BAYESIAN UPDATING OF THE LONG-TERM CREEP DEFORMATIONS IN CONCRETE CONTAINMENT VESSELS B. Sudret, M. Berveiller, Electricité de France, R&D Division, Site des Renardières - F-77818 Moret-surLoing, FRANCE F. Perrin, M. Pendola, Phimeca Engineering S.A., 1 Allée Alan Turing, F-63170 Aubière, FRANCE

ABSTRACT Delayed strains in concrete containment vessels is a major concern for Electricité de France. Codified models for predicting creep and shrinkage are not accurate in the long term. However, containement vessels are continuously monitored so that measures of creep are available. The paper aims at computing the evolution in time of a confidence interval on the creep strains. An a priori interval is obtained using a probabilistic model for creep together with an inverse FORM algorithm. The latter is then modified in order to introduce the measurement data and to update the confidence interval. Keywords : durability analysis / concrete creep / Bayesian updating / inverse reliability problem / FORM

1.

INTRODUCTION

interval of the total strain, as well as the continuous updating of the latter by introducing measurement data.

Long-term creep deformation in concrete containment vessels is a major concern for Electricité de France, which is currently exploiting a fleet of 58 pressurized water reactors. Indeed, the concrete containment vessel is designed so as to keep its structural integrity in case of a severe accident in the reactor. The long term creep strains tend to relax the tensioning of the prestressing cables. In order to assess the safety of the vessel, it is thus important to accurately predict these strains. It is well accepted that creep is caused by various chemical and chemical phenomena at various scales of space and time, taking place especially in the hardened cement paste. If short and mediumterm predictions are rather accurate, the long term behaviour is still difficult to capture in models (see e.g. Benboudjema (2002), Le Pape et al. (2003)). On the other hand, structures like containment vessels are monitored all along their service life, meaning that a set of strain measurement data is available. Incorporating this data in the predictions is the aim of the paper. We present indeed a framework for efficiently compute the 95 %-confidence

2. COMPUTATION OF RESPONSE FRACTILES BY INVERSE FORM 2.1

PROBLEM STATEMENT

Let us consider a mechanical model M which allows to compute a scalar time-dependent response quantity y(t) as a function of input parameters gathered in a vector x and time t: y(t) = M(x, t)

(1)

In real life problems, the parameters of such a model are not well known and may be modelled as a random vector X, whose joint probability density function (PDF) is denoted by fX (x). As a consequence, the response becomes a random process Y (t). Note that Y (t) is a particular kind of process, since it is a function of time and random variables, as seen from Eq.(1). 1

scheme such as Monte Carlo simulation is used for this purpose, some instabilities may appear in the iterative solving scheme due to sampling. In order to solve the inverse reliability problem efficiently, Der Kiureghian et al. (1994) proposed an algorithm based on the First Order Reliability Method (FORM). Associating to the target probability of failure Pfc a target reliability index β c = −Φ−1 (Pfc ), where Φ is the standard normal cumulative distribution function (CDF), the algorithm solves the following problem:

Of interest here is the evolution in time of the yet-to-be-determined response PDF fY (y, t) of Y (t), and more precisely α-fractiles of the latter, which are denoted by yα (t). In practice, one is interested in establishing ranges of variation of the response quantity, e.g. a 95 % confidence interval. This means that the e.g. 2.5 % and 97.5 % fractiles are of interest in this particular case. Introducing the following function: Pf (y, t) = P (Y (t) ≤ y)

(2)

the α-fractiles of the response can be obtained as the solution of the following problem: Find yα (t) :

2.2

Pf (yα (t), t) = α

Find θ :

where PF ORM (.) means that the probability of failure is computing using FORM analysis (see e.g. Ditlevsen and Madsen (1996)). The algorithm proposed by Der Kiureghian et al. (1994) is a modification of the HLRF algorithm usually used in FORM analysis, see details in the referred paper.

(3)

INVERSE FORM ANALYSIS

In the context of structural reliability analysis, one usually defines a limit state function g(X) depending on the input random vector X, in such a way that positive values correspond to the safe domain (denoted by Ds ) and negative values to the failure domain (denoted by Df ). The limit state surface is defined in the space of parameters as the set of points satisfying g(X) = 0. The probability of failure is then defined as follows: Z Pf = P (g(X) ≤ 0) = fX (x) dx (4)

2.3 APPLICATION TO THE COMPUTATION OF FRACTILES Combining Eqs.(1),(2),(6), the α-fractiles of the response quantity may be obtained by appling the inverse reliability algorithm using the following limit state function: g(X, t) = M(X, t) − yα (t)

Df

P (g(X, θ) ≤ 0) = Pfc

(7)

together with a target reliability index β c = Φ−1 (α). As FORM analysis is all the more accurate since the obtained probability of failure is small, Eq.(7) should be used for lower fractiles, e.g. 2.5 %. For upper fractiles (e.g. α = 97.5 %), the opposite limit state function should be preferred, together with a target reliability index β c = Φ−1 (1 − α).

