.31 for n = 6 and n = 16(αo = 0.05 , hc = 0.75).

hydration period te, which is evaluated from (Bazant and Najjar, 1972):

te = to +

∫ t

to

βT βh dt [s] (3.32)

where to is the time of first exposure of the concrete, and βT and βh are functions that reflect

the dependence of te on temperature T and pore relative humidity h. These semi-empirical

functions are given by (Bazant and Najjar, 1972):

βT = exp

[Uh

R

(1

Tref

− 1

T

)](3.33)

βh =[1 + (7.5− 7.5h)4

]−1(3.34)

where

Uh = activation energy of hydration (kJ/mol)

R = gas constant (8.314×10−3 kJ/K·mol)

Tref = reference temperature, taken as 296 K

T = actual temperature in the concrete (K)


The ratio Uh/R has been evaluated by Saetta et al. (1993) according to:

Uh

R= 4600

[30

(T − 263)

]0.39

[K] (3.35)

Once the degree of hydration te has been evaluated, function F3(te) is calculated from (Saetta

et al., 1993):

F3(te) = 0.3 +

√13

te(3.36)

3.3.2 Adsorption isotherm

The concrete moisture distribution given in terms of pore relative humidity h (Eq. 3.29) only

refers to the thermodynamic state of the pore water, but it does not provide the amount of

free water ωe present in the capillary system (Elkey and Sellevold, 1995). However, because

the moisture diffusion process in concrete is so slow, the various phases of water in the

concrete pores (vapour, capillary water and adsorbed water) are almost in thermodynamic

equilibrium at any time (Bazant and Najjar, 1972; Bazant and Thonguthai, 1978). Thus

the pore relative humidity and evaporable water content at a constant temperature can be

related by sorption isotherms.

The Brunauer-Skalny-Bodor model (Brunauer et al., 1969), also called BSB model or

the three-parameter BET model (the latter named after Brunauer, Emmett and Teller),

has been selected here to define the adsorption isotherm that relates h to ωe. Empirical

expressions for the three parameters of this model have been developed by Xi et al. (1994)

from a parametric analysis of available adsorption test data over the whole range of pore

relative humidity (0 ≤ h ≤ 1). The adsorption isotherm according to the BSB model is

given by:

ωe =C k Vm h

(1− kh) [1 + (C − 1) kh][g of water/g of cementing material] (3.37)

where parameters C, k, and Vm (monolayer capacity) are evaluated from the following ex-


pressions:

C = exp

(855

T

)(3.38)

k =(1− 1/n)C − 1

C − 1, 0 < k < 1 (3.39)

n =

(2.5 + 15

te

)(0.33 + 2.2w/cm) if te > 5 days and 0.3 < w/cm < 0.6,

5.5 (0.33 + 2.2w/cm) if te ≤ 5 days,

0.99(2.5 + 15

te

)if w/cm ≤ 0.3,

1.65(2.5 + 15

te

)if w/cm ≥ 0.6.

(3.40)

Vm =

(0.068− 0.22

te

)(0.85 + 0.45w/cm) if te > 5 days and 0.3 < w/cm < 0.6,

0.024 (0.85 + 0.45w/cm) if te ≤ 5 days,

0.985(0.068− 0.22

te

)if w/cm ≤ 0.3,

1.12(0.068− 0.22

te

)if w/cm ≥ 0.6.

(3.41)

where

T = temperature in the concrete (K)

te = equivalent hydration age (days), given by Eq. 3.32

w/cm= water-to-cementitious ratio of the concrete

The expressions given by Eqs. 3.38 to 3.41 were developed by Xi et al. (1994) by best-fitting

of experimental data. Figure 3.9 illustrates the effect of varying te and w/cm on the selected

adsorption isotherm.

Even though the adsorption isotherm given by Eq. 3.37 was based on test data for cement

paste (Xi et al., 1994), it will be used here to simulate vapour adsorption in concrete, since

experimental work done on this area is very limited. Furthermore, the expression given by

Eq. 3.37 has been assumed to be valid for both adsorption (wetting process) and desorption

(drying process) conditions. It is well known that adsorption and desorption curves of typical

moisture isotherms follow different paths, the former giving lower values of ωe for a given

value of h, and hysteresis loops are observed under wetting and drying cycles. A proper


0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.0 0.2 0.4 0.6 0.8 1.0


ω

e (

g/g

)

T = 25oCw/cm = 0.4

5 days

7 days28 days

180 days

(a) Influence of te.

0.00

0.06

0.12

0.18

0.24

0.30

0.36

0.0 0.2 0.4 0.6 0.8 1.0


ω

e (

g/g

)

T = 25oCt e = 28 days

0.3

0.4

0.5

0.6

(b) Influence of w/cm.

Fig. 3.9: Water vapour adsorption isotherms.


simulation of this phenomenon would require the formulation of hysteresis rules for wetting

and drying conditions, which is beyond the scope of this work. Xi et al. (1994) justified

the use of the BSB model to both processes by arguing that the shape of both curves is

approximately the same (even though they give different values).

The pore relative humidity in concrete is influenced by the concentration and types of

dissolved ions present in the pore solution (Elkey and Sellevold, 1995). In spite of the fact

that capillary pores contain a highly concentrated ionic solution, it is assumed here that

the water entering concrete and present in the pore solution is pure water, thus neglecting

the effect of ionic concentrations. For the same conditions, the change of relative humidity

above a concentrated ionic solution is different from that above pure water. This implies that

sorption isotherms would be different depending on the type of solution of the capillary pores.

The empirical data that relate free water content to levels of relative humidity in concrete is

very limited and, therefore, the assumption that water in concrete is pure becomes necessary.

The moisture capacity term in Eq. 3.29, ∂ωe/∂h, is obtained as the slope of the equilib-

rium adsorption isotherm given by Eq. 3.37, i.e.,

∂ωe

∂h=CkVm + ωeh [1 + (C − 1) kh]− ωeh (1− kh) (C − 1)

(1− kh) [1 + (C − 1) kh](3.42)

where ∂ωe/∂h is given in g of water/g of cementing material. As shown in Fig. 3.10, the

moisture capacity is not a constant, as it was assumed in previous models of drying of

concrete (Bazant and Najjar, 1971; Bazant and Thonguthai, 1978). The curve representing

the moisture capacity first drops at low values of h, then it remains almost constant with

increasing pore relative humidity, and it finally increases again at high values of h. According

to Xi et al. (1994), the transition from the initial drop to the constant region may correspond

to the limit of the monolayer capacity Vm, whereas the transition from the constant region

to the final increase may correspond to the beginning of capillary condensation.


In order to obtain a solution for Eq. 3.29, the pore relative humidity distribution in the

concrete at time t = 0 as well as the atmospheric relative humidity level existing at the

external environment in contact with the concrete need to be specified. Moisture transfer


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0


Mo

istu

re c

apac

ity

T = 25oCt e = 28 daysw/cm = 0.4

∂ω∂

e

h

Fig. 3.10: Moisture capacity ∂ωe/∂h.

from the external environment to the concrete surface (wetting) or viceversa (drying) is

evaluated from (Bazant and Thonguthai, 1978; Saetta et al., 1993; Akita et al., 1997):

Jsm = Bh (h− hen) [m/s] (3.43)

where

Jsm = moisture flux normal to the concrete surface (m/s)

Bh = surface moisture transfer coefficient (m/s)

h = pore relative humidity at the concrete surface

hen = relative humidity in the surrounding environment

When the moisture flux Jsm in Eq. 3.43 is positive, moisture is transferred from the concrete

surface to the environment (h > hen), which is representative of drying conditions of the

concrete surface. When the value of Jsm is negative, moisture is being transferred from the

environment to the concrete surface (hen > h), which simulates situations of wetting of the

concrete. A perfectly sealed concrete surface where Jsm = 0 can be simulated by assuming

Bh = 0, whereas perfect moisture transmission at the surface (h = hen) is a limiting case for

Bh →∞ (Bazant and Thonguthai, 1978).


Moisture diffusion into or from the concrete surface is believed to depend on the number

and size of capillary pores open to the surface, which in turn depend on the initial mix

proportions of the concrete at hand (Akita et al., 1997). Thus the surface moisture transfer

coefficient Bh depends on the type of concrete being modelled. There is not much information

available in the literature as to the range of values this parameter takes for various concrete

types and different environmental conditions. Akita et al. (1997) have reported Bh to range

between 2.43× 10−7 and 4.17× 10−7 m/s after fitting their experimental data on drying of

concrete to calculated values.

To simulate the seasonal variation of the daily average atmospheric relative humidity

hen, a sinusoidal variability in time of this parameter has been considered (Saetta et al.,

1993; Tang and Nilsson, 1996b), with the periodicity established over the period of one year

(Fig. 3.7(b)). The variable hen is thus evaluated from the following expression:

hen =hmin + hmax

2+hmin − hmax

2sin(2πt) (3.44)

where

hmin = minimum daily average atmospheric relative humidity in a year

hmax = maximum daily average atmospheric relative humidity in a year

t = time (years)

3.4 Heat transfer

In order to take into account the effects of temperature on the mode of transport of chloride

ions and moisture in concrete (Eq. 3.7), the temperature distribution throughout the concrete

also needs to be determined. Heat flow in concrete due to a temperature gradient can be

quantified by Fourier’s heat conduction law, which is given by:

q = −λ(∂T

∂x+∂T

∂y

) [W/m2] (3.45)

where

q = conduction heat flux (W/m2)

λ = thermal conductivity in concrete (W/m·C)


T = temperature in the concrete at depths x and y (C)

The negative sign in Eq. 3.45 indicates that heat is transferred in the direction of decreasing

temperature. Although the thermal conductivity in concrete λ slightly increases with water

content and is affected by temperature, variations are small and it can be assumed con-

stant (Bazant and Thonguthai, 1978). Typical values of thermal conductivity for saturated

concrete range between 1.4 and 3.6 W/m·C (Neville, 1981).

The temperature profile in concrete is determined by applying the energy conservation

requirement, i.e.,

%c cq∂T

∂t=

∂

∂x

(λ∂T

∂x

)+

∂

∂y

(λ∂T

∂y

) [W/m3] (3.46)

where

%c = density of concrete (kg/m3)

cq = concrete specific heat capacity (J/kg·C)

T = temperature in the concrete at depths x and y after time t (C)

For the sake of simplicity, it has also been assumed that the density %c and specific heat

capacity cq of concrete remain constant with changes in moisture content and temperature

(Saetta et al., 1993). The common range of values of the specific heat capacity for ordinary

concrete is between 840 and 1170 J/kg·C (Neville, 1981).


The solution of Eq. 3.46 depends on the temperature level existing at the boundaries of the

reinforced concrete member. Heat transfer across the concrete surface is calculated from:

qs = BT (T − Ten)[W/m2] (3.47)

where

qs = convective heat flux across the concrete surface (W/m2)

BT = convection heat transfer coefficient (W/m2·C)

T = temperature at the concrete surface (C)


Ten = temperature in the surrounding environment (C)

When the convective heat flux qs in Eq. 3.47 is positive, heat is transferred from the concrete

surface to the environment (T > Ten), whereas when the value of qs is negative, heat is

transferred from the environment to the concrete surface (Ten > T ).

The convection heat transfer coefficient BT depends on the conditions at the boundary.

There is a lack of information regarding the range of values this parameter takes for various

concrete types exposed to different environmental conditions. Bazant and Thonguthai (1978)

reported BT to be 0.07 W/m2·C after obtaining a good fit from experimental data. This

value was used by Saetta et al. (1993) when coupling heat transfer to chloride ingress in

concrete in their numerical model. However, much higher values (6.2–9.3 W/m2·C) were

reported by Khan et al. (1998), who carried out an experimental study to observe the effects

of different curing conditions and early form stripping on the temperature development in

structural concrete members.

To simulate the seasonal variation of the daily average environmental temperature Ten, a

sinusoidal variability in time of this value has been considered (Saetta et al., 1993; Tang and

Nilsson, 1996b), with the periodicity established over one year (Fig. 3.7(c)). The variable

Ten is thus evaluated from the following expression:

Ten =Tmin + Tmax

2+Tmin − Tmax

2sin(2πt) [C] (3.48)

where

Tmin = minimum daily average temperature in a year (C)

Tmax = maximum daily average temperature in a year (C)

t = time (years)

In order to obtain a solution for Eq. 3.46, the temperature distribution in the concrete

at time t = 0 also needs to be specified.

3.5 Kinetics of corrosion reactions

Once the chloride concentration at the reinforcing steel reaches the specified threshold value,

a difference in electrical potential between anodic and cathodic areas is established, and


Ecorr

IcorrIoc Ioa

Ecell

Potential of steel

Fe Fe+++2e-anodic polarization curve

O2+H2O+2e- 2OH-cathodic polarization curve

Eco

ηc=Ec-Eco

ηa=Ea-Eao

Eao

log current

Fig. 3.11: Evans diagram for the anodic and cathodic processes taking place on the reinforcingsteel surface.

corrosion is initiated. Due to this differential in electrical potential, iron is oxidized to ferrous

ions at the anode and oxygen is reduced releasing hydroxyl ions at the cathode according to:

Anode half-cell reaction: Fe → Fe++ + 2e− (3.49)

Cathode half-cell reaction: O2 + 2H2O + 4e− → 4OH− (3.50)

It is assumed here that the main reaction taking place at the cathode is oxygen reduction,

which predominates in neutral and alkaline solutions (hydrogen evolution, 2H+ +2e− → H2,

prevails in acid conditions).

As the corrosion cell is formed, a current flows through each of the half-cells, shifting

each of their potentials away from the reversible or equilibrium potential. This is known

as polarization. The kinetics of the anode and cathode half-cell reactions depend on the

degree of deviation from equilibrium. The relationship between the half-cell potential and

the current flowing through it forms the polarization curve of the electrode, known as the

Evans diagram (see Fig. 3.11). The shape of these curves gives an idea of the morphology of

the corrosion attack, the rate-controlling step of the corrosion process, and the equilibrium

corrosion current or corrosion rate (Rosenberg et al., 1989). The corrosion potential Ecorr as

well as the corrosion current Icorr are defined by the intersection of the anodic and cathodic

branches of the polarization curves shown in Fig. 3.11.

The shapes of the anodic and cathodic polarization curves depend on the kind of po-

larization process taking place. There are four types of polarization: ohmic or resistance


polarization, concentration polarization, activation polarization, and crystallization polar-

ization. Only the effects of concentration and activation polarization will be taken into

consideration here. Concentration polarization, also known as diffusion polarization, oc-

curs when the electrode reaction is controlled by slow mass transfer of reactants from the

solution to the electrode surface or of reaction products from the electrode surface to the

solution. Activation polarization arises when there are activation energy barriers hindering

the transfer of charge across the electrode/solution interface.

In the model presented here, the rates of iron oxidation at the anode and oxygen reduction

at the cathode are related to the equilibrium potential of each half-cell reaction through

Butler-Volmer kinetics (Harker et al., 1987; Walton and Sagar, 1987; Naish et al., 1990;

Kranc and Sagues, 1994). The kinetics of oxidation of iron are given by:

ia = ioa exp

(2.3

E − Eoa

βa

) [A/m2] (3.51)

where

ia = current density of the iron oxidation reaction (A/m2)

ioa = exchange current density for iron dissolution, i.e., the equilibrium current at

which iron dissolves and deposits at the same rate, taken as 3.75× 10−4 A/m2

(Kranc and Sagues, 1994)

E = potential at the concrete pore solution immediately next to the steel surface

(V)

Eoa = equilibrium potential of the anodic reaction (V)

βa = activation tafel slope for the anodic reaction (V)

Equation 3.51 assumes that the anodic reaction is subject only to activation polarization, i.e.,

the kinetics of iron oxidation are limited by the slower step in the transfer of electrochemical

charge. It is assumed here that ions present in the concrete pore solution are at an activity of

one and standard conditions prevail. Because of this assumption, the equilibrium potential

Eoa is taken as the standard electrode potential, which is equal to -0.682 V vs. SCE. Reported

values for the anodic tafel slope range from 0.06 to 0.524 V (Kranc and Sagues, 1994; Yalcyn

and Ergun, 1996). Yalcyn and Ergun (1996) observed that βa increases over time when

chloride ions are present in the concrete, from 0.338 V at 1 day to 0.480 V at 90 days. An


average value of 0.409 V will be used here. Similar to the case of the anodic reaction, the

kinetics of oxygen reduction are given by:

ic = iocCo

Cso

exp

(2.3

Eoc − E

βc

) [A/m2] (3.52)

where

ic = current density of the oxygen reduction reaction (A/m2)

ioc = exchange current density for the cathodic reaction, taken as 1.25× 10−5 A/m2

(Kranc and Sagues, 1994)

Co = dissolved oxygen concentration at the steel surface (kg/m3 of solution)

Cso = dissolved oxygen concentration at the external concrete surface (kg/m3 of so-

lution)

Eoc = equilibrium potential of the cathodic half-cell reaction, taken as the standard

electrode potential, i.e., 0.160 V vs. SCE

βc = activation tafel slope for the cathodic reaction

Equation 3.52 takes into consideration both activation and concentration polarization effects

for the oxygen reduction reaction. The kinetics of the cathodic reaction can also be limited

by depletion of oxygen at cathodic sites. Reported values for the cathodic tafel slope range

from 0.160 to 0.394 V (Naish et al., 1990; Kranc and Sagues, 1994; Yalcyn and Ergun, 1996).

