International Forum on Engineering Decision Making Second IFED Forum, April 26-29, 2006, Lake Louise, Canada

Introducing spatial variability in the lifetime assessment of a concrete beam submitted to rebars’ corrosion Sudret, B. Electricité de France, R&D Division, Site des Renardières - F-77818 Moret-sur-Loing, FRANCE Defaux, G. and Pendola, M. Phimeca Engineering S.A., 1 Allée Alan Turing, F-63170 Aubière, FRANCE

Abstract The influence of spatial variability of parameters appearing in a model of concrete carbonation (and resulting rebars corrosion) is studied. The concept of damaged 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 formulae for the mean and standard deviation of the damaged length. Results are validated using Monte Carlo simulation of the input random fields (after proper discretization) and resulting damaged length. Keywords : durability analysis / random fields / concrete carbonation / rebars corrosion / space-variant reliability /

1 Introduction Probabilistic models of concrete degradation have been intensively studied in the past ten years [1, 2]. They allow to compute the point-in-space probability of failure, without any account of spatial variability. Recently, papers devoted to the residual strength of beam structures submitted to environmental attacks have been published [3, 4]. In this communication, we are not interested in the structural behaviour, but more specifically in establishing a global indicator of the ageing of the concrete structure. Neither the point-in-space- nor the global probability of failure is suited to this goal. As an alternative, we introduce here the concept of damaged length. The model used for representing concrete carbonation and the resulting rebars corrosion is first detailed. Then the damaged length is defined, and its mean and standard deviation derived analytically. The Monte Carlo simulation of the latter is then developped. Both approaches are finally illustrated on an application example.

1

2 Problem statement 2.1 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. Two different mechanisms, namely chloride ingress and concrete carbonation, lead to 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. 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 . 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 Groups 5.1 & 5.2 [5]. The simplified version retained in the present paper reads: r 2 C0 DCO2 xc (t) = t (1) a 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 =

a e2 2 C0 DCO2

(2)

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, 6µm/year. If a constant annual corrosion rate is supposed, the expression of the rebars diameter as a function of time eventually reads:  φ0 if t ≤ Tinit φ(t) = (3) max[φ0 − 2 icorr κ(t − Tinit ) , 0] if t > Tinit

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, 2

• 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) > 0 is formulated in such a way that Ds = {x, g(x) > 0} defines the safe state of the structure, Df = {x, g(x) ≤ 0} defines the failure state. The limit state surface is defined by the set of points satisfying g(x) = 0. Denoting by fX (x) the joint probability density function of random vector X , the probability of failure of the structure is : Z Pf = P (g(X) ≤ 0) = fX (x) dx (4) Df

In the present paper, the random parameters are those appearing in Eqs.(2),(3), 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 }

(5)

The failure criterion is defined at a given time instant by the fact that the residual rebars diameter (Eq.(3)) becomes smaller than a prescribed fraction (1 − λ) of its initial value: g(X, t) = φ(t) − (1 − λ) φ0

(6)

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). Remarking that Eq.(3) rewrites: φ(t) = min (φ0 , φ0 − 2 icorr κ(t − Tinit )) (7) for reasonable values of t that do not lead to “negative” values of φ(t), Eq.(6) becomes: g(X, t) = min (λ φ0 , λ φ0 − 2 icorr κ(t − Tinit ))

(8)

Thus the probability of failure may be interpreted as that of a series system: Pf (t) = P ({λ φ0 ≤ 0} ∪ {λ φ0 − 2 icorr κ(t − Tinit ) ≤ 0})

(9)

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) = P (λ φ0 − 2 icorr κ(t − Tinit ) ≤ 0)

(10)

Following these remarks and using (2)), the limit state function (6) is defined from now on as: g(X, t) = λ φ0 − 2 icorr κ(t −

3

a e2 ) 2 C0 DCO2

(11)

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. 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 properties 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 Eq.(11) 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) = P (g(X(s), t) ≤ 0)

(12)

where X(s) is a multidimensional random field generalizing Eq.(5). In the sequel, it is supposed that the input vector random field X(s) is stationary (with respect to space). This assumption, which is not so strong in practical applications, implies that Pf (s, t) is actually independent of s. The point-in-space probability of failure is not of practical interest for estimating the state of deterioration of the beam. Neither is the global probability of failure Pfg (t) defined by the space-variant reliability problem [6, 7]:   [ Pfg (t) = P (∃ s ∈ [0, L] , g(X(s), t) ≤ 0) = P  g(X(s), t) ≤ 0 (13) 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, without any severe consequence on the residual strength of the beam.

