Optimized Maintenance Strategy for Concrete Structures Affected by Cracking due to Reinforcement...

ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 11 0-820

Optimized Maintenance Strategy for Concrete Structures Affected by Cracking due to Reinforcement Corrosion by Hua-Peng Chen and Amir M. Alani

An approach for predicting the development of concrete cracking because of steel reinforcement corrosion and for determining riskcost-benefit optimized maintenance strategies during the service life of concrete structures is presented in this paper. Crack evolution in the concrete cover due to expansive corrosion products is investigated at different stages during crack propagation across the cover from the reinforcing bar-concrete interface to the concretefree surface. Crack growth over time is predicted analytically by means of a softening-cohesive model for cracked concrete. Then, a stochastic deterioration model based on a stochastic process, such as a gamma process, is adopted to assess the actual state of the deteriorating structure and to evaluate the probability of failure associated with concrete cracking. Optimal repair planning and maintenance strategies during the service life are determined by balancing the cost for maintenance and the risk of failure. The results from the worked example show that the proposed approach can correctly predict concrete crack growth over time and effectively provide a risk-cost-benefit optimized repair strategy during the service life of the structure affected by bar corrosion.

Keywords: concrete cracking; optimized maintenance strategy; reinforcement corrosion; stochastic deterioration process; structural performance degradation.

INTRODUCTION The ability of concrete structures to fulfill their intended

functions, such as serviceability and durability, can be compromised because of deterioration. One of the major causes of performance degradation in reinforced concrete (RC) structures is severe environments-for example, because of the ingress of chlorides, which penetrate into the concrete cover and initiate chemical reactions leading to reinforcement corrosion. Reinforcement corrosion, which consumes original steel reinforcing bars and provides much lighter rust products, basically creates expansive layers at the interface between the reinforcement and the surrounding concrete. This will cause cracking in the cover once the hoop tensile stress in the concrete exceeds its tensile strength. The sectional loss in the reinforcement due to corrosion and the cracking or even spalling of the surrounding concrete directly influence the serviceability and durability of concrete structures. Therefore, condition assessment based on the prediction of crack evolution in the concrete cover is of great importance to define a timely repair schedule and to prevent the premature failure of RC structures.

Many investigations have been undertaken during the last decade regarding the influence of reinforcement corrosion and concrete cracking on the performance of RC structures (Ahmed et al. 2007; Liu and Weyers 1998; Pantazopoulou and Papoulia 2001; Vidal et al. 2004; Yuan et al. 2007). Studies have been conducted to assess the influence of bar corrosion on concrete cracking (Andrade et al. 1993; Castel et al. 2000; Chen and Xiao 2012; Coronelli and Gambarova 2004; Mullard and Stewart 2011; Torres-Acosta and Sagues 2004) and to predict the residual life of structures (Bhar-

ACI Structural Journal/March-April 2013

gava et al. 2007; Rodriguez et al. 1997; Torres-Acosta and Martinez-Madrid 2003; Torres-Acosta et al. 2007; Williamson et al. 2008). The smeared crack model has been employed for cracked concrete to investigate crack development due to the pressure generated at the bar-concrete interface (Gambarova et al. 1998; Stewart and Suo 2009). The time-dependent reliability analysis has been adopted to evaluate resistance degradation over time (Stewart and Rosowsky 1998; Zhang et al. 2009) and to optimize maintenance strategy during service life (Chen and Alani 2012; Van Noortwijk and Frangopol 2004). However, research on the prediction of crack development in concrete due to reinforcement corrosion is limited, with specific reference 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 for the evaluation of the probability of failure associated with concrete cracking to avoid premature structural failures and determine the optimized maintenance strategy during service life.

RESEARCH SIGNIFICANCE Any optimization of risk and cost decisions about repair

and maintenance of RC structures in aggressive environments requires the accurate prediction of crack development in the concrete cover subjected to reinforcement corrosion during the entire service life. The method presented in this paper is a significant contribution to correctly predict crack growth in the concrete cover over time and to effectively determine the optimal repair time by optimizing the balance between the risk of a structural failure and the maintenance -cost. The proposed approach considers realistic concrete properties, such as tension softening of cracked concrete and cracked-band formation in the surrounding concrete, and adopts the stochastic deterioration model based on a stochastic process-such as a gamma process-to evaluate the risk of failure associated with concrete cracking induced by bar corrosion during the service life. This study also investigates the influence of corrosion rate on crack development and risk-cost-benefit optimized repair planning, and concludes that the corrosion rate is the dominant factor affecting concrete crack growth and optimal repair time. The proposed approach provides an effective tool to assess the actual state of deterioration in concrete structures and allows asset managers to make rational decisions for timely repairs from the predicted corrosion-induced crack growth.

ACI Structural Journal, V. 110, No.2, March-April2013. MS No. S-2011-104.R2 received October 13, 2011, and reviewed under Institute

publication policies. Copyright © 2013, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author's closure, if any, will be published in the January-February 2014 AC/ Structural Journal if the discussion is received by September I, 2013.

