Probabilistic modeling of structural deterioration of reinforced concrete beams under saline environment corrosion

STRUCTURAL SAFETY

Structural Safety 30 (2008) 447–460 www.elsevier.com/locate/strusafe

Probabilistic modeling of structural deterioration of reinforced concrete beams under saline environment corrosion R.E. Melchers a, C.Q. Li a

b,*

, W. Lawanwisut

c

Centre for Infrastructure Performance and Reliability, The University of Newcastle, Australia b Department of Civil Engineering, The University of Greenwich, London, UK c Cho Lawanasphat Partnership, Thailand

Received 18 April 2005; received in revised form 8 December 2006; accepted 8 February 2007 Available online 11 April 2007

Abstract For existing reinforced concrete structures exposed to saline or marine conditions, there is an increasing engineering interest in their remaining safety and serviceability. A significant factor is the corrosion of steel reinforcement. At present there is little field experience and other data available. This limits the possibility for developing purely empirical models for strength and performance deterioration for use in structural safety and serviceability assessment. An alternative approach using theoretical concepts and probabilistic modeling is proposed herein. It is based on the evidence that the rate of diffusion of chlorides is influenced by internal damage to the concrete surrounding the reinforcement. This may be due to localized stresses resulting from external loading or through concrete shrinkage. Usually, the net effect is that the time to initiation of active corrosion is shortened, leading to greater localized corrosion and earlier reduction of ultimate capacity and structural stiffness. The proposed procedure is applied to an example beam and compared to experimental observations, including estimates of uncertainty in the remaining ultimate moment capacity and beam stiffness. Reasonably good agreement between the results of the proposed procedure and the experiment was found.  2007 Elsevier Ltd. All rights reserved. Keywords: Probability; Reinforcement; Concrete; Beams; Corrosion; Capacity; Salinity

1. Introduction For existing reinforced concrete structures exposed to saline or marine conditions there is an increasing engineering interest in their remaining safety and serviceability, including the expected remaining life. The methods available to assess this tend to be experimental, intrusive and mainly empirical rather than theoretical [20,5]. Typical techniques include estimation of chloride concentration levels and of reinforcement corrosion rates. These ‘material property’ measurements must be ‘translated’ into engineering expectations for the remaining life of a structure. It is this aspect that is still the most challenging and usually involves considerable difficulty, uncertainty and empiricism. *

Corresponding author. E-mail address: [email protected] (C.Q. Li).

