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Applications of Statistics and Probability in Civil Engineering – Kanda, Takada & Furuta (eds) © 2007 Taylor & Francis Group, London, ISBN 978-0-415-45211-3
Bayesian updating of the long-term creep strains in concrete containment vessels using a non intrusive stochastic finite element method Marc Berveiller, Yann Le Pape & Bruno Sudret Electricité de France, R&D Division, Site des Renardières, Moret-sur-Loing, France
Frédéric Perrin Phimeca Engineering S.A., Aubière, France LaMI, Université Blaise Pascal et Institut Français de Mécanique Avancée, Campus des Cézeaux, 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, containment 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. Two different models for predicting the delayed strains are compared. Keywords:
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Durability analysis, concrete creep, Bayesian updating, inverse reliability problem, FORM.
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INTRODUCTION
Electricité de France currently exploits a fleet of 58 pressurised water reactors. Their containment is ensured by two concrete vessels. The inner containment is made of reinforced prestressed concrete. 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. The prediction of short and medium terms is rather accurate but there are some difficulty to predict long-term strains. However, containment vessels are monitored all along their service life. This provides information on the real behaviour of the structure. The aim of this paper is to incorporate monitoring data for the long-term prediction of the strains, and to compute a confidence interval of the delayed strains. Two different models of prediction of creep strains are presented and compared in this study. The first one is analytical and the second one is implemented in a finite element code.
PRESENTATION OF THE UPDATING METHOD
In this section, we present a method for predicting the confidence interval of a time-dependant model response and how to take into account monitoring data to predict an update confidence interval [SBPP06, PSP06]. 2.1 Problem statement Let us consider a time-dependent model response S(X , t) and Sα (t) the target α-fractile. In real life problems, the parameters of such model are not well known and may be modelled as random vector X . In practice, one is interested to have ranges of variation of the response quantity, e.g. a 95% confidence interval. This means that 2.5% and 97.5% fractiles of the model response are of interest. Denoting by:
where M (X , t) = S(X , t) − y. The problem of finding α-fractiles of S(X , t) is equivalent to solve for Sα (t) the
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2.3 Presentation of the non intrusive regression method
following equation:
As the finite element model is quite large, a response surface model has to be used to minimise the time cost of computation. The expansion onto the polynomial chaos [GS91, Ber05] is relevant for having a stochastic surface response. The non intrusive regression method only requires deterministic finite element analyses and an analytical post-processing of the results. The non intrusive method presented in this communication is based on a least square minimisation between the exact solution and the solution approximated using the polynomial chaos [Isu99, Ber05]. First the input random variables (gathered in a random vector X whose joint PDF is prescribed) are transformed into a standard normal vector ξ. If these M variables are independent, the one-to-one mapping reads:
Let us consider [χi ]1in , n measurements of the response quantity for different instants t = ti , 1 i n. Theses n observations are associated to the measurement events {Hi = 0}:
The updated failure probability is given by the conditional probability:
[Mad85] gives an expression for the updated reliability index in the context of FORM analysis, function of the measurements:
where is the standard normal CDF and {Fi (Xi ), i = 1, . . . , M } are the marginal CDF of the Xi ’s. Suppose now that we want to approximate a response quantity S by the truncated series expansion:
where {βi } denotes the reliability index vector associated to events {Hi 0}, [ρHi Hj is the correlation matrix between the margin Hi and Hj and {ρM (t)Hi } is the correlation vector between margin Hi and M (t). The update failure probability is obtained by:
where {j , j = 0, . . . , P − 1} are P multidimensional Hermite polynomials of ξ whose degree is less or equal than p. Note that the following relationship holds:
where denotes the standard normal cumulative distribution function. 2.2
Resolution of the problem
Let us denote by {ξ (k) , k = 1, · · · , n} n outcomes of the standard normal random vector ξ. For each outcome ξ (k) , the isoprobabilistic transform yields a vector of input random variables X (k) (Eq.(7)). Using the finite element model, the response vector S (k) can be computed. Let us denote by {s(k),i , i = 1, . . . , Nddl } its components. Using Eq.(8) for the i-th component, one gets:
The computation of 2.5% and 97.5% fractiles is equivalent to solve Eq.(2) for each time instant t and α = 0.025 (resp. α = 0.975). This problem can be considered as a root-finding problem. Thus the bisection or the Newton-Raphson methods can be used. However, these methods require numerous evaluations of the function, which may be computationally expensive if the cost of solving each single reliability problem is large. Moreover, if a sampling computation scheme such as Monte Carlo simulation is used for this purpose, some instabilities may appear in the iterative solving scheme due to asmpling. In order to solve the inverse reliability problem efficiently, [DZL94] proposed an algorithm based on the First Order Reliability Method (FORM). This method is used in our study. The computation of the updated fractiles is carried out using a modified inverse FORM algorithm thaht incorporates the updating formula (5). Details can be found in [PSP06, PSP07].
where (sji ) are coefficients to be computed. The response regression method consists in finding for each degree of freedom i = 1, . . . , N the set of coefficients that minimises the difference:
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εds (t, td ) is the drying shrinkage; εbc (t, tl ) is the basic creep corresponding to the creep of concrete when insulated from humidity changes; • εdc (t, td , tl ) is the drying creep.