229

Hua-Peng Chen is a Senior Lecturer of civil engineering in the Department of Civil Engineering at the University of Greenwich, Chatham Maritime, Kent, UK. He

received his PhD in /998from the University of Glasgow, Glasgow, UK. His research

inlerests include stochastic deterioration modeling, pe1Jormance of concrete infrastructure in aggressive environments, time-dependent reliability analysis, health moni

toring of existing structures, and risk assessment and optimized maintenance strategy.

Amir M. Alani is a Professor of tunnel and bridge engineering in the Department

of Civil Engineering at the University of Greenwich. He received his PhD in /986 ftvm the University of Science and Technology ofMontpellie~; Montpellie~; France. His

research interests include construction materials, foundation engineering, structural condition assessment, and repair and maintenance of buildings.

Cruck Propagation

Phase

' ' ' ' ' Wllhout ' StructurolRepair

' \ ServlceabWI.y

' Llm~ \ Residual Life Phase

Time

Fig. ]-Resistance deterioration process and lifetime extension maintenance strategy of concrete structures affected by reinforcement corrosion.

REINFORCEMENT CORROSION AND STRENGTH DETERIORATION

Deterioration process The process of resistance degradation and the maintenance

strategy for extending the lifetime in RC structures affected by bar corrosion is described in Fig. 1. Three phases are considered in the process: the crack-initiation phase, crackpropagation phase, and residual-life phase. The crack-initiation phase starts from the time of construction and ends when the corrosion-induced cracking initiates at the bar-concrete interface. In general, the time period of the crack-initiation phase is very short compared to the entire service life of the given structures. After cracking occurs at the interface, the crack evolution inside the concrete cover can be accelerated due to the increase of concrete permeability and the performance of the concrete structures decreases with time. Due to further corrosion of the reinforcement, cracks propagate and widen inside the concrete cover and then reach an unacceptable level for serviceability and durability. The residual-life phase starts from the serviceability limit until the ultimate limit is reached, when structural collapse is likely to occur.

Effective maintenance planning is necessary to keep the deteriorating structures safe and to keep the structures serviceable (Guettala and Abibsi 2006). If no repair is carried out before the serviceability limit state is over, additional sectional losses in the reinforcing bars and spalling or delamination inside the concrete cover will occur, and the strength of the concrete structures will deteriorate further until reaching the ultimate limit or collapse state. A series of structural repairs should therefore be planned to control crack development in the concrete and to increase the resistance of the deteriorating structures before reaching the serviceability limit state. The repair strategy for the deteriorating structures should be determined on the basis of the

230

predictions of crack development in the concrete cover and the statistical estimation of failure probability associated with concrete cracking.

Reinforcement corrosion The buildup of the products of bar corrosion depends on

the level of oxidation and generates an expansive layer on the steel surface (Marcotte and Hansson 2007). Liu and Weyers (1998) reported that steel bars may expand by as many as six times their original volume. A porous zone of thickness d0 may exist around the original reinforcing bar surface with an initial radius r0 = Db/2 and should be completely filled before the surrounding concrete is deformed by the expansive corrosion products.

The increase of volume per unit length due to bar corrosion can be obtained from the volume of rust minus the volume of the original bar consumed, as well as the porous zone around the interface. An expansion rs in the radial direction is then generated on the surrounding concrete by the volume increase of the corroded reinforcement. By neglecting the deformation of the original steel and that of the corrosion products, the relationship of volume increase can be expressed as follows

7t[(r. +r +d )z -r.zJ = M,- Ms =a M (1) D s 0 0 m r

P, Ps

where Mr and Ms are the mass of rust products and the consumed mass of the original bar, respectively; p,. = 3600 kg/m3 (224.64 lb/ft3) and Ps = 7850 kg/m3 (489.8 lb/ft3) are the density of the rust determined from experimental results and that of original steel, respectively; and am is an empirical coefficient taken as am = 2.05 x 10-4 (Liu and Weyers 1998). The mass of rust Mr in kg/m (0.67 lb/ft) over time is estimated from

I

Mr = (mcf~1tDbicou(-c)d-c)2 (2)

where t is the duration of corrosion (year); me is an empirical coefficient taken as me = 2.1 X 10-2 (Pantazopoulou and Papoulia 2001); and icorr represents the mean annual corrosion current per unit length at the surface area of the bar in A/m2 (645jlA/in.2), which can be measured on site based on the determination of the polarization resistance (Andrade and Alonso 2001). A recent study by Jang et al. (2011) suggests that the corrosion rate can be affected by the crack width of the cover concrete, according to a linear relationship

(3)

where ieq is the equivalent corrosion current of cracked concrete; c8 is the crack geometry factor with a value ranging from 0.067 to 0.206 (Jang et al. 2011); D and Do are the diffusion coefiicients of intact concrete and in ti'ee solution in m2/s (1150 in.2/s), respectively, with a typical value of D0

= 2.032 X 10-9 m2/s (0.315 X I0-9 in.2/s) at 25°C (77°F); and w is the equivalent crack width in m (39.4 in.), calculated from the total crack width minus the threshold crack width of approximately 80 jlm (0.0032 in.).


