MODELLING OF CONCRETE BEHAVIOUR AT HIGH TEMPERATURE : state-of-the-art report. 9783030119942, 3030119947

Table of contents :
Front Matter ....Pages i-xxvii
Scope (Alain Millard, Pierre Pimienta)....Pages 1-5
Introduction (Alain Millard)....Pages 7-13
Engineering Modelling (Sven Huismann, Matthias Zeiml, Manfred Korzen, Alain Millard)....Pages 15-25
Advanced Modelling (Fekri Meftah, Francesco Pesavento, Colin Davie, Stefano Dal Pont, Matthias Zeiml, Manfred Korzen et al.)....Pages 27-65
Constitutive Parameters (Fekri Meftah, Colin Davie, Stefano Dal Pont, Alain Millard)....Pages 67-98
Conclusion (Alain Millard, Pierre Pimienta)....Pages 99-100

Citation preview

RILEM State-of-the-Art Reports

Alain Millard Pierre Pimienta Editors

Modelling of Concrete Behaviour at High Temperature State-of-the-Art Report of the RILEM Technical Committee 227-HPB

RILEM State-of-the-Art Reports

RILEM STATE-OF-THE-ART REPORTS Volume 30 RILEM, The International Union of Laboratories and Experts in Construction Materials, Systems and Structures, founded in 1947, is a non-governmental scientific association whose goal is to contribute to progress in the construction sciences, techniques and industries, essentially by means of the communication it fosters between research and practice. RILEM’s focus is on construction materials and their use in building and civil engineering structures, covering all phases of the building process from manufacture to use and recycling of materials. More information on RILEM and its previous publications can be found on www. RILEM.net. The RILEM State-of-the-Art Reports (STAR) are produced by the Technical Committees. They represent one of the most important outputs that RILEM generates—high level scientific and engineering reports that provide cutting edge knowledge in a given field. The work of the TCs is one of RILEM’s key functions. Members of a TC are experts in their field and give their time freely to share their expertise. As a result, the broader scientific community benefits greatly from RILEM’s activities. RILEM’s stated objective is to disseminate this information as widely as possible to the scientific community. RILEM therefore considers the STAR reports of its TCs as of highest importance, and encourages their publication whenever possible. The information in this and similar reports is mostly pre-normative in the sense that it provides the underlying scientific fundamentals on which standards and codes of practice are based. Without such a solid scientific basis, construction practice will be less than efficient or economical. It is RILEM’s hope that this information will be of wide use to the scientific community.

More information about this series at http://www.springer.com/series/8780

Alain Millard Pierre Pimienta •

Editors

Modelling of Concrete Behaviour at High Temperature State-of-the-Art Report of the RILEM Technical Committee 227-HPB

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Editors Alain Millard Service d’Études Mécaniques et Thermiques Commissariat à l’énergie atomique et aux énergies alternatives (CEA) Saclay, France

Pierre Pimienta Centre scientifique et technique du bâtiment (CSTB), Safety, Structures and Fire Université Paris-Est Marne la Vallée, France

ISSN 2213-204X ISSN 2213-2031 (electronic) RILEM State-of-the-Art Reports ISBN 978-3-030-11994-2 ISBN 978-3-030-11995-9 (eBook) https://doi.org/10.1007/978-3-030-11995-9 Library of Congress Control Number: 2018968107 © RILEM 2019 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Permission for use must always be obtained from the owner of the copyright: RILEM. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedication and Acknowledgements To the memory of Pr. Ulrich Schneider This state-of-the-art report is dedicated to the memory of Pr. Ulrich Schneider. Ulrich was the chairman of the RILEM 227-HPB Technical Committee. He mapped out our path and shared with us his impressive, deep and extensive scientific expertise and his human value. He has left to the scientific community an immense and rich heritage.

