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Civil Engineering and Environmental Systems Vol. 24, No. 2, June 2007, 165–178

Stochastic evaluation of the damage length in RC beams submitted to corrosion of reinforcing steel B. SUDRET*†, G. DEFAUX† and M. PENDOLA‡ †Electricité de France, R&D Division, Site des Renardières - F-77818 Moret-sur-Loing, France ‡Phimeca Engineering S.A., 1 Allée Alan Turing, F-63170 Aubière, France (Received 12 July 2006; in final form 1 December 2006) The influence of spatial variability of parameters appearing in a model of concrete carbonation (and resulting rebars corrosion) is studied. For this purpose, random fields are introduced in order to represent the input parameters of the model. The concept of damage length is defined as the portion of a beam structure for which a durability failure criterion is attained. Analytical derivations inspired from time-variant reliability methods lead to tractable formulæ for the mean and standard deviation of the damage length. Results are validated using Monte Carlo simulation (MLS) of both the input random fields (after proper discretization) and the resulting damage length. The histogram of the damage length, which is obtained as a byproduct of the MCS, is finally commented. Keywords: Extent of damage; Ageing indicator; Rebars corrosion; Time-variant reliability; PHI2 method; Space-variant reliability

1.

Introduction

Probabilistic models of concrete degradation have been intensively studied in the past ten years, see e.g. Engelund and Sorensen (1998), Stewart and Rosowsky (1998), Val et al. (1998), Vu and Stewart (2000). The main degradation mechanisms involve chemical reactions between some constituents of concrete and the environment. For instance, carbon dioxide may penetrate by diffusion into the pores of concrete, leading to concrete carbonation. The subsequent pH change in concrete makes corrosion of the reinforcing bars start, once the superficial concrete layer where chemical properties have changed has attained the latter. In the early papers, authors have focused on the prediction of the initiation time for corrosion and/or the estimation of the residual strength of structures, including the influence of pitting corrosion (Liu and Weyers 1998, Val et al. 1998, Vu and Stewart 2000, Bhargava et al. 2005, 2006,a,b). The models used allow to compute a point-in-space probability of failure, without any account of spatial variability. Recent advances in this field have pointed out the necessity of modelling the spatial variability of the model parameters in order to get better predictions. *Corresponding author. Email: [email protected]

Civil Engineering and Environmental Systems ISSN 1028-6608 print/ISSN 1029-0249 online © 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10286600601159305

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For instance, Stewart (2004) shows that the structural reliability of RC beams for strength limit states may be overestimated when ignoring the spatial variability. Moreover, as already pointed out by Stewart (2005), modelling the spatial variability is mandatory in order to be able to characterize, not only the probability of degradation in one point, but also the extent of damage. The need for spatially variable models in order to compare maintenance strategies of concrete structures has been recently addressed in Li et al. (2004), Stewart (2006), Stewart et al. (2006). Following the above authors, the focus of the present article is not on the behaviour and strength of the deteriorated structure, but more specifically on establishing a sound probabilistic description of the extent of damage (related to a prescribed local failure criterion), which may be interpreted as a global indicator of the ageing of the concrete structure. RC beams are considered, thus the introduction of the damage length (which is another name for the extent of damage in this case), whose probabilistic characteristics are to be examined. The model used for representing concrete carbonation and the resulting rebars corrosion is first detailed in section 2. The model is intentionally simple, since the main goal of the article is to properly deal with spatial variability and its consequences. Then the damage length is defined, and its mean and standard deviation derived analytically (section 3). In order to check the accuracy of the proposed formulæ, a Monte Carlo-based scheme is proposed to simulate the mean and standard deviation of the damage length. This requires the prior discretization of the input random fields, which is carried out using the expansion optimal linear estimation (EOLE) method (Li and Der Kiureghian 1993). Both approaches are finally illustrated on an application example (section 4).