0167-4730/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2007.02.002

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Rather than make observations and apply extrapolations for an already deteriorated structure, a more desirable approach is to predict the likelihood and likely extent of future deterioration and the associated structural behavior. Ideally, these would be calibrated to actual field observations. Of primary structural engineering interest are the likely future rate of deterioration and the associated loss of structural (ultimate) strength and (flexural) stiffness. Experience shows that for ageing concrete structures, the most important deterioration mechanism is reinforcement corrosion [1,7]. The deterioration of the concrete is known to be comparatively small. The rate of influence of reinforcement corrosion on structural behavior initially depends significantly on the rate of ingress of chlorides and also on the associated time to initiation of reinforcement corrosion. These are a direct function of the concentration of chlorides on (or near) the concrete surface and the permeability of the concrete surrounding the reinforcement. The latter, in turn, is a function of localized micro-cracking and thus localized concrete strains. These result primarily from applied external forces (loading), strains due to concrete shrinkage and cyclic strains such as due to temperature variations. Once corrosion has initiated, the subsequent rate of localized deterioration depends largely on the rate of reinforcement corrosion. Detailed discussion of the various aspects involved in the mechanics of the initiation of reinforcement corrosion, its interaction with (perhaps with additive-modified) concrete properties, the influence of external agents and current research on deterioration modeling is available in the literature (e.g., [6,29,33,2,3,35]) and need not be reviewed here. Herein, the emphasis is on the effect reinforcement corrosion has on concrete cracking and the effect this has, in turn, on the progression of corrosion and hence the structural response (i.e. capacity and deformation) of reinforced concrete structures. In a previous paper a method was proposed to estimate localized micro-cracking and increased permeability due to applied loading and hence to account for the loss of ultimate strength and bending stiffness for reinforced concrete beams [26]. This approach is briefly reviewed herein. It provides the basis for estimating the mean ultimate strength and mean bending stiffness remaining at any point in time. It also provides the basis for the estimation of the probabilistic uncertainty associated with these quantities. The outcome then provides second-moment probabilistic models for the uncertain remaining ultimate strength and bending stiffness. As is known, probabilistic models are essential to structural reliability assessment. A simple example illustrates the method of application. As is well known, the build-up of corrosion products eventually can lead to (longitudinal) cracking and spalling of concrete and loss of bond between the concrete and reinforcement. These are destructive phenomena and are usually evident only after the occurrence of very significant reinforcement corrosion. Such levels of deterioration are not generally acceptable during the service life of a structure [18]. The present paper is concerned only with the structural response in bending prior to such phenomena becoming evident. Within this scope, the main concern is with the effect of stresses caused by external loading on the ingress of chlorides and the effect this has on overall structural response. It does not deal with the stresses and cracking directly caused by the (expanding) corrosion products. 2. Estimation of structural response 2.1. Experimental approaches The most common approach to obtain estimates of structural deterioration has been through laboratory experimentation. A useful summary of the various research efforts is given by [35]. In most of these, load testing was conducted only after corrosion of reinforcing bars had occurred. However, in practice loading and corrosion (initiation and active) occur simultaneously and this has been observed to have an accelerating effect on corrosion process and hence on deterioration (e.g., [40,4]). The majority of tests employed DC current corrosion acceleration mainly to achieve observable results in a reasonable time-frame. However, DC current tends to promote ‘uniform’ corrosion over the surface of the bars, at least in the short-term, whereas chlorides in natural conditions tend to produce localized or pitting corrosion, at least initially [7]. Pitting produces little corrosion product and therefore much less cracking and spalling. Conversely, pitting can be very serious for local loss of steel cross-section. This calls into question the validity of DC current corrosion acceleration to obtain estimates of strength and stiffness deterioration. A

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further issue is how the results from accelerated tests can be related to realistic timescales. The problem is complicated because diffusion processes (e.g. for chlorides) are involved and these are known to be very difficult to scale in a simple manner. An alternative is to achieve limited acceleration through increasing the severity of the environment [16,17]. To relate these back to real time scales, the experimental results were converted (approximately) by comparing them with longer-term calibration tests [19,15]. This tends to allow directly for the effect of the accelerated migration of chloride ions [14] and to a limited extent for the changes in the physical and the diffusion properties of concrete with time [34]. It has been proposed also that observations from laboratory studies can be extended in time through correlating parameters of interest to critical influencing factors, for example, correlating load capacity to an indicator such as pit depth [35]. However, considerable scatter has been observed in the correlations, indicating the limitations of this approach. 2.2. Development of corrosion with time It has previously been proposed that the theoretical estimation of structural response should be based on the migration of chloride ions from the environment through the concrete to the steel surface [26]. Provided proper account is taken of internal micro-cracking and the effect this has on the diffusion of chloride and hydroxyl ions [13,10], this should provide an estimate of the time before significant reinforcement corrosion commences. The amount of corrosion loss as a function of time can be expressed for reinforcement in concrete through a bi-linear model of the typical actual behavior Fig. 1. Various versions have been given by different authors [36,7,20,10]. In the model shown in Fig. 1, 0–ti represents the time taken for the chloride ions to diffuse through the concrete cover, to reach the reinforcement and to attain a sufficiently high level of concentration to overcome the protective effect provided by the (high) local pH and the presence of hydroxyl ions [7]. The time taken for this to occur is known usually as the ‘initiation time’ ti, even though it involves a limited amount of corrosion, as shown schematically in Fig. 1 by the relationship for typical behavior. Since the period 0–ti involves diffusion processes, it is evident that it is a function also of the concentration C0(x) of chlorides at location x(0 6 x 6 L) along the outer surfaces of the concrete beam and this itself is a function of a number of climatic and local surface influences. It will be assumed in the following that the concentration C0(x) is known at all points on the surface of the structure, although in practice this may be a major source of uncertainty. In many cases it is then assumed, conservatively, that active, structural significant, corrosion commences after ti. In practice there may be a significant delay between the arrival of chlorides at the reinforcement and the commencement of significant corrosion. However, in the following, this aspect will be incorporated in ti. The accepted way to estimate the initiation time, ti, is through the solution of Fick’s diffusion equation. This assumes that the concrete is homogeneous. However, this is not the case and micro-cracking smaller than about 0.2–0.3 mm permits greater penetration of chlorides in the region of the cracks, thus effectively raising corrosion loss

