Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2023.40.0048 Acta Polytechnica CTU Proceedings 40:48–53, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague BENCHMARK VERCORS 2022: BLIND PREDICTION OF TIME-DEPENDENT BEHAVIOR OF CONCRETE CONTAINMENT BUILDING WITH LOW AND HIGH-FIDELITY MODELS Štěpán Krátký∗, Petr Havlásek Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: stepan.kratky@fsv.cvut.cz Abstract. The VERCORS program aims to acquire an extensive experimental dataset collected to provide a solid basis for numerical modeling of concrete containment buildings (CCBs). In order to cover the entire life-span of a real containment, the measurements are done on 3× smaller mock-up which leads to 9-fold acceleration of all processes related to drying. The goal for the participants of the third VERCORS benchmark was to predict the behaviour of the CCB based on standard laboratory measurements on VERCORS concrete. The previous paper [1] presented the calibration procedure of material models for moisture transport and time-dependent behavior of concrete and summarized the results obtained with a computationally efficient low-fidelity model (LFM). The present paper compares the responses of the LFM and a high-fidelity model (HFM) with a detailed geometry of the entire containment and presents a comparison with the experimental data collected over the last 8 years on the VERCORS mockup. Keywords: Concrete, creep, shrinkage, drying, cyclic temperature, modeling. 1. Introduction In order to prolong the lifespan of concrete contain- ment buildings of nuclear reactors by 50 %, detailed knowledge of their time-dependent behaviour is essen- tial. This implies the ability to accurately predict the evolution of various quantities over 60 years. Such a task can be accomplished by computational model- ing with advanced constitutive models. The main task of the first phase of the VERCORS 3rd benchmark was to predict the behavior of the concrete containment building (CCB) depicted on Figure 1, in particular the evolution of strain, displacement and relative humid- ity at prescribed locations. For a smoother evaluation of the benchmark and simpler comparison of the re- sults among the participants, the organizers provided a detailed finite element mesh of the CCB. This paper is organized as follows. The main results of the previous paper [1] are summarized first. Next, the low- and high-fidelity numerical models (LFM and HFM) are described and their features are presented. Finally, the responses of LFM and HFM are compared with the experimental response. 2. Previous outcomes 2.1. Laboratory experiments The provided dataset constitutes from the results of conventional short-term measurements (mean uniax- ial compressive strength fcm = 48.7 MPa, Young’s modulus E = 34.3 GPa, uniaxial tensile strength ft = 4.4 MPa) and the following long-term experi- ments: • Basic creep and autogenous shrinkage, • total creep and drying shrinkage at relative humid- ity of the environment henv = 0.5, • moisture loss, • porosity and aging sorption isotherm. Every experiment in this list was conducted under both room (20 °C) and elevated (40 °C) temperature. This data set was used for calibration of constitutive models. 2.2. Finite element modeling OOFEM [3] solver was used to run all numerical simulations. A weakly coupled computational ap- proach was adopted to model the influence of temper- ature and relative humidity on the structural response. Moreover, to further simplify the problem, no cross- coupling between the heat and moisture transport subproblems was assumed in the blind stage of the benchmark. This is partially due to the absence of relevant data. 2.3. Calibration of material models The constitutive models for heat and moisture trans- port (Bažant-Najjar model [4]) and time-dependent behavior of concrete (modified MPS model [5]) were calibrated using axis-symmetric models of laboratory specimens. The experimental data sets were provided by EDF (Électricité de France) at the beginning of the benchmark. Successful calibration of the consti- tutive models is demonstrated by excellent fits of the data and can be found in the final thesis of the first author [6]. 