2. 2.1

Problem statement Point-in-space model of degradation

The reinforcement bars in concrete structures are initially protected from corrosion by a microscopic oxide layer formed at their boundary due to the strong alkalinity of the pore solution. Concrete carbonation induces a decrease of the pH of the pore solution, which leads to dissolving the protective layer. Then the corrosion of the reinforcement starts. The volume of corrosion products causes tensile stresses that may be sufficiently large to cause internal micro-cracking and eventually spalling (Bertolini et al. 2004). Concrete carbonation is a complex physico-chemical process that includes the diffusion of CO2 into the gas phase of the concrete pores and its reaction with the calcium hydroxyl Ca(OH)2 . This reaction can be written schematically as: Ca(OH)2 + CO2 −→ CaCO3 + H2 O

(1)

As the high pH of uncarbonated concrete is mainly due to the presence of Ca(OH)2 , it is clear that the consumption of this species will lead to a pH drop, which can attain a value of 9 when the reaction is completed. In this environment, the oxide layer that protected the reinforcement bars is attacked and corrosion starts. In practice, CO2 penetrates into the concrete mass by diffusion from the surface layer. Thus a carbonation front appears that moves into the structure. A model for computing the carbonation depth xc is proposed by the CEB Task Group 5.1 & 5.2 (1997). The simplified version retained in the present article reads:  2C0 DCO2 xc (t) = t (2) a

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where DCO2 is the coefficient of diffusion of carbon dioxide in dry concrete, C0 is the carbon dioxide concentration in the surrounding air and a is the binding capacity, i.e. the amount of carbon dioxide necessary for complete carbonation of a concrete volume. It is supposed that corrosion immediately starts when carbonation has attained the rebars. Denoting by e the concrete cover, the time necessary for corrosion to start, called initiation time, reads: Tinit =

ae2 2C0 DCO2

(3)

If generalized corrosion is considered, the loss of metal due to corrosion is approximately uniform over the whole surface. In this case, Faraday’s law indicates that a corrosion current density corresponds to a uniform corrosion penetration of κ = 11, 6 μm/year. If a constant annual corrosion rate icorr is supposed, the expression of the rebars diameter as a function of time eventually reads:  φ(t) =

if

t ≤ Tinit

max[φ0 − 2 icorr κ(t − Tinit ), 0] if

t > Tinit

φ0

(4)

2.2 Point-in-space reliability problem Structural reliability analysis aims at estimating the probability of failure of a structure due to uncertainties in the material properties, loads, geometry, etc. It requires: • the probabilistic modelling of the parameters involved in the deterministic model, that is the definition of random variables (probability density function and associated parameters) gathered in a vector X, • the definition of a failure criterion by means of a limit state function defined in the space of parameters. Classically, the limit state function g(x) is defined in the space of parameters in such a way that: • Ds = {x, g(x) > 0} is the safe domain of the structure, • Df = {x, g(x) ≤ 0} is the failure domain. In a reliability context, it does not necessarily mean the breakdown of the structure, but the fact that certain requirements of serviceability or safety limit states have been reached or exceeded. • the limit state surface is defined by the set of points satisfying g(x) = 0. Denoting by fX (x) the joint probability density function (PDF) of random vector X, the probability of failure of the structure is:  Pf = Prob(g(X) ≤ 0) =

Df

fX (x) dx

(5)

In the present article, the random parameters are those appearing in equations (3), (4), namely the coefficient of diffusion of carbon dioxide DCO2 , the surface carbon dioxide concentration C0 , the binding capacity a, the concrete cover e, the rebars initial diameter φ0 and the corrosion current density icorr : X = {DCO2 , C0 , a, e, φ0 , icorr }

(6)

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Figure 1.

Evolution in time of the rebars diameter and associated state of damage.