Yalcyn and Ergun (1996) also observed a slight increase in βc when chlorides are present in

concrete, as opposed to a concrete without chlorides, and reported a range of 0.309–0.394

V. A value of 0.350 V will be used here.

The reinforcing steel is freely corroding when the polarization taking place at the anode

and the cathode is enough to make their polarization curves meet as illustrated in Fig. 3.11.

At this point, the corrosion system develops a new equilibrium corrosion potential Ecorr and

corrosion current Icorr. The intersection of the two electrode polarization curves represents

conditions at which the anodic and cathodic currents are equal (but opposite in polarity)

and no net current flows across the steel/concrete interface, i.e.,

Icorr = ia · Aa = ic · Ac (3.53)


where Aa and Ac denote the areas where the anodic and cathodic reactions take place,

respectively. By substituting Eqs. 3.51 and 3.52 into the above expression, an estimate of

the equilibrium corrosion potential can be obtained from:

Ecorr =1

βa + βc

[βa · βc

2.3ln

(ioc

ioa

Co

Cso

Ac

Aa

)+ βcE

oa + βaE

oc

](3.54)

The new equilibrium corrosion potential Ecorr lies somewhere between the equilibrium poten-

tials of the anodic and cathodic processes. The resulting corrosion rate depends not only on

the magnitude of the driving force (the difference between the two equilibrium potentials),

but also on the kinetics of the anodic and cathodic processes (Eqs. 3.51 and 3.52).

A better representation of the corrosion process would be obtained by computing the

steady-state current flow in the concrete pore solution (Eq. 2.34), where the influence of the

concrete electrical resistivity is considered. However, because the model presented here is

based on a cross-section of a reinforced concrete member where the reinforcing steel is consid-

ered to be active (i.e., chlorides have reached the threshold concentration), the condition of

conservation of electric charge cannot be enforced. To do so, the model should be formulated

on a longitudinal reinforced concrete member. By assuming axial symmetry, Eq. 2.34 could

then be solved by means of a two-dimensional finite element model formulated in cylindrical

coordinates.

3.6 Oxygen transport

When corrosion begins, the oxygen needed for the cathodic reaction is drawn from the supply

initially available in concrete. However, this initial supply soon gets exhausted, and oxygen

must be drawn from the surrounding environment for corrosion to proceed. Therefore,

oxygen transport within concrete needs to be modelled in order to determine the amount

of dissolved oxygen reaching the steel/concrete interface. Following a similar approach as

in the previous cases, conservation of mass for the amount of dissolved oxygen in concrete

requires:∂Co

∂t= −∂Jox

∂x− ∂Joy

∂y

[kg/(m3 · s)

](3.55)

where


Co = dissolved oxygen concentration in the concrete pore solution at depths x and

y after time t (kg/m3 of pore solution)

Jox = flux of oxygen due to diffusion in the x-direction (kg/m2·s)

Joy = flux of oxygen due to diffusion in the y-direction (kg/m2·s)

The flux of oxygen in concrete due to diffusion Jo is also given by Fick’s 1st law of

diffusion, and it is expressed as:

Jo = −Do

(∂Co

∂x+∂Co

∂y

) [kg/(m2 · s)

](3.56)

where Do is the oxygen diffusion coefficient (m2/s).

Substituting Eq. 3.56 into the mass conservation equation given by Eq. 3.55 and expand-

ing terms yields:

∂Co

∂t=

∂

∂x

(Do

∂Co

∂x

)+

∂

∂y

(Do

∂Co

∂y

) [kg/(m3 · s)

](3.57)

Equation 3.57 is the governing equation describing the process of oxygen transport in con-

crete.

3.6.1 Oxygen diffusion coefficient

The rate of oxygen diffusion through concrete Do is significantly affected by the extent

to which concrete is saturated with water (see Fig. 3.12). In wet concrete, oxygen will

primarily be diffusing in solution, while in partially-saturated concrete, oxygen will diffuse

partly through the gas phase and partly through the pore solution, with a reported diffusivity

through water 104–105 times less than through the gas phase (Harker et al., 1987; Lopez and

Gonzalez, 1993). For oxygen to be consumed in the cathodic reaction, however, it has to be

in a dissolved state. To take into account the dependence of the oxygen diffusivity Do on

the degree of water saturation ωe/ωsat of the concrete, the logarithm of the oxygen diffusion

coefficient is reduced linearly with increasing degree of pore saturation according to:

log (D2o) = log (D1

o)−[8.15 ·

(ω2

e − ω1e

ωsat

)](3.58)

where D1o and D2

o (m2/s) are the concrete oxygen diffusivities at free water contents ω1e and

ω2e , respectively, and ωsat is the amount of evaporable water under saturation conditions (m3


-14

-12

-10

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1

Degree of pore saturation, ω e /ω sat

log

( Do

)

Bazant (1979a)

Harker et al. (1987)

Balabanic et al. (1996)

Fig. 3.12: Dependence of Do on the concrete moisture content.

of water/m3 of concrete). The expression given by Eq. 3.58 has been obtained by linear

regression from oxygen diffusivities reported in the literature for different moisture contents

(Fig. 3.12).


In order to solve Eq. 3.57, the initial conditions of the reinforced concrete member as well as

the conditions existing at the boundaries need to be defined. The amount of oxygen initially

available in concrete is given by:

Co (t = 0) = 0.005 kg/m3 of pore solution (3.59)

The value of Co (t = 0) refers to concrete mixed with typical fresh water, which normally

contains 0.005 kg/m3 of dissolved oxygen (Bazant, 1979a).

The boundary condition for the oxygen transport equation at the external surface is

obtained from:

Cso = 8.576× 10−3 kg/m3 of solution (3.60)

where Cso is the oxygen concentration at the external concrete surface (kg/m3 of solution).


The value of Cso represents the quantity of dissolved oxygen when atmospheric air is saturated

(Harker et al., 1987; Kranc and Sagues, 1994; Yokozeki et al., 1997).

It is assumed that the oxygen supply from the concrete and the surrounding environment

equals the amount of oxygen consumed by the cathodic reaction and by the oxidation of

ferrous hydroxide at the anode. Thus the sum of oxygen fluxes due to diffusion and convection

at the steel/concrete interface must be equal to the sum of the rates of oxygen consumption

at the cathode and the anode. The rate of oxygen consumption at the cathode J co can be

determined from:

J co = − ic

z F

[mol/(m2 · s)

](3.61)

where

J co = rate of O2 consumption at the cathode (mol/m2·s)

ic = current density of the cathodic reaction, given by Eq. 3.52 (A/m2)

z = ionic valence of the ion going into solution

F = Faraday’s constant (= 96,486.7 C/mol of electrons)

The negative sign in Eq. 3.61 denotes that O2 is being consumed as opposed to being pro-

duced. According to Eq. 3.50, 1 mol of O2 (32 g/mol) is consumed at the cathode to yield

4 moles of OH−. By substituting the corresponding values for z (= 4) and F into Eq. 3.61

and converting moles into kg, J co is evaluated from:

J co = −8.291× 10−8 ic

[kg/(m2 · s)

](3.62)

The hydroxyl ions released at the cathode combine with the ferrous ions released at the

anode to form ferrous hydroxide according to:

Fe++ + 2OH− → Fe(OH)2 (3.63)

The rate of ferrous hydroxide production at the anode Jfh is evaluated from:

Jfh =iaz F

=ia

2 × 96, 486.7× 0.089845

kg

mol= 4.656× 10−7 ia

[kg/(m2 · s)

](3.64)

where

Jfh = rate of Fe(OH)2 production at the anode (kg/m2·s)


ia = current density of the anodic reaction, given by Eq. 3.51 (A/m2)

Ferrous hydroxide is further oxidized into ferric hydroxide, known as hydrated red rust,

according to:

4Fe(OH)2 + O2 + 2H2O → 4Fe(OH)3 (3.65)

It is assumed here that the main corrosion product formed is Fe(OH)3. As seen in Eq. 3.65,

4 moles of Fe(OH)2 (359.38 g) combine with 1 mol of O2 (32 g/mol) to form 4 moles of

hydrated red rust or Fe(OH)3. The rate of oxygen consumption at the anode Jao is given by:

Jao = − 32

359.38Jfh = −0.089 Jfh = −4.144× 10−8 ia

[kg/(m2 · s)

](3.66)

where Jfh is given by Eq. 3.64 and ia is given by Eq. 3.51.

Thus the flux of oxygen reaching the steel surface needs to be equated to the sum of the

rates of oxygen consumption at the cathode and the anode, i.e.,

J co + Ja

o = −8.291× 10−8 ic − 4.144× 10−8 ia[kg/(m2 · s)

](3.67)

Since ic = Aa/Ac · ia, the above expression becomes:

J co + Ja

o = −(

8.291× 10−8 · Aa

Ac

+ 4.144× 10−8

)ia

[kg/(m2 · s)

](3.68)

where ia is given by Eq. 3.51.

3.7 Rate of rust production

According to Eq. 3.65, 1 mol of Fe(OH)2 (89.845 g/mol), produce 1 mol of Fe(OH)3 (106.845

g/mol). The rate of production of hydrated red rust (Fe(OH)3) at the anode is estimated

from (Bazant, 1979a):

Jr =106.845

89.845Jfh = 1.189 Jfh = 5.536× 10−7 ia

[kg/(m2 · s)

](3.69)

where

Jr = rate of hydrated red rust production at the anode (kg/m2·s)

Jfh = rate of ferrous hydroxide production at the anode, given by Eq. 3.64 (kg/m2·s)


ia = current density of the anodic reaction, given by Eq. 3.51 (A/m2)

It should be noted that the rate of rust production given by Eq. 3.69 is assumed to be pres-

sure independent for simplicity. In reality, however, pressure applied around the corroding

reinforcing bar can slow down the kinetics of the electrochemical reactions.

Chapter 4

Mechanical Effects of Corrosion

4.1 Introduction

Corrosion damage in reinforced concrete structures is usually manifested by rust-staining of

the surface and cracking and spalling of the concrete cover. The transformation of metallic

iron into corrosion products is accompanied by an increase in volume due to the lower density

of iron oxides compared to that of iron metal itself. Depending on the oxidation state, iron

can expand as much as six times its original volume (see Fig. 1.2). This increase in volume

exerts tensile stresses in the concrete surrounding the rebars, and when these stresses exceed

the concrete tensile strength, cracking and spalling of the concrete cover occur.

Prediction of the damage caused by corroding reinforcing bars requires the solution of

the state of stress in the surrounding concrete. To do so, the volume expansion of the

corrosion products can be introduced by means of applying a uniform internal pressure

around the steel/concrete interface. To determine the cracking resistance of the concrete

cover, a concrete ring model treated as a thick-wall cylinder with internal pressure was used.

The approach adopted here follows the model proposed by Tepfers (1979) to analyze the state

of stress in concrete due to bond forces. Modified versions of this type of model have been

implemented by Noghabai (1995) and Seki and Oya (1996) to study the splitting behaviour

of concrete due to reinforcing steel corrosion and its effect on bond capacity.

In this study, concrete that is cracked due to effects other than corrosion was not consid-

ered. It was assumed that the final corrosion product formed is hydrated red rust Fe(OH)3

87

Mechanical effects of corrosion 88

(Bazant, 1979b), whose volume is four times larger than that of steel as shown in Fig. 1.2.

4.2 Estimate of expansion

In order to calculate the increase in volume due to the formation of corrosion products, the

amount of hydrated red rust produced at anodic sites needs first to be determined.

The rate of rust production Jr calculated

initial section

attackpenetration

d∆d/2

cracks

corrosionbuild-up

Fig. 4.1: Increase in volume due to accumula-tion of corrosion products (adapted from An-

drade and Alonso (1996a)).

from Eq. 3.69 is given per unit surface area

of the reinforcing bar. The mass of hydrated

red rust formed per unit length of rebar, mr, is

therefore given by:

mr = Jr · (t− ti)︸︷︷︸tp

·πd [kg/m] (4.1)

where Jr is the rate of rust production at the

anode (kg/m2·s), given by Eq. 3.69, t− ti is the

time elapsed since corrosion initiation (s), and

d is the rebar diameter (m). Equation 4.1 assumes that the rate of production of hydrated

red rust Jr is constant up to the time level t considered. As time proceeds and hydrated red

rust precipitates around the reinforcing bar perimeter, the accumulated mass of hydrated

red rust at a given time will result from the cumulative sum of Eq. 4.1 applied to each time

step. The mass of steel consumed to produce mr is calculated from the ratio of molecular

weights of metal iron and hydrated red rust (Bazant, 1979b), i.e.,

ms = 0.523mr [kg/m] (4.2)

where ms is the mass of steel consumed in the corrosion process, also given per unit length

of rebar (kg/m).

The increase in volume ∆V around the reinforcing bar due to accumulation of corrosion

products (see Fig. 4.1) is calculated from:

∆V =mr

%r

− ms

%s

= mr

(1

%r

− 0.523

%s

) [m2

](4.3)

where


∆V = increase in volume per unit length of rebar due to formation of hydrated red

rust (m2)

%s = density of steel (7.85× 103 kg/m3)

%r = density of hydrated red rust, which is one-fourth of %s (1.96× 103 kg/m3)

It is assumed here that all rust products formed create a pressure around the reinforcing bar

and therefore contribute to the volume increase given by Eq. 4.3. Although the geometry of

corrosion build-up is usually irregular and it is very difficult to determine, in this model rust

is considered to be equally distributed along the rebar perimeter as illustrated in Fig. 4.1.

The volume change per unit volume or “dilation” is thus given by:

εv =∆V

V=

4mr

πd2

(1

%r

− 0.523

%s

)(4.4)

which is related to the internal pressure pi created around the steel/concrete interface by:

εv =2 pi

Er

(1 + νr) (1− 2νr) (4.5)

where

Er = elastic modulus of rust, Er = 3(1− 2νr)Kr

Kr = bulk modulus of rust

νr = Poisson’s ratio of rust

Due to the lack of information on the elastic properties of rust, these are assumed to be

equal to those of liquid water (Molina et al., 1993), i.e., νr = 0.499 and Kr = 2 GPa .

4.3 Concrete ring model

The analysis of the state of stress in concrete due to the expansion of hydrated red rust

was done by means of a concrete ring model (Tepfers, 1979). The radial stresses on the

concrete resulting from the volume increase of hydrated red rust can be regarded as a uniform

pressure acting on a thick-walled concrete ring, whose thickness is determined from the

thinnest concrete cover around the rebar giving the shortest crack path from the bar to the

free surface, i.e., the smallest of cx or cy in Fig. 4.2. The concrete ring around the rebar


σr

σr+dσr

σt

σt

pi

d cx

cy

r

Fig. 4.2: Concrete ring model.

approximates the effect of the surrounding concrete. In fact, the stresses acting on the cover

of the concrete cylinder will approximately correspond to the actual stresses, whereas the

stresses inside the ring will be lower (Bazant, 1979b; Tepfers, 1979). Since the appearance

of cover cracks will be determined by the stresses on the cylinder cover, the concrete ring

model can be used for analysis purposes (Tepfers, 1979).

