SPECIAL SESSIONS

Performance based evaluation of corrosion inreinforced and pre-stressed concrete structures

Organizer: U. Schneck

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1 INTRODUCTION

The ability of concrete structures to fulfill their de-signed functions such as durability can be compro-mised because of performance deterioration. One of the major causes of performance degradation in rein-forced concrete structures such as motorway bridges and marine structures is severe environments. For example, because of the ingress of chlorides, chlo-rides penetrate into the concrete cover and initiate chemical reactions, leading to reinforcement corro-sion. Reinforcement corrosion consumes original steel rebar, generates much lighter rust products and creates expansive layer at the interface between the reinforcement and the surrounding concrete cover. As corrosion progresses, the expansive displacement at the interface generated by accumulating rust products causes tensile stress in the hoop direction within the concrete cover, leading to radial splitting cracks in the concrete. Consequently, the sectional losses in the rebar and the cracking or even the spall-ing of the surrounding concrete significantly affect the structural performance and eventually the re-maining service life of the concrete structures. Therefore, damage prognosis based on the prediction of corrosion-induced concrete crack growth is of great importance to define a timely maintenance strategy and to prevent the premature failure of con-crete structures (Frangopol et al. 2008; Straub and Faber, 2005).

Many investigations have been undertaken during last decade regarding the effect of rebar corrosion and concrete cracking on the performance of con-crete structures (Chen and Alani 2012; Liu and Weyers 1998; Pantazopoulou and Papoulia 2001; Vidal et al. 2004). Studies have been undertaken to assess the effect of rebar corrosion on concrete cracking (Andrade et al. 1993; Chen and Xiao 2012; Mullard and Stewart 2011; Torres-Acosta and Sagues 2004) and to predict the remaining useful re-sidual life (Bhargava et al. 2007; Torres-Acosta and Martinez-Madrid 2003). Reliability analysis associ-ated with limit states of a structural system is often utilised for assessing the safety of the structural sys-tems (Stewart and Rosowsky 1998; Melchers, 1999; Farrar and Lieven, 2007). However, research on the prediction of crack development in concrete due to reinforcement corrosion is limited, with specific ref-erence to such concrete properties as softening in tensile strength. Therefore, there is a need to develop an approach for crack growth prediction in concrete and then for the evaluation of structural reliability and remaining useful life associated with concrete cracking.

Reliability analysis associated with limit states of a structural system is often utilized for assessing the safety of the structural system at design stage. This approach however displays some limitations for the reliability analysis of existing structural systems, e.g. it usually ignores additional knowledge available from health monitoring; it often neglects the perfor-

ABSTRACT: Reinforcement corrosion could cause serious deterioration of the durability and reliability to aging concrete structures exposed to aggressive environments. The paper presents an approach for evaluating crack development in the cover concrete due to reinforcement corrosion and predicting the structural reliabil-ity and remaining useful life on the basis of the concrete evolution. The proposed approach estimates analyti-cally the time-to-crack at concrete cover surface and the crack width over time. The analytical estimates of cracking development are examined by existing experimental data. The crack width at cover surface is then adopted as a representative symptom associated with structural performance deterioration. The reliability function directly links to the chosen hazard function and is expressed as a function of the symptom. This symptom-based approach can be adopted for evaluating the current and future conditions and predicting the remaining lifetime. Finally, a case study is utilized to demonstrate the applicability of the proposed approach.

Corrosion-induced cracking evolution and reliabilityprediction of aging RC structures

H.-P. ChenDepartment of Civil Engineering, University of Greenwich, Chatham Maritime,Kent, UK

N. XiaoCollege of Civil Engineering and Architecture, Zhejiang University, Hangzhou,China

Life-Cycle and Sustainability of Civil Infrastructure Systems – Strauss, Frangopol & Bergmeister (Eds)© 2013 Taylor & Francis Group, London, ISBN 978-0-415-62126-7

532

mance deterioration over time; and it is unable to predict the safety and performance in the future. New investigations are therefore required on relia-bility-based reassessment of structural systems fo-cusing on updating structural reliability from moni-tored data. The symptom-based reliability is more appropriate than the traditional time-based reliability for existing structural members and system as the monitoring process can provide useful data (symp-toms) for further assessing current condition and predicting future performance (Cempel et al. 2000; Ceravolo et al. 2008).