The failure criterion is defined at a given time instant by the fact that the residual rebars diameter (equation (4)) becomes smaller than a prescribed fraction (1 − λ) of its initial value (see figure 1): (7) g(X, t) = φ(t) − (1 − λ) φ0 The value of λ (e.g. 1–5%) is selected from experimental evidence in such a way that the corresponding loss of cross section yields a given damage on the structure (e.g. spalling starts), see Broomfield (1997), Alonso et al. (1998). Note that this parameter could also be taken as a random variable in order to take into account model uncertainty on this threshold. Remarking that equation (4) rewrites: φ(t) = min(φ0 , φ0 − 2icorr κ(t − Tinit ))

(8)

for reasonable values of t that do not lead to ‘negative’ values of φ(t), equation (7) becomes: g(X, t) = min(λφ0 , λφ0 − 2icorr κ(t − Tinit ))

(9)

Thus the probability of failure may be interpreted as that of a series system: Pf (t) = Prob({λφ0 ≤ 0} ∪ {λφ0 − 2icorr κ(t − Tinit ) ≤ 0}

(10)

If the rebars diameter (which is in nature a positive definite quantity) is given a non-negative PDF, the above system event reduces to its second component. Thus: Pf (t) = Prob(λφ0 − 2icorr κ(t − Tinit ) ≤ 0

(11)

Following these remarks and using equation (3), the limit state function (7) is defined from now on as:   a e2 (12) g(X, t) = λφ0 − 2icorr κ t − 2 C0 DCO2 2.3 Introducing spatial variability The above limit state function allows to compute the evolution in time of the probability of failure Pf (t). Note that no specific time-variant reliability algorithm is needed for this purpose. Indeed, for any realization of the random vector X, the limit state function is non-increasing in time: the corrosion mechanism makes the rebars diameter monotonically decrease in time.

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Thus the time-variant reliability problem reduces to the time-invariant problem, and time t is treated as a dummy parameter in the study. The above model may be qualified as point-in-space or zero-dimensional. In order to introduce spatial variability of the input parameters in the analysis, suppose that the structure under consideration is a beam of length L, which is described by an abscissa s ∈ [0, L]. A reliability problem such as that defined by equation (12) may now be attached to each s ∈ [0, L]. This leads naturally to the introduction of random fields instead of random variables to describe the input parameters. The probability of failure at a given point-in-space (and a given time instant) now reads: Pf (s, t) = Prob(g(X(s), t) ≤ 0)

(13)

where X(s) is a multidimensional random field generalizing equation (6). In the sequel, it is supposed that the input vector random field X(s) is homogeneous (with respect to space). This assumption, which is not so strong in practical applications, implies that Pf (s, t) is actually independent of s. Note that this assumption may not hold if a large structure is considered (e.g. the mean value of the fields representing environmental parameters may vary significantly along a large bridge in a maritime environment). However, if the structure is decomposed into members that may be dealt with independently, the assumption of homogeneity along each member shall be valid. The point-in-space probability of failure is not of practical interest for estimating the state g of deterioration of the beam. Neither is the global probability of failure Pf (t) defined by the space-variant reliability problem (Der Kiureghian and Zhang 1999, Sudret et al. 2005): ⎛ ⎞ g Pf (t) = Prob(∃ s ∈ [0, L], g(X(s), t) ≤ 0) = Prob ⎝ g(X(s), t) ≤ 0⎠ (14) s∈[0,L]

This quantity indeed, which corresponds to the probability that corrosion appears at least in one point s, is likely to be close to one (when moderate damage such as cracking or beginning of spalling is considered), without any severe consequence on the residual strength of the beam. 2.4

Damage length

In this article, it is proposed to introduce the so-called damage length as an indicator of the state of deterioration of the structure. By definition, at each time instant t, the damage length Ld (t) is a random variable which measures the part of the beam where the failure criterion g(X(s), t) ≤ 0 is attained:  L Ld (t) = 1{g(X(s),t)≤0} (s) ds (15) 0

The support of this random variable is [0, L]. From equation (15), the expected damage length is:  L  L Ld (t) = E[Ld (t)] = E[1{g(X(s),t)≤0} (s)] ds = Pf (s, t) ds (16) 0

0

Using the homogeneity of the input random field, the latter equation rewrites: Ld (t) = L · Pf (t)

(17)

