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Durability Probabilistic Evaluation of RC Structures 

Subjected to Chloride Ion 

Han-Seung Lee1，Mohamed A. Ismail1,*, Mohd Warid Hussin2 
1School of Architecture & Architectural Engineering, Hanyang University, Ansan, S. Korea

 
2
Construction Research Centre (UTM CRC), Institute for Smart Infrastructure and Innovative 

 
Construction, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia 

*Corresponding Author Email: mismail@hanyang.ac.kr, phone: +82-31- 4005181, fax: +82-31- 4368169 

 

Abstract - Chloride attack on concrete structures is 

becoming a primary factor that deteriorates the 

durability of concrete structures.  For this reason, 

research has been conducted on chloride ion 

penetration and diffusion. This research produced an 

accurate durability life prediction through reliability 

assessments and proposes a prediction method for 

the chloride ion diffusion coefficient of a concrete 

applied assessment program for reliability. As a result, 

test materials were fabricated using different 

admixtures, and chloride ion diffusion coefficient was 

calculated by applying an RCPT test at each 

equivalent age. Based on the results, reliability 

prediction formulas were indicated through the 

reliability analysis for a durability life design using a 

Monte Carlo method. In addition, results were verified 

through comparisons and analysis using the proposed 

formula with the investigated data for chloride ion 

diffusion.  

 

Keywords - Evaluation of durability life, Chloride ion 

diffusion coefficient, Monte Carlo method. 
 

 

I. INTRODUCTION 

 

It is widely known that the ingress of chloride ions 

constitutes a major source of durability problems 

affecting reinforced concrete structures that are 

exposed to saline environments. Once a sufficient 

quantity of chloride ions has accumulated around the 

embedded steel, pitting corrosion of the metal is 

liable to occur unless the environmental conditions 

are strongly anaerobic. In the design of concrete 

structures, the influence of chloride ingress on 

service life must be considered [Metha 2006; 

Papadakis 2000; Nielsen et al. 2003; Han 2007; Song 

et al. 2007]. 

 

The common service life model for the chloride 

induced corrosion of reinforcing steel in concrete 

involves two time periods. The first is the time for 

chloride diffusion until a sufficient concentration of 

chlorides is available at the reinforcing bar depth to 

initiate corrosion. The second is the time for corrosion 

damage (from initiation to cracking and spalling of the 

cover concrete) to the end of functional service life.  

An apparent diffusion process, based on Fick’s 

second law, can be used to model the time for 

chloride to reach and initiate corrosion, where first 

repair and rehabilitation at reinforcing steel depths 

will take place. When solved for the condition of 

constant surface chloride and a one-dimensional 

infinite depth, Fick’s second law takes the following 

form [Kim et al. 2007]:  
 

0
( , ) (1 ( / 4 * * ))

C
C x t C erf x D t        (1) 

 

Where ( , )C x t  is chloride concentration at depth and 

time; 
0

C  is surface chloride concentration; CD  is apparent 

chloride ion diffusion coefficient; t  is time for diffusion; x  is 

concrete cover depth and erf  is statistical error function. 

When ( , )C x t  is set equal to the chloride corrosion 

initiation concentration and Eq. (1) is solved for t , the time 
for the diffusing chloride ions reaches rebar and initiates 

corrosion. However, for a given real condition, the values of 

( , )C x t ,
0

C  CD  and x  are random variables, each with 

their own statistical distributions, means, and variances. A 

solution to Eq. (1) for the time of diffusion should include 

the probabilistic nature of the input variables [Kirkpatrick et 

al. 2002; Enright and Frangopo 1998]. The time for 

corrosion damage to the end of functional service life is 

also a random variable and depends on the corrosion rate, 

concrete cover depth, reinforcing steel bar spacing, and 

size [Liu and Weyers 1998]. To predict the whole service 

life, the probabilistic nature of these variables should also 

be considered. 

 
One common modern statistical technique is called Monte 

Carlo simulation. Monte Carlo is a general class of 

repeated sampling methods, where a desired response is 

determined by repeatedly solving a mathematical model 

using values randomly sampled from probability 

distributions of the input variables [Kalos and Whitlock 

1986]. 

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II. SUMMARY OF CHLORIDE PENETRATION 

TEST 

 

A. Experimental Program 

Mineral components for improvement of chloride 

blocking property were tested in the same 

environment. To examine diffusion blocking 

properties, Rapid Chloride Penetration Test (RCPT) 

was done. Configurations of test are presented in 

Table 1. Water-binding material ratio has two levels: 

40% and 50%. Fly Ash (FA), Blast furnace Slag (BS), 

Silica Fume (SF) and Meta Kaolin (MK) were used in 

3 different levels.  