The internal pressure acting at the steel/concrete interface is balanced by rings of tensile

stresses in concrete as illustrated in Fig. 4.2. The tensile splitting strength of concrete will

be first reached at the inner surface of the concrete thick-wall cylinder, and at that point

a crack is initiated. However, the ultimate tensile load capacity of the concrete cylinder is

not reached yet. As the internal pressure is increased because of accumulation of hydrated

red rust around the reinforcing bar, the outer tensile rings will be stressed further up to the

concrete tensile splitting strength. The ultimate tensile load of the cylinder will be reached

when the crack propagates through the entire concrete cover. From an analytical point

of view, this process is treated in two stages: (1) an uncracked elastic stage, prior to the

initiation of the first crack at the steel/concrete interface; and, (2) a partly-cracked elastic

stage, during which the crack propagates through the concrete cover.

In the following, it was assumed that concrete is an homogeneous elastic material. It was

further assumed that the concrete ring is of sufficiently great length compared to its radius,

so that the strain in the axial direction is zero (i.e., εz = 0). This assumption simulates a


state of plane-strain and corresponds to stress conditions associated with a splitting mode

of failure.

4.3.1 Uncracked elastic stage

The stresses in a concrete thick-walled cylinder of inner radius d/2 and outer radius c+ d/2

subjected to a uniform internal pressure pi are given by (Timoshenko, 1956):

σr =(d/2)2 pi

(c+ d/2)2 − (d/2)2

(1− (c+ d/2)2

r2

)[MPa] (4.6)

σt =(d/2)2 pi

(c+ d/2)2 − (d/2)2

(1 +

(c+ d/2)2

r2

)[MPa] (4.7)

where

σr = radial stress in concrete, always compressive (MPa)

σt = tangential stress in concrete, always tensile (MPa)

d = rebar diameter (m)

c = concrete cover (m)

pi = internal pressure resulting from Fe(OH)3 accumulation at the steel/concrete

interface (MPa)

r = radius of the concrete ring where stresses are being calculated (m)

The tangential stress in concrete σt is maximum at the inner surface of the cylinder, i.e.,

where r = d/2. Thus,

σmaxt = pi

(c+ d/2)2 + (d/2)2

(c+ d/2)2 − (d/2)2[MPa] (4.8)

When the stress given by Eq. 4.8 exceeds the tensile splitting strength of concrete f ′sp,

an internal crack will start to form at the steel/concrete interface. The internal pressure

required for concrete to initiate a crack at the inner surface of the thick-wall cylinder is

therefore obtained by equating σmaxt to f ′sp in Eq. 4.8 and solving for pi, i.e.,

pi = f ′sp(c/d)2 + (c/d)

(c/d)2 + (c/d) + 0.5[MPa] (4.9)

The tensile splitting strength for normal strength concretes is evaluated from 0.4√f ′c MPa,

where f ′c is the cylinder compressive strength of concrete in MPa. This value is approximately


1.25 times the direct tensile strength f ′t , which is usually taken as 0.33√f ′c MPa. For

high strength concretes, the tensile splitting strength is taken equal to 0.59√f ′c MPa (ACI

Committee 363, 1984).

4.3.2 Partly-cracked elastic stage

As it has been seen previously, an internal crack will start at the inner surface of the concrete

cylinder when the peak tensile stress exceeds the tensile splitting strength of concrete. The

crack starting at this point will not penetrate through the concrete cover if the tensile load-

carrying capacity of the concrete ring has not been reached yet. At this stage, the concrete

ring has internal cracks where the tangential stresses have reached f ′sp (see Fig. 4.3). However,

concrete still behaves elastically in the outer part where the tangential stresses are still below

f ′sp. The internal pressure pi acting at the steel/concrete interface is now transferred through

the concrete teeth between cracks to the uncracked part of the cylinder. Since the inner area

of this uncracked ring is larger than that of the cylinder without any crack, the pressure at

this surface must therefore be reduced accordingly, i.e.,

pe = pid

2e[MPa] (4.10)

where pe is the pressure acting at the inner surface of the uncracked ring (MPa), and e is

the crack penetration length (m). The uncracked ring of concrete can now be treated as

a thick-wall cylinder in itself, with an inner radius equal to e and subjected to an internal

pressure pe (Fig. 4.3). The tangential stresses in this outer part of the concrete ring can be

evaluated similar to Eq. 4.7, i.e.,

σt = pid

2e

e2

(c+ d/2)2 − e2

(1 +

(c+ d/2)2

r2

)[MPa] (4.11)

The maximum tangential stress at the inner surface of this uncracked cylinder (r = e) is

calculated from:

σmaxt = pi

d

2e

(c+ d/2)2 + e2

(c+ d/2)2 − e2, where

d

2≤ e ≤ c+

d

2[MPa] (4.12)

The maximum stress given in Eq. 4.12 corresponds to the attainment of the concrete

tensile splitting strength, i.e., σmaxt = f ′sp. Equation 4.12 is only valid when the internal


d c

pi

pe

r

e

Fig. 4.3: Concrete ring with internal cracks.

pressure pi resulting from hydrated red rust accumulation exhibits a maximum for a value

of e within the bounds d/2 and c+ d/2. By differentiating pi/f′sp (Eq. 4.12) with respect to

e and equating the derivative to zero, the value of e for which the tensile load capacity of

the concrete ring is a maximum is obtained as:

e = 0.486

(c+

d

2

)[m] (4.13)

As a consequence, internal longitudinal cracks can develop and penetrate the concrete cover

to an optimum depth equal to:

Crack depth = e− d

2= 0.486 c− 0.257 d [m] (4.14)

When this depth is exceeded, the crack penetrates right through the concrete cover because

the tensile load capacity of the concrete ring is then reached. The minimum thickness of

the concrete cover c at which a crack is no longer internal but has penetrated through the

concrete cover is:

0.486 c− 0.257 d = 0 ⇒ c = 0.529 d (4.15)

The tensile stresses developed in the concrete ring when internal cracks have developed

are evaluated from Eq. 4.11 for e = 0.486 (c + d/2). The internal pressure pi required to

crack the cover is calculated from Eq. 4.12 with e given by Eq. 4.13 and σmaxt = f ′sp. The


resulting expression is given by:

pi = f ′sp(c/d) + 0.5

1.665[MPa] (4.16)

4.4 End of service life

Once the ultimate tensile load capacity of the reinforced concrete cylinder has been reached,

existing internal cracks penetrate through the concrete cover and appear at the free sur-

face. At this stage, the reinforced concrete member is assumed to have failed. Thus the

end of service life is defined here as the time required for cracks to penetrate through the

concrete cover all the way to the surface as a result of hydrated red rust accumulation at

the steel/concrete interface.

0

1

2

3

4

0 1 2 3 4 5 6

c/d

pi / f

' sp

elastic

partly-crackedelastic

crack initiatesat inner surface of

concrete ring

longitudinal crackgoes right throughthe concrete cover

(end of service life )

Fig. 4.4: Load-carrying capacity of concrete ring versus concrete cover thickness.

Figure 4.4 shows the plot of internal pressure pi normalized to f ′sp against the ratio of

concrete cover c to rebar diameter d for the relations given by Eqs. 4.9 and 4.16, which are the

expressions for the internal pressure required to initiate internal cracks at the steel/concrete

interface and to propagate such cracks to the concrete cover surface, respectively. For a

given value of c/d, as the internal pressure due to the accumulation of rust around the

rebar reaches the elastic curve in Fig. 4.4, internal cracks start to originate at the interface


between the steel and the concrete. As the corrosion products keep on accumulating around

the perimeter of the steel bars, the internal pressure pi also increases until it reaches the value

corresponding to internal cracks propagating through the entire concrete cover, represented

by the partly-cracked elastic curve in Fig. 4.4. At this point, cracks have appeared at the

free surface of the concrete, and the service life of the reinforced concrete member is assumed

to have ended.

4.5 Effect of corrosion on bond

Although the end of service life is defined here as the appearance of cracks at the concrete

surface, as seen in the previous section, it is recognized that reinforced concrete structures can

remain serviceable beyond the onset of cracking. In reality, the serviceability and ultimate

strength of a reinforced concrete structure will be directly affected by the weakening of

the bond action between reinforcement and concrete due to splitting of the concrete cover

(Guiriani et al., 1991; Cabrera, 1996; Gambarova and Rosati, 1997; Cairns, 1998).

Forces resulting from bond action are gen-

Fig. 4.5: Schematic representation of radialbond forces against concrete tensile rings (re-

produced from Tepfers (1979)).

erally transferred from the reinforcement to the

concrete by inclined compressive forces radiat-

ing out from the reinforcing bars and spread-

ing into the concrete. The radial components

of these bond forces are balanced by rings of

tensile stresses in the surrounding concrete (see

Fig. 4.5). The bond capacity of a reinforced

concrete element is therefore limited by the failure of these tensile rings when the concrete

cover cracks.

The radial component of bond stress pb is given by:

pb = τ tanϕ [MPa] (4.17)

where τ is the tangential bond component (MPa) and ϕ denotes the angle between the bond

forces and the reinforcing bar axis. This angle depends on the geometry of the reinforcing

bar ribs and the degree of crushing of the concrete in front of the ribs (Tepfers, 1979). The


tangential bond component τ is related to the change in the reinforcing steel stress along a

length dx of the rebar as illustrated in Fig. 4.6, i.e.,

τ =d

4· dσs

dx[MPa] (4.18)

where d is the reinforcing bar diameter (m) and σs is the axial stress carried by the rein-

forcement (MPa).

Corrosion-induced damage of reinforced

σsAs (σs + dσs) As

d

dx

τ

Fig. 4.6: Bond stresses along the reinforcement.

concrete causes a reduction on the steel-to-

concrete bond capacity due to the loss of

confinement provided by the cracked con-

crete. At an initial stage, the corrosion ex-

pansion around a corroding reinforcing bar increases the radial stresses between the rebar

and the concrete and, hence, the frictional component of bond (τ). However, further ac-

cumulation of corrosion products around the reinforcing bar leads to the development of

longitudinal cracking of the concrete cover and a consequent reduction in the resistance to

the bursting forces generated by the bond action. At this stage, confining action is ensured

only by the transverse reinforcement, although this confinement will decrease too if stirrups

are affected significantly by corrosion.

The residual bond capacity of a corroding reinforced concrete element could be estimated

by use of the equations developed in the previous sections. If the pressure originating from

accumulation of rust around the reinforcement is denoted by pc, the internal pressure applied

around the reinforcing bar required to crack the concrete cover will be given by the sum of

the radial bond component pb and the rust pressure pc:

pcr = pb + pc = τ tanϕ+ pc [MPa] (4.19)

where pcr is given by Eq. 4.16 and pc is calculated from Eq. 4.5 given an estimate of the

expansion due to rust accumulation. Thus the tangential component of bond will be reduced

according to:

τ =1

tanϕ(pcr − pc) [MPa] (4.20)

If confinement through transverse reinforcement is not provided, bond stresses will vanish

as soon as longitudinal cracks develop through the concrete cover due to reinforcement


corrosion, i.e., when pc = pcr. Based on experimental observations, Rodriguez et al. (1994)

developed an empirical relation between residual bond strength and corrosion similar to the

one given by Eq. 4.20, in which the corrosion attack penetration x as well as the confining

action provided by the stirrups are incorporated:

τ tanϕ = f ′sp(c/d) + 0.5

1.665(1− β · xµ) +

k As fy

s d[MPa] (4.21)

where β, µ, and k are empirical constants, As and fy are the cross-sectional area and yielding

stress of the stirrups, respectively, and s denotes the spacing between stirrups. Capozucca

(1995) also quantified the reduction in the tensile force carried by the reinforcement based

on the same type of considerations.

Chapter 5

Finite Element Formulation

5.1 Introduction

The mathematical model of the process of corrosion in reinforced concrete outlined in Chap-

ter 3 results in a set of four partial differential equations which describe chloride ingress into

concrete (Eq. 3.19), moisture and heat diffusion through concrete (Eqs. 3.29 and 3.46, re-

spectively), and oxygen transport to the reinforcement (Eq. 3.57). A closed-form solution of

this system of equations is not possible due to the dependence of the various material prop-

erties and boundary conditions on the physical parameters of the concrete and the time level

of exposure. Still, this set of equations can be solved numerically in space as a boundary-

value problem and in time as an initial-value problem by means of a two-dimensional finite

element formulation, in which appropriate boundary conditions are enforced to simulate sea-

sonal variations in exposure conditions. A time-step integration procedure (finite-difference

scheme) is applied to determine the variation in time of the different variables in concrete.

During the initiation stage, Eqs. 3.19, 3.29 and 3.46 are solved simultaneously in order

to obtain variations in time t and space (x, y) of the concentration of chlorides in the pore

solution Cfc, the level of pore relative humidity h, and the temperature distribution T within

the concrete. The concentration of total chlorides in concrete Ctc as well as the amount of

evaporable water ωe are also determined by means of chloride binding relationships and

adsorption isotherms, respectively. Once the concentration of chlorides in contact with the

reinforcing steel reaches a specified threshold value, Eq. 3.57 is added to the system of

98

Finite element formulation 99

Table 5.1: Correspondence between Eq. 5.1 and the governing differential equations.

Physical problem φ κ Jx Jy J ′x J ′

y

Chloride ingress Cfc 1 −D∗c

∂Cfc

∂x−D∗

c∂Cfc

∂yCfc · Jmx Cfc · Jmy

Moisture diffusion h ∂ωe

∂h−Dh

∂h∂x

−Dh∂h∂y

0 0

Heat transfer T %c cq −λ ∂T∂x

−λ ∂T∂y

0 0

Oxygen transport Co 1 −Do∂Co

∂x−Do

∂Co

∂y0 0

equations being solved to obtain the distribution of dissolved oxygen Co in concrete. An

estimate of the anodic current is obtained from Eq. 3.51 with E = Ecorr (Eq. 3.54).

The following sections present the mathematics underlying the finite element formulation

developed to solve the resulting system of partial differential equations.

5.2 Galerkin method

Because Eqs. 3.19, 3.29, 3.46, and 3.57 all have a similar mathematical structure, i.e.,

κ∂φ

∂t+∂Jx

∂x+∂Jy

∂y︸︷︷︸diffusion

+∂J ′

x

∂x+∂J ′

y

∂y︸︷︷︸convection

= 0 (5.1)

the numerical tool developed to solve one of them will be valid to solve any other of them. Ta-

ble 5.1 shows the correspondence between Eq. 5.1 and the different field equations presented

in Chapter 3.

Employing the Galerkin weighted residual method on the governing partial differential

equation given by Eq. 5.1 yields:∫Ω

Wi

(κ∂φ

∂t+∂Jx

∂x+∂Jy

∂y+∂J ′

x

∂x+∂J ′

y

∂y

)dΩ = 0 (5.2)

where Wi(x, y) is a weighting function and Ω represents the domain of the problem. Equa-

tion 5.2 is solved by dividing the domain Ω into elements, in which the field variable φ is

expressed in terms of the element nodal values according to:

φ(e) = [N ]Φ(e) (5.3)

where [N ] is the row vector containing the element interpolation functions associated with

each node, and Φ(e) is the vector containing the corresponding unknown nodal values (see


Fig. 5.1). The superscript (e) refers to element values. The Galerkin method uses as its

weighting functions the shape functions associated with the elements nodes (i.e., Wi = Ni,

the subscript i denoting the node to which the functions are associated). By expanding

terms, the element residual integral is therefore given by:∫A

[N ]Tκ∂φ

∂t− ∂

∂x

(D∂φ

∂x

)− ∂

∂y

(D∂φ

∂y

)− ∂φ

∂xDh

∂h

∂x− ∂φ

∂yDh

∂h

∂y

− φ

[∂

∂x

(Dh

∂h

∂x

)+

∂

∂y

(Dh

∂h

∂y

)]︸︷︷︸

∂ωe

∂h

∂h

∂t

dA = 0 (5.4)

Note that Eq. 5.4 is now integrated over the area A of the element under consideration. The

second-spatial derivatives in Eq. 5.4 can be replaced by first-derivative terms by applying

the product rule for differentiation and using Green’s theorem, i.e.,

−∫

A

[N ]T∂

∂x

(D∂φ

∂x

)dA = −

∮s

[N ]T D∂φ

∂xcos γ ds+

∫A

D∂[N ]T

∂x

∂φ

∂xdA (5.5)

where s denotes the element boundary and γ is the angle of the outward normal with respect

to the x-direction (Fig. 5.2). A similar procedure is applied to the term involving second-

spatial derivatives in y. Substituting Eq. 5.5 into Eq. 5.4 yields:∫A

κ [N ]T∂φ

∂tdA

−∮

s

[N ]TD

(∂φ

∂xcos γ +

∂φ

∂ysin γ

)ds+

∫A

D

(∂[N ]T

∂x

∂φ

∂x+∂[N ]T

∂y

∂φ

∂y

)dA

−∫

A

Dh [N ]T(∂h

∂x

∂φ

∂x+∂h

∂y

∂φ

∂y

)dA−

∫A

[N ]T∂ωe

∂h

∂h

∂tφ dA = 0 (5.6)

If the element nodal values of ∂φ/∂t are denoted by the vector Φ(e), a variation of

∂φ/∂t within the element can also be defined according to:

∂φ

∂t

(e)

= [N ]Φ(e) (5.7)

This assumption is commonly referred to as the consistent formulation, since both φ and

∂φ/∂t vary consistently with respect to x and y within an element (Segerlind, 1984).