The paper presents an approach for evaluating symptom-based reliability and remaining useful life of aging reinforced concrete structures from moni-tored data such as crack width data. The analytical predictions for the crack growth in concrete cover due to rebar corrosion are provided with considera-tion of the realistic bilinear softening curve for the cracked concrete. The predicted results are then as-sumed here as the measured symptom in the numeri-cal simulation study for damage prognosis in order to evaluate the hazard function and reliability. Based on continuous updating reliability, the remaining useful life and the upcoming structural performance are predicted.

2 PROBLEM MODELLING

The process of the resistance degradation of rein-forced concrete structures affected by steel rein-forcement corrosion is described in Fig. 1. Figure 1. Thick-walled cylinder model for cover concrete

cracking evolution due to reinforcement corrosion.

Three phases are considered in the process, i.e.

crack initiation phase, crack propagation phase and residual life phase. The crack initiation phase starts from the time of construction and ends at the time when the corrosion induced cracking initiates at the interface between the steel rebar and the concrete cover. After cracking occurs at the interface, the bond strength between the steel reinforcement and the surrounding concrete starts decreasing, and the

performance of the concrete structures gradually de-teriorates, eventually reaching the failure in deliver-ing structural functions.

2.1 Reinforcement corrosion

Steel rebar may expand by as many as six times its original volume, as suggested by Liu and Weyers (1998). The volume increase could be estimated if the mass of corrosion products and the mass of orig-inal steel consumed over time are available. The mass of rust products (kg/m) over the original radius of the rebar Rb (m) over time t (year) could be esti-mated from

tiRmtM corrbcr )( (1)

where cm is an empirical coefficient taken as 2101.2 cm (Pantazopoulou and Papoulia 2001);

corri represents the mean annual corrosion current per unit length at the surface area of the bar (A/m

2),

which can be measured on site based on the deter-mination of the polarization resistance (Andrade and Alonso 2001).

The corrosion penetration rate is defined as the ratio of the thickness of corrosion rust layer

rt over the origin radius of rebar Rb, namely

rb

r

b

r

R

M

R

t

2

2 (2)

where r is density of corrosion rust with an approx-

imate value of 3/3600 mkgr . To accommodate the volume increase due to cor-

rosion, the interface between the steel rebar and the surrounding concrete is to displace by a prescribed quantity over time t. By neglecting the deformation of the original steel and that of the corrosion prod-ucts as well as the porous zone around the interface, the expansion in radial direction is estimated from

)(2

)( tMR

tu r

b

m

b

(3)

where m is an empirical coefficient taken as 41005.2 m (Liu and Weyers 1998).

The prescribed displacement )(tub will be con-sidered as the internal boundary condition of the boundary-value problem for the evolution of cover concrete cracking. The method described above is based on the general assumption that reinforcement corrosion occurs uniformly and thus the expansion is uniform around the internal boundary of the concrete cover. Recent study by Jang and Oh (2010) suggests that in actual aggressive environments reinforcement corrosion may start from the places close to the free surfaces of the concrete cover and thus the rebar may not corrode uniformly in a cross section. How-ever, the difference in crack development between uniform expansion and non-uniform expansion is small in the case when the corrosion distribution co-

C

Steel rebar, Rb

Concrete

Equivalent

Corrosion Critical crack

front, rcr

External model

boundary, Rc

533

efficient (i.e. the ratio of the depth of non-uniform corrosion to that of uniform corrosion) does not ex-ceed 2. The uniform corrosion of reinforcement in concrete then could be utilised for the cases with rel-atively small corrosion distribution coefficients, as shown in many studies such as Chernin et al. (2010) and Pantazopoulou and Papoulia (2001).

