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Probabilistic description of concrete’s Thermo-Hydro-Mechanical behavior in Nuclear Containment Buildings – Application to the VeRCoRs mock-up
Salome-Meca Users’ Day 2018
20th March2018 – EDF Lab Paris-Saclay, France
http://fondation-grenoble-inp.fr/chaire/pereniti/
D. BOUHJITI, S. MICHEL-PONNELLE, B. MASSON, F. DUFOUR, J. BAROTH, M. BRIFFAUT
Salome-Meca Users’ Day 2018
VeRCoRs mock-up, EDF LAB Les Renardières
CONTENT
• Analysis framework
• Uncertainties and variability of concrete’s THM properties
• Tools to account for uncertainties in FE analysis of NCBs
• Examples – VeRCoRs gusset: Probabilistic modeling of early-age cracking Probabilistic modeling of saturation rate
• Conclusions
20th March2018 – EDF Lab Paris-Saclay, France
1 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o THM uncertainties
o Modeling tools & applications
o Conclusion
● Framework
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
The double-wall containment buildings (1300-1450 MWe NPP reactors)
[22 reactors]
• A reinforced and pre-stressed concrete inner wall tightness
• an outer reinforced concrete wall protection against external effects
Ambitions for EDF: • to demonstrate the good behavior of inner wall in situation of severe accident; • to have a better comprehension of the leakage and its evolution with the ageing of the structure
2 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o THM uncertainties
o Modeling tools & applications
o Conclusion
● Framework
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
How to answer such questions ?
Feed-back analysis and
NDT
Experimental studies
Numerical tools
VERCORS (1/3 scale)
3 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o THM uncertainties
o Modeling tools & applications
o Conclusion
● Framework
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
How to answer such questions ?
Engineering tools
Feed-back analysis and
NDT
Experimental studies
Numerical tools
z
D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o Modeling tools & applications
● THM uncertainties
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
THM uncertainties
C-S-H molecular model 4 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Modeling tools & applications
o Conclusion
o Framework
● THM uncertainties
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Cement, aggregates, voids
Cement, capillary voids
CSH, gel pores
Concrete = heterogeneous + multiphasic material
Variable mechanical + transfer properties
Time t
Space 𝑥𝑦𝑧
Mo
lecu
lar
scal
e
Mes
o s
cale
M
acro
sca
le
Homogeneous Equivalent Material (HEM)
[Ve
RC
oR
s co
ncr
ete
]
Variable properties + Variable THM boundaries
5 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Modeling tools & applications
o Conclusion
o Framework
● THM uncertainties
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Variable properties + Variable THM boundaries
[Inner temperature variation in a real NCB]
[Design of a full scale NCB]
[Prestressing losses variation in VeRCoRs]
6 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Modeling tools & applications
o Conclusion
o Framework
● THM uncertainties
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
There is a need for probabilistic coupling in order to account for uncertainties whilst modeling concrete’s behavior in NCBs ?
Variable structural response
[Early age cracks based on visual inspections]
• Variable spacing values • Variable crack openings at early age • Variable crack openings during pressurization
• Variable strain values • Variable ageing kinetic
• Variable thermal behavior • Variable hydration kinetic
[VeRCoRs gusset thermo-hydration]
[VeRCoRs gusset vertical strain]
D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Modeling tools and applications (Deterministic modeling)
7 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Deterministic modeling
DEFI_MATERIAU THER_HYDR
U4.43.01
THER_NON_LINE U4.54.02
Thermo-hydration calculations
Drying calculations
Mechanical calculations
Leakage calculations
Temperature Hydration rate
Water content Saturation rate
Relative Humidity
DEFI_MATERIAU SECH_GRANGER
U4.43.01
THER_NON_LINE U4.54.02
DEFI_MATERIAU MFRONT* U2.10.02
U4.43.01
STAT_NON_LINE U4.51.03
THM-L scheme
Strain/Stress Damage
Crack opening
DEFI_MATERIAU SECH_NAPPE*
U4.43.01
THER_NON_LINE U4.51.03
Pressure Leakage
*: Python-based developments
[Bouhjiti et al. 2017]
8 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Deterministic modeling Expected results
(c) (d)
(a) (b)
Crack initiation Crack initiation
(e) (f) (g) (h)
Test 0 Test 0 Test 1 Test 1
(c) (d)
(a) (b)
Crack initiation Crack initiation
(e) (f) (g) (h)
Test 0 Test 0 Test 1 Test 1
[Bouhjiti et al. 2018]
D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Modeling tools and applications (Spatial variability and random fields)
9 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
1/Stochastic modeling of cracking
The heterogeneousness of concrete and its effect on the spatial scattering of the mechanical properties is modeled by spatially correlated Random Fields.