By substituting Eqs. 5.3 and 5.7 into Eq. 5.6, and using the same spatial approximation


Y

Xi

kl

φ

Φ i Φ j

Φ k

Φ l

φ = + + +N N N Ni i j j k k l lΦ Φ Φ Φ

i

kj

(Xi,Yi)

(Xj,Yj)(Xk,Yk)

Φ i

Φ j

Φ k

φ = + +N N Ni i j j k kΦ Φ Φ

2bj

2a

Fig. 5.1: Linear triangular and bilinear rectangular elements.

for h and h, where h denotes ∂h/∂t, the following expression results:(∫A

κ [N ]T [N ] dA

)Φ(e) −

∮s

[N ]T D

(∂φ

∂xcos γ +

∂φ

∂ysin γ

)ds

+

(∫A

D

(∂[N ]T

∂x

∂[N ]

∂x+∂[N ]T

∂y

∂[N ]

∂y

)dA

)Φ(e)

−(∫

A

Dh [N ]T(∂[N ]

∂xh(e)∂[N ]

∂x+∂[N ]

∂yh(e)∂[N ]

∂y

)dA

)Φ(e)

−(∫

A

∂ωe

∂h[N ]T [N ] h(e) [N ] dA

)Φ(e) = 0 (5.8)

which written in matrix form results in:

[c(e)] Φ(e)+ I(e)+ [k(e)] Φ(e) = 0 (5.9)

where [c(e)], known as capacitance matrix, is given by:

[c(e)] =

∫A

κ [N ]T [N ] dA (5.10)

I(e), the interelement vector (Segerlind, 1984), is given by:

I(e) = −∮

s

[N ]T D

(∂φ

∂xcos γ +

∂φ

∂ysin γ

)ds (5.11)

where the integral is performed around the boundary of the element in a counterclockwise


Y

X

n

γ

s (boundary)

φ (x,y)

Fig. 5.2: Element boundary.

direction; and [k(e)], the element property matrix, is given by:

[k(e)] =

∫A

D [B]T [B] dA−∫

A

Dh [N ]T[∂[N ]

∂xh(e) ∂[N ]

∂yh(e)

][B] dA

−∫

A

∂ωe

∂h[N ]T [N ] h(e) [N ] dA (5.12)

From a mathematical point of view, matrix [k(e)] is similar to the element stiffness ma-

trix in mechanics applications. The matrix [B] appearing in Eq. 5.12 contains the element

interpolation gradient vectors and is evaluated from:

[B]T =

[∂[N ]T

∂x

∂[N ]T

∂y

](5.13)

As seen in Chapter 3, boundary conditions are enforced by specifying either the value of

φ at the boundary or the flux across it. When fluxes across the boundary are specified, the

boundary condition takes the following form:

−D(∂φ

∂xcos γ +

∂φ

∂ysin γ

)= Lφb −M (5.14)

where φb represents the value of φ at the boundary (unknown quantity), and L and M

are two constants. Table 5.2 shows the correspondence between L and M and the different

boundary conditions presented in Chapter 3. Equation 5.14 assumes the boundary flux going

into the body. Substituting Eq. 5.14 into Eq. 5.11 gives:

I(e) =

∮s

[N ]T (Lφb −M) ds (5.15)

where φb is given by Eq. 5.3. Substituting Eq. 5.3 into Eq. 5.15 results in:

I(e) =

∮s

[N ]T(L [N ] Φ(e) −M

)ds (5.16)


Table 5.2: Correspondence between L and M and the imposed boundary conditions.Diffusive term Convective term

Physical problem φb L M φb L M

Chloride ingress Cfc Bc Bc · Cen h Bh · Cen Bh · hen · Cen

Moisture diffusion h Bh Bh · hen — — —

Heat transfer T BT BT · Ten — — —

Oxygen transport

(steel surface)0 0 −kiaa — — —

ak = 8.291× 10−8Aa/Ac + 4.144× 10−8

which can be separated into:

I(e) =

(∮s

L [N ]T [N ] ds

)Φ(e) −

∮s

M [N ]T ds (5.17)

The first component of Eq. 5.17 adds to the element property matrix [k(e)] since it multiplies

Φ(e), whereas the second component constitutes the element environmental load vector

f (e), i.e.,

f (e) =

∮s

M [N ]T ds (5.18)

Note that the integrals in Eqs. 5.17 and 5.18 are only evaluated along the boundary of

elements where fluxes are specified.

When all the element contributions are combined using a direct stiffness procedure, the

following system of equations results:

[C] Φ+ [K] Φ − F = 0 (5.19)

Equation 5.19 represents a system of linear first-order differential equations in the time

domain. In order to obtain a numerical solution, Eq. 5.19 is integrated in time by means of

a finite-difference approximation (Segerlind, 1984; Jaluria, 1988), i.e.,

([C] + θ∆t [K]) Φt+∆t =

([C]− (1− θ) ∆t [K]) Φt + ∆t((1− θ) Ft + θ Ft+∆t

)(5.20)

where θ is a parameter ranging from 0 to 1 and ∆t denotes a time increment. Equation 5.20

evaluates physical quantities at time t+ ∆t, i.e., Φt+∆t, as a function of quantities at the


previous time step, i.e., Φt. Equation 5.20 is used to solve Eqs. 3.19, 3.29, 3.46, and 3.57,

all of which describe time-dependent problems.

5.3 Elements description

Since the integrals expressed in Eqs. 5.10, 5.12, and 5.17 only require first-spatial derivatives

of the element shape functions, the formulation has been based on a linear distribution of

the field variable φ with respect to x and y. The two elements used to discretize the region

of interest are the linear triangular and bilinear rectangular elements shown in Fig. 5.1.

For the type of problems being solved here, the area of interest (the concrete cover to the

steel reinforcement) is usually small compared to the whole cross section of the reinforced

concrete member. Unless the entire member is discretized and analyzed according to exter-

nal boundary conditions, which can be computationally expensive particularly for transient

problems (Damjanic and Owen, 1984), it is common practice to truncate the finite element

model at an arbitrary distance far enough away from the region of interest and to impose

the far field boundary condition at this limiting edge. If this “artificial” boundary lies on an

axis of symmetry, the condition imposed is zero normal flux (i.e., ∂φ/∂n = 0), otherwise the

far field boundary condition corresponds to the initial condition of the problem. A difficulty

that arises with simple truncation is the positioning of the remote boundary in order to

obtain an accurate solution. This problem has been addressed in finite element formulations

by treating the domain corresponding to the far field as infinite in extent and using what

is known as “infinite elements”, which extend the domain of a finite element to infinity so

that it remains unbounded (Bettes and Bettes, 1984). The use of infinite elements in a finite

element model allows for satisfactory results to be obtained from fewer elements than would

otherwise be required (Cook et al., 1989). The infinite element formulation has followed two

main approaches (Bettes and Bettes, 1984; Cook et al., 1989): (1) the use of decay func-

tions in the infinity direction which multiply the parent element standard shape functions;

and, (2) the mapping of elements from finite to infinite domains. The second approach has

been used here to simulate the medium surrounding the region of interest. Mapped infinite

elements have been successfully implemented in the past in the solution of both mechanics

(Bettes and Bettes, 1984; Marques and Owen, 1984) and transient thermal (Damjanic and


Owen, 1984) applications.

The linear triangular, bilinear rectangular, and mapped infinite elements used in the

formulation presented herein are described in the following sections.

5.3.1 Linear triangular element

The linear triangular element has straight sides with a node at each corner. There is one

degree of freedom associated to each node, field variable φ, which is approximated over the

triangular region by:

φ(e) = Ni Φi +Nj Φj +Nk Φk (5.21)

where Ni, Nj, and Nk are the shape functions associated with nodes i, j, and k, respectively,

and Φi, Φj, and Φk are the corresponding nodal values of φ. The shape functions for the

triangular element are given by:

Ni =1

2A(ai + bi x+ ci y) (5.22)

Nj =1

2A(aj + bj x+ cj y) (5.23)

Nk =1

2A(ak + bk x+ ck y) (5.24)

with

ai = Xj Yk −Xk Yj, bi = Yj − Yk, ci = Xk −Xj (5.25)

aj = Xk Yi −Xi Yk, bj = Yk − Yi, cj = Xi −Xk (5.26)

ak = Xi Yj −Xj Yi, bk = Yi − Yj, ck = Xj −Xi (5.27)

where (Xi, Yi), (Xj, Yj), and (Xk, Yk) are the nodal coordinates, and A represents the area

of the triangle, which is evaluated from:

A =1

2

∣∣∣∣∣∣∣∣∣1 Xi Yi

1 Xj Yj

1 Xk Yk

∣∣∣∣∣∣∣∣∣ (5.28)

5.3.2 Bilinear rectangular element

The bilinear rectangular element has straight sides with a node at each corner. It is 2b long

by 2a wide as shown in Fig. 5.3. Variable φ is approximated over the rectangular region by:


Y

X

y’

x’i j

kl2a

2b

Fig. 5.3: Bilinear rectangular element.

φ(e) = Ni Φi +Nj Φj +Nk Φk +Nl Φl (5.29)

where Ni, Nj, Nk, and Nl are the shape functions associated with nodes i, j, k, and l,

respectively, and Φi, Φj, Φk, and Φl are the corresponding nodal values of φ. The shape

functions for the rectangular element in terms of the local coordinate system x′ − y′ whose

origin is located at node i (Fig. 5.3) are given by:

Ni =

(1− x′

2b

) (1− y′

2a

)(5.30)

Nj =x′

2b

(1− y′

2a

)(5.31)

Nk =x′y′

4ab(5.32)

Nl =y′

2a

(1− x′

2b

)(5.33)

5.3.3 Mapped infinite elements

The two types of mapped infinite elements presented here are based on the bilinear rectan-

gular element described in Section 5.3.2 and illustrated in Fig. 5.4 in the natural coordinate

system ξ − η. These two infinite elements are the bilinear singly infinite element, which

extends to infinity in the ξ direction only as shown in Fig. 5.5(a), and the bilinear doubly in-

finite element, which extends to infinity in both the ξ and η directions as shown in Fig. 5.5(b).

This latter element is usually employed as a “corner” element in a mesh where transition


ξ

η

i j

kl

1 1

1

1

Ni =1

4(1− ξ)(1− η) (5.34)

Nj =1

4(1 + ξ)(1− η) (5.35)

Nk =1

4(1 + ξ)(1 + η) (5.36)

Nl =1

4(1− ξ)(1 + η) (5.37)

Fig. 5.4: Bilinear rectangular element in the natural coordinate system ξ−η with its correspondingshape functions.

is needed between regions extending to infinity along two different directions (Marques and

Owen, 1984).

As illustrated in Fig. 5.5(a), the singly bilinear infinite element is similar to the bilinear

rectangular element with the two nodes corresponding to ξ = +1 positioned at infinity. In

the case of the bilinear doubly infinite element, the nodes corresponding to both ξ = η = +1

are located at infinity (Fig. 5.5(b)). The geometry of both elements is interpolated according

to:

x(ξ, η) = [M ] X (5.38)

y(ξ, η) = [M ] Y (5.39)

where [M ] is the row vector containing the geometric mapping functions and X and Y

are the vectors containing the global x and y coordinates of the finite nodes, respectively.

Similar to the standard interpolation functions [N ], mapping functions [M ] are equal to one

at nodes to which they are associated and zero at the remaining finite nodes. However, these

functions grow without limit as a particular coordinate approaches infinity. The condition

that the geometrical mapping be independent of the selection of coordinate system requires

that the sum of the infinite element mapping functions equals to one (Bettes and Bettes,

1984).

Mapping functions for the bilinear singly and doubly infinite elements have been devel-

oped by Marques and Owen (1984) and are listed in Table 5.3. Note that the mapping

functions listed in Table 5.3 automatically place the nodes corresponding to ξ = +1 for the

singly infinite element and to ξ = η = +1 for the doubly infinite element at infinity. This


mapped elements innatural coordinates

infinite elements inglobal coordinates

ξ

η

ξ

η

poles ofexpansion

(a)

(b)

i

i

j

j

k

k

l

l

xp

xp

xp

xp

yp

yp

δ

Fig. 5.5: Mapped infinite elements: (a) bilinear singly infinite element; (b) bilinear doubly infiniteelement (adapted from Damjanic and Owen (1984)).

is why the coordinates of these nodes are not included in vectors X and Y in the geo-

metric mapping given by Eqs. 5.38 and 5.39. According to Marques and Owen (1984), it is

difficult to enforce simultaneously the singularity at ξ = +1 and the invariance requirement

under change of coordinate origin when forming a singly infinite element from the bilinear

rectangular element. To solve the problem, the authors suggest the inclusion of two extra

nodes for geometric mapping only (represented by the white nodes in Fig. 5.5(a)). Likewise,

three additional nodes are required to define the geometric mapping of the doubly infinite

element, as illustrated in Fig. 5.5(b).

The field variable φ is interpolated within the mapped infinite elements using the standard

shape functions of the parent finite element (the bilinear rectangular element in this case)

according to Eq. 5.3. If the value of φ can be manipulated so that it approaches zero at

infinity, the effect of the interpolation functions [N ] associated with those nodes located

at infinity (ξ = +1 for the singly infinite and ξ = η = +1 for the doubly infinite) can be

removed from the mathematical formulation, since, by doing so, the zero boundary condition

is automatically satisfied (Damjanic and Owen, 1984; Marques and Owen, 1984). This


Table 5.3: Mapping and shape functions for infinite elements (Marques and Owen, 1984).

Element Mapping functions Shape functions

Bilinear singly infinite Mi = −ξ (1−η)1−ξ

Ni = 14(1− ξ)(1− η)

Mj = (1+ξ) (1−η)2(1−ξ)

Nl = 14(1− ξ)(1 + η)

Mk = (1+ξ) (1+η)2(1−ξ)

Ml = −ξ (1+η)1−ξ

Bilinear doubly infinite Mi = 4ξη(1−ξ) (1−η)

Ni = 14(1− ξ)(1− η)

Mj = −2η (1+ξ)(1−ξ) (1−η)

Mk = (1+ξ) (1+η)(1−ξ) (1−η)

Ml = −2ξ (1+η)(1−ξ) (1−η)

reduces the size of the elements as well as the associated matrices and vectors and increases

the efficiency of the computational analysis. The bilinear singly infinite element can thus

employ only two nodes for the field variable description (nodes i and l in Fig. 5.5(a)), whereas

the bilinear doubly infinite element can employ only one (node i in Fig. 5.5(b)), i.e.,

φ(e) =

Ni φi +Nl φl when φ(ξ = +1) = 0 for the singly infinite,

Ni φi when φ(ξ = +1) = φ(η = +1) = 0 for the doubly infinite.

(5.40)

Note that the nodes used for interpolation of the field variable φ (black nodes in Fig. 5.5) are

not the same set of nodes used for geometrical mapping (black and white nodes in Fig. 5.5).

Since the standard shape functions [N ] are of lower degree than the mapping functions [M ],

the mapped infinite elements illustrated in Fig. 5.5 are superparametric.

The geometry and field variable expansions involved in the mapped infinite elements are

both referred to a set of points known as poles (Marques and Owen, 1984). The poles of

expansion are singular points about which the field quantity φ decays. These poles must be

located outside the infinite element as shown in Fig. 5.5, but their positioning is arbitrary

and depends on the geometric and physical characteristics of the problem. Once that their

positions are selected, the external nodes of the infinite elements are located halfway between

the poles and the first set of internal nodes (Bettes and Bettes, 1984; Damjanic and Owen,

1984; Marques and Owen, 1984).


In order to ensure continuity across the boundary between infinite and finite elements (or

adjacent infinite elements) the number and location of the connecting nodes must coincide.