B increasing corrosion

r0

active corrosion cracking of concrete

0 local corrosion initiation

A ti

exposure time

negligible local corrosion

Fig. 1. Corrosion loss–time behavior and simplified bi-linear model (bold line).

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the local permeability in the tension zone. The effect tends to be greater close to reinforcing bars. Usually, the effect of micro-cracking is allowed for by using an ‘equivalent diffusion coefficient’ Da0. For realistic reinforced concrete beams exposed for over 6 years under different applied stress levels, the effect of stress on concrete permeability, defined by Da(r), could be modeled empirically as Da ðrÞ ¼ Da0 ½1 þ F ðrÞ  Da0 ½1 þ 1:6  108 r3 

ð1Þ

where F(r) is an empirically determined function of the stress r (MPa) in the reinforcing bar [10]. In principle, there also should be an ‘ageing function’ in Eq. (1) to allow for the increase in local micro-cracking as a result of environmental cycling (dry-humid cycles, etc.) and mechanical actions. However, concrete permeability typically will decease with time because of the gradual continued hydration of most concrete cements [34]. Because of lack of adequate data for these two opposing effects, this aspect is not considered herein, although it is recognized as potentially important. Crack sizes greater than about 0.3 mm appear to have little effect on concrete permeability [11,9] although such cracks may expose the reinforcement to the direct influence of the environment. These situations typically are associated with poorly designed or badly deteriorated structures and are not considered in the following analysis. Most experimental observations show that the low rate of initial active corrosion gradually increases with time to a roughly steady rate, modeled as r0 herein. It is now accepted that after corrosion has been activated corrosion loss ro

c1

ro

c2 t i2

t i1

t

time

Fig. 2. Effect of reduced initiation time on total corrosion loss.

P

(a) D(x) x (b) ti (x) x t>ti

(c) A loss(x,t)

A max

x t
Aloss

ti

t

Fig. 3. Beam with permeability properties reflecting the bending moment diagram (BMD) and resultant variable time to initiation and steel loss. Amax demotes the maximum reinforcement area available and provides an upper limit on Aloss. The other symbols are defined in the text.

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the corrosion rate is controlled by oxygen, water and ion transport mechanisms. These appear not to be significantly influenced by the presence of small cracks [28,10]. 2.3. Theoretical modeling approach The central concept proposed earlier is that local permeability and hence initiation time in general will vary along the length of a (reinforced concrete) beam and that this will influence the ultimate strength and the stiffness of the beam [26]. Non-uniform internal actions, either resulting from applied loadings or as strains, will cause variation in the stress state along a beam and hence in local micro-cracking. In turn, this will affect local permeability and hence initiation time for active corrosion will, in general, vary along a beam. Moreover, these properties will change under constant loading and more severely so under greater loading. The net effect is that a loaded beam will tend to corrode earlier and apparently faster than an equivalent unloaded beam for a given level of surface chloride concentration C0. The effect of shorter initiation time on total corrosion loss is shown schematically in Fig. 2 at a generic beam cross-section. Since this paper deals with micro-cracking, it may be reasonable to assume that there will be little effect of cracking on the corrosion rate [7].