The thickness of rust layer t, is relatively small compared to the original radius of the bar r0 and could be estimated from the volume of rust over the perimeter of the original bar. The corrosion penetration rate t,lr0, defined as the ratio of the thickness of rust layer over the original, is then expressed as

(4)

To accommodate the volume increase due to bar corrosion, the bar-concrete interface is to be displaced by a prescribed quantity over time t. By neglecting the higher-order terms in Eq. (1) (assuming d0, rs << r0), the prescribed displacement at the interface over time rs(t) can be estimated from

r;(t) = ~M,.(t)-d0 2nr0

(5)

The aforementioned displacement will be considered as the internal boundary constraint for modeling concrete cracking, as shown in the following section. So far, bar corrosion and rust expansion have been assumed to be uniformly distributed at the bar-concrete interface. A recent study by Jang and Oh (2010), however, suggests that in actual aggressive environments, corrosion may start in the bars closest to the free surfaces (that is, where the cover is a minimum); hence, the bars may not be uniformly corroded in given cross sections. However, the difference in crack development and associated service life between uniform and nonuniform expansions is rather small, provided that the coefficient of corrosion distribution, defined as the ratio of the depth of nonuniform corrosion to that of equivalent uniform corrosion in Vidal et al. (2004), does not exceed 2. Hence, assuming uniform corrosion is acceptable in the cases with relatively small values of corrosion distribution coefficient, as shown in many studies (Chernin et al. 2010; Pantazopoulou and Papoulia 2001).

CORROSION-INDUCED CRACKING MODEL The development of concrete cracking because of bar corro

sion can be modeled according to the thick-walled cylinder adopted in many investigations (refer to, for instance, Chen and Xiao [2012], Chernin et al. [2010], and Gambarova et al. [1998]), as shown in Fig. 2. In this model, the reinforcing bar has an initial diameter Db embedded in concrete with a clear cover thickness C. The initial defects at the barconcrete interface, such as micropore structures, will affect crack development in cover concrete-in particular when the cover thickness is less than 100 mm (3.94 in.) (Zhang et al. 2010). Hence, a porous zone of thickness d0 may be assumed to account for the unevenly distributed voids at the interface. Initially, the porous zone is gradually filled with rust until the voids completely disappear. Any further buildup of expansive rust stresses the surrounding concrete and favors cracking starting from the interface Rb. Cracking then propagates through the cover up to the free surface of concrete Rc. In the end, the cover is completely cracked. According to the Cohesive Crack Model, concrete cracking can be modeled as a process based on tension softening, provided that crack width does not exceed a critical value (Bazant and Planas 1998). Cracked concrete, therefore, has a reduced crackwidth-related tensile strength, which vanishes above a critical


Co~crete c;orrosiot~ producm

Fig. 2-Sketch of idealization of cracking in cover concrete.

crack width. Above this value, no residual tensile strength is left, and cracks become discrete and their openings increase with the development of expansive rust.

Crack initiation The mass of rust required to completely fill the porous

zone of thickness d0 can be calculated from

(6)

The time taken for completely filling the porous zone TP can be estimated using Eq. (2), provided that the corrosion current icorr is available. Any further expansion of the rust will cause a stress state in the surrounding concrete.

The thick-walled cylinder model shown in Fig. 2 reduces the problem to a plane stress problem, as the normal longitudinal stress in the concrete is ignored (Chernin et al. 2010; Pantazopoulou and Papoulia 2001 ). Because the surrounding concrete remains intact and elastic at this stage, the classical elastic solution concerning the radial axis-symmetrical displacement u in thick-walled cylinders can be expressed as follows

(7)

where r is the radial distance within the cover; and C1 and C2 are coefficients to be determined by enforcing the boundary conditions. From the given radial displacement, the radial stress O'r and the hoop stresses cr8 can be calculated from

E E 1 cr = --c ---c -

r 1 - U I 1 + U 2 r2

E E 1 cra =--c~ +--c22

1-u 1+u r

(8a)

(8b)

where E and u are modulus of elasticity and Poisson's ratio of the intact concrete, respectively. Effective modulus Eef should be adopted to replace E in Eq. (8) and thereafter to consider the influence of concrete creep when cracks grow under sustained loading-that is, Eef = E/(1 + 9c)-where

231

9c is the creep coefficient of the cover concrete (Zhang et al. 201 0). Considering the boundary conditions at the reinforcing bar surface R" and the free surface of the concrete R·

(9a)

(9b)

the coefficients C1 and C2 can be determined. The hoop stress cr8 is therefore expressed as

cr = b x 1+-e- xr t ER [ R 2

) 9 (1-u)R/ +(1+u)Re2 r 2

,() (10)

The concrete at the bar surface starts cracking when the hoop stress cr8 at the bar surface reaches the tensile strength ;;. From Eq. (5) and (10), the required mass of rust can be calculated from

(11)

Similarly, the time required by cracking to start at the bar surface T; can be estimated from Eq. (2) on the basis of the calculated rust mass.