Based on the assumption that the steel rebar has uniform corrosion at the surface, the thick-walled cylinder model for cover concrete cracking can be considered as an axisymmetrical problem. The thick-walled cylinder model could be further treated as a plane stress problem because the normal tension-softening stress in the direction of longitudinal axis could be ignored (Pantazopoulou and Papoulia, 2001), although the approach discussed here can al-so be applied to a plane strain problem. Therefore, the hoop stress in the cylinder is typically a principle tensile stress whereas the radial stress is a principle compressive stress. When the hoop stress reaches the tensile strength of concrete, the radial splitting cracks propagate from the bond interface in ax-isymmetrical directions to the free surface of con-crete cover (Rc). As corrosion progresses the cover concrete becomes completely cracked in the cover.

2.2 Concrete cracking model

Concrete cracking could be modelled as a process of tensile softening if the cracking is considered as co-hesive and the crack width does not exceed a limited value (Bažant and Planas, 1998). In cohesive crack model for quasi-brittle materials such as concrete, the stress transferred through the cohesive cracks is assumed to be a function of the crack opening (Hillerborg et al., 1976). It has been shown that the bilinear softening curve has been accepted as rea-sonable approximations of the function for cracked concrete in tension. Figure 2. Bilinear softening for cohesive cracking in concrete.

The bilinear softening curve adopted in the pre-

sent study is shown in Fig. 2 and expressed as

)( bWaf tw (4)

where w is the tensile stress crossing cohesive cracks, ft is the tensile strength of concrete, and W is dimensionless variable that normalises actual crack width w(r) to a non-dimensional form and defined as

Ft GrwfW /)( in which FG is the fracture energy of concrete. The coefficients a and b in Eq. (4) for the bilinear softening curve are given by

1 craa , cr

cr

Wbb

)1( ,

if crWW 0 (5a)

)( cru

uu

WW

Waa

,

)( cru

u

WWbb

,

if ucr WWW (5b)

where coefficient , normalised critical crack width Wcr and normalised ultimate cohesive crack width Wu may be determined from experiments. In the CEB-FIB Model Code (CEB, 1990), the coefficient is given as 15.0 and Wcr and Wu could be evaluated from the maximum aggregate size of con-crete materials, with typical values of Wcr = 0.02~0.05mm and Wu = 0.1~0.3mm.

From the crack band theory for the fracture of concrete (Bažant and Planas, 1998), the total number of cracks cn separating cracking bands in concrete cover and appearing at cover surface (Rc) may be es-timated from ccc LRn /2 in which cL is mini-mum admissible crack band width estimated from

ac dL 3 in which ad is maximum aggregate size of concrete. The typical value of total crack number

cn in the thick-walled cylinder model for cover con-crete cracking is approximately three or four from the experimental data available.

The total hoop strain of the cracked concrete consists of fracture strain

f

and linear elastic strain between cracks

e

. The fracture strain is generated by a total number of cn cracks whereas the linear elastic strain between cracks is associated with the residual tensile hoop stress , defined as

r

W

E

fbl

r

rwn tcf

02

)(

(6a)

)( bWaE

f

E

te

(6b)

where material coefficient blnl chc 2/0 in which

chl is characteristic length 2

/ tFch fEGl defined in Hillerborg et al. (1976). The total hoop strain of the cracked concrete is then given by

r

WblbWa

E

f tef

0)( (7)

The radial displacement u of the cracked concrete is then calculated from

ft

Wcr Wu

ft

0

Ten

sile

str

ess

,

w

Normalised crack width, W

534

WblrbWaE

fru t

0)( (8)

and the radial strain is given by

dr

dWrlbbWa

E

f

dr

du t

r )()( 0 (9)

The reduction factor of residual tensile stiffness can be expressed as

rbWa

Wblfe

e

)(1

1

0

(10)