Boundary conditions at infinity

The use of mapped infinite elements is restricted to problems where the field variable φ tends

monotonically to the value at infinity with increasing distance (Bettes and Bettes, 1984;

Damjanic and Owen, 1984). It has been previously mentioned that when φ approaches zero

at infinity, the boundary condition is automatically satisfied by removing the influence of

nodes located at infinity in the finite element formulation (Eq. 5.40). However, for the type

of problems being solved here, the field variable at infinity does not always tend to zero

but to a constant initial value. For the latter case, the zero-boundary condition approach

can still be preserved if the initial value is substracted from the field variable φ (Bettes and

Bettes, 1984; Damjanic and Owen, 1984). The problem is then expressed in terms of a new

field variable ψ, which is defined as:

ψ(x, y, t) = φ(x, y, t)− φ(x, y, 0) for t > 0 (5.41)

where φ(x, y, 0) is the specified initial condition of the problem. Since φ(x, y, 0) is constant,

substitution of Eq. 5.41 into Eq. 5.19 yields:

[C] Ψ+ [K] Ψ − F = 0 (5.42)

which is identical in form to Eq. 5.19. Equation 5.42 can now be solved in terms of ψ with

ψ(x, y, 0) = 0, and the transient and spatial distribution of φ can be calculated from Eq. 5.41.

Note that external boundary conditions in the transformed problem (Eq. 5.42) also have to

be decreased by φ(x, y, 0).

Fluxes along any edge extending to infinity are incompatible with the assumption of φ

vanishing at infinity (Marques and Owen, 1984). Thus fluxes in infinite elements can only be

specified along the edge common to the finite elements to which they are connected, and they

can be assigned to the finite element instead of the infinite element for the sake of simplicity.

Therefore, Eq. 5.17, which results from specifying fluxes along the boundary of an element,

does not need to be evaluated for the mapped infinite elements described above.


5.4 Evaluation of element integrals

The integrals given by Eqs. 5.10, 5.12, and 5.17 have been evaluated in closed-form for the

linear triangular, bilinear rectangular, and mapped infinite elements described in Section 5.3

and are presented in the following sections.

5.4.1 Linear triangular element

Capacitance matrix

[c(e)] =

∫A

κ [N ]T [N ] dA =κA

12

2 1 1

1 2 1

1 1 2

(5.43)

where A is the area of the triangle (Eq. 5.28).

Property matrix

[k(e)] = [k(e)1 ]− [k

(e)2 ]− [k

(e)3 ] + [k

(e)4 ]

[k(e)1 ] =

∫A

D [B]T [B] dA =D

4A

b2i + c2i bibj + cicj bibk + cick

bibj + cicj b2j + c2j bjbk + cjck

bibk + cick bjbk + cjck b2k + c2k

(5.44)

where D is evaluated at the centre of the element, where its value is considered representative

of the whole element, and bi, bj, bk, ci, cj, and ck are given by Eqs. 5.25, 5.26, and 5.27.

[k(e)2 ] =

∫A

Dh [N ]T[∂[N ]

∂xh(e) ∂[N ]

∂yh(e)

][B] dA =

Dh

12A

m1 m2 m3

m1 m2 m3

m1 m2 m3

(5.45)

where Dh is the humidity diffusion coefficient as given in Eq. 3.30, also evaluated at the

centre of the element, and

m1 = (b2i + c2i )hi + (bibj + cicj)hj + (bibk + cick)hk (5.46)

m2 = (bibj + cicj)hi + (b2j + c2j)hj + (bjbk + cjck)hk (5.47)

m3 = (bibk + cick)hi + (bjbk + cjck)hj + (b2k + c2k)hk (5.48)


with hi, hj, and hk being the pore relative humidity values at each of the triangle nodes.

[k(e)3 ] =

∫A

∂ωe

∂h[N ]T [N ] h(e) [N ] dA

=∂ωe

∂h

A

60

6hi + 2hj + 2hk 2hi + 2hj + hk 2hi + hj + 2hk

2hi + 2hj + hk 2hi + 6hj + 2hk hi + 2hj + 2hk

2hi + hj + 2hk hi + 2hj + 2hk 2hi + 2hj + 6hk

(5.49)

where hi, hj, and hk are the triangle nodal values of ∂h/∂t. Once the nodal values of h are

known by solving Eq. 5.20, the nodal values of ∂h/∂t are calculated from Eq. 5.19, i.e.,

Φ = [C]−1 (F − [K] Φ) (5.50)

[k(e)4 ] =

∮s

L [N ]T [N ] ds =L

6lij

2 1 0

1 2 0

0 0 0

︸︷︷︸

side ij

+L

6ljk

0 0 0

0 2 1

0 1 2

︸︷︷︸

side jk

+L

6lik

2 0 1

0 0 0

1 0 2

︸︷︷︸

side ik

(5.51)

where lij, ljk, and lik are the lengths of sides ij, jk, and ik, respectively. Equation 5.51 is

only applied on those sides of the triangle where fluxes are specified.

Environmental load vector

f (e) =

∮s

M [N ]T ds =M

2lij

1

1

0

︸︷︷︸side ij

+M

2ljk

0

1

1

︸︷︷︸side jk

+M

2lik

1

0

1

︸︷︷︸side ik

(5.52)

As in the case of Eq. 5.51, Eq. 5.52 is only evaluated on the boundary sides of triangular

elements where fluxes are prescribed.


5.4.2 Bilinear rectangular element

Capacitance matrix

[c(e)] =

∫A

κ [N ]T [N ] dA =κ a b

9

4 2 1 2

2 4 2 1

1 2 4 2

2 1 2 4

(5.53)

where 2a and 2b are the height and width of the rectangle, respectively (Fig. 5.3).

Property matrix

[k(e)] = [k(e)1 ]− [k

(e)2 ]− [k

(e)3 ] + [k

(e)4 ]

[k(e)1 ] =

∫A

D [B]T [B] dA

=D

6ab

2(a2 + b2) −(2a2 − b2) −(a2 + b2) (a2 − 2b2)

−(2a2 − b2) 2(a2 + b2) (a2 − 2b2) −(a2 + b2)

−(a2 + b2) (a2 − 2b2) 2(a2 + b2) −(2a2 − b2)

(a2 − 2b2) −(a2 + b2) −(2a2 − b2) 2(a2 + b2)

(5.54)

[k(e)2 ] =

∫A

Dh [N ]T[∂[N ]

∂xh(e) ∂[N ]

∂yh(e)

][B] dA =

Dh

24ab

k11

2 k122 k13

2 k142

k212 k22

2 k232 k24

2

k312 k32

2 k332 k34

2

k412 k42

2 k432 k44

2

(5.55)


where Dh is the humidity diffusion coefficient as given in Eq. 3.30, and

k112 = (3a2 + 3b2)hi − (3a2 − b2)hj − (a2 + b2)hk + (a2 − 3b2)hl (5.56)

k122 = −(3a2 − b2)hi + (3a2 + b2)hj + (a2 − b2)hk − (a2 + b2)hl (5.57)

k132 = −(a2 + b2)hi + (a2 − b2)hj + (a2 + b2)hk − (a2 − b2)hl (5.58)

k142 = (a2 − 3b2)hi − (a2 + b2)hj − (a2 − b2)hk + (a2 + 3b2)hl (5.59)

k212 = (3a2 + b2)hi − (3a2 − b2)hj − (a2 + b2)hk + (a2 − b2)hl (5.60)

k222 = −(3a2 − b2)hi + (3a2 + 3b2)hj + (a2 − 3b2)hk − (a2 + b2)hl (5.61)

k232 = −(a2 + b2)hi + (a2 − 3b2)hj + (a2 + 3b2)hk − (a2 − b2)hl (5.62)

k242 = (a2 − b2)hi − (a2 + b2)hj − (a2 − b2)hk + (a2 + b2)hl (5.63)

k312 = (a2 + b2)hi − (a2 − b2)hj − (a2 + b2)hk + (a2 − b2)hl (5.64)

k322 = −(a2 − b2)hi + (a2 + 3b2)hj + (a2 − 3b2)hk − (a2 + b2)hl (5.65)

k332 = −(a2 + b2)hi + (a2 − 3b2)hj + (3a2 + 3b2)hk − (3a2 − b2)hl (5.66)

k342 = (a2 − b2)hi − (a2 + b2)hj − (3a2 − b2)hk + (3a2 + b2)hl (5.67)

k412 = (a2 + 3b2)hi − (a2 − b2)hj − (a2 + b2)hk + (a2 − 3b2)hl (5.68)

k422 = −(a2 − b2)hi + (a2 + b2)hj + (a2 − b2)hk − (a2 + b2)hl (5.69)

k432 = −(a2 + b2)hi + (a2 − b2)hj + (3a2 + b2)hk − (3a2 − b2)hl (5.70)

k442 = (a2 − 3b2)hi − (a2 + b2)hj − (3a2 − b2)hk + (3a2 + 3b2)hl (5.71)

with hi, hj, hk, and hl being the pore relative humidity values at each of the rectangle nodes.

[k(e)3 ] =

∫A

∂ωe

∂h[N ]T [N ] h(e) [N ] dA =

∂ωe

∂h

ab

36

k11

3 k123 k13

3 k143

k213 k22

3 k233 k24

3

k313 k32

3 k333 k34

3

k413 k42

3 k433 k44

3

(5.72)


with

k113 = 9hi + 3hj + hk + 3hl (5.73)

k123 = k21

3 = 3hi + 3hj + hk + hl (5.74)

k133 = k31

3 = hi + hj + hk + hl (5.75)

k143 = k41

3 = 3hi + hj + hk + 3hl (5.76)

k223 = 3hi + 9hj + 3hk + hl (5.77)

k233 = k32

3 = hi + 3hj + 3hk + hl (5.78)

k243 = k42

3 = hi + hj + hk + hl (5.79)

k333 = hi + 3hj + 9hk + 3hl (5.80)

k343 = k43

3 = hi + hj + 3hk + 3hl (5.81)

k443 = 3hi + hj + 3hk + 9hl (5.82)

where hi, hj, hk, and hl are the rectangle nodal values of ∂h/∂t evaluated from Eq. 5.50.

[k(e)4 ] =

∮s

L [N ]T [N ] ds =L b

3

2 1 0 0

1 2 0 0

0 0 0 0

0 0 0 0

︸︷︷︸

side ij

+La

3

0 0 0 0

0 2 1 0

0 1 2 0

0 0 0 0

︸︷︷︸

side jk

+L b

3

0 0 0 0

0 0 0 0

0 0 2 1

0 0 1 2

︸︷︷︸

side kl

+La

3

2 0 0 1

0 0 0 0

0 0 0 0

1 0 0 2

︸︷︷︸

side il

(5.83)


Environmental load vector

f (e) =

∮s

M [N ]T ds = M b

1

1

0

0

︸︷︷︸side ij

+M a

0

1

1

0

︸︷︷︸side jk

+M b

0

0

1

1

︸︷︷︸side kl

+M a

1

0

0

1

︸︷︷︸side il

(5.84)

Equations 5.83 and 5.84 are only evaluated on those sides of the rectangle where fluxes are

specified.

5.4.3 Mapped infinite elements

In order to evaluate the integrals expressed in Eqs. 5.10 and 5.12 for the mapped infinite

elements, the domain of integration dA must be written in terms of the natural coordinates

ξ and η, i.e.,

dA = dxdy = det[J ] dξdη (5.85)

where [J ] is the Jacobian matrix which defines the geometric mapping and is given by:

[J ] =

∂[M ]∂ξ

∂[M ]∂η

[X Y

](5.86)

where [M ] is the row vector containing the mapping functions given in Table 5.3, and X

and Y are the vectors containing the global x and y coordinates of the nodes defining the

element geometry, respectively (black and white nodes in Fig. 5.5). The geometric mapping

functions [M ] are therefore used to relate the coordinates in the global system x− y to the

natural system ξ − η through the computation of the Jacobian matrix [J ]. Matrix [B] in

Eq. 5.13, which contains the derivatives of the shape functions [N ] with respect to the global

coordinates x and y, is evaluated for mapped infinite elements according to:

[B]T =

[∂[N ]T

∂x

∂[N ]T

∂y

]= [J ]−1

[∂[N ]T

∂ξ

∂[N ]T

∂η

](5.87)

Closed-form solutions of Eqs. 5.10 and 5.12 for the mapped infinite elements described in

Section 5.3.3 follow.


Capacitance matrices

[c(e)] =

∫A

κ [N ]T [N ] dA =

κxp δ

1/3 1/6

1/6 1/3

for the singly infinite,

κxp yp for the doubly infinite.

(5.88)

where xp and yp are the distances to the poles of expansion and δ is the width of the singly

infinite element, as illustrated in Fig. 5.5.

Property matrices

[k(e)] = [k(e)1 ]− [k

(e)2 ]− [k

(e)3 ]

[k(e)1 ] =

∫A

D [B]T [B] dA =

D

18 xp δ

2δ2 + 18x2p δ2 − 18x2

p

δ2 − 18x2p 2δ2 + 18x2

p

singly infinite,

D3

(x2

p+y2p

xp yp

)doubly infinite.

(5.89)

where D is evaluated at the centre of the element.

[k(e)2 ] =

∫A

Dh [N ]T[∂[N ]

∂xh(e) ∂[N ]

∂yh(e)

][B] dA

=Dh

48xp δ

(3δ2 + 12x2p)hi + (δ2 − 12x2

p)hl (δ2 − 12x2p)hi + (δ2 + 12x2

p)hl

(δ2 + 12x2p)hi + (δ2 − 12x2

p)hl (δ2 − 12x2p)hi + (3δ2 + 12x2

p)hl


=Dh

8hi

(x2

p + y2p

xp yp

)for the doubly infinite. (5.90)


where Dh is the humidity diffusion coefficient as given in Eq. 3.30 and hi and hl are the pore

relative humidity values at each of the finite nodes of the infinite elements.

[k(e)3 ] =

∫A

∂ωe

∂h[N ]T [N ] h(e) [N ] dA

=

∂ωe

∂h

xp δ

24

3hi + hl hi + hl

hi + hl hi + 3hl


∂ωe

∂h14xp yp hi for the doubly infinite.

(5.91)

where hi and hl are the finite nodal values of ∂h/∂t.

5.5 Solution procedure

The implementation of Eq. 5.20 to each of the problems described in Chapter 3 results in

a system of nonlinear equations for the cases of moisture, chloride, and oxygen transport

in concrete. Since parameters λ, cq, and %c defined in Section 3.4 were assumed to be

constant, the solution of the equation governing heat transfer in concrete yields a system

of linear equations. The nonlinearity of the moisture and chloride problems arises from

the dependence of coefficients Dh and Dc on the unknowns h and Cfc, respectively. The

dependence of Dc on the free chloride concentration Cfc is only manifested when nonlinear

binding (Langmuir or Freundlich) is considered. The nonlinearity of the oxygen transport

problem is due to the dependence of the imposed boundary conditions at the steel/concrete

interface on the concentration of dissolved oxygen Co available at the reinforcing steel surface.

The algebraic nonlinear equation given by Eq. 5.20 was solved by means of the frontal

solver developed by Hinton and Owen (1977), which takes advantage of the symmetry of

the capacitance and most of the property matrices by only storing their upper triangle in a

one-dimensional array. In the problem describing chloride ingress into concrete, the property

matrices resulting from integration of the convective term are nonsymmetric (Eqs. 5.45, 5.55,

and 5.90), and thus the frontal solver proposed by Hinton and Owen (1977) was modified

for this particular case by storing the element matrices in their entirety.


1

12

23

34

45

56

67

78

92

105

115 116 118 120 122 123

2 3 4 5 6 7 8 9 10 11

22

33

44

55

66

77

88

102

104

X

Y40 mm 20 mm

φ/ n = 0

φ/ n = 0

Fig. 5.6: Mesh for a corner of a reinforced concrete member.

5.5.1 Numerical parameters

Several runs of the nonlinear system of equations describing moisture diffusion and chloride

and oxygen transport in concrete were done in order to ensure that the variables introduced

into the numerical scheme were independent of the obtained results. The numerical tests

were done on a two-dimensional mesh representing a corner of a reinforced concrete member

section (Fig. 5.6). The selection of the numerical parameters θ and ∆t is explained in the

following paragraphs.