Foreword

This state-of-the-art report represents a valuable step forward with respect to the available literature. Till now, other STARs were limited to a range of temperatures not so far from the standard ones, up to 80 °C, or considered mainly ordinary concrete with only some short descriptions of the behaviour of high-performance concrete. Furthermore, the existing reports or monographs do not cover the most recent progress in the field of computational modelling. In the last 20 years, the computer processing power has increased dramatically, enabling engineers to formulate and implement ever more sophisticated mathematical models, taking into account increasingly complex aspects of the physical and chemical behaviour of cementitious materials. The analysis of concrete behaviour when the material is exposed to high temperature is a crucial problem in several fields ranging from civil to nuclear engineering and it is of paramount importance with regard to spalling. Spalling, which can appear in various forms including explosive spalling, is especially dangerous for concrete structures because it can lead to a generalized collapse. Taking into account the technical and scientific progress in recent years allowing for a better understanding of the topic, we can state that classical numerical models should not be used for the simulation of the behaviour of concrete with respect to spalling. In this state-of-the-art report, the mathematical/numerical models are classified into two major groups: Engineering models, which are based on the prescriptions of the Eurocode and take into account essentially the thermo-mechanical aspects of the behaviour of the material at high temperature; Advanced models, referring to computational ones considering concrete as a heterogeneous material with one or more fluid phases flowing in its pores. For the first class of models, the report emphasizes their easy use for a large range of structural design. At the same time, their limits are clearly identified and described, especially as far as spalling is concerned. In the second class of models, a plethora of approaches is nowadays available in the literature based on different assumptions and taking into account different physicochemical processes. This makes their description rather difficult, so the vii

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Foreword

authors of the STAR have chosen a top-bottom approach. First, a very complete model based on multiphase porous media mechanics (i.e. MPMM), likely the most sophisticated one currently available, is presented and discussed. Then, a set of progressive simplifications are introduced to make this type of approach easier to use in practical applications. The structure of the STAR, the approach chosen for the description, and the clarity of the presentation make the report a must for all those who want to approach the rather complicated subject of heated concrete. Padua, Italy

Bernhard Schrefler Professor Emeritus

Acknowledgements

All the authors are extremely grateful to Katarzyna Mróz from Cracow University of Technology and Colin Davie from Newcastle University for their very important and efficient involvement in the formatting and review of the document.

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RILEM Technical Committee 227-HPB (Physical Properties and Behaviour of High-Performance Concrete at High Temperature)

Chairmen U. Schneider and P. Pimienta

Secretary Robert Jansson McNamee

Members Ma-Cruz Alonso, Spain Lars Boström, Sweden Stefano Dal Pont, France Colin Davie, UK Gerard Debicki, France Frank Dehn, Germany Ulrich Diederichs, Germany Roberto Felicetti, Italy Izabela Hager, Poland Sven Huismann, Germany Ulla-Maija Jumppanen, Finland Manfred Korzen, Germany Eddy Koenders, The Netherlands Jianzhong Lai, China Katarzyna Mróz, Poland Bas Lottman, The Netherlands

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RILEM Technical Committee 227-HPB …

Fekri Meftah, France Alain Millard, France Jean-Christophe Mindeguia, France Josko Ozbolt, Germany Francesco Pesavento, Italy Klaus Pistol, Germany Fabienne Robert, France Joao Paulo Correia Rodriguez, Portugal Jay Sanjayan, Australia Kosmas K. Sideris, Greece Benedikt Weber, Switzerland Frank Weise, Germany The members of the Technical Committee thank the following experts for their contribution to this report Takashi Horiguchi, Japan Sergei Leonovich, Republic Belarus Berenice Moreau, France Martin Schneider, Germany Yury Zaytsev, Russia Matthias Zeiml, Austria

RILEM Publications

The following list is presenting the global offer of RILEM Publications, sorted by series. Each publication is available in printed version and/or in online version.

RILEM PROCEEDINGS (PRO) PRO 1: Durability of High Performance Concrete (ISBN: 2-912143-03-9; e-ISBN: 2-351580-12-5; e-ISBN: 2351580125); Ed. H. Sommer PRO 2: Chloride Penetration into Concrete (ISBN: 2-912143-00-04; e-ISBN: 2912143454); Eds. L.-O. Nilsson and J.-P. Ollivier PRO 3: Evaluation and Strengthening of Existing Masonry Structures (ISBN: 2-912143-02-0; e-ISBN: 2351580141); Eds. L. Binda and C. Modena PRO 4: Concrete: From Material to Structure (ISBN: 2-912143-04-7; e-ISBN: 2351580206); Eds. J.-P. Bournazel and Y. Malier PRO 5: The Role of Admixtures in High Performance Concrete (ISBN: 2-912143-05-5; e-ISBN: 2351580214); Eds. J. G. Cabrera and R. Rivera-Villarreal PRO 6: High Performance Fiber Reinforced Cement Composites—HPFRCC 3 (ISBN: 2-912143-06-3; e-ISBN: 2351580222); Eds. H. W. Reinhardt and A. E. Naaman PRO 7: 1st International RILEM Symposium on Self-Compacting Concrete (ISBN: 2-912143-09-8; e-ISBN: 2912143721); Eds. Å. Skarendahl and Ö. Petersson PRO 8: International RILEM Symposium on Timber Engineering (ISBN: 2-912143-10-1; e-ISBN: 2351580230); Ed. L. Boström PRO 9: 2nd International RILEM Symposium on Adhesion between Polymers and Concrete ISAP ’99 (ISBN: 2-912143-11-X; e-ISBN: 2351580249); Eds. Y. Ohama and M. Puterman PRO 10: 3rd International RILEM Symposium on Durability of Building and Construction Sealants (ISBN: 2-912143-13-6; e-ISBN: 2351580257); Eds. A. T. Wolf