In some cases, the limit state function is depending on a deterministic parameter θ, whose value is to be determined in such a way that the related probability of failure takes a target value Pfc . The so-called inverse reliability problem reads: Find θ :

PF ORM (g(X, θ) ≤ 0) = Φ(−β c ) (6)

(5)

This problem can be considered as a rootfinding problem associated to the equation P (g(X, θ) ≤ 0) − Pfc = 0. Thus the bisection or the Newton-Raphson methods can be used. However, these methods require several evaluations of the function, which may be computationally expensive if the cost of solving each single reliability problem (i.e. compute P (g(X, θ0 ) ≤ 0) for a given θ0 ) is large. Moreover, if a sampling computation

3. UPDATING THE PROBABILITY OF FAILURE 3.1

INTRODUCTION

Let us consider a mechanical system S0 for which a scalar response quantity may be monitored in time. 2

Let y(t) = M(x, t) be a model for this response quantity. If the model was perfectly representing the mechanical system, and if we perfectly knew the true value of the parameters describing this particular system S0 (i.e. the realization x0 of the input random vector the particular system is corresponding to), and if finally we had a perfect measurement device at hand, then the measured value y˜(t) of the response quantity would exactly be equal to the predicted value : y˜(t) = M(x0 , t). In real life problems, none of these three conditions is satisfied. Indeed:

• Suppose the analyst has performed N measurements of the response quantity at time instants {tj , j = 1, ... N}. These measurements are represented by N random variables Y˜j :

• No model can ever represent the full complexity of a natural phenomenon. A model is rather a mathematical abstraction of reality that reveals predictive at a given scale of time and space description;

• These measurements are consistent with the model output corresponding to the (unknown) realization of the input parameters (possibly including the realization of the model error if any) :

Y˜j = y˜j + εj

εj ∼ N (0, σj2)

(8)

where the random variables {εj , j = 1, ... N} are independent zero-mean normally distributed random variables with variance σj2 and where {˜ yj , j = 1, ... N} are the observed values.

y˜j = M(x0 , tj )

• There is no perfect measurement device and there will never be. Thus neither the input parameter corresponding to the particular system S0 nor the monitored response quantity can be exactly measured.

j = 1, ... N

(9)

• As x0 is not known, the following measurement events {Hj = 0} are introduced, where: Hj = Y˜j −M(X, tj )

However the following quite general assumptions shall be made:

j = 1, ... N (10)

3.3 CONDITIONAL PROBABILITY OF FAILURE

• The physical model may be complemented with a model error, which may be e.g. additive or multiplicative. This corresponds to introducing an additional random variable. Without any loss of generality, it can be considered that this random variable is including in the set X, and thus in the model itself.

The experimental measurements constitute a source of information that may be introduced in the framework of reliability analysis in order to get better predictions. Practically speaking, the probability of failure of a mechanical system may be updated to the conditional probability of failure, when “conditional” should be understood with respect to the measurement events in Eq.(10). The computation of conditional probabilities of failure is treated in details by Ditlevsen and Madsen (1996, Chap. 13). The main results are reported in the sequel. In the context of first order reliability analysis, the updated reliability index β upd (t) corresponding to the conditional probability P (g(X, t) ≤ 0 | H1 = 0 ∩ . . . ∩ HN = 0) is given by:

• The result of the process of measurement is modelled by a random variable, whose mean value is the measured value and whose standard deviation is related to the accuracy of the device. Usually this variable is supposed to be Gaussian distributed.

3.2 MODELLING THE MEASUREMENT DATA

β0 (t) − z T · R · β β upd (t) = q 1 − (z T · R · z)2

The following notation is introduced: 3

(11)

where β0 (t) is the original reliability index associated to {g(X, t) ≤ 0}, β is the vector of the reliability indices associated to the events {{Hj ≤ 0}, j = 1, ... N}, R is the matrix of generic term Rij = αi · αj and z is a vector of generic term zj = α0 · αj . In the latter equations, the αvectors correspond to the usual unit normal vector to the limit state surface at the design point, the subscript “0” referring to the reliability problem {g(X, t) ≤ 0}, and the subscript “j” referring to the problems {{Hj ≤ 0}, j = 1, ... N}.

ted to shrinkage and creep. The models used in the sequel are based on the French Standard B.P.E.L (1991) and modifications done by Granger (1995). A similar study was presented in Heinfling et al. (2005).