Finite-difference weighting factor, θ

Solutions to Eq. 5.20 obtained for θ ≥ 0.5 are known to be unconditionally stable for a

constant time step ∆t (Segerlind, 1984; Cook et al., 1989). The Crank-Nicolson method

in which θ = 0.5 is usually preferred in numerical applications since its asymptotic rate of

convergence is ∆t2. However, its implementation is frequently characterized by oscillations

around the correct solution. In fact, it was observed that large oscillations occurred in the

solution of the chloride transport equation (Eq. 3.19) when sudden changes took place in

the imposed boundary conditions; the application of de-icing salts during the winter time

is represented by a step function as illustrated in Fig. 3.7(a). Oscillations were numerically

damped by increasing θ to 2/3. These oscillations were also reduced by linearly increasing

and decreasing the value of Cen with a slope of 10% in the step function shown in Fig. 3.7(a).


Time step, ∆t

Since the solution of Eq. 5.20 is unconditionally stable for θ = 2/3, the choice of time step

was based on accuracy and computing efficiency considerations only.

Initially, the nonlinear solution of Eq. 5.20 was iterated at each time step by evaluating

coefficients from the values of φ obtained at a previous iteration. The iterative procedure

was terminated when a predefined convergence criterion was reached at the given time step.

The convergence of the iterative scheme was defined in terms of the normalized value of the

change in the computed variable φ between two iterations, i.e.,∣∣∣∣φi+1 − φi

φi

∣∣∣∣ ≤ ε (5.92)

where φi+1 and φi refer to the approximation of the solution after the (i + 1)th and ith

iterations, respectively, and ε is a specified convergence parameter. To increase the rate of

convergence in a given time step, a successive underelaxation method was employed:

φi+1 = ω φi+1 + (1− ω)φi (5.93)

where ω is a constant known as the relaxation factor in the range 0 < ω < 1.

Following the above procedure, the solution to Eq. 3.29 (moisture diffusion) was obtained

by equating the time step to 1 day, 2 days, and 5 days. Figure 5.7 shows the calculated

moisture profiles along diagonal 1–104 of the mesh shown in Fig. 5.6 after 120 days and 1

year of exposure for the different time steps considered (hmax and hmin were 90% and 65%,

respectively, and h (t = 0) was 100%). As it can be observed from the figure, the difference

in the solutions is more significant at early stages of exposure (Fig. 5.7(a)), with a 13%

difference with ∆t = 5 days compared to ∆t = 1 day; however, these differences tend to

decrease as time proceeds. In fact, it was observed that after 5 years of exposure the profiles

were almost identical regardless of the chosen time step. For the linear problem of heat

transfer (Eq. 3.46), the different values selected for ∆t did not affect the obtained solution,

which resulted to be the same in the three cases even at an early age.

In spite of the fact that the influence of ∆t on the solution of Eq. 3.29 tended to disap-

pear as time proceeded, it was observed that the number of iterations required to achieve

convergence for ε = 10−4 increased significantly as ∆t increased (see Fig. 5.8). Thus a value

of 1 day was chosen initially as the time step.


0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Concrete cover (mm)

Po

re r

elat

ive

hu

mid

ity,

h

120 days∆t = 1 day

∆t = 2 days (9%)

∆t = 5 days (13%)

(a) 120 days of exposure.

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Concrete cover (mm)

Po

re r

elat

ive

hu

mid

ity,

h

1 year

∆t = 1 day

∆t = 2 days (3%)

∆t = 5 days (5%)

(b) 1 year of exposure.

Fig. 5.7: Influence of time step ∆t on the calculated pore relative humidity profile.


0

20

40

60

80

100

0 40 80 120 160 200 240 280 320 360

Time (days)

Nu

mb

er o

f it

erat

ion

s

∆t = 1 day

∆t = 2 days

∆t = 5 days

Fig. 5.8: Dependence of the number of iterations needed for convergence on the time step ∆t.

The relaxation factor ω used in Eq. 5.93 was varied over the whole range from 0 to 1 in

the numerical tests described. Figure 5.9 shows the dependence of the number of iterations

required for convergence to be achieved (ε = 10−4) on ω. The optimum value of ω for which

convergence was faster was found to be 0.1, which means that more weight is put on the

solution of variable φ obtained at a previous time step. However, it was noticed that the

problem of chloride transport with nonlinear binding posed convergence problems, especially

at the points where de-icing salts are no longer applied (Cen = 0 in the step function). It is

for this reason that the problem was then treated as a linear one evaluating Dh and Dc from

humidities and chloride concentration values obtained at a previous time level without any

iterations. To minimize the error, the time step ∆t was then reduced to 12 hours.

5.5.2 Solution algorithm

The solution algorithm of the problem proceeds as follows (see Fig. 5.10):

1. Input of the general geometry of the structure, material properties, chloride threshold

concentration, and initial and boundary conditions.

2. During the initiation stage:


0

5

10

15

20

0 40 80 120 160 200 240 280 320 360

Time (days)

Nu

mb

er o

f it

erat

ion

s

= 0.1

= 0.5

= 0.9

ω

ωω

Fig. 5.9: Dependence of the number of iterations needed for convergence on the relaxation factor

ω.

(a) Solution of the temperature distribution T (x, y) throughout the concrete (Eq. 3.46).

(b) Solution of the pore relative humidity level h (x, y) in concrete (Eq. 3.29). The

amount of evaporable water ωe (x, y) is determined from adsorption isotherms

(Eq. 3.37).

(c) Solution of the concentration of chlorides dissolved in the pore solution Cfc (x, y)

(Eq. 3.19). The concentration of total chlorides in concrete Ctc (x, y) is determined

by means of chloride binding relationships (Eqs. 3.10, 3.13, 3.14, 3.15, and 3.16).

(d) If the total chloride concentration at the reinforcing bar has reached the critical

value, corrosion is assumed to have initiated (Step 3). If not, time is incremented

by ∆t = 12 hours and Step 2 is repeated.

3. During the propagation stage:

(a) In addition to the solution of the equations presented in the initiation stage,

estimate of the anodic current ia (Eq. 3.51).

(b) Solution of the dissolved oxygen profiles Co (x, y) throughout the concrete (Eq. 3.57).


(c) Estimate of the amount of hydrated red rust accumulating at the steel/concrete

interface (Eq. 4.1).

(d) Conversion of accumulating Fe(OH)3 into an equivalent uniform internal pressure

applied around the steel/concrete interface (Eq. 4.5).

(e) Evaluation of the state of stress in the surrounding concrete by means of an elastic

analysis (Eqs. 4.9 and 4.16). If internal cracks have reached the external surface

of concrete, the service life of the reinforced concrete member is assumed to have

ended. If internal cracks have not propagated through the entire concrete cover,

the entire procedure (Steps 2 and 3) is then repeated for the following time step.


Yes

Yes

No

No[Cl-] at steel > threshold?

Input data:* geometry* material properties* [Cl-] threshold* initial conditions* boundary conditions

End of analysis

Evaluate:1) anodic current2) oxygen profile

Evaluate:1) heat profile2) moisture profile3) chloride profile

Increment time step

Solve forstate of stresses

in concrete

Is concrete surface cracked?

Pro

paga

tion

stag

eIn

itiat

ion

stag

e

Fig. 5.10: Solution algorithm for proposed computational service life model.

Chapter 6

Analytical Results

6.1 Introduction

The theoretical model described in Chapters 3 and 4 and the finite element formulation

presented in Chapter 5 were implemented in a computer program written in Fortran77. The

numerical performance of the mathematical model was examined by studying the influence

of the different model parameters and different mechanisms considered on each of the stages

illustrated in Fig. 3.1, namely chloride penetration into concrete, oxygen transport to the

reinforcing steel, and cracking of the concrete cover due to corrosion products accumulation

around the reinforcement. The results of the numerical analyses are presented in the following

sections.

6.2 Finite element model

The numerical simulations were based on a linear strip of reinforced concrete with a cover to

the reinforcement of 50 mm, as illustrated in Fig. 6.1. The finite element mesh was only sub-

ject to flux boundary conditions along the left side. This problem simulates one-dimensional

flow, and it represents the “semi-infinite medium” for which an analytical solution for the

case of constant diffusivity and constant surface concentration exists (Eq. 2.6). Besides the

desire to compare the finite element solution of the problem to closed-form solutions readily

available, the geometry of the mesh was kept simple in order to better identify trends in the

126

Analytical results 127

1 2 3 4 5 6 7 8 9 10 11

x φ/ n = 0

φ/ n = 0

10x5.0=50mm

poles ofexpansion

Fig. 6.1: Finite element mesh simulating a linear semi-infinite concrete strip.

mathematical formulation.

In order to determine the influence of the element size on the results, the concrete cover

was discretized by using fifty 1× 1 mm, twenty 2.5× 2.5 mm, and ten 5× 5 mm rectangular

elements, respectively. The “far field” (i.e., x > 50 mm) was approximated by one singly

infinite element in the three cases. Figure 6.2 shows the resulting free chloride profiles for a

concrete with Dc = 1.0 × 10−12 m2/s (assumed to be constant over time, i.e., the effect of

temperature, time of exposure and chloride binding has been ignored) and immersed in 0.5M

chloride solution after an exposure period of one year, five years, and 10 years, respectively.

Also shown in the graphs are the corresponding profiles obtained from the closed-form solu-

tion given by Eq. 2.6, which assumes both the chloride surface concentration and diffusion

coefficient to be constant over time (Crank, 1975). The numerical solutions obtained for the

different meshes are identical regardless of the element size used. The 5× 5 mm mesh gives

the same results as the other two more refined meshes, and the computation time is highly

decreased as less elements are needed in the analysis. Note that the numerical approxima-

tion to the diffusion problem with bilinear rectangular elements (a linear approximation of

the concentration of chlorides within the element) compares very well with the analytical

solution given by Eq. 2.6. However, as the period of exposure increases (see Fig. 6.3), the

singly infinite element used to approximate the “far field” boundary condition tends to un-

derestimate the buildup of chlorides at the reinforcement level as compared to the analytical

solution. This can already be observed after 10 years of exposure, although the discrepancy

is more significant at 25 years (Fig. 6.3). This trend was further studied by plotting the

free chloride history at x = 50 mm over a 100 years given by the 5 × 5 mm finite element

mesh solution and Eq. 2.6 (see Fig. 6.4). Comparison between the analytical and numerical

solutions confirms the lower buildup of chlorides at the reinforcing steel of the latter; how-


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

Analytical solution (Eq. 2.6)1x1 mm2.5x2.5 mm5x5 mm

1 year

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

5 years

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

10 years

Fig. 6.2: Calculated free chloride profiles after (a) 1 year, (b) 5 years, and (c) 10 years of exposure,respectively, for meshes with 1× 1 mm, 2.5× 2.5 mm, and 5× 5 mm rectangular elements.


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

Analytical solution (Eq. 2.6)

5x5 mm

25 years

Fig. 6.3: Free chloride profiles after 25 years of exposure given by the 5 × 5 mm finite elementmesh solution and Eq. 2.6.

0.0

5.0

10.0

15.0

20.0

0 20 40 60 80 100

Years

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)


5x5 mm

50 mm

Fig. 6.4: Free chloride build-up at x = 50 mm over an exposure period of 100 years.


ever, the rate at which both solutions increase with time seems to be similar. According to

Damjanic and Owen (1984) bilinear infinite elements do not give sufficient accuracy in areas

near to the finite edge of the infinite domain, and the authors suggest the use of higher order

elements instead to get better approximations.

6.3 Chloride penetration

The response of the proposed chloride ingress model to its different input parameters was

investigated by considering two different cases: a saturated concrete exposed to a chloride

solution (pure diffusion), and a partially-saturated concrete exposed to a chloride solution

and also subjected to wetting and drying cycles (diffusion and convection).

It was assumed that the concrete taken as a reference case in the numerical examples

had w/cm of 0.3 and cementitious content of 450 kg/m3 of concrete. The material param-

eters used in the numerical analyses are shown in Table 6.1. The reference temperature

and age at which the humidity and chloride diffusivities were assumed to be measured were

23C and 120 days, respectively. Nonlinear chloride binding was considered by assuming a

Freundlich isotherm between bound and free chlorides; the binding constants shown in Ta-

ble 6.1 are representative of OPC cement pastes with w/cm = 0.3 and 40% slag replacement

level (Martın-Perez, Zibara, Hooton, and Thomas, 2000). The concrete was assumed to be

exposed for the first time after 14 days.

The sensitivity of the model was evaluated by varying a single parameter while maintain-

ing the others constant. Each input parameter was studied independently. The mesh used

in the finite element analyses is shown in Fig. 6.1.

6.3.1 Diffusion

To simulate diffusion as the main mechanism of chloride ingress, the maximum and minimum

environmental humidities (hmax and hmin in Eq. 3.44) were kept constant at 100%. Thus

the moisture content of the concrete cover remained at saturation conditions at all times.

Parameters that affect chloride diffusion in concrete and were investigated in this study

were the surface chloride transfer coefficient Bc, the temperature distribution throughout


Table 6.1: Material parameters used in the numerical simulations.

Heat transfer

λ 1.4 W/m·C

%c cq 1.932 ×106 J/m3·C

BT 0.07 W/m2·C

T (t = 0) 23C

Moisture diffusion

Dh 10−11 m2/s

Bh 1.0 m/s

h (t = 0) 1.0

Chloride transport

Dc 10−12 m2/s

Bc 1.0 m/s

m 0.15

Freundlich binding α = 1.05 , β = 0.36

Cfc (t = 0) 0.0 kg/m3 solution

Cen 17.73 kg/m3 solution


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

Bc = 1 m/s

Bc = 3E-07 m/s

Bc = 6 m/s

1 year

5 years

25 years

Fig. 6.5: Influence of Bc on the calculated free chloride profiles.

the concrete cover, the time of exposure, and the binding properties of the used cementitious

system. The results of the numerical analyses of the reference case (Table 6.1) and the

cases where the mentioned model parameters were varied are shown in the following. Also

shown are some comparisons of the performance of the model with respect to experimentally

obtained chloride profiles found in the literature.

Influence of Bc

The effect of the surface chloride transfer coefficient Bc on chloride penetration profiles

due to ionic diffusion was studied by varying its value from 3.0 × 10−7 m/s (a mean value

reported by Akita et al. (1997) to fit experimental results on drying of concrete) to 6 m/s

(maximum value used by Saetta et al. (1993) in their chloride ingress model). As it can

be seen from Fig. 6.5, significant changes in the value of Bc do not affect the resulting free

chloride penetration profiles at all, regardless of the time of exposure. The only difference

observed was that equilibrium between the exposed surface and the external solution took

few more time steps for lower values of Bc; however, note that the time step was set to 1/2

day making this difference not really significant.

Although the value given to Bc did not show any influence on the calculated profiles


due continuous exposure to a chloride solution (pure diffusion case), it was observed that

large values of Bc led to problems in the numerical solution when the chloride solution was

applied assuming the step function shown in Fig. 3.7(a). However, the problem was solved

by keeping Bc rather small (1× 10−5 m/s), so that equilibrium between the exposed surface

and the external solution would take a little longer when external chlorides were suddenly

applied to or removed from the concrete surface.

Influence of temperature

The effect of temperature on chloride diffusion was studied by running an example where

the seasonal temperature was allowed to vary sinusoidally with Tmax = 30C (summer) and

Tmin = −10C (winter) and another example where the temperature was kept constant at

23C during the entire year. Input parameters used to solve the equation describing heat

transfer in concrete are listed in Table 6.1. The resulting free and total chloride profiles

after an exposure period of 1 year, 5 years, and 25 years are shown in Fig. 6.6. As it can be

observed from the graph, the temperature distribution in concrete greatly affects the resulting

chloride profiles. The dependence of the chloride diffusion coefficient on temperature follows

the Arrhenius’ relation given in Eq. 3.7 (Fig. 3.2(a)), and Dc becomes almost negligible

for temperatures lower than 0C. Thus most of the chlorides penetrate the concrete during

the warm months of the year. For the case of constant temperature, parameter Dc is not

increased or decreased according to temperature (Dc,ref was assumed to be determined at

23C), and chlorides penetrate the concrete at a constant rate with respect to T during the

entire year. The effect of temperature is therefore important to be considered in calculations

involving structures that are likely to be exposed to very different seasonal temperatures

along the year. However, if the annual temperature range is not too large, an average

temperature value typical of a 12-month period can be used in the analysis.

Influence of time of exposure

The effect of the time of exposure on chloride diffusion was studied here by comparing the

case where Dc was allowed to decrease with time according to Eq. 3.8 (Fig. 3.2(b)) to that

of constant diffusivity with respect to time. The parameter m was calculated from Fig. 3.3,


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

Variable temperature (-10 to +30 C)

Constant temperature (23 C)

1 year

5 years

25 years

(a) Free chloride concentration profiles.