Crack propagation Numerous investigations on the Cohesive Crack Model

have been conducted, and the experiments show that the shapes of the softening curve do not vary very much among the different mixtures of ordinary concrete (Bazant and Planas 1998). Furthermore, when crack width becomes significantly large, the tensile strength of cracked concrete becomes negligible and of no significance. To obtain analytical solutions for the crack growth induced by bar corrosion, a linear softening (Hillerborg et a!. 1976) is adopted to model the smeared cracking in the cover

(12)

where O"w is the tensile strength of cracked concrete; Wcr is the critical normalized crack width (no residual tensile strength); and W is the normalized actual crack width w(r), defined as

where GF is the fracture energy of concrete. The total hoop strain Ee of the cracked concrete consists of the fracture strain Ee', generated by a total number of nc cracks, and the linear elastic strain between cracks E8e, associated with the residual tensile hoop stress O"w. The total hoop strain Ee can then be calculated from

232

Ef = ncw(r) =!, nJ,.h W 9 2n;r E 2n;r

(13a)

(13b)

where lc11 is characteristic length introduced originally. by Hillerborg et a!. (197 6) as

EGF [eh=y

nc is the total number of the cracks in the cover, estimated from

2nR n =--e

c L c

where the spacing of crack bands Lc is approximately three times the maximum aggregate size to consider the heterogeneity of the cover concrete; refer to Bazant and Planas (1998). The radial displacement u in the cracked concrete is then expressed as

(14)

where

l =newer l 0 21( ch

The secant tensile stiffness of cracked concrete in the hoop direction £ 8 is reduced gradually from the initial stiffness E due to tension softening in the cracked concrete. The reduction ratio of the residual tensile stiffness p can be defined as

f.l. _ £ 8 _ E8 e _ f, r [ W ) f-'--- --X- 1--

£ E9 e + E/ E U ~·r (15)

The governing equation for cracked concrete modeled as anisotropic elastic continuum (Pantazopoulou and Papoulia 2001) can be expressed as

(16)

By introducing the radial displacement u given in Eq. (14), the governing equation can now be rewritten in terms of the normalized crack width Was

d 2 W I dW (/0 -r) dr2 +(/0 -3r)-;- dr = 0 (17)


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

W = D [ 1 1 Ito -rl] t -l21n-- +D2

lo(lo-r) o r (18)

Two boundary conditions are required to determine two constants D 1 and D2 . To estimate the time required by the cracks to propagate up to the concrete surface across the cover, crack width is assumed to be zero at the cover surface Re. From Eq. (14), the displacement boundary condition at bar surface Rb, given in Eq. (9a), can be rewritten as the boundary condition for the normalized crack width. These two boundary conditions are summarized as

(19a)

W =W. = -r t -R X cr I ( E ) W , =R, b J; s ( ) h (to _ Rb ) (19b)

The coefficients D 1 and D2 in Eq. (18) can now be expressed as a function of Wb, and the normalized crack width can then be rewritten herein as

where the crack width coefficient obe(Re, r) is defined as

o R r = (Rc -r) +_!_In Re lzo -rl be ( c' ) l (f - R )(f - ) { 2 It - R I o o cor o ro c

(21)

and the coefficient Obe(Re, Rb) is obtained from Obe(Re, r) when r = Rb. Considering the free surface of the concrete described in Eq. (9b), the radial stress cr, at the free surface can be obtained from the theory of anisotropic elastic continua for the plane stress problem

cr,l,=~ =~(dulr=R +u~lr=R )=o (22) 1-u dr ' r '

Substituting the radial displacement given in Eq. (14) and the normalized crack width given in Eq. (20), the normalized crack width at the reinforcing bar surface Wb can be expressed as

Consequently, by using Wb, and from Eq. (19b) and (5), the mass of rust required for extending the cracks throughout the cover surface is determined from

M(T -T)=[l+(lo-Rh) Wb]xR J; xnDb (24) r c P R W bE a

h a m


Again, the time required by the cracks to fully penetrate the cover Te can be estimated from Eq. (2). From Eq. (24) and (4), the corresponding critical corrosion penetration fer can be determined.