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The above equation has a straightforward interpretation: the fraction of the beam length which is damaged (Ld (t)/L) is, in the mean, equal to the point-in-space probability of failure. This remark has two important consequences: • the point-in-space probability of failure (equation (11)) is worth to be computed since it has the practical interpretation given above. • it is not necessary to introduce the complex formalism of random fields when one is interested only in the mean value of Ld (t). Only the description of the input random variables gathered in X is required. • even if homogeneous random fields were introduced in the modelling, the mean damage length is independent of the correlation structure of these fields. This is a valuable result since this kind of data (shape of autocorrelation coefficient functions and correlation lengths) is difficult to obtain in practice. In order to better capture the probabilistic content of Ld (t), it is useful to study the variance of this quantity. Following results by Koo and Der Kiureghian (2003) in the context of firstpassage problem in time-variant reliability analysis, the variance of the damage length reads: 2

Var[Ld (t)] = E[L2d (t)] − Ld (t)

(18)

where:  E[L2d (t)] =

0

L



L

Prob(g(X(s1 ), t) ≤ 0 ∩ g(X(s2 ), t) ≤ 0) ds1 ds2

(19)

0

It is to be noticed that the function of (s1 , s2 ) in the above integral actually depends only on |s1 − s2 | when the input random field is homogeneous. Thus it is an even function of (s1 − s2 ). It may be shown in this case that the above two-dimensional integral may be reduced to a one-dimensional integral. Thus:  Var[Ld (t)] = L2

1

2

Prob(g(X(0), t) ≤ 0 ∩ g(X(Lv), t) ≤ 0)(2 − 2v) dv − Ld (t)

(20)

0

where the dummy integration variable v = (s2 − s1 )/L has been introduced in equation (19), as shown in details in Appendix A.

3.

Numerical implementation

3.1 Analytical approach We consider here the computation of mean and standard deviation of the damage length from equations (17), (20). As shown in the previous section, equation (17) reduces to solving a time-invariant point-in-space reliability problem. The first order reliability method (FORM) is used for this purpose (Ditlevsen and Madsen 1996, Melchers 1999, Lemaire 2005).

3.1.1 FORM for parallel systems. In order to evaluate equation (20), it is observed that the probability under the integral is nothing but the parallel system failure probability associated to

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the events {g(X(0), t) ≤ 0} and {g(X(Lv), t) ≤ 0}. This probability may be computed using the FORM method applied to parallel systems: Prob((g(X(0), t) ≤ 0) ∩ g(X(Lv), t) ≤ 0) ≈ 2 (−β(0, t), −β(Lv, t), ρg (Lv, t))

(21)

where 2 (x, y, ρ) is the binormal CDF, β(0, t) (respectively β(Lv, t)) is the point-in-space reliability index at abscissa s = 0 (resp. abscissa s = Lv) and ρg (Lv, t) = α(0, t) · α(Lv, t) is the dot product of the α-vectors obtained by FORM. Note that β(Lv, t) is actually independent of v when homogeneous fields are considered. Equation (21) is practically evaluated using the PHI2 method as in time-variant reliability analysis (Andrieu-Renaud et al. 2004, Sudret 2007a). This method works as follows (the description uses here the terms ‘point-in-space’ instead of ‘point-in-time’ since random fields are considered): • a point-in-space FORM analysis is carried out at s = 0, replacing the random fields by a set of random variables having the same PDF, say X (1) . • another point-in-space FORM analysis is carried out at s = Lv, replacing the random fields by another set of random variables having the same PDF, say X (2) . If some components of X(s) are actually random variables (i.e. fully correlated fields), the same variables are used in both analysis. The other components of X (2) (corresponding to true random fields) are correlated to the corresponding components of X (1) using the random field autocorrelation matrix. • the reliability indices and α-vectors obtained by both analysis are reported in equation (21). Note that no discretization of the random fields is required in this case. Finally, equation (20) may be evaluated using a Gaussian quadrature formula. The principle of Gaussian integration is to approximate the integral by a weighted summation:  0

1

 h(u) du =



1

h −1

1+s 2



  K ωi ds 1 + si ≈ h 2 2 2 i=1

(22)

where {(ωi , si ), i = 1, . . . , K} are the integration weights and points (Abramowitz and Stegun 1970). 3.2