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

Concrete cover (mm)

To

tal c

hlo

rid

es ,

Ctc

(% w

t o

f ce

men

titi

ou

s)

1 year5 years

25 years

(b) Total chloride concentration profiles.

Fig. 6.6: Effect of temperature on chloride penetration due to diffusion.


which for a concrete of w/cm = 0.3 is 0.15. Note that this value may not be significant for

the type of concrete being modelled here; however, it illustrates the effect of parameter m

on the chloride penetration profiles. Figure 6.7 shows the calculated free and total chloride

profiles after an exposure period of 1 year, 5 years, and 25 years, respectively. The effect of

t on the resulting profiles becomes more significant as the time of exposure increases, but it

seems that it does not exhibit any influence during the first year of exposure. The reason

for this is that the value of Dc = 1.0 × 10−12 m2/s was assumed to be achieved after 120

days, and it is increased according to Eq. 3.8 in the numerical analysis for the time previous

to that point. Thus more chlorides are allowed to penetrate during that period compared to

the case where Dc is constant since first exposure.

Influence of chloride binding

The effect of the binding isotherm nature on the chloride concentration profiles was studied

by analysing three different cases: one in which binding was neglected; a second one in which

a linear relationship between bound and free chlorides was assumed (α = 0.07); and, a third

one in which nonlinear binding (Freundlich) was taken into account (α = 1.05 and β = 0.36).

The input parameters to the binding isotherms were evaluated from chloride binding data

obtained at the University of Toronto (Martın-Perez et al., 2000) for OPC cement pastes

with w/cm = 0.3 and 40% slag replacement level (see Fig. 6.8). The binding constants of

each of the isotherms described in Section 3.2.3 were obtained by applying the least-squares

method to the data shown as black squares in Fig. 6.8 and by assuming that the concrete

had a porosity of 8% and a cementitious content of 450 kg/m3. Note that these values

depend on the units used for Cfc and Cbc, which are kg/m3 of pore solution and kg/m3

of concrete, respectively. The Freundlich isotherm was used to describe nonlinear binding

in the numerical simulations since it fit the experimental results better than the Langmuir

relationship.

The results of the numerical analyses are shown in Figs. 6.9 and 6.10, where free and

total chloride concentration profiles after an exposure period of 1 year, 5 years and 25 years

are plotted with depth, respectively. As expected, the calculated penetration depth of free

chlorides is lower when Freundlich binding is considered, since the decrease of Dc at low


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

Time of exposure ignored

Dc decreased with time

1 year

5 years

25 years

(a) Free chloride concentration profiles.

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

Concrete cover (mm)

To

tal c

hlo

rid

es ,

Ctc

(% w

t o

f ce

men

titi

ou

s)

1 year

5 years

25 years

(b) Total chloride concentration profiles.

Fig. 6.7: Effect of time of exposure on chloride penetration due to diffusion.


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 20 40 60 80 100Free chlorides, C fc

Bo

un

d c

hlo

rid

es, C

bc

Linear isotherm(α = 0.07)

Freundlich isotherm (α = 1.05, β = 0.36)

Langmuir isotherm(α = 0.39, β = 0.07)

Fig. 6.8: Idealized binding isotherms for a concrete with 40% slag replacement level and w/cm =0.3. (It was assumed that the porosity was 8% and the cementitious content was 450 kg/m3 of

concrete).

concentration values of Cfc is much more significant in this case (see Fig. 3.5). For the same

reason, the free concentration profile due to linear binding is slightly lower than that due to

Freundlich binding at high values of Cfc, since the linear relation between bound and free

chlorides provides for more bound chlorides at high concentrations of Cfc. The decrease in

penetration depth due Freundlich binding becomes more significant as the time of exposure

increases. Note that the penetration depth after an exposure period of 25 years for Freundlich

binding (40 mm) is the same as that after an exposure period of 5 years when no binding is

taken into account. Thus consideration of the binding properties of the cementitious system

used has a significant effect on the calculated chloride profiles due to ionic diffusion.

The shapes of the total chloride profiles when Freundlich binding is considered are a

result of the nonlinear dependency of the corresponding binding capacity (i.e., the slope of

the chloride binding curve) on the level of free chloride ions. The binding capacity of the

Freundlich isotherm is very high and changes very quickly at low concentration values of

dissolved chloride ions, whereas it decreases significantly and becomes almost constant as

the concentration of Cfc increases (Fig. 6.8). In general, chloride profiles obtained from field

samples resemble more the curves obtained when no binding or linear binding are considered


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

No binding

Linear binding

Freundlich binding

1 year

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

5 years

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

25 years

Fig. 6.9: Effect of chloride binding on free chloride profiles after (a) 1 year, (b) 5 years, and (c) 25years of exposure, respectively.


0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

Concrete cover (mm)

To

tal c

hlo

rid

es ,

Ctc

(% w

t o

f ce

men

titi

ou

s)

No binding

Linear binding

Freundlich binding

1 year

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

Concrete cover (mm)

To

tal c

hlo

rid

es ,

Ctc

(% w

t o

f ce

men

titi

ou

s)

5 years

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

Concrete cover (mm)

To

tal c

hlo

rid

es ,

Ctc

(% w

t o

f ce

men

titi

ou

s)

25 years

Fig. 6.10: Effect of chloride binding on total chloride profiles after (a) 1 year, (b) 5 years, and (c)25 years of exposure, respectively.


rather than when Freundlich binding is taken into account. According to Sandberg (1998)

the nonlinear binding relationship observed in laboratory equilibrium tests is not relevant

for concrete structures exposed in the field, where equilibrium conditions do not prevail and

the decrease of hydroxyl ions in the pore solution due to leaching causes more chlorides to

be bound. The author observed that the relation between bound and free chlorides in field

concrete is almost linear because of the effect of counterdiffusing OH−. This may be the

reason of the discrepancy between the shapes of calculated profiles using Freundlich bind-

ing and measured profiles in the field. Tang and Nilsson (1996a) addressed this issue by

considering the dependence of chloride binding on the pH of the pore solution and formulat-

ing hydroxyl ions counterdiffusion. Their numerical results compared reasonable well with

measured chloride profiles.

By looking at Fig. 6.10, the amount of total chlorides for nonlinear binding is higher

than the ones for linear or no binding cases. Note that Freundlich binding results in higher

total chloride values at the exposed concrete surface compared to the other two cases, even

though the same external boundary conditions apply to all of them. This emphasizes the

importance of defining the nature of the chloride threshold concentration in terms of free or

total chlorides. At first, it seems to be more appropriate to define the threshold value in terms

of free chlorides since these are the ones that penetrate through the concrete cover and initiate

corrosion. However, some authors may argue that this approach is not conservative enough

since loosely bound chlorides could potentially become aggressive (Glass and Buenfeld, 1995).

Profiles in terms of total chlorides can be misleading in this respect. Higher profiles here

mean more chloride ions being bound along the diffusion path.

Comparison with some experimental results

To view the ability of the model to predict real chloride penetration data, the numerical

performance of the model was compared against some measured chloride profiles obtained

from the literature. Figure 6.11 shows as black squares the experimentally obtained free

chloride profile for an OPC cement paste of w/cm = 0.5 exposed to 1M NaCl solution for a

100 days reported by Sergi et al. (1992). The authors found that the relationship between

bound and free chlorides for their specific cementitious system was best approximated by


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc (

M)

Experimental results

Numerical results


w/cm = 0.5, 100 days

D c = 4.28x10-11 m2/s

Fig. 6.11: Comparison between free chloride profiles calculated by the model and measured bySergi et al. (1992).

a Langmuir relation with α = 1.67 and β = 4.08. The reported units for bound and

free chlorides were mmol/g of cement and mol/l, respectively. The measurement of the

effective diffusion coefficientDc was not part of their study; instead, the authors used a finite-

difference model and a least-squares criterion to find the diffusivity for which the calculated

theoretical profile best fitted the experimental data, and their result was Dc = 4.28× 10−11

m2/s. This value, however, does not truly represent the effective diffusion coefficient at

t = 100 days but the average diffusivity value over the entire period of exposure. The

experimental results reported by Sergi et al. (1992) were reproduced here by the proposed

numerical model by using this value of Dc and assuming it to be constant with respect to

time (temperature was also kept constant at 25C simulating experimental conditions). The

good comparison between experimental and numerical results illustrated in Fig. 6.11 does

not mean the good predictability of the model but rather its ability to reproduce results

reported by others. Also shown in Fig. 6.11 is the result obtained by applying Eq. 2.6 with

the same value of diffusivity. The results emphasize the importance of including chloride

binding in the calculations.

The model was also compared against the field data reported by Sandberg (1998) shown


0.0

2.0

4.0

6.0

8.0

0 10 20 30 40 50

Concrete cover (mm)

To

tal c

hlo

rid

es ,

Ctc

(% w

t o

f ce

men

titi

ou

s)

Field, 1 year

Calculated, 1 year

Field, 5.1 years

Calculated, 5 years


D c = 4.5x10-12 m2/s

SRPC, w/cm = 0.4

Fig. 6.12: Comparison between free chloride profiles calculated by the model and reported bySandberg (1998).

in Fig. 6.12. The chloride profiles shown in Fig. 6.12 correspond to a submerged marine re-

inforced concrete structure made out of SRPC concrete with a w/cm = 0.4 and cementitious

content of 420 kg/m3. Following the same procedure as Sergi et al. (1992), the author deter-

mined the value of Dc by best fit of the theoretical solution (linear binding was considered)

to the measured chloride profiles and found it to be 4.5×10−12 m2/s after an exposure period

of 216 days. The decrease of Dc with time was assumed in the computation by specifying the

age factor m to 0.1. Linear binding was also considered with a binding constant α = 0.31;

this value was obtained from a similar concrete whose binding isotherm was reported by the

same author. The value of the chloride concentration at the surface was allowed to increase

according to reported data. From the figure it can be observed that the model underesti-

mates the depth of penetration, especially after 5 years of exposure. It is believed that the

reason for these differences is that this concrete does not truly exhibit linear binding prop-

erties, since the profile given by Eq. 2.6, which does not consider chloride binding, resembles

more the field profile. This result emphasizes the importance of determining the value of Dc

from laboratory measurements as an intrinsic material property rather than from modelling

assumptions and curve fitting to measured profiles.


0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 10 20 30 40 50

Concrete cover (mm)

Po

re r

elat

ive

hu

mid

ity

, h

Dh=1E-11

Dh = 1E-10

1 year5 years

1 year

5 years

Fig. 6.13: Calculated humidity profiles.

6.3.2 Diffusion and convection

The mechanism of chloride convection due to moisture flow in concrete was examined by

subjecting the concrete with material properties listed in Table 6.1 to 7 days cycles of wetting

and drying, respectively. The atmospheric relative humidity hen was assumed to follow a

step function with a time period of 7 days, during which hen was either hmax = 1.0 (wetting

conditions) or hmin = 0.7 (drying conditions). The temperature was kept constant at 23C so

that the effect of moisture movement would be more evident. Chloride binding was neglected

in the numerical simulations since it had not been considered in the analytical formulation

for the convective term.

Figure 6.13 shows the calculated humidity distributions along the concrete depth for

Dh = 1.0× 10−11 and Dh = 1.0× 10−10 m2/s after an exposure period of 1 year and 5 years.

The pore relative humidity profiles seem to suggest that there is very little wetting of the

surface, whereas drying takes place along the entire concrete cover. Moisture movement was

represented here by pore relative humidity diffusion, which is a very slow process compared to

water absorption by a partially-saturated concrete cover. The mechanism of water absorption

through capillarity is thus not well described by Eq. 3.29 and its corresponding boundary


conditions; however, it adequately simulates the drying phase. Because of this reason, the

convective movement of chloride ions into concrete is not well represented by the proposed

formulation when cycles of wetting and drying of the concrete cover occur. However, another

simulation was run in which the concrete was assumed to be partially-saturated at time t = 0

(h(t = 0) = 0.8) and subjected to wetting conditions along the entire period of exposure

(hmax = hmin = 1.0). Figure 6.14 shows the calculated free chloride profiles after an exposure

period of 1 year and 5 years for the case of pure diffusion and diffusion and convection

combined. From the figure it is observed that the penetration depth is higher for the case

when convection is considered, although the difference is not very significant between both

cases.

6.3.3 Effect of environmental conditions

To simulate conditions in the field, a numerical simulation was run by assuming that chlorides

coming from de-icing salts were only applied during four months in a year. The atmospheric

relative humidity hen was assumed to follow a step function with a time period of 7 days,

during which hen was either hmax = 1.0 (wetting conditions) or hmin = 0.7 (drying condi-

tions). The temperature was allowed to vary sinusoidally over the year with Tmax and Tmin

equal to 30C and −10C, respectively. The analysis was assumed to start in the middle of

the fall as illustrated in Fig. 3.7.

Figures 6.15 and 6.16 show the calculated free chloride concentration Cfc profiles for the

different seasons of the year after an exposure period of 1 year, 5 years and 25 years. The

shape of the computed free chloride profiles close to the surface depend significantly on the

season of the year (environmental boundary conditions). As it is observed from the graphs,

most of the chloride penetration results during the spring and summer months. Once de-

icing salts are no longer applied, the chloride flux across the surface is set to zero, since no

washing effects due to rain or snowfall are taken into account; however, moisture flow due to

wetting and drying cycles and a higher chloride diffusivity due to an increase in temperature

during these months contribute to push the penetration front towards the reinforcing steel.

Note that there is a larger increase in the penetration depth from the spring to summer

during the first year of exposure; the chloride concentration gradient is higher at this time.


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

Diffusion

Diffusion+convection (Dh=1.0E-10)

1 year

(a) Free chloride concentration profile after 1 year.

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

5 years

(b) Free chloride concentration profile after 5 years.

Fig. 6.14: Effect of convection on chloride penetration.


However, the penetration rate of chlorides is slower after 5 and 25 years; the reduced rate of

penetration is mainly due to the effect of time on Dc (expressed in the model by factor m).

6.4 Active corrosion

The stage corresponding to active corrosion was assumed here to be linked to the polarization

characteristics of the anode and cathode and oxygen availability at the steel surface (see

Chapter 3). The numerical performance of the model in this stage was tested by analysing

the resulting oxygen profile in a concrete with a w/cm = 0.3 with properties listed in

Table 6.1 and oxygen diffusivity Do = 10−12 m2/s at 100% moisture saturation. Note that

the oxygen diffusivity is increased according to Eq. 3.58 depending on the actual moisture

saturation level of the concrete. Thus the moisture distribution along the concrete cover

will affect the amount of oxygen available at the reinforcing steel. Environmental humidity

values were varied between 100% and 70% following a step function over a period of 7 days.

It is observed from the results shown in Fig. 6.17 that the amount of dissolved oxygen in

concrete does indeed depend on the moisture distribution along the concrete cover. As the

cover partially dries out with increasing time of exposure, the oxygen penetration depth is

greater due to an increase in its diffusivity. The time taken for this increase in penetration

depth will depend on the initial moisture state of the concrete. It can also be inferred from

the profile shown in Fig. 6.17 that all the oxygen reaching the steel surface is being consumed

by the corrosion reactions. If the corrosion current is too high, oxygen at the cathode will be

depleted, bringing the concentration of dissolved oxygen at the steel surface down. When this

happens, the corrosion rate is diminished because of mass transfer limitations, thus allowing

the concentration of dissolved oxygen to build up again. It was observed that after 5 years

of exposure the corrosion current remained quite low, a little bit higher than the exchange

current density for iron dissolution (3.8 × 10−4 A/m2), suggesting that all the polarization

is taking place in the cathode and is limited by oxygen diffusion towards the steel.

A more complete representation of the corrosion process would be obtained by solving

the continuity equation (Eq. 2.34) along a reinforced concrete member, where distinct areas

could be treated as anodes or cathodes depending on the passivity state of the reinforcing

steel and where the electrical resistivity of the pore solution would have a direct effect. The


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

winter

1 year 5 years 25 years

(a) Free chloride concentration profile during the winter.

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

spring

1 year 5 years 25 years

(b) Free chloride concentration profile during the spring.

Fig. 6.15: Effect of the winter and spring seasons on chloride penetration.


0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

summer

1 year

5 years

25 years

(a) Free chloride concentration profile during the summer.