The stiffness in radial direction rE is assumed to equal the initial stiffness E of concrete because the radial stress is typically in compression for the prob-lem considered. By using the approxima-tion

rr , the stress and strain relations for the boundary-value problem could be rewritten as

)(1 2

rr

E (11)

The governing equation for cracked concrete modelled as anisotropic elastic continuum (Panta-zopoulou and Papoulia 2001) can be expressed as

01

22

2

r

u

dr

du

rdr

ud (12)

By introducing the radial displacement u given in Eq. (8), the governing equation can now be rewritten in terms of the normalised crack width W as

01

)3()( 02

2

0 dr

dW

rrl

dr

Wdrl (13)

The general solution of the above differential equation can be expressed as

201 ),( CrlCW (14)

where constant coefficients 1C and 2C in the gen-eral solution can be determined from two boundary conditions of the boundary-value problem. ),( 0 rl is a crack width function associated with material coefficient 0l and radius r within the concrete cover is defined as

r

rl

lrllrl

0

2

000

0 ln1

)(

1),( (15)

Once the normalised crack width is obtained from Eq. (14), the actual crack width and crack growth at various stages can be evaluated.

3 CRACK EVOLUTION

3.1 Time-to-crack on concrete cover

To estimate the time required by the cracks to prop-agate up to concrete surface across the cover, crack front with width of zero is assumed at the cover sur-face ( cR ). From Eq. (8), the displacement boundary condition at bar surface ( bR ) can be rewritten as the boundary condition for the normalised crack width. In general, the thick-walled cylinder is divided into two zones, a cracked outer ring where crack width does not exceed critical value ( ccr Rrr

) and a

cracked inner ring where crack width exceeds criti-cal value (

crb rrR ). For the cracked outer ring,

considering the critical crack width at the internal boundary (

crr ), the boundary conditions for the cracked outer ring are

crrrWW

cr

| , 0| cRrW (16)

From Eq. (14), the normalised crack width within the cracked outer ring is then given by

cr

c

cr

cr

cr

c

crcr

WRlrl

RlrlW

),(),(

),(),(

00

00

(17)

Assuming the surface at the concrete cover is free, i.e. free surface for the external boundary of the thick-walled cylinder mode

0 cRrr (18)

the radial stress r at the free surface can be ob-tained from Eq. (11) in which =1. Therefore, the critical crack front crr between the outer ring and the inner ring can be obtained from

)(

1

1

1),(),(

0

00

c

cr

c

c

cr

cr

cr

RlRRlrl

(19)

For the cracked inner ring, where the crack width exceeds the critical value, from Eq (8) the normal-ised crack width at its internal boundary is given by

b

u

b

tb

uu

u

b Ratuf

E

RlbtW )(

)(

1)(

0

(20)

where material coefficient u

chc

ublnl 2/0 . The

boundary conditions for the cracked inner ring are

)(| tWWu

bRr b , crrr

WWcr

| (21)

Therefore, the normalised crack width within the cracked inner ring is given by

cr

cr

u

b

u

u

b

uu

b

cr

u

b

u

cr

uu

WrlRl

rlRlW

rlRl

rlrlW

),(),(

),(),(

),(),(

),(),(

00

00

00

00

(22)

Meanwhile, the condition of radial stress continu-ity at the critical crack boundary ( crr ) between the cracked outer ring and the cracked inner ring gives

535

crcr rrrrrr || (23)

By using the radial stress described in Eq. (11), the normalised crack width at the internal boundary at the time to cracking on cover surface ( cT ) is ob-tained from

)()],(),()[(

)],(),()[(1)(

000

000cru

y

cr

cr

cr

cr

cr

cr

u

b

u

cr

u

crc

u

b WWrlrlrl

rlRlrlWTW

(24)

From Eq. (24), the corresponding displacement at the internal boundary of the thick-walled cylinder at time cT can be determined from Eq. (20). By using Eq. (3), the mass of rust products at time cT is calcu-lated from

m

bt

c

u

bb

uu

b

u

cr

R

E

fTWRlbRaTM

2)()()( 0 (25)

Consequently, the time to cracking cT can be es-timated from Eq. (1). It can be seen that the time to cracking is a function of concrete cover dimensions, material properties of cover concrete and reinforce-ment corrosion rate.