2.4 Damaged length In this paper, it is proposed to introduce the so-called damaged length as an indicator of the state of deterioration of the structure. By definition, at each time instant t, the damaged 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: Z L

Ld (t) =

0

1g(X(s),t)≤0 (s) ds

(14)

The support of this random variable is [0, L]. From (14), the expected damaged length is: Z L Z L   Ld (t) = E [Ld (t)] = E 1g(X(s),t)≤0 (s) ds = Pf (s, t) ds 0

(15)

0

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

(16)

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: 4

• the point-in-space probability of failure (Eq.(10)) is worth to be computed since it has a practical interpretation. • 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. 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 et al. [8] in the context of first-passage problem in time-variant reliability analysis, the variance of the damaged length reads:   2 Var [Ld (t)] = E L2d (t) − Ld (t)

where: E



L2d (t)



=

Z

0

LZ L 0

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

(17)

(18)

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 stationary. 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 (see details in Appendix A). Thus: Z 1 2 2 Var [Ld (t)] = L P (g(X(0), t) ≤ 0 ∩ g(X(Lv), t) ≤ 0) (2 − 2 v) dv − Ld (t) (19) 0

3

Numerical implementation

3.1 Analytical approach We consider here the computation of mean and standard deviation of the damaged length from Eqs.(16),(19). As shown in the previous section, Eq.(16) reduces to solving a time-invariant point-inspace reliability problem. The first order reliability method (FORM) is used for this purpose [9]. In order to evaluate Eq.(19), note that the probability under the integral is nothing but the parallel system failure probability associated to 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: P (g(X(0), t) ≤ 0 ∩ g(X(Lv), t)) ≈ Φ2 (−β(0, t), −β(Lv, t), ρg (Lv, t))

(20)

where Φ2 (x, y, ρ) is the binormal CDF, β(0, t) (resp. 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 stationary fields are considered. Eq.(20) is practically evaluated using the PHI2 method as in time-variant reliability analysis [10, 11]. This method works as follows (the description uses here the terms “point-in-space” instead of “pointin-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) . 5

• 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 Eq.(20). Note that no discretization of the random fields is necessary in this case. Finally, Eq.(19) is evaluated using a Gaussian quadrature formula.

3.2 EOLE method for random field discretization Monte Carlo simulation (MCS) is another way to compute the mean and standard deviation of the damaged length, or its full PDF. This requires the simulation of trajectories of the field after a proper discretization. The expansion optimal linear estimation method (EOLE) [12] 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 stationary 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 Gaussian vector whose mean value µχ and covariance matrix Σχ χ read: µiχ = µ(si )

(21)

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

(22)

The optimal linear estimation (OLE) of random variable H(s) onto the random vector χ reads:
ˆ H(s) ≈ H(s) = µ + Σ′Hχ (s) · Σ−1 (23) χ χ · χ − µχ where (.)′ denotes the transposed matrix and ΣHχ (s) is a vector whose component are given by: ΣjHχ (s) = Cov [H(s) , χj ] = Cov [H(s) , H(sj )] = σ 2 ρ(s , sj )

(24)

Let us now consider the spectral decomposition of the covariance matrix Σχχ : Σχχ φi = λi φi

i = 1, . . . , N

(25)

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

N p X

λi ξi φi

(26)

i=1

where {ξi , i = 1, ... N } are independent standard normal variables. Substituting for (26) in (23) and using (25) yields the EOLE representation of the field : ˆ H(s) =µ+

N X ξ √ i φi′ · ΣH(s)χ (s) λi i=1

6

(27)

The series can be further truncated after r ≤ N terms, the eigenvalues λi being sorted first in descending order. The variance of the error for the EOLE discretization reads: r h i X
2 1 ˆ Var H(s) − H(s) = σ2 − φ′i · ΣH(s) χ (s) λi

(28)

i=1

The above equation allows to check that the grid density (i.e. number of points N ) and number of terms r are large enough to attain a prescribed accuracy in the discretization. Details on how choosing these parameters are given in [13]. As seen in the application example, they depend on the form of the autocorrelation coefficient function and the related fluctuation length.