Fig. 4. Algorithm for estimating the ultimate moment capacity with corroding reinforcement. The symbols are defined in the text and in Fig. 3. Note x defines distance along the beam and L is the length of the beam. BMD denotes ‘bending moment diagram’.

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In actual beams there will be some variability (randomness) associated with the precise starting time for corrosion initiation as a result of localized variability in material and other properties. Similarly, variability would be expected in the rate at which active corrosion will occur once it commences. These variability aspects are considered below. The effect of increasing loading P and hence local internal action Q(x) on the permeability and the subsequent initiation time can be seen in Fig. 3. Initially, under zero loading, the beam may be considered to have theoretically perfect concrete having diffusion coefficient D(x)jt=0 = Da0 for all points x along the beam. However, as the loading increases, the effect of micro-cracking caused by the stress r(Q, x) is to modify the diffusion coefficient to Da(x) at each cross-section x according to Eq. (1). As a direct result, the corrosion initiation time, ti(Q, x), will be a function of location x as well as loading. The variation of local initiation times will cause the total amount of corrosion and hence loss of local reinforcing steel area (Aloss) that will occur at each cross section to vary, reflecting the variation of Da. The relationship

Fig. 5. Algorithm for estimating beam deflection with corroding reinforcement. The symbols are defined in the text and in Fig. 3. Note x defines distance along the beam and L is the length of the beam. BMD denotes ‘bending moment diagram’.

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Table 1 Material and other properties and their uncertainty estimates for experimental beams Property

Units

Mean

COV

Reference

Steel yield strength fy

MPa

400

Negligible

[17]

Steel elastic modulus

MPa

200,000

Negligible

[17]

Cover to reinforcement

mm

31

0.2

[17] [32]

Concrete strength fc0

MPa

45.80 (w/c = 0.45) 29.20 (w/c = 0.60)

0.135

[17] [37]

Concrete elastic modulus Ec

MPa

37,222 (w/c = 0.45) 29,720 (w/c = 0.60)

0.12

[17] [22]

Average surface chloride concentration C0

% Weight of concrete

0.306

0.25

Initial apparent chloride diffusion coefficient Da0

m2/s

1.55 · 1011

0.4

[17] Appendix [17] Appendix

is sketched in Fig. 3. Since Eq. (1) is non-linear, typically the relationships for ti and Aloss also will be increasingly non-linear as the loading (and hence the stress level) increases. The initiation time, ti(Q, x), may be estimated using Fick’s diffusion laws even though it is well known that they only approximately represent the actual diffusion/migration processes for chlorides in concrete. Concrete inhomogeneity, micro-cracking, variability, incomplete saturation, discontinuous moisture and chloride exposure are some of the factors involved [7]. In addition, due to a variety of reasons, the chloride level C0 will seldom be uniform or retain its maximum value precisely at the actual concrete surface. There has been considerable discussion in the literature suggesting that it should be represented as water soluble chlorides even though in practice acid soluble levels are typically measured. A recent review suggests that the latter is probably more relevant as chlorides can re-dissolve [12]. The corrosion of the reinforcing bars may be assumed to be essentially uniform around, but varying along, the reinforcing bars, sometimes significantly so. This is of course an idealization based on the observation that significant pitting corrosion becomes important only for highly advanced reinforcement corrosion, which is beyond the scope of the paper. The ultimate bending capacity of the beam at any point x along the beam may be estimated from Mult(t) = fult(Anet(t)), where Anet(t) is the net area of reinforcement at x. The function fult( ) is provided by standard reinforced concrete design or analysis formulae (e.g. [27]). It follows that the ultimate moment can be obtained as a function of time of exposure t. Of course, for the ultimate strength of a beam, the loss of reinforcement at the most highly stressed section is important. Stiffness deterioration usually manifests itself through beam deflections. For the beam depicted in Fig. 3 this is its central deflection. To estimate it the bending stiffness for each cross-section is used in integration along the beam. For reinforced concrete this implies that the equivalent cross-sectional area must be estimated using an accepted procedure (e.g. [8]). Consistent with the assumption about the distribution of cracks in relation to their effect on diffusion, it will be taken that the distribution of cracking caused by applied loading is homogeneous in the region of high bending moment. This ignores any concentration of cracking that may occur in practice in such a zone but is considered a reasonable first approximation since the required integration along the beam to estimate stiffness and hence deflection tends to smooth any localized effects. Finally, it is noted that both the sectional stiffnesses and the deflection will be functions of time t. The ultimate strength (in bending) and its deterioration can now be described functionally as M ult ðQ; tÞ ¼ M ult ðQ; C 0 ; Da0 ; F ðrÞ; r0 ; B; tÞ