EOLE method for random field discretization

Monte Carlo simulation (MCS) is another way to compute the mean and standard deviation of the damage length, or its full PDF. This requires the simulation of trajectories of the field after a proper discretization. The EOLE method (Li and Der Kiureghian 1993) is used for this purpose (it is assumed that the components of the vector input random field are independent). The method is based on the pointwise regression of the original random field with respect to selected values of the field, and a compaction of the data by spectral analysis. Let us consider a homogeneous scalar Gaussian random field H (s) defined by its mean value μ, its standard deviation σ and its autocorrelation coefficient function ρ(s1 , s2 ) (which depends on a single parameter |s1 − s2 | due to stationarity). Let us consider a grid of points {s1 , . . . , sN } in [0, L]. Let us denote by χ the random vector {H (s1 ), . . . , H (sN )}. By construction, χ is a

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Gaussian vector whose mean value μχ and covariance matrix  χ χ read: μiχ = μ(si )

  χ χ i,j = Cov[H (si ), H (sj )] = σ 2 ρ(si , sj )

(23) (24)

The optimal linear estimation (OLE) of random variable H (s) onto the random vector χ reads: H (s) ≈ Hˆ (s) = μ +  H χ (s) ·  −1 χ χ · (χ − μχ )

(25)

where (·) denotes the transposed matrix and  H χ (s) is a vector whose component are given by: j

 H χ (s) = Cov[H (s), χj ] = Cov[H (s), H (sj )] = σ 2 ρ(s, sj )

(26)

Let us now consider the spectral decomposition of the covariance matrix  χ χ :  χ χ φ i = λi φ i

i = 1, . . . , N

(27)

This allows to linearly transform the original vector χ : χ = μχ +

N 

λi ξi φ i

(28)

i=1

where {ξi , i = 1, . . . , N} are independent standard normal variables. Substituting for equation (28) in equation (25) and using equation (27) yields the OLE representation of the field: N ξi Hˆ (s) = μ + (29) √ φ i ·  Hχ (s) (OLE expansion) λi i=1 The series can be further truncated after r ≤ N terms, the eigenvalues λi being sorted first in descending order. This gives the EOLE expansion: Hˆ (s) = μ +

r ξi √ φ i ·  Hχ (s) λi i=1

(EOLE expansion)

(30)

The pointwise variance of the error for the EOLE discretization reads: Var[H (s) − Hˆ (s)] = σ 2 −

r 1 (φ i ·  H (s)χ (s))2 λ i i=1

(31)

The above equation allows to check that the grid density (i.e. number of points N ) and the number of terms r are large enough to attain a prescribed accuracy in the discretization. Details on how choosing these parameters are given in Sudret and Der Kiureghian (2000). They depend on the form of the autocorrelation coefficient function and the related fluctuation length. 3.3 MCS of the damage length Once the input random fields have been discretized using EOLE, a realization (or trajectory) of the field ho (s) (i.e. a usual function of s ∈ [0, L]) is obtained by simulating a set of r standard normal random variables, say {ξ1o , . . . , ξro }, and substituting for them in equation (29).

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Figure 2. Algorithm for computing the damage length by MCS.

Computationally speaking, a trajectory is stored in an array of size Q + 1 corresponding to the evaluation of (29) at selected points:   r ξio o h (s) ≡ μ + (32) √ φ i ·  Hχ (s)(j L/Q) , j = 0, . . . , Q λi i=1 These trajectories are generated for all the random fields appearing in equation (12). Then the limit state function in equation (12) is evaluated in each point of the same grid and stored in an array {gj = g(xo (j L/Q), t), j = 0, . . . , Q}. The damage length is then obtained by screening these values : if, for a given j ∈ [0, Q − 1], gj ≤ 0 and gj +1 ≤ 0, then it is supposed that the failure criterion is negative for any s ∈ [j L/Q, (j + 1)L/Q] and the damage length is increased by L = L/Q. If g changes its sign between j L/Q and (j + 1)L/Q, a linear interpolation allows to compute the portion of the latter interval where g is negative. Of course Q has to be selected in such a way that the trajectories do not change sign more than once within an interval of the form [j L/Q, (j + 1)L/Q]. The above procedure is detailed in the algorithm presented in figure 2. Note that another estimate of the damage length could be obtained as Ngj ≤0 · L, where Ngj ≤0 ∈ [0, Q] is the number of negative values in the array {gj , j = 0, . . . , Q}. This kind of estimate, which is straigthforward when the midpoint technique is used to discretize the random fields, has been used e.g. by Vu and Stewart (2005). It has been observed in the present work that this estimate may be too crude to accurately compute the damage length from a trajectory, thus the algorithm in figure 2 has been used.