0.0

5.0

10.0

15.0

20.0

0 10 20 30 40 50

Concrete cover (mm)

Fre

e ch

lori

des

, C

fc

(kg

/m3 p

ore

so

luti

on

)

fall

1 year

5 years

25 years

(b) Free chloride concentration profile during the fall.

Fig. 6.16: Effect of the summer and fall seasons on chloride penetration.


0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

0 10 20 30 40 50 60

Concrete cover (mm)

Dis

solv

ed o

xyg

en ,

Co

winterspringsummerfall

3 months

66

912 5 years

Fig. 6.17: Dissolved oxygen concentration profiles for a cathode-to-anode ratio equal to 10.

model as it is now neglects this aspect due to limitations of the problem geometry solved

here (cross-section of a reinforced concrete member). Equation 2.34 cannot be enforced in

this type of problem since electric currents mainly move along the reinforcing steel, and this

dimension was ignored in the formulation.

6.5 Onset of cracking

Given an estimate of the corrosion rate, the time to onset of cracking tp of the concrete cover

after reinforcing steel depassivation can be calculated from Eq. 4.1, i.e.,

tp =mr

Jr πd[s] (6.1)

where mr is the mass of hydrated red rust formed per unit length of rebar (kg/m) required

to crack the concrete cover (obtained from Eq. 4.3), Jr is the rate of rust production at the

anode (kg/m2·s), given by Eq. 3.69, and d is the rebar diameter (m). The accumulation of

corrosion products around the reinforcing bar is simulated by means of an internal pressure pi

(Eq. 4.5) acting against the steel/concrete interface. When this pressure equals the pressure

needed for internal cracks to propagate through the concrete cover (Eq. 4.16) the member is

assumed to have failed.


Because of the dependence of the corrosion rate on temperature and moisture content

(Lopez and Gonzalez, 1993; Liu and Weyers, 1996; Newhouse and Weyers, 1996), it is

common practice to use an annual average corrosion rate in calculating the rate of rust

production Jr. For the analyses presented here, the anodic current density ia, assumed to

be the corrosion rate, was varied from 0.1 to 10 µA/cm2. According to Andrade and Alonso

(1996b) corrosion rates above 0.5 µA/cm2 correspond to a moderate level of corrosion whereas

corrosion rates higher than 1 µA/cm2 are associated with high corrosion levels (see Fig. 2.18).

Figures 6.18(a) and (b) show the resulting times to first cracking of the concrete cover

for various corrosion rates (log scale) assuming hydrated red rust Fe(OH)3 and ferrous hy-

droxide Fe(OH)2 to be the main corrosion products formed, respectively. Although the

exponential decay in time to first cracking with increasing corrosion rate is captured, the

obtained cracking times are somewhat lower compared to reported values (Liu and Weyers,

1996; Newhouse and Weyers, 1996). However, note that the results depend on the assumed

corrosion product formed as well as its specific density compared to steel. When it is assumed

that Fe(OH)2 is the main corrosion product formed and it occupies double the volume of

steel (usual assumptions in this type of calculations), the times to onset of cracking at the

concrete surface are increased significantly. Thus a proper evaluation of corrosion levels and

their associated damage states requires knowledge on the type of corrosion products formed.

According to the results, the model is sensible to c/d values for corrosion rates lower than 1

µA/cm2 (moderate levels of corrosion), whereas higher corrosion rates correspond to much

lower cracking times regardless of the concrete cover-to-diameter ratio.

The low cracking times as compared to reported values in the field is due to limita-

tions in assumptions adopted in the model, such as that (1) all corrosion products formed

create a pressure around the reinforcing bar, (2) the build-up around the rebar is created

by the same type of corrosion product, (3) cracked concrete does not exhibit any residual

strength, and (4) confinement effects due to transverse reinforcement or external loading are

neglected. Newhouse and Weyers (1996) and Andrade et al. (1993) observed that corrosion

products diffuse away from the steel as much as 5 cm, thus not all the corrosion products

formed will exert a pressure against the concrete. In fact, once internal cracks appear at

the steel/concrete interface (Eq. 4.9), corrosion products could easily diffuse through the

newly formed cracks releasing the build-up pressure against the concrete cover, and thus


0

1

2

3

4

5

6

0.1 1 10

Corrosion rate, i corr (µ A/cm2)

Tim

e to

cra

ckin

g (

year

s)

c/d = 2

c/d = 4

c/d = 6

Fe (OH)3, f' c =45 MPa

(a) Fe(OH)3 assumed to be main corrosion product (%r = %s/4).

0

2

4

6

8

10

12

14

0.1 1 10


Tim

e to

cra

ckin

g (

year

s)

c/d = 2

c/d = 4

c/d = 6


(b) Fe(OH)2 assumed to be main corrosion product (%r = %s/2).

Fig. 6.18: Time to first cracking of the concrete cover versus corrosion rate.


delaying the time to onset of cracking at the surface. This aspect is not considered by the

model. Furthermore, Liu and Weyers (1996) and Weyers (1998) suggest that the rate of rust

production may not be a linear relationship of the product of the corrosion rate and time

as stated by Faraday’s law (Eqs. 4.1 and 6.1), because iron ions have to diffuse through the

rust layer built around the reinforcing bars before any further oxidation can take place. As

a result, the authors proposed a nonlinear relation between corrosion build-up and time to

cracking, but it was not used here because it was not consistent with the units used.

According to Reinhardt (1992) and Noghabai (1995), the strength-based model devel-

oped by Tepfers (1979), used here to model the splitting behaviour of concrete due to rust

accumulation around the reinforcing bars, fails to provide a consistent assessment of the

material behaviour since qualities of fractured concrete are ignored. Both Reinhardt (1992)

and Noghabai (1995) modified Tepfer’s model by incorporating the softening behaviour char-

acteristic of cracked concrete in the partly-cracked stage. The model was used to estimate

cracking of concrete due to radial forces generated by bond and was able to better reproduced

experimental results.

Regardless of the times obtained to onset of cracking, first-cracking of the concrete cover

as a limiting damage criterion may be not be an appropriate indicator of unsatisfactory resid-

ual structural capacity. Furthermore, cracking times reported in the literature are seldom

related to corresponding cracking widths. Models as the one illustrated in Fig. 2.19 may be

more realistic in representing the actual process of deterioration taking place in an existing

structure, where deterioration beyond the onset of cracking is considered too. A better rep-

resentation of the damage process may be to estimate the time to require the concrete cover

to spall off once longitudinal cracking is present.

After the propagation of an internal crack through the entire concrete cover, the pressure

pi generated by corrosion products build-up around the reinforcing bar will be sustained

until part of the cover spalls off. By assuming a 45-plane of failure with respect to the

external concrete surface, as illustrated in Fig. 6.19, the concrete cover will not spall off

until the tensile strength of section AB has been exceeded (Tepfers, 1982). By establishing

equilibrium in the vertical direction, the internal pressure pi corresponding to the attainment


45o

A

Bpi

c

d

longitudinal crack

f’t

Fig. 6.19: Splitting failure pattern of the concrete cover.

of tensile strength at AB is given by:

pi = 1.8f ′t (c/d+ 0.5) [MPa] (6.2)

Equation 6.2 provides a limiting value for the internal pressure pi before failure at the

surface AB occurs. Figure 6.20 shows the plot of internal pressure pi normalized to f ′t against

the ratio of concrete cover c to rebar diameter d for the expression in Eq. 6.2 as well as for

the relations given by Eqs. 4.9 and 4.16 in Chapter 4. As it can be observed in Fig. 6.20,

the resistance of the concrete cover against the pressure exerted by the corrosion products

goes well beyond the appearance of longitudinal cracks at the external surface (denoted by

the partly-cracked elastic curve in Fig. 6.20). This resistance would be further increased

if confinement due to the presence of transverse reinforcement and to the residual tensile

strength of splitted concrete were taken into account in the formulation.

Figure 6.21 shows the times corresponding to the propagation stage versus corrosion rates

when Eq. 6.2 is used as a limiting criterion. As expected, times to assumed “end-of-service-

life” are much higher than results previously presented. Note than the effect of cracking

other than that induced by corrosion build-up has not been considered in the analysis. For

a realistic estimation of the deterioration rate of in-service structures, loading and stress

conditions in the reinforced concrete members with corroding steel would also have to be

taken into account (Philips and Hooton, 1998).


0

2

4

6

8

10

12

0 1 2 3 4 5 6

c/d

pi / f

' t

elastic

partly-crackedelastic

concrete coverspalls off

Fig. 6.20: Load-carrying capacity of concrete cover versus cover-to-diameter ratio.


0

2

4

6

8

10

12

14

16

0.1 1 10


Tim

e to

sp

allin

g (

year

s)

c/d = 2

c/d = 4

c/d = 6


(a) Fe(OH)3 assumed to be main corrosion product (%r = %s/4).

0

5

10

15

20

25

30

35

40

0.1 1 10


Tim

e to

sp

allin

g (

year

s)

c/d = 2

c/d = 4

c/d = 6


(b) Fe(OH)2 assumed to be main corrosion product (%r = %s/2).

Fig. 6.21: Time to spalling of the concrete cover versus corrosion rate.

Chapter 7

Closure

7.1 Conclusions

A general model for service life prediction associated with chloride-induced corrosion was

proposed in this study. The mathematical formulation followed a two-stage sequence: an

initiation stage, during which chloride ions penetrate the concrete towards the reinforcing

steel, and a propagation stage, where the state of damage in concrete resulting from corrosion

build-up at the rebar is assessed. From the analytical study performed on the computational

model, the model for chloride ingress proved to be promising as a prediction tool for chloride

profiles and time-to-depassivation estimates of reinforced concrete structures exposed to

chloride environments, since it showed sensitivity to almost all relevant parameters involved

in the process.

Comparisons between a closed-form solution of the governing partial differential equation

and the numerical solution provided by the proposed model showed that the use of linear

interpolation of the field variables over the finite elements is suitable to obtain an accurate

solution of the problem along the concrete cover. However, the bilinear infinite elements

developed to simulate the “far field” response tend to underestimate the build-up of chlorides

near the finite edge of the infinite elements. This is believed to be due to the low order of

approximation of the field variable over this domain, and, according to the literature on the

subject, a higher order element formulation would improve the accuracy of the results.

Numerical simulations using the chloride diffusion model showed that the calculated

156

Closure 157

chloride profiles were insensitive to values given to the chloride transfer coefficient Bc but

most sensitive to the temperature distribution in the concrete, time of exposure and binding

characteristics of the cementitious system used. The effect of temperature distribution in

concrete on chloride ingress is significant on structures that are likely to be exposed to

very different environmental temperatures throughout the year. The assumption of constant

temperature during the period of exposure can underestimate or overestimate (depending on

the value assumed) the chloride penetration depth at a given time. It is therefore important

to consider the effect of temperature in the formulation.

The decrease in chloride penetration due to the effect of time of exposure on chloride

diffusion was found to be significant even for low values of parameter m (a value of 0.15

was used in the numerical simulations); however, experimental evidence of the effect of t

on chloride diffusivity has shown that concretes having supplementary cementing materials

such as fly ash and slag exhibit higher values of m, making the effect of t on Dc even more

relevant for chloride penetration calculations.

A sensitivity analysis of the chloride binding problem on chloride penetration profiles

highlighted the need to determine the binding properties of a specific cementitious system,

since these have a great influence of the “apparent diffusivity” of chloride ions in concrete.

From the results it is concluded that the nature of the binding relationship assumed for

modelling purposes has a significant implication on service life predictions, i.e., calculations

with no binding result in higher free chloride profiles, and therefore underestimate the time to

corrosion initiation of a reinforced concrete structure exposed to chloride salts. Also, it was

observed that the shape of the calculated profiles using Freundlich binding isotherms did not

resemble profiles obtained from field data. The shape of profiles when no binding or linear

binding were considered was found to be more realistic of what has been observed in the field.

This is believed to be due to the lack of control of boundary conditions and the variation of

the local pH of the pore solution in concretes exposed in the field. This confirms observations

by others in which binding isotherms determined from in-service concretes followed a linear

relationship instead of a nonlinear one, the latter being more prevalent under equilibrium

conditions well established in laboratory measurements. It is therefore important to assess

the binding properties of cementitious systems along with pH variations so that more realistic

estimations of the time to depassivation of reinforcing steel are obtained.

Closure 158

From the numerical analyses on chloride penetration due to diffusion and convection,

it was concluded that the model is not able to simulate chloride ingress under alternate

wet/dry cycles. Whereas the moisture diffusion equation describes drying of concrete in a

realistic way, moisture diffusion is too slow to simulate the quick capillary absorption process

that takes place in partially-saturated concretes under wetting conditions. It was observed

that when cycles of wetting and drying are applied to an initially saturated concrete, a net

outflow of moisture occurs (drying of the concrete cover) and, consequently, there is not an

increase in the chloride penetration front due to inwards moisture movement. However, the

model was able to show some increase in the penetration depth when wetting conditions

were applied to an initially unsaturated concrete.

The numerical analyses presented in the last chapter also showed the influence of envi-

ronmental conditions on the shape of the computed chloride profiles, given the assumption

that de-icing salts were only applied during a specified period of the year. This emphasizes

the importance of correctly accounting in the formulation the conditions at the boundaries

if realistic profiles are to be expected.

Because of limitations associated with the problem geometry considered in this study

(cross-section of a reinforced concrete member), the model describing active corrosion does

not identify all the relevant factors governing the kinetics of corrosion, such as the concrete

resistivity. The corrosion model proposed in this thesis was based under the assumption

that the kinetics of corrosion are limited by oxygen diffusion through the concrete cover

(concentration polarization). Under the given assumption, the analysis confirmed that the

corrosion rate remains low as long as oxygen diffusivity is low, which is a function of the

porosity as well as moisture content of the concrete. However, depending on the moisture

state of the concrete cover, resistivity can be a determining factor in the kinetics of the

corrosion process, and this aspect is not considered in the present formulation. Corrosion

is a very complex phenomenon which is not very well understood yet; in particular, there

seems to lack a general theory that explains the relation among the main parameters of the

process, and this makes its quantification a difficult task.

The mechanical model used to assess the damage in concrete due to accumulation of

corrosion products proved to be sensitive to important parameters, such as the cover-to-

diameter ratio and the type of corrosion product formed. Nevertheless, the calculated times

Closure 159

to appearance of cracks at the concrete surface were lower than reported values. The onset

of longitudinal cracking as an end-of-service-life criterion may be not a realistic target for

assessment of the remaining life of in-service structures. In fact, a better representation of

the problem is the quantification of corrosion-induced damage in terms of crack widths or

residual bond capacity.

7.2 Recommendations for future work

Based on the findings of this study, the following recommendations for future work are listed:

Numerical model

• study of the effect of higher-order (quadratic and cubic) infinite elements in the ap-

proximation of the field variable near the finite edge of the “far field” compared to

available analytical solutions based on a semi-infinite medium.

• generalization of the model to a three-dimensional formulation. Inclusion of the third

dimension enables the solution of the continuity equation (Eq. 2.34), which is a better

representation of the corrosion process.

Chloride ingress model

• verification of the chloride diffusion model for a wide range of concrete types and

environmental conditions.

• further refinement of equations describing the process of capillary absorption in con-

crete. This also includes a better quantification of the critical parameters involved as

well as measurements of field moisture profiles.

• implementation of hysteresis curves describing moisture adsorption/desorption in ce-

mentitious systems.

• generalization of the model to situations where chlorides penetrate the concrete due to

pressure gradients (permeation).

Closure 160

• study of exposure conditions of reinforced concrete highway structures exposed to

de-icing salts for better definition of the boundary conditions, such as intensity and

duration of chloride applications and wet/dry cycles.

• consideration of temperature and pH of the pore solution on the binding properties of

a specific cementitious system. This also requires modelling of hydroxyl ions diffusion.

• consideration of chloride binding on the convection term and how this affects the

resulting chloride profile.

• study of the effect of concrete cracking on transport properties.

Corrosion model

• solution of the continuity equation.

• study of the role of chloride ions on the kinetics of the corrosion process.

• consideration of different proportions of corrosion products formed and their diffusion

away from the steel through the concrete pores and cracks.

• study of the effect of concrete cracking on corrosion rate.

• further investigation on the interaction between corrosion expansion and incurred dam-

age.

Mechanical model

• inclusion of the softening behaviour of cracked concrete and confinement provided by

the reinforcement and loading in the model assessing corrosion-induced damage.

• generalization of the state of stress in concrete and reinforcement to include in-service

conditions.

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