Spatial variability of parameters
Theory: Karhunen-Loève decomposition
𝐱 = 𝛍 + λi 𝛗𝐢 𝐱 𝛏𝐢
N
i=1
Centered reduced normal vector of random realizations
Spatially correlated random realizations
Vector of mean values (spatially constant or not)
Covariance matrix: C x, 𝑦 = σ2 ρac x, 𝑦 , lac
Autocorrelation function
[Ghanem et al. 1991]
10 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
1/Stochastic modeling of cracking Spatially Discretized Random Field - MidPoint
A given mesh Considered spatial discretization for RF generation
Geometrical boundaries
• Wider domain to prevent boudary effects associated with correlation
• Discretization: 4 elements per lac (3 at least for representativeness)
Linear Projection
Random values assigned per element
• Non-intrusive • Within a given element all Gauss
points have the same random value !
𝐥𝐚𝐜 𝐥𝐚𝐜
𝐥𝐚𝐜 𝐥𝐚𝐜
[Der Kiureghian A. 1988]
11 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
1/Stochastic modeling of cracking User-defined inputs - Coefficient of variation or variance - Fluctuation length (related to auto-correlated length) - Auto-correlation function - RF nature: Gaussian or Lognormal
Preprocessing Generate centroids table containing Elements’ Ids & coordinates coor.libr TAB=POST_ELEM() IMPR_TABLE(TABLE=TAB, FORMAT=’ASTER’, NOM_PARA=(‘Lieu’, ‘CDG_X’, ‘CDG_Y’, ‘CDG_Z’)
Mesh.med
Processing 0/ In file.comm call libraries: Import math;Import random;Import numpy as np; Import time;From scipy.interpolate import griddata 1/ Read coor.libr and define the extended domain for RF generation 2/ Generate the centered and reduced random values on the extended domain 3/ Define the covariance matrix and derive its Eigen values and vectors 4/ Calculate the autoccoraletd RF values on the extended domain 5/ Linearly project the RF on the considered mesh.rmed using coor.libr
12 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
1/Stochastic modeling of cracking
Random cracking patterns at early-age
Spatial scattering of Young’s modulus property !
[Bouhjiti et al. 2018]
D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Uncertainties propagation using alternative stochastic approaches to MCM (Non-intrusive strategies)
13 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
2/Stochastic modeling of drying Water content
The variation of concrete’s behavior at the structural scale is modeled using non-intrusive stochastic approaches: Quadrature-Polynomial Chaos method
Descriptive equation of drying: dSrdt
− div A e𝐁𝐂𝟎grad(Sr) = 0
B*C0 considered as a random input
Aiming at an explicit: Srmoy
= function B ∗ C0 ? MC at low cost • CDF of Sr
moy
• Pf = P(Srmoy
≤ Srlim)
After a limited number of model evaluations for a limited number of values
Carefully select the integration points to obtain a representative response surface
14 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Water content
What B*C0 value ? Quadrature method
Hyp: B*C0 follows a Lognormal distribution
𝑵𝒑𝒈 ↗ : Computational cost ↗ Accuracy↗
What projection basis? Hermite Polynomials
He−1 = He0 = 1
Hen≥1 x =x Hen−1 x − n Hen−2 x
n!