 
Table 1. Dimensions of various fins 

 

Water- binding material ratio (%) Admixture type 
Admixture 

replacement ratio 
(%) 

Measurement 
item 

Measurement aging  
(day) 

40, 50 

Non-blending Designation - 

Compressive 
strength 

 

Chlorine ion 
diffusion 

coefficient 

7, 28, 56, 91 

Fly ash FA 10, 20, 30 

Blast-furnace 
slag 

BS 30, 50, 70 

Silica fume SF 5, 10, 15 

MetaKaolin MK 5, 10, 15 

 

Table 2 shows composition of the plain concrete, 

without adding mineral components, to accomplish 

slump 18 ± 2.5cm and air entrained quantity, high 

efficiency AE water reducing agent. For the test of 

chloride penetration, RCPT, which was proposed by 

Tang & Nilsson, was referred to as illustrated in Fig. 1. 

Accordingly, each side of the cell is filled with 0.3M of 

sodium hydroxide (NaOH) as anode and 3% of 

sodium chloride (NaCl) as cathode. After that 30V was 

applied to the cell. The test was maintained for 8 

hours; then 0.1 N water solution of silver nitrate 

(AgNO3) was sprayed after the split test body. As a 

result, the color of affected area changed. The colored 

depth was measured by Vernier calipers. The chloride 

ion diffusion coefficient was calculated using Eq. (2) 

based on the measured results. 

 

t

xax

zFU

RTL
D

dd


   

( )
2

1(2
0

1

c

c
erf

zFU

RTL
a

d



)              (2) 

 

 

 

 

 

 

 

Where: 

 

Table 2. Composition of the plain concrete 

 

Water-
cement 
ratio (%) 

Fine 
aggregate 
ratio (%) 

Unit weight (kg/m3) 

wat
er 

cement 
fine 

aggregate 
coarse 

aggregate 

40 45.6 158 395 793 954 

50 47.7 158 316 861 951 

D diffusion coefficient (m2/s) 

z atomic value of ion (z=1 for chloride ion) 

F Faraday constant (96,481.04 J/Vmol) 

U 
Voltage differences between positive and negative pulse 

(V) 

R gas constant (8.314 J/Kmol) 

T solution temperature (K) 

L specimen thickness (m) 

d
x

 
penetration depth of chloride ion (m) 

t test sustaining time 

erf error function 

d
c

 

chloride ion density at the section changed in color by 

AgNO3 

0
c

 
chloride ion density of cell located in a negative pole 

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Fig .1. Diagram of chloride ion diffusion test equipment 

 

Fig. 2 demonstrates the procedure of rapid chloride 

penetration test to examine chloride ion diffusion 

coefficient of chloride ion. 

 

 
 

 
 

Fig .2. Procedure of rapid chloride penetration test 

 

III. EVALUATION OF PROBABILISTIC 

DURABILITY LIFE 

 

A. In the latent period 

 
Corrosion in RC structure assumed to have occurred 

by the attack of salt. In this case, chloride 

concentration depth is determined by Eq. (3); it is 

also assumed that if chloride concentration is over 

the limits of (1.2 kg/m3) at steel bar position then it is 

the end of service for the structure. In this study, 

durability of the structure in sea environment was 

statistically analyzed by Monte Carlo method. Table 3 

shows the factors of statistical analysis. D0 is 

determined by RCPT, Dm is calculated by Eqs. (4a) 

and (4b) is used with chloride ion diffusion coefficient 

of chloride on the 28th day. 
 

tD
C

C
erftx

m

s

cr



)1(2)(

1
       (3) 

 

)(
1

00

t

t

n

D
D

m


 (t<tc)       (4a) 

 

n

m
t

t

n

n

t

t
DD )](

1
1[

00

0


 (t  tc)    (4b) 

 
Where: 

 
Table 3. Factors of statistical analysis 

 

Factor Average Standard deviation 

D0 Test result 1.2 

N Test result 0.08 

d(mm) 40 7 

Ccr(kg/m
3) 1.2 0.24 

Cs(kg/m
3) 9 1.8 

 

B. In the progress period 
 

Progress period means the period that starts after the 

beginning of corrosion and until the crack begins. In 

this period, if the corrosion quantity is larger than 

corrosion quantity when structure cracked, service 

year will be end. The amount of corrosion in steel 

bars can be calculated using Eq. (5). 