Fully cracked cover As soon as cracking reaches the concrete-free surface, the

cover is completely cracked. To determine two coefficients D1 and D2 in the general solution given in Eq. (18), two boundary conditions associated with the normalized crack widths at the bar surface Wb and at the concrete-free surface We have to be introduced

(25a)

(25b)

where We will be determined from the free surface condition and Wb will be determined from Eq. (19b). The normalized crack width can now be expressed as

(26)

Poisson's effect associated with the hoop strain in Eq. (22) can be ignored because the surrounding concrete cover is completely cracked. From Eq. (22), the normalized crack width at the concrete surface We is given by the following expression

(27)

The normalized crack width at the bar surface Wb over time can be expressed from Eq. (19b) and (5) as

(28)

Therefore, the crack width at the cover surface We as a function of time can now be determined from Eq. (27). It is interesting to see that the crack width at the concrete surface reaches the critical value at the same time when the crack width at the bar surface becomes critical. The time taken for cracks to become critical in concrete Ter can be estimated from Eq. (28) in which Wb = Wer·

Discrete cracks The Cohesive Crack Model is no longer appropriate

because no residual tensile strength is left in the cracked concrete, as the cracks in the concrete have reached their critical value. At the critical time Ten the front of rust Rer is

a M (T -T) R = R + -(T ) = R + m r cr p cr b Y', cr b D

1C b (29)

233

The increase in volume of corrosion products around the bar after the critical time Tcr is

(30)

The expansion of rust after the critical time Tc,. implies an increase of the thickness of the rust layer

(31)

The width of the cracks in the cover increases due to the progressive expansion of the rust layer. The average increase of the width in the individual cracks within the cover past the critical time can be determined from the increase of the perimeter of the front of the corrosion products. The width of the individual cracks is independent of the radius r at this stage

(32)

Should icorr be kept constant during the time period required by crack growth, the equivalent crack width over time t, defined as the cumulated crack width over the cover (Zhang et a!. 2009), would be as follows

for 0 ~ t ~ Tc

for Tc -< t ~ 1'.:,.

fort:>- Tc,

(33)

where ac and be are the coefficients concerning the propagation of cohesive cracks in the propagation stage described in Eq. (27)

(34a)

and ad and bd are the coefficients concerning the growth of the discrete cracks in the growth stage described in Eq. (32)

(35a)

(35b)

234

It is found that the crack growth induced by bar corrosion at time t follows a square root law, provided that the current at the bar interface remains constant throughout the corrosion process.

STRATEGY FOR OPTIMIZING MAINTENANCE ON RISK-COST-BENEFIT BASIS

Probabilistic analysis can be a useful tool for assessing the performance of deteriorating concrete structures affected by the corrosion of the reinforcement during their service life. The deterioration of the resistance because of concrete cracking can be modeled as a stochastic process because of its uncertainty. In this study, the stochastic deterioration process and the time-dependent reliability approach (Van Noortwijk and Frangopol 2004) are used to estimate the probability of failure associated with concrete cracking. The optimal maintenance decision can then be made by balancing the costs required for maintenance against the risk of failure.

Gamma Process Model The Gamma Process Model has been increasingly used

for modeling a stochastic deterioration process in optimizing maintenance strategy (Van Noortwijk 2009). The gamma process is a stochastic process with independent non-negative increments having a gamma distribution with a given average of deterioration rate. Hence, this model seems appropriate for modeling crack growth inside concrete cover, which occurs randomly in time under continuous use. From the definition of the gamma process (Van Noortwijk and Frangopol 2004 ), the probability density function of crack growth at time t (t :2: 0) can be described as follows

fw(l>(w) = Ga(w lll(t),A)

l A n(IJ

-n(t)- 1 -!.w fi - > O ---w e , orw_ = f(ll(t))

0 , elsewhere

(36)

where A is scale parameter with A :>- 0 , which could be estimated from statistical estimation methods such as a Maximum Likelihood Method by maximizing the logarithm of the likelihood function of the increment of the parameter (Van Noortwijk 2009);

f(ll(t)) = fo~ vn(IH e-v dv

for l'J(t) :>- 0 is the gamma function; and l'J(t) is the shape function and can be obtained from the expected crack growth discussed in the previous section as

ll(t) = Aw(t) (37)

Here, the following definition is given for the failure of an RC structure in serviceability and durability: an RC structure fails as a serviceable and durable structure when its equivalent crack width exceeds a certain allowable value for a serviceability limit problem at time TL. The allowable crack width of the concrete cover wL may vary in accordance with the requirements for the serviceability limit-for example, 0.3 mm (0.12 in.)-for aggressive environments (Zhang et a!. 2009). Also, cracks in cover concrete due to


Table 1-Concrete material properties

Parameter

Specified strength

Compressive strength

Tensile strength

Modulus of elasticity

Poisson's ratio

Fracture energy

Critical crack width

Stewart and Rosowsky ( 1998). lPantazopoulou and Papoulia (2001).