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PRO 11: 4th International RILEM Conference on Reflective Cracking in Pavements (ISBN: 2-912143-14-4; e-ISBN: 2351580265); Eds. A. O. Abd El Halim, D. A. Taylor and El H. H. Mohamed PRO 12: International RILEM Workshop on Historic Mortars: Characteristics and Tests (ISBN: 2-912143-15-2; e-ISBN: 2351580273); Eds. P. Bartos, C. Groot and J. J. Hughes PRO 13: 2nd International RILEM Symposium on Hydration and Setting (ISBN: 2-912143-16-0; e-ISBN: 2351580281); Ed. A. Nonat PRO 14: Integrated Life-Cycle Design of Materials and Structures—ILCDES 2000 (ISBN: 951-758-408-3; e-ISBN: 235158029X); (ISSN: 0356-9403); Ed. S. Sarja PRO 15: Fifth RILEM Symposium on Fibre-Reinforced Concretes (FRC)— BEFIB’2000 (ISBN: 2-912143-18-7; e-ISBN: 291214373X); Eds. P. Rossi and G. Chanvillard PRO 16: Life Prediction and Management of Concrete Structures (ISBN: 2-912143-19-5; e-ISBN: 2351580303); Ed. D. Naus PRO 17: Shrinkage of Concrete—Shrinkage 2000 (ISBN: 2-912143-20-9; e-ISBN: 2351580311); Eds. V. Baroghel-Bouny and P.-C. Aïtcin PRO 18: Measurement and Interpretation of the On-Site Corrosion Rate (ISBN: 2-912143-21-7; e-ISBN: 235158032X); Eds. C. Andrade, C. Alonso, J. Fullea, J. Polimon and J. Rodriguez PRO 19: Testing and Modelling the Chloride Ingress into Concrete (ISBN: 2-912143-22-5; e-ISBN: 2351580338); Eds. C. Andrade and J. Kropp PRO 20: 1st International RILEM Workshop on Microbial Impacts on Building Materials (CD 02) (e-ISBN 978-2-35158-013-4); Ed. M. Ribas Silva PRO 21: International RILEM Symposium on Connections between Steel and Concrete (ISBN: 2-912143-25-X; e-ISBN: 2351580346); Ed. R. Eligehausen PRO 22: International RILEM Symposium on Joints in Timber Structures (ISBN: 2-912143-28-4; e-ISBN: 2351580354); Eds. S. Aicher and H.-W. Reinhardt PRO 23: International RILEM Conference on Early Age Cracking in Cementitious Systems (ISBN: 2-912143-29-2; e-ISBN: 2351580362); Eds. K. Kovler and A. Bentur PRO 24: 2nd International RILEM Workshop on Frost Resistance of Concrete (ISBN: 2-912143-30-6; e-ISBN: 2351580370); Eds. M. J. Setzer, R. Auberg and H.-J. Keck PRO 25: International RILEM Workshop on Frost Damage in Concrete (ISBN: 2-912143-31-4; e-ISBN: 2351580389); Eds. D. J. Janssen, M. J. Setzer and M. B. Snyder PRO 26: International RILEM Workshop on On-Site Control and Evaluation of Masonry Structures (ISBN: 2-912143-34-9; e-ISBN: 2351580141); Eds. L. Binda and R. C. de Vekey PRO 27: International RILEM Symposium on Building Joint Sealants (CD03; e-ISBN: 235158015X); Ed. A. T. Wolf