5.1 MODEL FOR CREEP DEFORMATIONS The total strain tensor ε can be decomposed into the elastic, creep and shrinkage components: ε(t, td , tl ) = εel (t) + εas (t, td ) + εds (t, td )

4. COMPUTATION OF UPDATED FRACTILES

+ εbc (t, tl ) + εdc (t, td , tl ) where:

In the above section, it was shown how to compute an updated reliability index (and associated probability of failure) as soon as measurements of the response quantity of interest are available. Moreover, it was shown in Section 2 how fractiles of the response quantity may be evaluated by an inverse FORM algorithm. In this section, both methods are put together in order to compute updated fractiles of the response, say yαupd(t). The “updated” counterpart of Eq.(3) is: Find yαupd(t) :

Pfupd (yαupd(t), t) = α

• td (resp. tl ) denotes the time when drying starts (resp. the time of loading, i.e. cable tensioning in the present case); • εel (t) is the elastic strain; • εas (t, td ) is the autogeneous shrinkage, corresponding to the shrinkage of concrete when insulated from humidity changes; • εds (t, td ) is the drying shrinkage;

(12)

• εbc (t, tl ) is the basic creep corresponding to the creep of concrete when insulated from humidity changes;

This problem may be solved by adapting the inverse FORM algorithm. This corresponds to solve the “updated” counterpart of Eqs.(6), Eq.(7):

• εdc (t, td , tl ) is the drying creep.

Find yαupd(t) : PF ORM g(X, yαupd(t)) ≤ 0 | H1 = 0 ∩ . . . ∩ HN = 0) = α (13)

The following models taken are used for each component. The elastic strains are related to the stress tensor σ by Hooke’s law:

The original inverse FORM algorithm by Der Kiureghian et al. (1994) is taken as is, except that the current reliability index β (k) at iteration k is replaced by Eq.(11). Note that matrix R and does not change from one iteration to the next (it may computed and stored once and for all), in contrary to vector z.

5.

(14)

εel =

1 + ν el ν el σ− (tr σ)1 Ei Ei

(15)

where Ei is the elastic Young’s modulus (measured at t = tl ) and ν el is the Poisson’s ratio. The autogeneous and drying shrinkage are modelled by (time unit is the day in the sequel):

APPLICATION EXAMPLE

εas (t, td ) = εas ∞

In this application example, we consider the current part of the (cylindrical) containment vessel submit-

εds (t, td ) = εds ∞ 4

t − td 1 (16) 50 + t − td 100 − RH t − td (17) 1 2 /4 + t − t 50 45Rm d

ds In these equations, εas ∞ (resp. ε∞ ) is the asymptotic autogeneous shrinkage (resp. the asymptotic drying shrinkage), RH is the relative humidity in %, Rm is the drying radius (half of the containment wall thickness, in cm) and 1 is the unit tensor, meaning that these strains are isotropic. The basic creep is modelled by:   νc 1 + νc bc ε (t, tl ) = 3500 σ − (tr σ)1 Ei Ei   √  2.04 t − tl √ 0.1 + (tl − td )0.2 22.4 + t − tl (18)

5.2 RANDOM VARIABLES AND MEASUREMENT DATA

In a prestressed concrete containment vessel, the stress tensor in concrete may be regarded as bi-axial in the current zone, i.e. having a vertical compo0 nent σzz = 9.3 MPa and an orthoradial component 0 σθθ = 13.3 MPa. The drying radius, which is equal to half of the wall thickness, is 0.6 m. The cable tensioning is supposed to occur two years after the casting (tl − td = 2 years). Due to the presence of reinforcing bars and prestressed cables, the above equations for creep and shrinkage (initially obtained for unreinforced concrete) are corrected where ν c is the so-called creep Poisson’s ratio. The by a multiplicative factor λ = 0.82 obtained from drying creep is modelled by: the design code and experimental results. The other parameters are supposed to be random. The related
tr σ/2 ds εdc (t, td , tl ) = 3200 ε (t, td ) − εds (tl , td ) 1 data is given in Table 1. No correlation was considEi (19) ered between the input random variables.

Table 1: Probabilistic input data Parameter Concrete Young’s modulus Poisson’s ratio Creep Poisson’s ratio Relative humidity Maximal autogeneous shrinkage strain Maximal drying shrinkage strain † coefficient of variation.