 

corr
tOHFeFIspeW  ])([)2()(

1

3         (5) 

 

(
5.1

),(025.0 tdCI          (6) 

 

Ccr 
critical chloride content ion density at the beginning of 

corrosion (kg/m3) 

Cs chloride ion density at the surface of concrete (kg/m3) 

Dm average chloride ion diffusion coefficient until time t (m2/s) 

Do chloride ion diffusion coefficient at t0 (m2/s) 

tC 
perpetual time of chloride ion diffusion coefficient 

(assumed as 30 years) 

t time lapse (years) 

n time dependent coefficient 

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i

m

s
C

tD

d
erfCtdC 


 ))

2
(1(),( ))    (7) 

Where: 

 

If the shield thickness is considered with factor k, Eq. 

(5) will be expressed as Eq. (8). Eq. (9a) and Eq. (9b) 

considers diameter and shield thickness to get factor 

k. C(d, t) is calculated by Eq. (7); Dm is calculated by 

(4a) and (4b) as a latent period. Ci assumed as 0 and 

steel bar diameter is assumed as 13 mm. If corrosion 

calculated by Eq. (8) is larger than Wcr, which is 

calculated by Eq. (10), it is assumed that its service 

year has ended. As a process of the latent period, 

possibility of crack is calculated in the progress 

period. 

 

W = k * W (spe)        (8) 

k = 1.0 (3*D  d)       (9a) 

k = 3 * D/d (3*D < d)        (9b) 

D

d
Wcr

3

02.0
          (10) 

Where: 

 

 

IV. TEST ANALYSIS   

 

The 28th day compressive strength of this study is 

found to be between 30~40 MPa, as shown in Fig 3, 

whereas Fig. 4 the chloride ion diffusion coefficient of 

plain concrete for W/C is between 0.4 and 0.5. The 

estimation of the chloride ion diffusion coefficient 

DPC, can be produced using Eq. (11). 

 

DPC = 0.547e3.95W/C        (11) 

 
 

Fig .3. Compressive strength development with time 

 

 

Fig .4. Diffusion coefficient of plain concrete W/C 0.4 and 

0.5 

 

 

Fig .5. Diffusion coefficient according to time passage (W/C 

0.4 and 0.5) 

 
 

0

10

20

30

40

50

60

7 28 56

Curing Duration(Days)
C

o
m

p
re

ss
iv

e 
S

tr
en

g
th

(M
p

a
)

40

45

50

55

30.4M Pa

45.7M Pa

y = 0.547e
3.95W/C

0

1

2

3

4

5

0.3 0.4 0.5 0.6

W/C

D
if

fu
si

o
n

 C
o

e
ff

ic
ie

n
t,

 D
m

P
C

(*
1

0
-1

2
m

2
/s

)

0

1

2

3

4

0 20 40 60 80 100

Time (days)

D
P

C
  
(*

1
0

 -
1
2
m

2
/s

e
c
)

W/C 40%

W/C 50%

28 days 91 days

I corrosion current density (μA/cm2) 

C(d,t) 
chloride ion density at the shield d(cm) and time 

t(year) (kg/m3) 

W(spe) amount of steel bar corrosion(g/cm2) 

F Faraday constant (96500C/mol) 

Fe(OH)3 molecular weight of Fe(OH)3 (Ⅲ) (106.9g/mol) 

tcorr 

time lapse (years) from the beginning of the time 

that exceeds the corrosion occurred critical 

chloride content ion density 

Ci 
chloride ion density in the early stage 

(kg/m3)(=0) 

Dm 
Average chloride ion diffusion coefficient at t 

(year) (m2/s) (Eqs. 3a, 3b) 

W corrosion speed of steel bars 

k shield parameter that affects corrosion speed 

D steel bar diameter (cm) 

d shield thickness (cm) 

Wcr corrosion quantity when cracking begins(g/cm2) 

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In this study, chloride ion diffusion coefficient 

according to time passage (Fig. 5) was determined by 

RCPT and Eqs. (4a) and (4b). With this result, 

variation of diffusion coefficient was calculated. Fig. 6 

shows diffusion coefficient ratio when the plain 

concrete was replaced by admixture. According to Fig. 

6, equation for determining the diffusion coefficient of 

each admixture was suggested in Table 4. 