4. 4.1

Numerical results Probabilistic input data

In order to illustrate the concept of damage length described above, we consider a concrete beam of length L = 5 m. This beam is reinforced by a single longitudinal reinforcing bar whose initial diameter is modelled by a lognormal random variable φ0 . The concrete cover e(s) is a one-dimensional Gaussian random field. This allows to model the imperfections in placing the rebars into the falsework. The parameters describing the carbonation process, namely {DCO2 , C0 , a} are modelled by random variables. Furthermore, the corrosion current density icorr (s) is modelled by a one-dimensional Gaussian random field. The parameters describing these six input quantities are gathered in table 1. The numerical values are chosen here for the purpose of illustrating

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Table 1.

Probabilistic input data.

Parameter

Type of PDF

Rebars’ diameter φ0 Diffusion coefficient DCO2 Surface concentration C0 Binding capacity a Concrete cover e(s) Corrosion current density icorr (s)

Gaussian lognormal lognormal lognormal Gaussian Gaussian

†

Mean value

Coefficient variation(%)

A.c.f†

10 mm 5.10−8 m2 /s 6.2 10−4 kg/m3 80 kg/m3 3.5 cm 2 μA/cm2

10 30 30 30 20 25

– – – – ρ(s1 , s2 ) ρ(s1 , s2 )

Autocorrelation coefficient function, see equation (33).

the approach. They do not come from experimental investigations. Nevertheless, the order of magnitude of the various values have been chosen consistently with the existing literature (e.g. CEB Task Group 5.1 & 5.2 (1997)). The threshold in the limit state function equation (12) corresponds to λ = 5%. 4.2 Discretization of the input random fields The concrete cover and corrosion current density are modelled by mutually independent Gaussian random fields whose mean value and standard deviation are given in table 1. Their autocorrelation coefficient function are of exponential square type, i.e.: ρ(s1 , s2 ) = e−π(s1 −s2 )

2

/2

(33)

In this expression,  denotes the scale of fluctuation of the random field, according to the definition by Vanmarcke (1983). The scale of fluctuation is equal to 1 m for both fields. The random fields are discretized using the EOLE method. The Random Field discretization toolbox of FERUM (Der Kiureghian et al. 2006) has been used. This Matlab-based software is available at http://www.ce.berkeley.edu/FERUM and documented in Sudret and Der Kiureghian (2000). A regular grid consisting in N = 36 points over [0, 5 m] was used. A number r = 12 points was retained in the spectral decomposition. This allows to get a mean relative discretization error on the field variance less than 1%, as shown in figure 3.

Figure 3. Variance discretization error for the input random fields modelling the concrete cover and the corrosion current density.

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4.3

175

Damage length results

A total number of 10,000 simulations was run. The mean value and standard deviation of the damage length obtained by the analytical formulation (a 5-point integration scheme is used in evaluating equation (20)) and by MCS are reported in table 2 and plotted in figure 4. These results show an almost perfect agreement between the approaches, thus validating the analytical derivations presented in section 2.4. The largest discrepancy is observed for the first time instant, for which it is likely that the MCS results are not accurate enough (10,000 samples for evaluating a probability of failure in the range of 10−4 ). It is interesting to note that the coefficient of variation of the damage length is rather large (e.g. it is greater than 100% as long as time is smaller than 50 years). However it decreases in time, from about 900% at t = 15 years to 75% at t = 60 years. These numbers confirm that an analysis which limits to evaluating the mean value of the damage length could be strongly inaccurate. The large coefficient of variation is in relationship with the peculiar shape of the histogram of the damage length, as it can be derived from MCS. As an example, the histogram at t = 35 years is plotted in figure 5. Table 2. Mean value and standard deviation of the damage length obtained by the analytical formulation and by MCS. Time (years) 15 20 25 30 35 40 45 50 55 60