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PRO 28: 6th International RILEM Symposium on Performance Testing and Evaluation of Bituminous Materials—PTEBM’03 (ISBN: 2-912143-35-7; e-ISBN: 978-2-912143-77-8); Ed. M. N. Partl PRO 29: 2nd International RILEM Workshop on Life Prediction and Ageing Management of Concrete Structures (ISBN: 2-912143-36-5; e-ISBN: 2912143780); Ed. D. J. Naus PRO 30: 4th International RILEM Workshop on High Performance Fiber Reinforced Cement Composites—HPFRCC 4 (ISBN: 2-912143-37-3; e-ISBN: 2912143799); Eds. A. E. Naaman and H. W. Reinhardt PRO 31: International RILEM Workshop on Test and Design Methods for Steel Fibre Reinforced Concrete: Background and Experiences (ISBN: 2-912143-38-1; e-ISBN: 2351580168); Eds. B. Schnütgen and L. Vandewalle PRO 32: International Conference on Advances in Concrete and Structures 2 vol. (ISBN (set): 2-912143-41-1; e-ISBN: 2351580176); Eds. Ying-shu Yuan, Surendra P. Shah and Heng-lin Lü PRO 33: 3rd International Symposium on Self-Compacting Concrete (ISBN: 2-912143-42-X; e-ISBN: 2912143713); Eds. Ó. Wallevik and I. Níelsson PRO 34: International RILEM Conference on Microbial Impact on Building Materials (ISBN: 2-912143-43-8; e-ISBN: 2351580184); Ed. M. Ribas Silva PRO 35: International RILEM TC 186-ISA on Internal Sulfate Attack and Delayed Ettringite Formation (ISBN: 2-912143-44-6; e-ISBN: 2912143802); Eds. K. Scrivener and J. Skalny PRO 36: International RILEM Symposium on Concrete Science and Engineering—A Tribute to Arnon Bentur (ISBN: 2-912143-46-2; e-ISBN: 2912143586); Eds. K. Kovler, J. Marchand, S. Mindess and J. Weiss PRO 37: 5th International RILEM Conference on Cracking in Pavements— Mitigation, Risk Assessment and Prevention (ISBN: 2-912143-47-0; e-ISBN: 2912143764); Eds. C. Petit, I. Al-Qadi and A. Millien PRO 38: 3rd International RILEM Workshop on Testing and Modelling the Chloride Ingress into Concrete (ISBN: 2-912143-48-9; e-ISBN: 2912143578); Eds. C. Andrade and J. Kropp PRO 39: 6th International RILEM Symposium on Fibre-Reinforced Concretes— BEFIB 2004 (ISBN: 2-912143-51-9; e-ISBN: 2912143748); Eds. M. Di Prisco, R. Felicetti and G. A. Plizzari PRO 40: International RILEM Conference on the Use of Recycled Materials in Buildings and Structures (ISBN: 2-912143-52-7; e-ISBN: 2912143756); Eds. E. Vázquez, Ch. F. Hendriks and G. M. T. Janssen PRO 41: RILEM International Symposium on Environment-Conscious Materials and Systems for Sustainable Development (ISBN: 2-912143-55-1; e-ISBN: 2912143640); Eds. N. Kashino and Y. Ohama PRO 42: SCC’2005—China: 1st International Symposium on Design, Performance and Use of Self-Consolidating Concrete (ISBN: 2-912143-61-6; e-ISBN: 2912143624); Eds. Zhiwu Yu, Caijun Shi, Kamal Henri Khayat and Youjun Xie