3.2 Concrete cover crack growth

After the crack front reaches the external surface, the concrete cover becomes completely cracked. A sin-gle cracked zone is considered for the thick-walled cylinder in general and the crack width over the con-crete cover is assumed to exceed the critical value. The boundary conditions for this case are given by

u

bRr WWb| , cRr WW

c| (26)

From Eq. (14), the normalised crack width within the cracked concrete cover is expressed as

c

c

u

b

u

u

b

uu

b

c

u

b

u

c

uu

WRlRl

rlRlW

RlRl

RlrlW

),(),(

),(),(

),(),(

),(),(

00

00

00

00

(27)

To determine the unknown cW , the free surface condition at the external boundary ( cR ) described in Eq. (18) is adopted. The radial stress at the external boundary can be calculated from Eq. (11) by using the stiffness reduction factor in Eq. (10). Poisson’s effect associated with the hoop strain can be ignored because the surrounding concrete cover is complete-ly cracked. From the free surface condition at the ex-ternal surface (Rc), the normalised crack width at the concrete surface Wc is given by

u

b

c

u

b

u

c

u

c

u

b

u

c

u

b

u

c

u

c

c WRlRlRlR

W

WRlRlRlR

W

)],(),()[(1

)],(),()[(1

000

000

(28)

where the normalised crack width at the internal boundary

u

bW is calculated from Eq. (20). When cracks in the cover concrete reach the ultimate cohe-sive width, the cracks become cohesionless and no residual strength exists in the cracked cover con-crete. From Eq. (28), it can be seen that cracks at both the internal and external boundaries reach the ultimate cohesive width at the same time ( uT ). The displacement at the internal boundary at time uT is calculated from Eq. (10). The mass of rust products at time uT is obtained from

ut

cru

u

ub lE

f

WW

WTu 0)(

(29)

Similarly, from Eq. (3) the mass of rust products at time uT is calculated from

m

bt

cru

u

u

ur

R

E

f

WW

WlTM

2)( 0

(30)

The time at the end of cohesive cracking stage uT can be then estimated from Eq. (1).

4 DAMAGE PROGNOSIS

The symptom-base reliability was proposed by Cempel et al. (2000) initially for diesel engines, and recently further extended to be used for civil engi-neering structures (Ceravolo et al. 2008). In symp-tom-based system performance evaluation, the relia-bility is assumed to be dependent of measurable quantities, i.e. symptom. The system and its compo-nents fail to meet the designed requirements when a symptom exceeds a given limit value Sl. Symptom reliability R(S) for a critical system in operation is defined as the probability associated with a symptom S.

For reinforced concrete structures subject to ag-gressive environments, the corrosion-induced crack-ing of the cover concrete can be used for a symptom to evaluate the symptom-based reliability. This symptom-based reliability is then adopted for dam-age diagnosis for reinforced concrete structures af-fected by reinforcement corrosion, when the evolu-tion of the symptom is available.

In general, there are two basic system lifetime distribution models used in symptom based reliabil-ity analysis, Weibull and Frechet models. The Weibull model is often used for structural capacity deterioration caused by structural damage (Ceravolo et al. 2008). In this study, the Weibull model adopt-ed for concrete cracking evolution is assumed as

/1)])~(

1ln([~

bl t

wt

w

w (31)

where α>0 is the scale parameter; γ > 0 is the shape parameter; tb is the system’s design life; the initial

536

symptom is lwS 0 in which wl is allowable con-

crete crack width. The symptom hazard function for the adopted Weibull model is expressed as

1]~

[)~(

ll w

w

wwh (32)

The symptom reliability is then calculated from

])~

(exp[)(exp)~(~

0

l

w

w

wdhwR

(33)

The remaining useful life for the reinforced struc-tures can be predicted from

b

l

brul tw

wtwRt ])

~(exp[)~(

(34)

In the case when the hazard function is propor-tional to the single measured system symptom w~ , the hazard function associated the difference be-tween the predictions and the real measured data may be expressed as

l

m

w

wwhLwhLwh

~)~()~(),~( 00 (35)

where L is a logistic vector defined as lwwL /~ .