Symbol

F,

!c

f,

E

'll

GF

Wa

reinforcing bar corrosion cause significant deterioration of the resistance of an RC structure. The load capacity and bond strength could reduce to approximately 80% of their initial values to an RC element when the corrosion penetration rate reaches approximately 2%, corresponding to crack width of approximately 0.3 mm (0.12 in.) (Torres-Acosta and Martinez-Madrid 2003; Bhargava et al. 2007). Thus, the lifetime distribution of time to failure associated with crack width is given by

(38)

where

in which x :2: 0 and ll >- 0 is the incomplete gamma function. The probability of failure per unit time at the i-th time interval can then be computed from

P1 '= F(t;)- F(t1_

1) , fori= 1, 2, 3 ... (39)

When the corrosion-induced crack width in the cover gradually reaches an unacceptable level, the probability of failure increases and the potential loss caused by the failure may become unacceptably high. The requirement for repair therefore becomes critical to reduce the risk of structural failure and to avoid the unacceptable possible loss.

Maintenance optimization Maintenance can be modeled as a discrete time renewal

process, whereby the renewals bring a structure back to its original condition. Two typical types of maintenance are often used: preventive maintenance before failure and corrective maintenance after failure (Chen and Alani 2012; Van Noortwijk 2009). Through timely and effective maintenance, deterioration can be delayed so that failure in terms of serviceability and durability is postponed and the structural lifetime is extended. In this study, the maintenance model based on risk-cost-benefit balanced criteria is adopted to optimize maintenance strategy in terms of repair


Evaluation Value Reference

- 34.5 MPa (5.00 ksi) . 1.03F, 35.5 MPa (5.15 ksi)

0.69-.JJ, 4.1 MPa (0.59 ksi)

4400/c 0,516 27.8 GPa (4032 ksi)

- 0.18 ' Gfo(J;"' r 88 N/m (0.50 lbf/in.) ' fcmo

kGF 0.1 mm (0.004 in.) ' J;

time. Because the planned lifetime of most concrete structures is yery long compared with the possible renew cycle length, the strategy for risk-cost-benefit optimized maintenance during the lifetime can be approximately considered over an unbounded time horizon. From the renew reward theory (Van Noortwijk 2009), the expected discounted costs at different dates over time intervals (0, k] are related to the preventive maintenance cost Cp, the corrective maintenance cost CF, and the expected renew cycle length

( ~ I ) *( k ) La p, Cv + 0: I - L P, C,, c (k) = t~l 1~ 1

cJ l-[( f a' p,) + a" (I -± P1 )] 1~ 1 l~l

(40)

where k = 1, 2, 3 ... represents the number of time intervals to be determined; a= (1 + rt1 is the. discount factor per unit time; and r is discount rate per unit time. The optimal maintenance time interval k* is then obtained by minimizing the expected discounted costs over lifetime.

NUMERICAL EXAMPLE An RC structure exposed to an aggressive environment

with a service life of 60 years is used in the following to demonstrate the applicability of the proposed strategy for risk-cost-benefit optimized maintenance. A typical bar and its cover concrete of the RC members, as shown in Fig. 2, are taken for the investigation of the crack growth due to reinforcing bar corrosion and for the evaluation of the optimal repair strategy. The typical bar of a diameter of 12 mm (0.47 in.) is embedded into a concrete member with an average clear cover thickness of 39 mm (1.54 in.). The concrete has a specified compressive strength of 34.5 MPa (500 ksi), as shown in Table 1. Concrete fracture energy GF is estimated from the compressive strength, where the maximum aggregate size is assumed to be 25 mm (0.98 in.). The critical crack width Wcr (cohesive crack opening at zero residual strengtg) is a function of the fracture energy and the given maximum aggregate size. The total crack number nc is calculated from nc = 2nRcfLc = 4 in the case in question, which agrees with the experimental results in Torres-Acosta and Sagues (2004). A porous zone with a thickness of do = 12.5 jlm (0.0005 in.) (Liu and Weyers 1998) and a concrete creep coefficient of 8c = 1 (Zhang et al. 2010) are considered in the calculations. Various values of corrosion rate are adopted to investigate

235

0.10 ..-----------------------,

~ • 0.08

~ ~ :: 0.06 It . . "II ~ 0.04

B

! Q 0.02

-+-- Prediction by lhis sludy

:lK Tom:!r-Acostu. & Sa8J.1ea 2004, Long anode lenglb

fl. Torre .. Acosta & Sague• 2004

0 Torre .. Acosla & Sagues 2004

+ Torres-AeoSia & Sagues 2004

*

+ fl. 0

0.00 -1-----'0~--------~---~----1

Cover-CO-bn dl1meter

Fig. 3-Critical corrosion penetration tcr of steel reinforcement as function of concrete cover-to-reinforcing-bar diameter ratio C;Db, compared with available experimental data in Torres-Acosta and Sagues (2004). (Note: 1 mm = 0.0394 in.)