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PRO 43: International RILEM Workshop on Bonded Concrete Overlays (e-ISBN: 2-912143-83-7); Eds. J. L. Granju and J. Silfwerbrand PRO 44: 2nd International RILEM Workshop on Microbial Impacts on Building Materials (CD11) (e-ISBN: 2-912143-84-5); Ed. M. Ribas Silva PRO 45: 2nd International Symposium on Nanotechnology in Construction, Bilbao (ISBN: 2-912143-87-X; e-ISBN: 2912143888); Eds. Peter J. M. Bartos, Yolanda de Miguel and Antonio Porro PRO 46: ConcreteLife’06—International RILEM-JCI Seminar on Concrete Durability and Service Life Planning: Curing, Crack Control, Performance in Harsh Environments (ISBN: 2-912143-89-6; e-ISBN: 291214390X); Ed. K. Kovler PRO 47: International RILEM Workshop on Performance Based Evaluation and Indicators for Concrete Durability (ISBN: 978-2-912143-95-2; e-ISBN: 9782912143969); Eds. V. Baroghel-Bouny, C. Andrade, R. Torrent and K. Scrivener PRO 48: 1st International RILEM Symposium on Advances in Concrete through Science and Engineering (e-ISBN: 2-912143-92-6); Eds. J. Weiss, K. Kovler, J. Marchand, and S. Mindess PRO 49: International RILEM Workshop on High Performance Fiber Reinforced Cementitious Composites in Structural Applications (ISBN: 2-912143-93-4; e-ISBN: 2912143942); Eds. G. Fischer and V. C. Li PRO 50: 1st International RILEM Symposium on Textile Reinforced Concrete (ISBN: 2-912143-97-7; e-ISBN: 2351580087); Eds. Josef Hegger, Wolfgang Brameshuber and Norbert Will PRO 51: 2nd International Symposium on Advances in Concrete through Science and Engineering (ISBN: 2-35158-003-6; e-ISBN: 2-35158-002-8); Eds. J. Marchand, B. Bissonnette, R. Gagné, M. Jolin and F. Paradis PRO 52: Volume Changes of Hardening Concrete: Testing and Mitigation (ISBN: 2-35158-004-4; e-ISBN: 2-35158-005-2); Eds. O. M. Jensen, P. Lura and K. Kovler PRO 53: High Performance Fiber Reinforced Cement Composites—HPFRCC5 (ISBN: 978-2-35158-046-2; e-ISBN: 978-2-35158-089-9); Eds. H. W. Reinhardt and A. E. Naaman PRO 54: 5th International RILEM Symposium on Self-Compacting Concrete (ISBN: 978-2-35158-047-9; e-ISBN: 978-2-35158-088-2); Eds. G. De Schutter and V. Boel PRO 55: International RILEM Symposium Photocatalysis, Environment and Construction Materials (ISBN: 978-2-35158-056-1; e-ISBN: 978-2-35158-057-8); Eds. P. Baglioni and L. Cassar PRO 56: International RILEM Workshop on Integral Service Life Modelling of Concrete Structures (ISBN 978-2-35158-058-5; e-ISBN: 978-2-35158-090-5); Eds. R. M. Ferreira, J. Gulikers and C. Andrade PRO 57: RILEM Workshop on Performance of cement-based materials in aggressive aqueous environments (e-ISBN: 978-2-35158-059-2); Ed. N. De Belie PRO 58: International RILEM Symposium on Concrete Modelling— CONMOD’08 (ISBN: 978-2-35158-060-8; e-ISBN: 978-2-35158-076-9); Eds. E. Schlangen and G. De Schutter