ð2Þ

where the terms have been defined above and the vector B contains the dimensions and other properties of the beam. A similar expression applies for bending stiffness but with a different functional relationship and different contents in B. Both expressions are not explicit because they involve algorithmic evaluation. Fig. 4 shows the algorithm for ultimate moment capacity and Fig. 5 the algorithm for deflection estimation, both using the procedure outlined above.

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3. Probabilistic deterioration modeling To develop a probabilistic model for ultimate strength deterioration and for stiffness deterioration, the functional relationships equation (2) and associated algorithms (Figs. 4 and 5) need to account also for uncertainty information in the variables involved. There are essentially two levels of uncertainty to be considered: (i) that associated with the knowledge (or lack of it) for each input variable such as C0(x), Da0, Q(x, t) and the properties in B, and (ii) the uncertainty associated with functional relationships such as F(r) and the conventional structural engineering relationships for ultimate moment and for cross-sectional stiffness and its integration for deflection. Table 1 sets out estimates for these uncertainties based on the (limited) information in the literature. To represent uncertainty about a variable, it is customary when using probability theory to let such variables be random variables and to associate with them a probability density function based on data and other information. Assume that the random variables can be collected in the vector X. Previous research and existing data-bases make this possible for a number of random variables, as shown in Table 1 and the Appendix. For others there is insufficient information at this time. However, it is consistent with structural reliability theory to let their uncertainty be represented in a second-moment form, that is, through the expected or mean value lX i and the standard deviation rX i of the random variable Xi. In principle the relationships that operationalize Fig. 4 can be used to estimate the uncertainty that should be associated with the ultimate moment capacity. Similarly, the relationships for Fig. 5 can be used to estimate the uncertainty in the structural stiffness or the deflection of a beam. In structural reliability theory, this is done using the so-called ‘limit state function’ which would be an explicit or implicit statement equivalent to the algorithms involved. A number of techniques can be applied for this purpose. However, it is considered that in this case the non-linear nature of a number of the underlying processes indicates that analysis through a Monte Carlo technique is likely to be the most efficient approach. Monte Carlo methods are well known and need not be explained here in detail [22]. It is worthy to note that for variables modeled as random (i.e. the X) and hence assumed to have a (joint) prescribed probability distribution function (fX( )), a sample value ^xi is selected for each random variable Xi, typically using a computer technique. This set of sample values plus the various deterministic values (collected in the vector v) are then used to evaluate the function of interest (say Eq. (2) for ultimate moment capacity) to provide a sample value ^ ult ¼ Mult ð^ M x; vÞ. This process is repeated for different sets of sample values, each such set drawn from the collection of probability distributions. The set of resulting sample outcomes allows a probability density function to be postulated and a mean and a standard deviation to be estimated. Evidently, this process can be repeated for different exposure periods t, with the result that the variation of the mean and the standard deviation can be obtained as a function of t. 4. Example application of proposed procedure There are almost no independent experimental or theoretical results with which to compare the proposed procedure. As noted in Section 2, most experimental investigations have been concerned with unrealistic cases (non-concurrent deterioration and loading and DC current impressed corrosion). Moreover, there is seldom sufficient replication of the tests to allow estimates of variability in response (ultimate moment, deflection). However, a small number of experimental observations for simple cantilevers tested under similar environmentally accelerated corrosion conditions are available [17]. In these tests the deterioration of the ultimate bending capacity and the increase in tip deflection were obtained as a function of time. The cantilevers were of realistic scale (Fig. 6) and were subject to environmentally accelerated laboratory testing involving a faster than normal wetting-drying rate. The latter may affect the mechanics of chloride transport. Comparison to naturally exposed samples showed that the accelerated results could be related to real time through a simple multiplier [15]. In the following, only the natural time results will be used. (The cantilevers were actually tested also for the effect of bond slip but this effect has been removed.) Importantly, the multiple results allowed estimation of the variability of these output quantities [17]. These are summarized in Table 2. It is well known that concrete strength increases as a result of the slow increase in cement hydration with time, and also that the permeability of concrete slowly reduces with time. However, because the comparison of