48 https://doi.org/10.14311/APP.2023.40.0048 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 40/2023 Benchmark VERCORS 2022: Response to Ambient Conditions Figure 1. Section of the CCB VERCORS mock- up [2], dimensions in [mm]. The calibration procedure is briefly outlined in the following sections and the identified values of mate- rial parameters for the Vercors concrete are listed in Tables 1 and 2. 2.3.1. Basic creep and autogenous shrinkage Under hygrally sealed conditions and at constant room temperature, the MPS model reduces to the basic creep compliance function of the B3 [7] model whose parameters q1–q4 were initially estimated from the composition of the concrete mixture. These parame- ters were slightly adjusted to obtain a more accurate fit of the experimental data. The measured strain in the creep experiment was compensated for the autogenous shrinkage measured on a companion specimen. Since the experiment began at the age of 90 days, the recorded (incremental) value of autogenous shrinkage was very small, ≈ 50 × 10−6. Parameter Value q1 9.0 × 10−6 MPa−1 q2 70.0 × 10−6 MPa−1 q3 25.0 × 10−6 MPa−1 q4 6.0 × 10−6 MPa−1 ksh 1.0 × 10−3 k3 10 kT m 6.5 Table 1. Identified parameters of the MPS material model. Parameter Value C1 28.2 mm2 · day−1 α0 0.055 hc 0.7 n 10 f 1.08 mm · day−1 Table 2. Identified parameters of the material model Bažant–Najjar for moisture diffusion. Autogenous shrinkage per se was not considered in the analysis. 2.3.2. Drying and drying shrinkage Parameters of the Bažant-Najjar model for moisture diffusion [4] were determined by hand fitting and were calibrated simultaneously with the parameter ksh of the MPS model which links the rates of axial drying shrinkage and relative humidity. The response of the models was checked against the measured evolution of drying shrinkage and moisture loss at room tem- perature. To back-calculate the moisture loss from the relative humidity, the moisture capacity was set to 130 kg/m3 which approximately corresponds to the slope of the measured desorption isotherm. 2.3.3. Drying creep Parameters for the drying creep, creep at elevated temperature, and transitional thermal creep were de- termined last. Drying creep is controlled chiefly via parameter k3. The experimental data on cylinders al- lowed to identify parameter kT m which is responsible for the creep rate at elevated temperature. How- ever, under temperature cycles this value would have caused overestimated compliance. Unfortunately, the experimental data set did not comprise sufficient infor- mation to allow for calibration of the last parameter which damps concrete creep at subsequent thermal cycles, kT c. 2.3.4. Heat transfer There were no relevant data for the identification of parameters related to heat conduction; therefore, typical values for concrete were adopted: heat ca- pacity c = 1000 J · kg−1 · K−1, heat conductivity k = 1.7 W·m−1·K−1, and surface factor a = 8 W·m−2·K−1. 49 Štěpán Krátký, Petr Havlásek Acta Polytechnica CTU Proceedings Figure 2. LFM – structural model. Figure 3. LFM – moisture transport model. 2.3.5. Damage Except for the pressure tests, the stress state in the CCB is chiefly biaxial compression in plane of the cylinder and dome. The out-of-plane stress is far below the strength limit. Since the benchmark results do not cover places with significant stress concentrations, the nonlinear behavior does not need to be considered. For this reason, concrete damage (neither compressive nor tensile) was not considered in the first stage of the benchmark. 2.4. Low-fidelity model The enormous computational demands of the high- fidelity model made it impossible to tune up the boundary conditions and the remaining parameters related to transient thermal creep, which could not be identified from the experimental data. Therefore, a computationally efficient numerical model was devel- oped to verify the response to the ambient conditions. The resulting low-fidelity model represents a small segment of the wall in the mid-height of the CCB to eliminate the interaction with more rigid parts of the structure (foundation slab and dome stiffener). The finite element meshes for the individual subprob- lems – structural analysis, moisture transport, and heat transfer – are shown in Figures 2, 3 and 4. The dimensions of the computational models are derived Figure 4. LFM – heat transfer model. 