10.00 ..--:.:-r<d«oii-:-:--:-by-.,.:-,-ludy:-------.---- ----- -.... :1 1lt.!rm~r;. .. MII1ilu-Midriif lOOl,Rit

+

001-l------~~---~---------~ 0.001 0.010

Corrotlo•peuetradoa nte 0100

Fig. 4-Equivalent crack width w as function of corrosion penetration rate t,lr0, compared with experimental data from accelerated and natural corrosion tests in Torres-Acosta and Martinez-Madrid (2003) and Vidal et al. (2004 ). (Note: 1 mm = 0.0394 in.)

3.0 .,....------------ -----------,

X Corrosion cummt icorr = 0,3

IJ Corrosion cuuent icorr = 0,5

l:J. Conosion cunenl icon = I .0

o Corrosion current icorr = 2.0

0 Corrosion cunent icorr = 5.0

10 20 30 Time, year

40 50 60

Fig. 5-Equivalent crack widths at cover surface (we) over time t for various mean annual corrosion currents icorr per unit length (unit: j.JA/cm2

). (Note: 1 mm = 0.0394 in.; 1 J1A1 cm2 = 6.45 j.JA/in. 2)

its influence on crack evolution in the cover and the optimal repair time associated with concrete cracking.

The results in Fig. 3 show the critical corrosion penetration tcr at the time required by cracks to reach the free surface as a function of concrete cover-to-reinforcing-bar diameter ratio (C!Db). To minimize the influence of localized steel

236

0.8

i ~ 06 '; . ~ :zs ~ 0.4

0,2

10 20

X Acceplablelim~ 0.3mm(0.012in)

o Acceplable liml 0.4mm (0 0 16in)

6 Acceplablelimil 0.5mm(0.020in)

30

Time, year

40 50 60

Fig. 6-Lifetime distribution of time to failure F(t) for various acceptable crack width limits.

corrosion on the critical corrosion penetration, experimental results in Torres-Acosta and Sagues (2004) with a coverto-anode length ratio of less than 0.18 are adopted in the study. Then, the results predicted by the proposed method are plotted for comparison with the available experimental data. From Fig. 3, the predicted critical corrosion penetration increases as the C/Db increases, agreeing well with the previous experimental results. The results in Fig. 4 are for the equivalent crack width w over time as a function of the corrosion penetration rate t,.lr0 . 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 corrosion agree well with the available experimental data.

To investigate the effect of the corrosion current icorr on crack evolution, various values of the corrosion rate ranging from 0.3 to 5.0 j.t.A/cm2 (1.94 to 32.25 j.t.A/in.2) corresponding to low-to-high corrosion intensities are adopted to predict the crack width at the concrete ~urface, as shown in Fig. 5. The corrosion rate increases with the development of crack width when the equivalent crack width exceeds the threshold value. The results indicate that the crack width at the concrete surface largely depends on the corrosion rate icorr· The predicted value of crack width is three times as much when the corrosion rate increases from 1.0 to 5.0 j.t.A/ cm2 (6.45 to 32.25 j.t.A/in.2).

The deterioration of the structural performance in terms of serviceability and durability (measured by the growth of the equivalent crack width) is modeled as a gamma process. The results of the lifetime distribution of time to failure F(t) are shown in Fig. 6 for different acceptable limits-for example, WL = 0.3, 0.4, and 0.5 mm (0.012, 0.016, and 0.020 in.), respectively. As expected, the probability of failure associated with the equivalent crack width depends on the given acceptable limit, with a higher probability of failure for a lower acceptable level at any given time. The probability of failure increases steadily over time and reaches approximately 50% at the time when the expected equivalent crack width is equal to the given acceptable limit. As shown in Fig. 5, crack width in the cover is significantly affected by the mean annual corrosion current icorr· The probability of failure is thus strongly related to the corrosion rate, as indicated in Fig. 7, where various corrosion rates ranging from 0.3 to 5.0 j.t.A/cm2 (1.94 to 32.25 j.t.A/in.2) are considered. The probability of failure rapidly goes to unity in the case of


e .!!

08

~ 0.6 0

~ :E .! 0.4

E 0.2

10 20 30

Time, year

X icorr = 0 ,3

0 icorr=0 ,5

.0. icorr::;: I ,0

D icorr=2.0

<> Lcorr=S.O

40 50 60

Fig. 7-Lifetime distribution of time to failure F(t) for various mean annual corrosion currents icorr (unit: pA!cm2

).

(Note: 1 pA!cm2 = 6.45 pA/in. 2)

relatively high values of the corrosion rate, with an estimated time of approximately 5 years to approach 100% probability of failure when icorr = 5.0 jlA/cm2 (32.25 jlA!in.2

).