Orthonormal functions with respect to Gaussian distribution
Hnor =Hen𝑛! 𝑛≥1
= 1, 𝑥,𝑥2 − 1
2,𝑥3 − 3𝑥
6,…
𝐒𝐫,𝐧=𝟑𝐦𝐨𝐲
𝛇, 𝐭 = 𝐚𝟎 𝐭 + 𝐚𝟏 𝐭 𝛇 + 𝐚𝟐 𝐭𝛇𝟐 − 𝟏
𝟐+ 𝐚𝟑 𝐭
𝛇𝟑 − 𝟑𝛇
𝟔
𝜁 =log 𝐵∗𝐶0 −𝜇𝐵𝐶0
𝜎𝐵𝐶0 gaussian
[Sudret et al. 2000]
2/Stochastic modeling of drying
[Baldeweck 1999]
15 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Water content
Exact Response Surface
𝑁𝑝𝑔 ≥ 4 ‼
𝑅2 ≥ 99,96 % 1 − 𝑄2 ≤ 3. 10−3
PC-based Response surface
2/Stochastic modeling of drying
16 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
Water content
𝐒𝐫,𝐧=𝟑𝐦𝐨𝐲
𝛇, 𝐭 = 𝐚𝟎 𝐭 + 𝐚𝟏 𝐭 𝛇 + 𝐚𝟐 𝐭𝛇𝟐 − 𝟏
𝟐+ 𝐚𝟑 𝐭
𝛇𝟑 − 𝟑𝛇
𝟔
MC
Metamodel (based on 4 calculations)
Accessible quantities • Mean • Statistical moments • Failure probabilities
Drying
2/Stochastic modeling of drying
17 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Conclusion
o Framework
o THM uncertainties
● Modeling tools & applications
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
3/Stochastic modeling of leakage rates
18 D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Framework
o THM uncertainties
o Modeling tools & applications
● Conclusion
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
• Non-intrusive methods used to: Generate RFs representative of the spatial scattering of a given material property Stochastically describe concrete THM-L behavior (early age + delayed ageing)
• Tools/Methodology used and validated based on the VeRCoRs mock-up
Thermo-hydration calculations
Drying calculations
Mechanical calculations
Leakage calculations
0,5 hr – 225 steps
0,5 hr –179 steps
2 dy –634 steps
0,5 hr – 160 steps
Number of calls to describe probabilistically the leakage rate: 768 (using 30 14-core nodes //)
Parametric analysis [U2.08.07] – Uncertainties propagation [OpenTurns]
Salome-Meca Users’ Day 2018
Contact: [email protected] / [email protected]
Thank you for listening!
‘’ The probabilistic modeling of concrete’s behavior allows better assessment of NCBs ageing and aims at improving the prediction of their lifespan and the planning of
maintenance and repair operations ‘’
D. BOUHJITI et al. – Probabilistic description of concrete’s THM behavior in NCBs
o Framework
o THM uncertainties
o Modeling tools & applications
o Conclusion
Salome-Meca Users’ Day 2018 20th March2018 – EDF Lab Paris-Saclay, France
• References:
o VeRCoRs project: Corbin M., Garcia M. Benchmark VERCORS report 2015 (https://www.conference-service.com/EDF-VeRCoRs-Benchmark-2018/welcome.cgi)
o Bouhjiti D. E-M., Baroth J., Dufour F., Masson B. Sensitivity analysis of cracking in large reinforced concrete structures: Case of the VeRCoRs mock-up. CFM 2017
o Bouhjiti D. E-M., Boucher M., Briffaut M., Dufour F., Baroth J., Masson B. Accounting for realistic Thermo-Hydro-Mechanical Boundary Conditions whilst modeling the ageing of concrete in Nuclear Containment Buildings: Model validation and sensitivity analysis. Eng. Struct 2018
o Bouhjiti D. E-M., Baroth J., Dufour F., Masson B. Prediction of air permeability in large RC structures using a stochastic FE THM modeling strategy. Euro-C 2018
o Sudret B., DER KIUREGHIAN A. Stochastic Finite Element Methods and Reliability. A State-of-the-Art Report. Report UCB/SEMM-2000/08.
o Der Kiureghian A., Ke. Second order reliability approximations. Eng. Mech. 1988. o Baldeweck H. Methodes aux elements finis stochastiques - Application a la geotechnique. These
de doctorat 1999. o …