0 10 20 30 40 0 400 800 1200 1600 2000 Te m p e ra tu re , T [ °C ] Time, t [day] Int T2 Ext T2 Int T1 Ext T1 Figure 5. Ambient conditions – temperature pre- scribed on the inner and outer faces of the concrete wall. from the spacing of the prestressing tendons in the vertical and circumferential directions. 2.5. Results A comprehensive description and discussion of the results can be found in the previous paper [1]. The most significant discrepancy in the structural behavior was associated with the response to the (simplified) ambient conditions, in particular temperature (pro- gram T2 in Figure 5). Although a higher creep rate can be anticipated under cyclic temperature, the re- sulting strains were several times higher than under T1 program with bilinear temperature history. This response was considered as unrealistic. To damp the excessive sensitivity to subsequent temperature cycles, the parameter kT c was introduced and set to kT m/20. 3. High-fidelity model In order to obtain compatible results with the bench- mark specifications, a full-lifespan simulation was fi- nally run on the HFM. To detect unintentional mis- takes which are likely to be made when modeling large structures with complex boundary conditions, the similarity of the HFM response at the mid-height was subsequently cross-checked with the LFM. 50 vol. 40/2023 Benchmark VERCORS 2022: Response to Ambient Conditions Figure 6. HFM – structural model of concrete con- tainment building. Figure 7. HFM – structural model of vertical and dome tendons. 3.1. Structural model Structural HFM (Figures 6, 7 and 8) with 635 328 de- grees of freedom is composed of the concrete body (quadratic brick elements) and prestressing tendons (linear truss elements); conventional concrete rein- forcement is neglected due to its low influence on the global behavior. Fully fixed boundary conditions are assigned to the bottom face of the substructure. This simplification was acceptable because of the extreme stiffness of the substructure in comparison with the superstructure. As no specific prestressing schedule was provided, prestressing of all tendons was set to day 278+45 of the simulation. The value of eigenstrain assigned to every tendon is constant over its length and thus does not reflect the direction of prestress- ing; the prestress losses due to friction are evaluated according to Eurocode 2. Due to the instantaneous deformation of the concrete structure, even the initial distribution of prestress is not uniform. Gradual pre- stress relaxation is described by the fib Model Code 2010/Eurocode 2 (Annex D) approach which uses the concept of equivalent time. The material constants adopted comply with Class 2 reinforcement (wire or strands with low relaxation). Characteristic strength is defined as 1870 MPa and the Young modulus is 190 GPa. The effects of elevated temperature or tem- Figure 8. HFM – structural model of hoop tendons. 0 20 40 60 80 100 0 400 800 1200 1600 2000 R e la ti v e h u m id it y , R H [ % ] Time, t [day] Int henv3 Ext henv3 Int henv2 Ext henv2 Int henv1 Ext henv1 Figure 9. Ambient conditions – relative humidity prescribed on the inner and outer faces of the concrete wall. perature fluctuations on the relaxation rate are not considered. The periods with the increased value of inner over- pressure, which would have led to the development of tensile cracks, are not considered in the analysis as the influence on the long-term behavior is negli- gible because the cracks completely close once the overpressure vanishes. 3.2. Moisture and heat transfer Models for moisture and heat transfer with 1 061 450 degrees of freedom represent the concrete body discretized by linear brick elements. The bound- ary conditions are assigned to the inner and outer face of CCB. Similarly to LFM, the real history of ambient con- ditions provided by EDF, which was measured on VERCORS mock-up, was simplified into several pro- grams with different complexity. Unrealistic response of the constitutive model to cyclic temperature was prevented by the parameter kT c which in turn allowed to adopt more complex history of ambient tempera- ture (Int/Ext T2 in Figure 5) and thus to get closer to the actual measurements. Ambient relative humidity was defined by henv3 shown in Figure 9. 