These coefficients are solution of the following linear system:
• •
The following models taken are used for each component. The elastic strains are related to the stress tensor σ by Hooke’s law:
where Ei is the elastic Young’s modulus (measured at t = tl ) and νel is the Poisson’s ratio. The autogeneous and drying shrinkage are modelled by (time unit is the day in the sequel):
Note that the P × P matrix on the left hand may be evaluated once and for all. The crucial point in this approach is to properly select the regression points, i.e. the outcomes {ξ (k) , k = 1, . . . , n}. Note that n ≥ P is required so that a solution of (12) exist. [Isu99, Ber05] choose for each input variable the (p + 1) roots of the (p + 1)-th order Hermite polynomial, and then built (p + 1)M vectors of length M using all possible combinations. Then they select n outcomes {ξ (k) , k = 1, . . . , n} out of these (p + 1)M possible combinations. [Sud06] presents a method to obtain the minimum number of points that makes the matrix on the lefthand side invertible. When the finite element model provides a time dependent result, the regression is carried out at each time instant under consideration (the matrix in Eq.(12) being inverted only once). 3
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:
PRESENTATION OF CONSTITUTIVE MODELS where νc is the so-called creep Poisson’s ratio. The drying creep is modelled by:
3.1 Analytical model for creep strains This analytical model comes from the French standard [BPE93], slightly modified by [Gra96]. The total strain tensor ε can be decomposed into the elastic, creep and shrinkage components:
A similar study was presented in [HCV05]. 3.2 Presentation of the finite element model for the delayed strain of concrete in containment vessel
where: • 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;
The constitutive law, implemented in the EDF Finite Element Code, Code_Aster1 , is the result of the previous works by Granger [Gra96] and Benboudjema 1
This code can be downloaded for free at http://www.codeaster.org
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[Ben02]. The total strain is a function of the temperature T , the hydration β, the relative humidity h and the macroscopic stress σ. The conventional strain rate decomposition reads:
GIBI FECIT
with εe : elastic strain, dependent of σ; εth : thermal dilation/contraction; εas : autogenous shrinkage, being a function of the hydration degree β ∈ [0; 1]; εds : drying shrinkage, dependent of the drying process controlled by the evolution of h; εbc : basic creep, naturally being a function of σ, but also, of the hydrous state of the material – basic creep of pre-dried specimen exhibits some dependency with the equilibrated relative humidity–; εdc : drying creep. The model proposed by Bažant and Chern is assumed [BC85]. In the sealed specimens (autogenous shrinkage and basic creep), the stress state remains homogeneous. Therefore, the computation is done analytically. Conversely, the drying specimens (loss of weight, drying shrinkage and creep) exhibit a humidity gradient responsible of an heterogeneous stress state. Their analysis requires a numerical simulation performed with Code_Aster. The calibration of the parameters is performed on the basis of the previous experimental results and follows the procedure:
Figure 1. Mesh of the Representative Structural Volume.
far enough from the equipment hatch and the tendon buttresses, so that homogeneous strain states may be applied in the directions of the prestressing. The thickness of the wall is discretized to account for the gradient of humidity. Single elements cover the vertical and the tangential direction due to the total strain homogeneity. The prestressing is introduced by external forces. Iterations on the effective applied prestress are computed over the non linear calculation to account for the loss of prestress induced by the concrete creep. The temperature, humidity and prestress evolutions follow the scenario: 15◦ C-60%RH inside and outside during building period, 35◦ C-45%RH inside the containment building when the reactor is in-service. The prestressing is applied graduously following the building stages. Figure 1 presents the mesh used in the sequel.
1. the drying process is modeled through a non linear thermal analogy, i.e. the water diffusion coefficient D is a function of the water content C, i.e.: D(C) = a · exp (b.C). With the back-analysis of the time-evolution of the specimen loss of weight, the parameters a and b are retrieved. 2. The autogenous shrinkage is fitted on a simple hyperbolic law: εas = Kas β1, with β = t/(t + t1/2 ). 3. The basic creep constitutive law assumes the full uncoupling of the spherical and the deviatoric strains. It requires the knowledge of the strain tensor derived from the experimental test. 4. The drying shrinkage is assumed to be propor˙ tional to the loss of water content: ε˙ ds = −Kds C1. The influence of the auto-induced creep due to the stress gradient was found insignificant while calibrating Kds . 5. The drying creep is finally modeled once all the other parameters are known. The intrinsic drying ˙ creep constitutive equation reads ε˙ dc = |h|σ/η dc .