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Fig. 6. Experimental beam (Beams are made of normal concrete with water cement ratios of 0.45 and 0.6. ‘‘Y’’ denotes reinforcing bars with fy = 400 MPa and ‘‘R’’ denotes those with fy = 250 MPa): (a) dimension and reinforcement layout and (b) cross-section A-A.

the theoretical prediction will be to test results that have been converted from an accelerated test, it is not appropriate herein to allow for this characteristic. Instead, a constant value equivalent to the average concrete permeability (Da0) observed during the test program is used for the present example. The corrosion rate r0 (mm/yr) is represented by corrosion current density (icorr in lA/cm2) with 1 lA/cm2 approximately equivalent to a corrosion loss of 0.011 mm/yr. The experimental program reported by [17] allows this to be expressed as a function of time. For example, for specimens with w/c = 0.45, an empirical fit to the experimental data found that the relationship could be represented by 2

icorr ¼ 0:3683LnðtÞ þ 1:1305ðlA=cm Þ

ð3Þ

where Ln denotes the natural logarithm and t is the time during which active corrosion is occurring, i.e. since corrosion initiation. It might be noted that there is a clear experimental evidence, from a variety of sources, Table 2 Comparison of coefficients of variation Time (years)

COV of ultimate bending moment w/c = 0.45 a

0 3.65 6.08 8.51 a

From [17].

COV of tip deflection

w/c = 0.60

Test

Calculated

0.10 0.06 0.14

0.06 0.06 0.06 0.07

Test 0.03 0.05 0.23

a

w/c = 0.45 Calculated 0.06 0.06 0.06 0.06

a

w/c = 0.60

Test

Calculated

0.07 0.11 0.13

0.16 0.20 0.21 0.22

Testa

Calculated

0.09 0.11 0.12

0.11 0.12 0.12 0.12

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that the ‘corrosion rate’ is actually not a constant value but that it actually changes (usually reduces) with increased exposure time [25]. In this example, it is assumed that apart from the properties listed in Table 1, the cross-sectional properties may be treated as deterministic. Similarly, the internal action(s) due to the applied loading are treated as random variables. This approach is realistic since (i) load variability typically is much greater than any variability in material properties and dimensions and (ii) it is known that matters such as concrete cover are subject to considerable variability in practice. The estimated mean of the ultimate moment capacity is shown in Fig. 7a as a function of exposure time for the case of w/c ratio = 0.45. The upper and lower 90% confidence intervals and the experimental values also are shown. The results for w/c ratio = 0.60 were found to be generally similar and are not given here. Although the experimental values fall well within the 90% confidence intervals, it does appear that the theoretically derived mean and confidence trends are over-estimates. This perception is supported by estimates of the loss of ultimate moment based on the loss of reinforcement cross-section using the (slowly reducing) corrosion rate derived from Eq. (3). The reasons for this are not clear, but given the shape of the curves in Fig. 7a most likely it is the result of underestimation in the computational scheme of the time to initiation. This would underestimate the theoretical corrosion loss compared with that which actually occurred in the experiments. Evidently, there is room for improvement of the relationships such as Eq. (1) and in the level of simplification made in the theory. This includes the possible inclusion of the effect of concrete creep, which has been ignored herein since it is likely to have only a secondary influence relative to the effect of loading. Fig. 7b shows that the coefficient of variation (COV) derived from theory for the ultimate moment capacity is approximately constant with exposure time. Point estimates for the coefficient of variation obtained from the experimental program (Table 2) are shown. It is clear that these are much higher, presumably because the latter were estimated from a very limited number of samples [17].