To find an optimal value of the repair time, the cost defined in Eq. (40) is minimized with respect to the value of time interval k. Only relative values of the preventive maintenance cost Cp and the corrective maintenance cost CF are needed in the calculations. The corrective maintenance cost CF for an RC bridge with 60 years of design life, including traffic delay costs, is taken herein as CF = £1440/tonne (in pound sterling per 1000 kg, 1 £"' $1.56, $2246/short ton), as indicated in Concrete Bridge Development Group (1995). The preventive maintenance cost Cp = £216/tonne ($372/ short ton) is adopted in this study, giving Cp/CF = 0.15. Figure 8 shows the results for the expected relative costs as a function of repair time for various acceptable limits, where the annual discount rate of 5% is considered. The results show that the optimal repair times are 5.7 years for WL = 0.3 mm (0.012 in.), 8.9 years for WL = 0.4 mm (0.016 in.), and 12.3 years for wL = 0.5 mm (0.020 in.), respectively. In Fig. 9, the optimal repair time is plotted as a function of mean annual corrosion current icorr for various acceptable crack width limits. It is obvious that the corrosion rate has a significant impact on the optimal repair time-from 34.7 years for icorr = 0.3 jlA/cm2 (1.94 jlA/in.2) to 1.7 years for icorr = 5.0 jlA/cm2 (32.25 jlA/in.2) in the case of acceptable crack width limit WL = 0.4 mm (0.016 in.).

Figure 10 shows the influence of the preventive maintenance cost Cp on the optimal repair time, where the preventive maintenance cost ranges from Cp = 0.1CF to 0.5CF. The acceptable crack width WL is set at 0.3 mm (0.012 in.) in this case and the mean annual corrosion current per unit length is assumed to be icorr = 1.0 jlA/cm2 (6.45 jlA/in.Z) in the calculations. It can be seen that the value of the optimal repair time increases when the preventive maintenance cost goes up from 4.9 years for Cp = 0.1CF to 12.3 years for Cp = 0.5Cp. The results also show that earlier repairs are necessary to reduce the risk of failure if the preventive maintenance cost is relatively low. In the case with a higher preventive maintenance cost, the optimal repair time can be delayed, but it should be earlier than the expected time to failure if failure criteria is controlled by crack width.

CONCLUSIONS Different phases in the deterioration process of RC struc

tures because of corrosion-induced concrete cracking are


3000 r--.-------------- ------, 2500

"' 2000 ..r 8 J 1500

l " " 1000

500

X Acceptable limit 0.3mm(O.Ol2in)

o Acceptable limit 0.4mm (0,016in)

t> Acceptable limit 0.5mm (0.020in)

o~~r-~~~~~-r-r~~~~~~~~~ 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20

Time, year

Fig. 8-Expected costs over repair time interval with discounting of annual rate of 5% as function of repair time for various acceptable crack width limits, cost in pound sterling£. (Note: 1 £ ""'$1.56.)

6.0 .--------------------------,

s.o "a

--><-- Acceptable limit OJmm (0.012in)

i 4.0 -e-- Acceptable limit 0.4mm (0.016in)

.... --tr- Acceplablelimit O.Smm (0.020in)

~ e 3.0 . 0 'II

2.0 E a 1.0

0.0 10 15 20 25 30 35 40 45 50

Time, yen

Fig. 9-0ptimal repair time as function of mean annual corrosion current icorr for various acceptable crack width limits. (Note: 1 pA!cm2 = 6.45 pA!in.2

)

4800

4000

"' i 3200 ... 1 2400

" ., 1600

800

X CP=O.lxCP

0 CP=0,2xCF

a CP= 0.3xCF

+ c,~0.4xc;

C CP=0.5xCP

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20

Time, year

Fig. 1 0-Expected costs with discounting as function of repair time for various preventive maintenance costs Cr, cost in pound sterling£, WL = 0.3 mm (0.012 in.), and icorr = 1.0 pA/cm2 (6.45 pA!in.2

). (Note: 1£ ==$1.56.)

discussed and the time period for each phase is evaluated from the proposed approach. The development of concrete cracking caused by bar corrosion at various stages is investigated. The Stochastic Deterioration Model is then employed to evaluate the probability of failure associated with the growth of cracking across the cover concrete. The optimized strategy for the repair time is illustrated on the basis of the minimization of the balance between the maintenance costs

237

and the risk of failure in serviceability and durability. The influence of corrosion rate on the cracking development in the cover and the optimal repair time are discussed in terms of optimized risk and cost.

On the basis of the results from the numerical example involving an RC structure, the following conclusions can be drawn: 1) the proposed analytical model can correctly predict the growth of concrete cracking because of bar corrosion at different stages of crack propagation. The numerical results are in good agreement with the available experimental data and indicate that the corrosion rate is the dominant factor affecting the crack growth across the cover; 2) the proposed deterioration model based on the gamma process and strictly related to crack growth makes it possible to evaluate the lifetime distribution of time to failure for the deteriorating structure; 3) the optimal maintenance strategy during the service life of a structure affected by bar corrosion can be determined by optimizing the balance between the risk of failure and the maintenance costs; and 4) the optimal repair time largely depends on the corrosion rate, the acceptable crack width limit, and the preventive maintenance cost. Further efforts are needed to include the effect of nonuniform bar corrosion and the crack-induced strength deterioration due to bar corrosion in the optimizing repair strategy.

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