The system reliability with consideration of differ-ence between measured data and predicted results is rewritten as

))~(ln(

~1)(|),~( 000

wRw

wSRLwR

l

LL

(36)

It is obvious that the symptom reliability will change depending on the difference between the measured and predicted crack widths, www m

~~~ .

5 NUMERICAL EXAMPLE

A steel reinforced concrete structure exposed to an aggressive environment with a service life of 75 years is now utilized to demonstrate the applicability of the proposed damage prognosis approach. The re-inforcing steel bars of a diameter of 16mm are em-bedded into concrete structural elements with an av-erage clear cover thickness of 39mm. The material properties for the cover concrete utilized in this study are taken as compressive strength

MPafc 5.31 , tensile strength MPaft 3.3 , elastic modulus of concrete GPaE 27 , Poisson’s ra-tio 18.0 , concrete creep coefficient assumed here 1 , density of corrosion rust products

3/3600 mkgr , density of steel 3/7850 mkgs , and coefficient 57.0 (Liu and Weyers, 1998). Other material properties adopted in the predictions are evaluated from the given concrete properties with assumed maximum aggregate size mmda 25 , such as fracture energy mNG f /83 , total crack number 4cn , critical crack width mmwcr 05.0 , and ultimate cohesive crack width mmwu 2.0 .

Figure 3. Critical corrosion penetration (tcr) of steel reinforce-

ment as a function of concrete cover-to-rebar diameter ratio

(C/Db), compared with the available experimental data in

Torres-Acosta & Sagues (2004).

The results in Fig. 3 show the critical corrosion

penetration (tcr) at the time required by cracks to reach the fee surface as a function of concrete cover-to-rebar diameter ratio (C/Db). In order to minimise the influence of localised steel corrosion on the criti-cal corrosion penetration, experimental results in Torres-Acosta and Sagues (2004) with cover-to-anode length ratio less than 0.18 are adopted in the study. Then, the results predicted by the proposed method are plotted for comparing with the available experimental data. From Fig. 3, the predicted critical corrosion penetration increases as the cover-to-rebar diameter ratio increases, agreeing well with the pre-vious experimental results.

Figure 4. Equivalent crack width ( w~ ) as a function of corro-

sion penetration rate (tr/r0), compared with the experimental

data from accelerated and natural corrosion tests in Torres-

Acosta and Martinez-Madrid (2003) and Vidal et al. (2004).

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 1 2 3 4 5

Crit

ica

l co

rro

sio

n p

en

etr

ati

on

, m

m

Cover-to-bar diameter

Prediction by this study

Torres-Acosta & Sagues 2004, Long anode length

Torres-Acosta & Sagues 2004, Ref [4] - [7].

Torres-Acosta & Sagues 2004, Ref [8].

Torres-Acosta & Sagues 2004, Ref [9].

0.01

0.10

1.00

10.00

100.00

0.001 0.010 0.100

Eq

uiv

ale

nt

cra

ck

wid

th,

mm

Corrosion penetration rate

Prediction by this study

Torres-Acosta & Martinez-Madrid 2003,Columns

Torres-Acosta & Martinez-Madrid 2003,Beams

Torres-Acosta & Martinez-Madrid 2003

Torres-Acosta & Martinez-Madrid 2003, Cylinders

Rodriguez et al. 1997, Pull out

Torres-Acosta et al. 2003, Slabs

Torres-Acosta et al. 2007, Beams

Vidal et al. 2004, Natural corrosion

537

The results in Fig. 4 are for the equivalent crack width ( w~ ) over time as a function of the corrosion penetration rate ( 0/ rtr ). The predicted results are then compared with previous experimental investigations obtained from accelerated or natural corrosion tests in concrete (Torres-Acosta and Martinez-Madrid 2003; Vidal et al. 2004). Here again, the predicted results for the crack growth in the cover due to cor-rosion agree well with the available experimental da-ta.

Fig. 5 gives results for lifetime evolution of cor-rosion-induced concrete cracking predicted by the Weibull model, which are compared with the ‘meas-urements’ simulated by the analytical solutions, in the case with corrosion rate of 2.33µA/cm