3.3 Monte Carlo simulation of the damaged 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 Eq.(27). Computationally speaking, a trajectory is stored in an array of size Q + 1 corresponding to the evaluation of (27) at selected points: ( ) r o X ξ √i φi′ · ΣH(s)χ (jL/Q) , j = 0, ... Q (29) ho (s) ≡ µ + λi i=1 These trajectories are generated for all the random fields appearing in Eq.(11). Then the limit state function in Eq.(11) is evaluated in each point of the same grid and stored in an array {gj = g(xo (jL/Q), t), j = 0, ... Q}. The damaged length is then obtained by screening these values : if, for a given j ∈ [0, Q], gj ≤ 0 and gj+1 ≤ 0 it is supposed that the failure criterion is negative for any s ∈ [jL/Q , (j + 1)L/Q] and the damaged length is increased by L/Q. If g changes its sign between jL/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 twice within an interval of the form [jL/Q , (j + 1)L/Q].

4 Numerical results 4.1 Probabilistic input data In order to illustrate the concept of damaged length described above, we consider a concrete beam of length L = 5 m. This beam is reinforced by a 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. In contrary, 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.

7

Table 1: Probabilistic input data Parameter Type of PDF Mean value Coef. Var. Rebars’ diameter φ0 Gaussian 10 mm 10 % Diffusion coefficient DCO2 lognormal 5.10−8 m2 /s 30 % Surface concentration C0 lognormal 6.2 10−4 kg/m3 30 % Binding capacity a lognormal 80 kg/m3 30 % Concrete cover e(s) Gaussian 3.5 cm 20 % 2 Corrosion current density icorr (s) Gaussian 2 µA/cm 25 % † Autocorrelation coefficient function 2 2 2 2 ρe (s1 , s2 ) = e−π(s1 −s2 ) /ℓe , ℓe = 1 m and ρi (s1 , s2 ) = e−π(s1 −s2 ) /ℓi , ℓi = 1 m

A.c.f† − − − − ρe (s1 , s2 ) ρi (s1 , s2 )

4.2 Point-in-space results The point-in-space probability of failure, obtained by freezing the spatial coordinate s (i.e. replacing the two random fields by two Gaussian random variables) is computed by FORM analysis for various instants. The results are reported in Figure 1, after multiplying by the total length L in order to get the expected damaged length (see Eq.(16)). The results obtained by FORM are confirmed by Monte Carlo simulation (of the point-in-space problem) using 100,000 samples.

Expected damaged length(m)

2.5

FORM 2

1.5

1

0.5

0 10

20

30

40

50

60

Time (years)

Figure 1: Evolution in time of the expected damaged length

4.3 Damaged length results Both random processes are discretized using the EOLE method. 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 %. A total number of 10,000 simulations was run. The mean value and standard deviation of the damaged length obtained by the analytical formulation (a 5-point integration scheme is used in evaluating Eq.(19)) and by Monte Carlo simulation are reported in Figure 2(a). These results show a perfect agreement between the approaches, thus validating the analytical derivations presented in Section 2.4. 8

(a) Numerical results

1.0

1.5

2.0

Standard deviation Eq.(19) MCS 0.061 0.055 0.211 0.190 0.457 0.449 0.748 0.744 1.033 1.026 1.281 1.281 1.477 1.470 1.619 1.618 1.711 1.707 1.760 1.755

0.5

Mean value Eq.(16) MCS 0.007 0.006 0.049 0.044 0.165 0.161 0.366 0.363 0.639 0.640 0.960 0.971 1.305 1.306 1.654 1.669 1.993 2.012 2.313 2.331

0.0

Time (years) 15 20 25 30 35 40 45 50 55 60

0

1

2

3

4

5

(b) Histogram of the damaged length (t = 35 years)

Figure 2: Mean value and standard deviation of the damaged length obtained by the analytical formulation and by Monte Carlo simulation

As an example, the histogram of the damaged length at t = 35 years is plotted in Figure 2(b). This shows the peculiar form of the PDF of the damaged length, which is mix between a discrete density at the boundaries (P (Ld = 0) and P (Ld = 5) are finite quantities) and a continous density in between.

5 Conclusions The concept of damaged length was introduced as an 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 damaged length are presented and compared. The first method relies upon analytical derivations and techniques inspired from timevariant reliability analysis. It avoids the discretization of the random fields, which is a nice feature. It is very efficient but only provides the two first statistical moments of the damaged length. The second method is based on the simulation of the damaged length after proper discretization of the random fields. It is computationnally demanding. The results obtained by both methods compare very well. However, the second method provides the histogram of the damaged length, from which it is observed that the PDF of the latter may be complex. The study of the properties of this PDF is currently in progress.

A

Proof of Eq.(20)

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

x+y −1 L y−x L 9

(30)

The integral rewrites: L2 I= 2

Z

1

−1

Z

a(v)/2

f˜(L v) du dv

(31)

−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: Z 1 2 I=L f˜(L v)(2 − 2v) dv (32) 0

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