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RESULTS Random variables and measurement data
Tables 1 and 2 present the different random input parameters used in both models. No correlation was considered between the input random variables. Note that the coefficient of variation in Table 2 are chosen by expert judgment. A set of 38 values for the total orthoradial strain are available. They have been obtained between 1,598 and 4,753 days after the concrete drying process has started. Each strain measure has a standard deviation of 25.10−6 , meaning that the measured value is supposed to lie within a range of ±50.10−6 at a confidence level of 95%. 4.2 Results for the analytical model
On the basis of the previously calibrated parameters, some numerical computations [LPTMP05] are performed on a so-called Representative Structural Volume (RSV). This RSV is located approximately at an equal distance from the dome and the raft and
We choose to expand the response of the model onto an 2-order polynomial chaos. As there are 6 random variables, we have 28 coefficients to compute which requires 42 finite element analysis. Figure 2 compares a priori and a posteriori results for the analytical and
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Table 1. Analytical model: Probabilistic input data. Parameter
Notation
Type of distribution
Mean
COV †
Concrete Young’s modulus Poisson’s ratio Creep Poisson’s ratio Relative humidity Maximal autogeneous shrinkage strain Maximal drying shrinkage strain
Ei νel νc RH εas ∞ εds ∞
Lognormal Beta [0,0.5] Beta [0,0.5] Beta [0,100%] Lognormal Lognormal
33,700 MPa 0.2 0.2 40% 90.10−6 526.10−6
7.4% 50% 50% 20% 10% 10%
† coefficient of variation. Table 2.
Finite Element model: Probabilistic input data.
Parameter
Notation
Type of distribution
Mean
COV †
Concrete Young’s modulus Dessication creep parameter Hydrates stiffness at mesoscopical scale Hydrates stiffness at microscopical scale Stiffness associated to water capacity to transmit charges Adsorb water viscosity Free water viscosity
Ei ETA_FD K_RS K_IS K_RD
Lognormal Lognormal Lognormal Lognormal Lognormal
33,700 MPa 5.80E+09 Pa.s 6,00E+10 Pa 3,00E+10 Pa 3,402E+10 Pa
7.4% 20% 20% 20% 20%
ETA_RD ETA_ID
Lognormal Lognormal
4,082E+17 Pa.s 2,329E+18 Pa.s
20% 20%
† coefficient of variation.
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order 2 a priori 2,5% order 2 a priori 50% order 2 a priori 97,5% order 2 a posteriori 2,5% order 2 a posteriori 50% order 2 a posteriori 97,5% exact a priori 2,5% exact a priori 50% exact a priori 97,5% exact a posteriori 2,5% exact a posteriori 50% exact a posteriori 97,5% monitoring data
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Figure 2. Analytical model : results for the approximated model with an order 2 polynomial chaos and for the exact model.
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4500
4000
3500
3000 Delayed strain
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1500 FE a priori 2,5% FE a priori 50% FE a priori 97,5% FE a posteriori 2,5% FE a posteriori 50% FE a posteriori 97,5% monitoring data
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Figure 3. Finite element model : results for the approximated model with an order 2 polynomial chaos and for the exact model.
and the 95%-confidence interval at 60 years for the a posteriori model is equal to 364.10−6 . The main difference between a priori and a posteriori results is the reduction of the confidence interval. The a posteriori model can be used to reduce the conservatism of the method compared to the a priori model.
approximate models onto the polynomial chaos. We can see that the approximate model is close to the exact model. Thus a 2-order polynomial chaos seems to have the best ratio accuracy-efficiency. It will be used for the finite element model. The a priori model does not fit well the monitoring data, and at time 60 years there is a 500.10−6 confidence interval around the wide median. The a posteriori is close to the monitoring data and have a 120.10−6 confidence interval around the median. We can conclude that the a posteriori can be used for predict short-term delayed strains and to reduce the conservatism of the method compared to the a priori model. 4.3
4.4 Comparison of the analytical model and the finite element model Figure 4 compares analytical and finite element results. As mentionned above, a priori results are better with the finite element model than the analytical one. Both a posteriori models are accurate for shortterm prediction compared to the monitoring data, but there is a big difference in the median curve for delayed strain at 60 years between the analytical (2319.10−6 ) and the finite element model (3489.10−6 ). The 95%confidence interval at 60 years is more important for the finite element model (364.10−6 ) than for the analytical model (120.10−6 ). This is due to the difference between coefficients of variation of the input parameters for both models. As input parameters of both models are different, we can not take the same coefficient of variation and compare directly different results.