Mean value

Standard deviation

Equation (17)

MCS

Equation (20)

MCS

0.007 0.049 0.165 0.366 0.639 0.960 1.305 1.654 1.993 2.313

0.006 0.044 0.161 0.363 0.640 0.971 1.306 1.669 2.012 2.331

0.061 0.211 0.457 0.748 1.033 1.281 1.477 1.619 1.711 1.760

0.055 0.190 0.449 0.744 1.026 1.281 1.470 1.618 1.707 1.755

Figure 4. Mean value and standard deviation of the damage length vs. time - Comparison between the MCS results (section 3.3) and the analytical results (equations (17), (20)).

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Figure 5. Histogram of the damage length (t = 35 years). (NB : the probability spike at Ld = 0 is not at scale.)

This figure shows that the probability measure of the damage length is a mix between a discrete probability function at the boundaries (Prob(Ld = 0) and Prob(Ld = 5) are finite quantities) and a continuous probability density function in between. Thus it is strongly non Gaussian. This should be given further attention, especially if one is interested in studying some failure criterion based on this damage length (as done by Vu and Stewart (2005), who considered the damage surface as Gaussian under simplifying assumptions). The evolution in time of the histograms as well as that of the probability spikes is studied in Sudret (2007b).

5.

Conclusions

The concept of damage length was introduced as a global indicator of the state of deterioration of beam structures. The passage from zero-dimensional probabilistic deterioration models to this damage length requires the introduction of random fields in the analysis. Two methods for computing the mean and standard deviation of the damage length are presented and compared. The first method relies upon analytical derivations and techniques inspired from time-variant reliability analysis. It avoids the discretization of the random fields, which is a valuable feature. It is very efficient but provides only the first two statistical moments of the damage length. The second method is based on the simulation of the damage length after proper discretization of the random fields using the EOLE method. It is computationally demanding. The results obtained by both methods compare very well. However, the second method provides the histogram of the damage length, from which it is observed that the PDF of the latter may be complex. It appears in particular that probability spikes corresponding to a ‘fully sound’ or ‘fully damaged’ beam exist. The approximation of the damage length by a

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Gaussian random variable thus seems inappropriate. The study of the properties of this PDF is currently in progress. As a conclusion, the paper presents a framework that makes a rigorous transition between zero-dimensional degradation models and the associated extent of damage. Although the carbonation-induced corrosion model is simple in the application example, this framework could be easily applied to more complex models, including chloride-induced corrosion models and/or pitting corrosion. References Abramowitz, M. and Stegun, I.A., (Editors), Handbook of Mathematical Functions, 1970 (Dover Publications, Inc.). Alonso, C., Andrade, A., Rodriguez, J. and Diez, J., Factors controlling cracking of concrete affected by reinforcement corrosion. Mater. Struct., 1998, 31, 435–441. Andrieu-Renaud, C., Sudret, B. and Lemaire, M., The PHI2 method: a way to compute time-variant reliability. Rel. Eng. Sys. Safety, 2004, 84, 75–86. 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Appendix A:

Proof of equation (20)

Suppose f (x, y) ≡ f˜(x − y), where f˜(z) is an even function of its argument. Assume the LL integral I = 0 0 f (x, y) dx dy exists and is to be computed. The following mapping is used: x+y −1 L y−x v= L

u=

The integral rewrites:



L2 I= 2



1 −1

a(v)/2

f˜(Lv) du dv

(A1)

(A2)

−a(v)/2

Since the integrand does not depend anymore on u, the integration with respect to u provides a(v). Moreover, due to the fact that f˜ is even, the integral with respect to v is twice that computed over [0, 1]. Finally, for v ≥ 0, it is easy to show that a(v) = 2 − 2v. Thus:  I =L

2 0

1

f˜(Lv)(2 − 2v) dv

(A3)