Mean of Bending Moment (kNm)

a 12 10 8 6 Experiment (w/c=0.45) Calculated (w/c=0.45) Upper Confidence Limit Lower Confidence Limit

4 2 0 0

b

2

4 6 Time (year)

8

10

8

10

0.2 Experiment (w/c=0.45) Calculated (w/c=0.45)

COV

0.15

0.1

0.05

0 0

2

4 6 Time (year)

Fig. 7. Deterioration of ultimate bending moment capacity (w/c = 0.45) (a) mean function with 90% confidence limits and (b) coefficient of variation (COV) function.

a

30

Mean of Deflection (mm)

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25

457

20 15 10

Experiment (w/c=0.45) Calculated (w/c=0.45)

5

Upper Confidence Limit Lower Confidence Limit

0 0

b

2

4 6 Time (year)

8

10

81

0

0.35 Experiment (w/c=0.45)

0.3 Calculated (w/c=0.45)

COV

0.25 0.2 0.15 0.1 0.05 0 0

2

4

6

Time (year)

Fig. 8. Deterioration of beam deflection (w/c = 0.45): (a) mean function with 90% confidence limits and (b) coefficient of variation (COV) function.

For the beam tip deflection Fig. 8a shows the estimated mean value as a function of exposure time for the case of w/c ratio = 0.45. Also shown are the upper and lower 90% confidence intervals and the experimental values. In this case there is a close match between experimental beam tip deflections and computed mean values. Fig. 8b shows the change in the coefficients of variation (COV) estimated from theory for the beam tip deflection as a function of exposure time. As before, point estimates for the coefficient of variation obtained from the experimental program (Table 2) are shown. Also as before, these are rather different from the theoretical values, presumably because the experimental values were estimated from a very limited number of samples [17]. 5. Discussion The above example illustrates that estimates can be made of corrosion induced deterioration of structural capacity and of response for reinforced concrete structures under applied loading. Moreover, estimates could be given for the mean value as well as the variance of the responses as a function of time. In principle this should allow extrapolation with a reasonably high degree of confidence. The procedure is based on (i) the idealized corrosion loss–time models of Fig. 1 and (ii) the increase of permeability brought about by applied loading in actual structures. Previous studies have considered the influence of corrosion and its initiation as separate events from applied loading and have also been largely empirical. The example given shows that the proposed approach to estimate the corrosion loss of reinforcing steel provides calculated results that correspond broadly with the few experimental observations that can be used for comparison (Figs. 7 and 8). As expected, the mean value of the ultimate bending moment capacity reduces with time within reasonable confidence limits and, as noted in the previous study [26], this is not strongly dependent on water to cement (w/c) ratio. Similarly, the overall trend in deterioration of the expected value