To estimate possibility of failure according to the time 

dependent coefficient variation, durability failure 

possibilities according to the service year was 

calculated when SF15%, time dependent coefficient 

n=0, 0.4 and 0.8 (Fig. 7). Chloride ion diffusion 

coefficient  can calculated by Eq. (12) when the 

concrete property according to the time dependent is 

considered. According to Eq. (13), induced by Eq. (12) 

and test result, the variation of time dependent 

parameter [n] is shown in Fig. 8. 

  
n

t

t
DotD )()(

0
           (12) 

 
)(0

)(log
tD

t

t
Don                     (13) 

 

Fig .6. Type of admixture and substitution ratio 

 

Table 4. Diffusion coefficient of each admixture 

 

Component Equation(m2/sec) 

BS DPC·e
-0.016·SF 

FA DPC·e
-0.005·SF 

SF DPC·e
-0.016·SF 

MK DPC·e
-0.016·SF 

 

Fig .7. Durability failure possibilities according to the time 

dependent coefficient 

 

 

Fig .8. Time dependent parameter by type and quantity of 

mineral admixture 

 

Probabilistic durability analysis for the latent period 

and the progress period were shown in Fig. 9. 

Meanwhile fly ash does not have any efficiency to 

improve durability, wherea blast furnace slag, silica 

fume and MetaKaolin do improve durability. 

 

 

Fig .9. Possibility of durability failure by type and quantity of 

mineral admixture 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P
la

in

M
K

5

M
K

10

M
K

15

F
A

10

F
A

20

F
A

30
SF

5

SF
10

SF
15

B
S3

0

B
S5

0

B
S7

0

Type of Mineral admixture

T
im

e
  
d

e
p

e
n

d
e
n

t 
 p

a
r
a

m
e
te

r
n

W/B 40

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Fig. 10 shows the durability failure possibility 

according to shield thickness. If the shield thickness of 

steel bar is increased, distance from concrete surface 

to the steel is also increased. Consequently, 

resistance of chloride ion is increased and durability 

failure possibility is decreased. Fig. 11 shows 

durability failure possibility in latent period according 

to time. It is also replaced by SF 15% and BS 70% is 

most efficient to improve durability. The point of the 

possibility of durability failure will be 10% in the 80th 

year. 

 
Fig. 12 shows durability failure possibility when 

admixture type and replacement ratio are different. 

According to this graph, the deviation of admixture 

type is smaller than the deviation of replacement ratio. 

Durability failure possibility is also approximately 30% 

smaller. 

 

 

Fig .10. Durability failure possibility according to shield 

thickness 

 

 

Fig .11. Durability failure possibility according to time 

 

 

Fig .12. Durability failure possibility of admixture type and 

replacement variation 

 

Fig. 13 presents the result of durability failure 

possibility according to the shield thickness, while Fig. 

14 shows durability failure possibility in progress 

period according to time. In the latent period, nothing 

satisfied the durability failure possibility under 10% 

when shield thickness was 40mm and over 100 years. 

However, when replaced by silica fume 15% and blast 

furnace slag 70%, the durability failure possibility was 

reduced to 9 % in the progress period, which satisfied 

possibility under 10%. Other parameters have 

durability failure possibility over 30%. 

 

Fig .13. Durability failure possibility according to shield 

thickness 

 

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Fig .14. Durability failure possibility according to time 

 

V. CONCLUSION 

 

In this paper, based on experimental results of 

chloride diffusion coefficients and Monte Carlo 

method, the service of concrete structure 

incorporating different  mineral components, such as 

fly ash (FA), blast furnace slag (BS), silica fume (SF) 

and metakaolin (MK) was predicted. The proposed 

model incorporates the statistical nature of chloride 

induced corrosion of reinforced concrete and can be 

used to evaluate the time of first repair and 

rehabilitation of concrete structures.  

 

This study obtained a DPC equation with a W/C using 

the chloride ion diffusion coefficient and time 

dependent coefficient (n) calculated by the 

experiment. It proposes an estimation equation for 

the chloride ion diffusion coefficient according to the 

blending ratio of admixtures, using the chloride ion 

diffusion coefficient of the plain concrete and 

admixture blended concrete. Large differences of 20, 

40, and 160 years at a 10 % probabilistic durability 

failure rate were evident by varying the time 

dependent index to 0.0, 0.4, and 0.8 under the same 

conditions of the probabilistic durability life test.  It 

was clear that the time dependent index significantly 

affected the evaluation of the durability life. Thus, it is 

necessary to precisely calculate the time dependent 

index. 

 

ACKNOWLEDGEMENT 

 

This work was supported by Sustainable Building 

Research Center Hanyang University, which was 

funded by the SRC/ERC program of MOST (#R11-

2005-056-04003-0). This paper is an extended 

abstract that was published in Korea by AIK. 

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[3] Erik P. Nielsen, Mette R. Geiker (2003) Chloride 

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