2. The

simulated “measurements” indicate that the crack width at the concrete cover surface cw increases ab-ruptly when crack front reaches the free cover sur-face at the time of 1.83 years due to sudden release of energy. After the time to cracking, the crack width at the cover surface cw increases gradually and becomes ultimate cohesive width at the time of 42.9 years. From the results, the scale parameter α=1.8 and shape parameter γ =2.0 of the Weibull model are determined to best match the simulated ‘measurements’.

Figure 5. Comparison of symptom evolution by analytical so-

lution and Weibull model.

The symptom reliability varying over time and

influenced by monitoring symptom data is plotted in Fig. 6. The reliability decreases with time due to the evolution of corrosion-induced concrete cracking. When the real monitored data are different from the predictions by the Weibull model, the symptom reli-ability will be affected by the difference. The results also show the symptom reliability decreases when monitored crack width exceeds the prediction. On the other hand, the reliability increases when the monitored data is below the prediction. This shows that the corrosion-induced crack width has influence on the symptom reliability, and the reliability needs to be adjusted on the basis of real measurements.

Figure 6. Symptom based reliability varying over time and in-

fluenced by monitoring symptom data.

The remaining useful life is then predicted from

the estimated symptom reliability, as shown in Fig. 7. From these results, it can be seen that the differ-ence between the predicted data and monitored symptom data has an obvious influence on the re-maining useful life, reducing from 45 year for the correct predictions to 33.5 year for the symptom dif-ference of 0.5 after 30 year service of the structure.

Figure 7. Estimated remaining useful lifetimes varying over

time and influenced by monitoring symptom data.

6 CONCLUSION

On the basis of the results from the numerical exam-ple involving the damage diagnosis for the concrete structures affected by reinforcement carrion, the fol-lowing conclusions are noted:

1) The proposed analytical model including real-istic softening concrete properties can correctly pre-dict the time-to-crack and the growth of rebar corro-sion-induced cover concrete cracking.

Tc

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30 35 40

Eq

uiv

ale

nt

cra

ck

wid

th,

mm

Time, year

Weibull model estimate

Analytical prediction

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30

Sy

mto

m r

eli

ab

ity

Time, year

dw/w0 = -0.5

dw/w0 = -0.2

dw/w0 = 0.0

dw/w0 = +0.2

dw/w0 = +0.5

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30

Rem

ain

ing

use

ful

life

, y

ea

r

Time, year

dw/w0 = -0.5

dw/w0 = -0.2

dw/w0 = 0.0

dw/w0 = +0.2

dw/w0 = +0.5

538

2) The Weibull distribution can be used for mod-elling concrete cracking evolution if appropriate shape and scale parameters are chosen.

3) The symptom reliability needs to be adjusted by the real monitored data to reflect the difference between the real monitored data and the analytical predicted results.

4) The remaining useful life largely depends on the time passed and the difference between the mon-itored and predicted data.

The present research focuses on reinforced con-crete cover cracking, however the bond strength at the interface between rebar and the surrounding con-crete should be considered for evaluating the per-formance of aging reinforced concrete structures in the future. also, special issues of reinforcement cor-rosion, such as strongly deviating local behavior, different stages, forms of corrosion products and the kinetics of corrosion should be considered in further studies.

REFERENCES

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Bažant, Z. P., Planas, J. 1998. Fracture and Size Effect in Con-crete and Other Quasibrittle Materials, CRC Press, Boca Raton, Florida, USA.

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