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of beam bending stiffness (measured as mean beam tip deflection) compares reasonably well with the experimental results. It is also not strongly dependent on w/c ratio. Sensitivity analysis on the effect of surface chloride concentration C0 and effective permeability of the concrete Da on structural response indicates that these also are not major input factors [26]. However, the initial corrosion rate r0 (Fig. 1) for reinforcement in concrete is an important parameter, even though there appears to be little reliable information about its uncertainty. Typically, it might be assumed that there is a high degree of uncertainty associated with it, based on field data for corrosion rates of steel in other circumstances [23]. However, more recent work has shown that this is more likely the result of poor discrimination between data sets and lack of appreciation that r0 is very unlikely to be a constant value with time. Usually, r0 will eventually decrease owing to the oxygen transport inhibiting effects of the rust [24,25]. In the example given above r0 was set by Eq. (3) at the value corresponding with the experimental observations. However, more generally r0 must be estimated in some way and this will add to the uncertainty in the estimated ultimate moment capacity and in the stiffness. There are obvious research needs in this area. In practical applications, account must be taken also of the changing value of Da0 as a result of the slowly increasing hydration of the cement. As noted, this was not considered herein as the comparison of theory with experiment was made using test results obtained from an expansion of an accelerated test. For that particular test the value of Da0 would have changed little. In summary, the importance of the procedure proposed herein lies in the view that it provides a reasonably realistic approach to modeling the structural response of reinforced concrete beams subject to chloride diffusion and reinforcement corrosion. It does this through considering the impact of internal damage within the concrete surrounding the reinforcement and the effect this has on concrete permeability. Moreover, it is seen that using simple Monte Carlo procedures and reasonable data for the uncertainties attached to the input variables, estimates can be made for the uncertainty to be attached to these mean value trends. The significance of the approach proposed here is that it provides a means for obtaining quantitative rather than qualitative information for the assessment of the remaining safe life and serviceability of reinforced concrete structures deteriorating through reinforcement corrosion. 6. Conclusion A procedure has been outlined for the estimation of the expected (mean) value and the variability (coefficient of variation) of the ultimate moment capacity and stiffness of reinforced concrete beams subject to reinforcement corrosion. The proposed technique relies on the modification of the permeability of the concrete in response to internal damage within the concrete surrounding the reinforcing bars caused by bending stresses. This affects the local time to initiation of corrosion and hence the amount of local corrosion loss. The proposed procedure was applied to one of the very few test results available for beams corroding under applied loading. It uses a Monte Carlo routine to estimate the variability to be associated with the ultimate moment capacity and the beam stiffness. It was found that the theory provided estimates of structural deterioration that are generally consistent with the experimental results. Acknowledgement Financial support from the Engineering and Physical Sciences Research Council (UK) Under Grants GR/ S30368/01 and EP/E00444X/01, and Australian Research Council with Grant No. LX0559653 is gratefully acknowledged. Appendix. Summary of statistical data The statistical data available in the literature for surface chloride content C0 and the apparent chloride diffusion coefficient Da0 are limited. A summary is given in Table A1 for Portland cement only. Table A2 reviews the statistical parameters used by other investigators to represent probabilistic properties. The COVs used by [39,38,31] were obtained from ensemble-average results and are much higher than would be expected for any one individual structure. The data reported by [21,5] apply to unstressed concrete.

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Table A1 Surface chloride content (C0) and apparent chloride diffusion coefficient (Da0) Source

[5] [21] [39]

a

Element/exposure condition

8 years UK coastal exposure of concrete block: 1 · 0.5 · 0.3 m 2 years (w/c = 0.60) 4 years (w/c = 0.60) Highway bridges deck soffits in UK Abutments Piers

C0 (%wt concrete)

Da0 (·1012) (m2/s)

Mean

COV

Mean

COV

0.306 3.35 4.43 0.13a 0.23a 0.19a

0.18 0.31 0.22 1.53 1.65 1.68

15.5 6.46 6.29

0.23 0.15

% by weight of cement.

Table A2 Statistical data used by others for probabilistic modeling Diffusion coefficient Da0 (mm2/year)

Source/location/distribution

Surface chloride content C0 (%wt concrete) Mean

COV

Mean

COV

[33]/highway bridges/normal [30]/highway bridges/normal

0.65 –

0.12 –

0.17 0.20

[38,31]/coastal zone/lognormal

C0(d) = 2.95; d < 0.1 km =1.15–1.81 · log10 (d); 0.1 < d < 2.84 kma

0.50

30 50 (good) 100 (average) 150 (poor) 1010+4.66w/cb

a b

0.75

d, distance (kg/m3). cm2/s.

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