51 Štěpán Krátký, Petr Havlásek Acta Polytechnica CTU Proceedings Analysis HFM LFM Linear Linear DOF (SM) 635 328 26 166 Memory consumption 34 GB - CPU i9-11900 RL AMD 5 3600 CPU threads used 8 1 Time steps 382 382 Time consumption (RT) 31.5 hours 0.7 hours Table 3. Comparison of the characteristics of HFM and LFM. 70 80 90 100 500 1000 1500 2000 2500 3000 R e la ti v e h u m id it y , R H [ % ] Time, t [day] Ambient hum. ext Ambient hum. int Measurement HFM LFM Figure 10. Evolution of relative humidity: LFM, HFM and experiment. 4. Results and discussion The original intention of the authors was not only to thoroughly evaluate the behaviour of the CCB but also to use the measurements to improve the behavior of the constitutive model of concrete and possibly also steel relaxation. At the very beginning of the benchmark, the participants were promised that af- ter the initial blind stage of the benchmark, they would be granted full access to the Cheops database with detailed information including hundreds of strain, displacement, humidity, and temperature sensors. Unfortunately, after months of work of the present authors, the organizers revealed data on a very limited fraction of the structure referred to as PACAR area with dimensions 2×2 meters approximately in the middle of the height of the cylinder whose response should be in a good agreement with the LFM model. Data of relative humidity were available for a single sensor, which can neither support the credibility of the computational models nor it can provide confidence about the accuracy of the experimental data. Com- parison of several characteristics of LFM and HFM models is summarized in Table 3. All future goals of further studies on nonlinear mod- els, evolution of damage, and the behaviour under increased pressure were thus dropped as there is in- sufficient experimental evidence. -1200 -1000 -800 -600 -400 -200 0 200 400 600 500 1000 1500 2000 2500 3000 Ta n g e n ti a l st ra in , ε t a n [ 1 0 -6 ] Time, t [day] Measurement HFM-INI. LFM-INI. LFM-T COMP. LFM-shr LFM-bc LFM-shr COMP. LFM-thermal strain Figure 11. Evolution of hoop strain close to the inner face of the wall. -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 500 1000 1500 2000 2500 3000 V e rt ic a l st ra in , ε v e r [1 0 -6 ] Time, t [day] Measurement HFM-INI. LFM-INI. LFM-T COMP. LFM-shr LFM-bc LFM-shr COMP. LFM-thermal strain Figure 12. Evolution of vertical strain close to the outer face of the wall. 4.1. Relative humidity Figure 10 compares the computed and measured evolu- tion of relative humidity in the middle of the concrete wall at the PACAR area. At first glance, it is ap- parent that the computed drying rate by both LFM and HFM is substantially faster than in the experi- ment. Additionally, it is striking that the measured data exhibit significant daily fluctuations which even reach 8 % which is unrealistic given the position of the sensor at the wall middepth. 4.2. Evolution of the strain in the PACAR area Figures 11 and 12 present the evolution of tangential and vertical strain. For clarity, Figure 13 shows the first 800 days in more detail. Lines “LFM-INI.” and “HFM-INI.” represent the initial and unprocessed FEM results. There is a good agreement between HFM and LFM with only subtle differences that might be attributed to the slightly different placement of virtual sensors and/or FEM discretization. This concordance allowed to present 52 vol. 40/2023 Benchmark VERCORS 2022: Response to Ambient Conditions -600 -400 -200 0 200 400 300 400 500 600 700 800 Ta n g e n ti a l st ra in , ε t a n [ 1 0 -6 ] Time, t [day] Measurement HFM-INI. LFM-INI. LFM-T COMP. LFM-shr LFM-bc LFM-shr COMP. LFM-thermal strain Figure 13. Evolution of hoop strain close to the inner face of the wall - detailed view. the results of LFM with several modifications intro- duced. First, the “LFM-thermal strain” curve was computed, which represents only the strain induced by variations in ambient temperature. Subsequently, this strain was subtracted from the original “LFM-INI” curve to produce data series “LFM-T COMP” which is in very good agreement with the strains provided in the VERCORS benchmark. In the vertical direction, the prediction is