Results for the finite element model
We choose to expand the response of the model onto a 2-order polynomial chaos. As there are 7 random variables, we have 36 coefficients to compute which requires 56 finite element analysis. Figure 3 presents a priori and a posteriori results obtained with the finite element model. The a priori median is close to the monitoring data. The 95%-confidence interval at 60 years for the a priori model is equal to 1103.10−6 . The a posteriori median fits well the monitoring data
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6000 An. a priori 2,5% An. a priori 50% An. a priori 97,5% An. a posteriori 2,5% An. a posteriori 50% An. a posteriori 97,5% FE a priori 2,5% FE a priori 50% FE a priori 97,5% FE a posteriori 2,5% FE a posteriori 50% FE a posteriori 97,5% mesures
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0
0
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Figure 4. Comparison of results for analytical and finite element models.
and to give a confidence interval on the delayed strains of concrete.
If a choice has to be made between the analytical model and the finite element model, the second one will be preferred because a priori results are better even if the confidence interval for the analytical model is smaller than for this model. The determination of parameters of input random variables is important because it influences the confidence interval and the 50%-fractile of the response.
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REFERENCES [BC85] Z.P. Bažant and J.C. Chern. Concrete creep at variable humidity: constitutive law and mechanism. Materials and Structures, 18(103):1–20, 1985. [Ben02] F. Benboudjema. 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 la Vallée, 2002. [Ber05] M. Berveiller. Eléments finis stochastiques: approches intrusive et non intrusive pour des analyses de fiabilité. PhD thesis, Université Blaise Pascal – Clermont Ferrand, 2005. [BPE93] Règles BPEL 91, règles techniques de conception et de calcul des ouvrages et constructions en béton précontraint suivant la méthode des états-limites, 1993. [DZL94] A. Der Kiureghian, Y. Zhang, and C. Li. Inverse reliability problem. J. Eng.Mech., 120:1154–1159, 1994. [Gra96] L. Granger. Assessment of creep methodologies for predicting prestressing forces in nuclear power plant containments. Technical Report ENSIGC9604A, 1996. [GS91] R-G Ghanem and P-D Spanos. Stochastic finite elements – A spectral approach. Springer Verlag, 1991.
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. Two different models of prediction (an analytical one and a finite element one) are presented and compared in this study. It appears that the finite element model gives better a priori results. The method presented in this paper allows to update both models
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[HCV05] G. Heinfling, A. Courtois, and E. Viallet. Reliability-based approach to predict the long-term behaviour of prestressed concrete containment vessels. In G. Pijaudier-Cabot, B. Gérard, and P. Acker, editors, Proceedings of the seventh International Conference CONCREEP, Creep, Shrinkage and Durability of Concrete and Concrete Structures, pages 323–328, 2005. [Isu99] S.S. Isukapalli. Uncertainty Analysis of TransportTransformation Models. PhD thesis, The State University of New Jersey, 1999. [LPTMP05] Y. Le Pape, E. Toppani, and S. MichelPonnelle. Analysis of the delayed behaviour of NPP containment building. In G. Pijaudier-Cabot, B. Gérard, and P. Acker, editors, Proceedings of the seventh International Conference CONCREEP, Creep, Shrinkage and Durability of Concrete and Concrete Structures, pages 353–358, 2005. [Mad85] Madsen, H.O. Model updating in firstorder reliability theory with application to fatigue crack growth.
In 2nd International Workshop on Stochastic Methods in Structural Mechanics, Pavia, Italy, 1985. [PSP06] F. Perrin, B. Sudret, and M. Pendola. Bayesian model response updating using an efficient reliability method. Struct. Safe., 2006. submitted for publication. [PSP07] F. Perrin, B. Sudret, and M. Pendola. Comparison of markov chain monte carlo simulation and a form based approach for bayesian updating of mechanical models. In Proc. 10th Int. Conf. on Applications of Stat. and Prob. in Civil Engineering, ICASP10, Tokyo, 2007. [SBPP06] B. Sudret, M. Berveiller, F. Perrin, and M. Pendola. Bayesian updating of the long-term creep deformations in concrete containment vessels. In Proc. AsRANET 3, Glasgow, Scotland, 2006. [Sud06] B. Sudret. Global sensitivity analysis using polynomial chaos expansion. Rel. Eng. Sys. Safe, 2006. submitted for publication.
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