Notation Ei ν el νc RH εas ∞ εds ∞

Type of distribution Lognormal Truncated normal [0,0.5] Truncated normal [0,0.5] Truncated normal [0,100%] Lognormal Lognormal

5

Mean 33,700 MPa 0.2 0.2 40 % 90.10−6 526.10−6

COV † 7.4 % 50 % 50 % 20 % 10 % 10 %

1600 a priori 2.5% a priori 50% a priori 97.5% measurements

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Figure 1: Prediction of the total orthoradial strain εzz and measurements

6

6.

A set of ten values for the total orthoradial strain are available. They have been obtained every 150 days between 1,150 and 2,500 days after the concrete drying process has started. Each strain measure has a standard deviation of 15.10−6, meaning that the measured value is supposed to lie within a range of ±30.10−6 at a confidence level of 95%.

CONCLUSION

The paper presents the use of an inverse FORM algorithm to efficiently compute fractiles of a response quantity of a model. When this model is depending on time, and when measurements of the output quantity are available, the fractiles of the latter may be updated using a slight modification of the algorithm. The approach is validated in the context of the prediction of long term strains in concrete containment vessels. It appears that a rather small number of measures is sufficient to get an accurate a posteriori model.

5.3 A PRIORI AND A POSTERIORI RESULTS Figure 1(a) presents the a priori 95 %-confidence interval predicted from the model (Eqs.(14)-(19)). The measured values are also reported together with their 95% confidence interval. It is obvious from this figure that the original model underestimates the total strain compared to the measurements. Figure 1(b) presents the a posteriori 95 %confidence interval predicted from the updating scheme presented above. Again, the measured values have been reported on the same figure. It clearly appears that the median updated model is close to the measurements and that the variance of the prediction (viewed here as the bandwith between the curves) has decreased. In order to judge the accuracy of the method, the a priori and updated models are used to predict the total orthoradial strain at t = 3, 968 days, for which a measured value (which is of course not introduced in the updating scheme) is available. Moreover, the influence of the size of the measures data set on the accuracy of the updated prediction is studied : Scheme #1 considers 3 measures at 1,503, 2,012 and 2,501 days. Scheme #2 considers 5 measures at 1,303, 1,601, 1,900, 2,201 and 2,501 days. Scheme #3 corresponds to the 10 measures mentioned above. Results are reported in Table 2. If we consider the median value, the a priori prediction is false by almost 37 %, whereas the updated value is close within 1 % to the measured value. Moreover, there is only a slight difference between the three updating schemes. This means that a important amount of information is contained in each measure and taken advantage of, even when e.g. one measure every two years is available.

ACKNOWKEDGEMENT The first author would like to thank Dr Yann Le Pape (Electricté de France, R&D Division) and Alexis Courtois (Electricté de France, Basic Engineering) from inspiring discussions about this work and part of the data.

References Benboudjema, F.: 2002, Modélisation des déformations différées du béton sous sollicitations biaxiales. Application aux enceintes de confinement de bâtiments réacteurs des centrales nucléaires, PhD thesis, Université de Marne-laVallée. B.P.E.L: 1991, Béton précontraint aux étatslimites, Eyrolles. Der Kiureghian, A., Zhang, Y. and Li, C.: 1994, Inverse reliability problem, J. Eng. Mech. 120, 1154–1159. Ditlevsen, O. and Madsen, H.: 1996, Structural reliability methods, J. Wiley and Sons, Chichester. Granger, L.: 1995, Comportement différé du béton dans les enceintes de centrales nucléaires, PhD thesis, Ecole Nationale des Ponts et Chaussées. 7

Heinfling, G., Courtois, A. and Viallet, E.: 2005, Reliability-based approach to predict the longtime behaviour of prestressed concrete containment vessels, in G. Pijaudier-Cabot, B. Gérard and P. Acker (eds), Proc. Concreep-7, "Creep, shrinkage and durability of concrete and con-

crete structures", pp. 323–328. Le Pape, Y., Benboudjema, F. and Meftah, F.: 2003, Numerical analysis of the delayed behaviour of French NPP double containments, in D. Owens and B. Suarez (eds), Proc. Computational Plasticity VII, Fundamentals and Applications.

Table 2: Comparison of updating data Orthoradial strain (10−6 ) Measured A priori Updated (Sch. #1) Updated (Sch. #2) Updated (Sch. #3)

8

ε2.5%

ε50% ε97.5% 1116 ± 30 564 699 863 1056 1109 1169 1064 1116 1175 1064 1109 1160