almost perfect, while in the hoop direction the experimental response is initially slightly underestimated, which can be explained by overesti- mated prestress losses due to friction. It is of interest that even though the drying rate is overestimated (Figure 10), the structural response seems to match. The remaining curves are added to illustrate the influence of the individual factors. Line “LFM-shr” represents only the influence of drying shrinkage (with- out prestressing). Basic creep of concrete and steel relaxation only were considered in “LFM-bc”. The significance of drying creep is illustrated in “LFM- shr COMP” in which both basic and drying creep were considered and the results are compensated for shrinkage (“LFM-shr”). Increased pressure during prescribed pressure tests are manifested by the spikes in the hoop strain. In the vertical direction, the spikes have opposite orientation than in the experiment; this is because no equivalent pressure was applied in the vertical direction and the response in Figure 12 is merely the consequence of the lateral pressure scaled by the Poisson-effect. 5. Conclusions The blind prediction submitted by the present authors shows satisfactory agreement with the experimental data even though the simulation completely neglected the construction sequence, the prestressing schedule was unknown, and the calibration of material param- eters was based entirely on laboratory experiments. The low-fidelity model which represents a periodic sec- tion of the CCB wall can substitute the high-fidelity model which covers the entire CCB if the wall be- havior is of interest. The computationally efficient low-fidelity model was essential for the estimation of material parameters which could not be calibrated from the laboratory experiments and subsequently for verifying the correct definition of the high-fidelity model. Simulation results for approximately 50 sensors over the structure were submitted to EDF for evaluation. Unfortunately, the extent of accessible data provided by the benchmark organizers does not allow further research. Acknowledgements The authors gratefully acknowledge financial support from the Czech Science Foundation (GA ČR), project number 21-03118S, and from the Grant Agency of the Czech Technical University in Prague, project number SGS22/030/OHK1/1T/11. References [1] Š. Krátký, P. Havlásek. Benchmark VERCORS 2022: Mechanical response of the prestressed concrete containment wall to ambient conditions. Acta Polytechnica CTU Proceedings 34:26–31, 2022. https://doi.org/10.14311/APP.2022.34.0026 [2] Direction Production Ingeniere, EDF. Project maquette vercors, plan general, 2012. [3] B. Patzák. OOFEM home page, 2000. [2022-10-13]. http://www.oofem.org [4] Z. P. Bažant, L. J. Najjar. Nonlinear water diffusion in nonsaturated concrete. Materials and Structures 5:3–20, 1972. https://doi.org/10.1007/BF02479073 [5] Z. Bažant, P. Havlásek, M. Jirásek. Microprestress-solidification theory: Modeling of size effect on drying creep. In N. Bicanic, H. Mang, G. Meschke, R. de Borst (eds.), Computational Modelling of Concrete Structures, pp. 749–758. CRC Press/Balkema, EH Leiden, The Netherlands, 2014. [6] Štěpán Krátký. Benchmark VERCORS 2022 – slepá predikce chování železobetonového kontejnmentu [in Czech; Benchmark VERCORS 2022 – blind prediction of mechanical response of reinforced concrete containment]. Bachelor thesis, Czech Technical University in Prague, 2021. [7] Z. P. Bažant, S. Baweja. Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3. In Adam Neville Symposium: Creep and Shrinkage – Structural Design Effects, pp. 85–101. 2000. 53 https://doi.org/10.14311/APP.2022.34.0026 http://www.oofem.org https://doi.org/10.1007/BF02479073 Acta Polytechnica CTU Proceedings 40:48–53, 2023 1 Introduction 2 Previous outcomes 2.1 Laboratory experiments 2.2 Finite element modeling 2.3 Calibration of material models 2.3.1 Basic creep and autogenous shrinkage 2.3.2 Drying and drying shrinkage 2.3.3 Drying creep 2.3.4 Heat transfer 2.3.5 Damage 2.4 Low-fidelity model 2.5 Results 3 High-fidelity model 3.1 Structural model 3.2 Moisture and heat transfer 4 Results and discussion 4.1 Relative humidity 4.2 Evolution of the strain in the PACAR area 5 Conclusions Acknowledgements References