RILEM Publications

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PRO 59: International RILEM Conference on On Site Assessment of Concrete, Masonry and Timber Structures—SACoMaTiS 2008 (ISBN set: 978-2-35158-061-5; e-ISBN: 978-2-35158-075-2); Eds. L. Binda, M. di Prisco and R. Felicetti PRO 60: Seventh RILEM International Symposium on Fibre Reinforced Concrete: Design and Applications—BEFIB 2008 (ISBN: 978-2-35158-064-6; e-ISBN: 978-2-35158-086-8); Ed. R. Gettu PRO 61: 1st International Conference on Microstructure Related Durability of Cementitious Composites 2 vol., (ISBN: 978-2-35158-065-3; e-ISBN: 978-2-35158-084-4); Eds. W. Sun, K. van Breugel, C. Miao, G. Ye and H. Chen PRO 62: NSF/ RILEM Workshop: In-situ Evaluation of Historic Wood and Masonry Structures (e-ISBN: 978-2-35158-068-4); Eds. B. Kasal, R. Anthony and M. Drdácký PRO 63: Concrete in Aggressive Aqueous Environments: Performance, Testing and Modelling, 2 vol., (ISBN: 978-2-35158-071-4; e-ISBN: 978-2-35158-082-0); Eds. M. G. Alexander and A. Bertron PRO 64: Long Term Performance of Cementitious Barriers and Reinforced Concrete in Nuclear Power Plants and Waste Management—NUCPERF 2009 (ISBN: 978-2-35158-072-1; e-ISBN: 978-2-35158-087-5); Eds. V. L’Hostis, R. Gens, C. Gallé PRO 65: Design Performance and Use of Self-consolidating Concrete— SCC’2009 (ISBN: 978-2-35158-073-8; e-ISBN: 978-2-35158-093-6); Eds. C. Shi, Z. Yu, K. H. Khayat and P. Yan PRO 66: 2nd International RILEM Workshop on Concrete Durability and Service Life Planning—ConcreteLife’09 (ISBN: 978-2-35158-074-5; ISBN: 978-2-35158-074-5); Ed. K. Kovler PRO 67: Repairs Mortars for Historic Masonry (e-ISBN: 978-2-35158-083-7); Ed. C. Groot PRO 68: Proceedings of the 3rd International RILEM Symposium on ‘Rheology of Cement Suspensions such as Fresh Concrete (ISBN 978-2-35158-091-2; e-ISBN: 978-2-35158-092-9); Eds. O. H. Wallevik, S. Kubens and S. Oesterheld PRO 69: 3rd International PhD Student Workshop on ‘Modelling the Durability of Reinforced Concrete (ISBN: 978-2-35158-095-0); Eds. R. M. Ferreira, J. Gulikers and C. Andrade PRO 70: 2nd International Conference on ‘Service Life Design for Infrastructure’ (ISBN set: 978-2-35158-096-7, e-ISBN: 978-2-35158-097-4); Ed. K. van Breugel, G. Ye and Y. Yuan PRO 71: Advances in Civil Engineering Materials—The 50-year Teaching Anniversary of Prof. Sun Wei’ (ISBN: 978-2-35158-098-1; e-ISBN: 978-2-35158-099-8); Eds. C. Miao, G. Ye and H. Chen PRO 72: First International Conference on ‘Advances in Chemically-Activated Materials—CAM’2010’ (2010), 264 pp., ISBN: 978-2-35158-101-8; e-ISBN: 978-2-35158-115-5; Eds. Caijun Shi and Xiaodong Shen PRO 73: 2nd International Conference on ‘Waste Engineering and Management —ICWEM 2010’ (2010), 894 pp., ISBN: 978-2-35158-102-5; e-ISBN: 978-2-35158-103-2, Eds. J. Zh. Xiao, Y. Zhang, M. S. Cheung and R. Chu

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PRO 74: International RILEM Conference on ‘Use of Superabsorsorbent Polymers and Other New Addditives in Concrete’ (2010) 374 pp., ISBN: 978-2-35158-104-9; e-ISBN: 978-2-35158-105-6; Eds. O.M. Jensen, M.T. Hasholt, and S. Laustsen PRO 75: International Conference on ‘Material Science—2nd ICTRC—Textile Reinforced Concrete—Theme 1’ (2010) 436 pp., ISBN: 978-2-35158-106-3; e-ISBN: 978-2-35158-107-0; Ed. W. Brameshuber PRO 76: International Conference on ‘Material Science—HetMat—Modelling of Heterogeneous Materials—Theme 2’ (2010) 255 pp., ISBN: 978-2-35158-108-7; e-ISBN: 978-2-35158-109-4; Ed. W. Brameshuber PRO 77: International Conference on ‘Material Science—AdIPoC—Additions Improving Properties of Concrete—Theme 3’ (2010) 459 pp., ISBN: 978-2-35158-110-0; e-ISBN: 978-2-35158-111-7; Ed. W. Brameshuber PRO 78: 2nd Historic Mortars Conference and RILEM TC 203-RHM Final Workshop—HMC2010 (2010) 1416 pp., e-ISBN: 978-2-35158-112-4; Eds. J. Válek, C. Groot and J. J. Hughes PRO 79: International RILEM Conference on Advances in Construction Materials Through Science and Engineering (2011) 213 pp., ISBN: 978-2-35158-116-2, e-ISBN: 978-2-35158-117-9; Eds. Christopher Leung and K.T. Wan PRO 80: 2nd International RILEM Conference on Concrete Spalling due to Fire Exposure (2011) 453 pp., ISBN: 978-2-35158-118-6; e-ISBN: 978-2-35158-119-3; Eds. E.A.B. Koenders and F. Dehn PRO 81: 2nd International RILEM Conference on Strain Hardening Cementitious Composites (SHCC2-Rio) (2011) 451 pp., ISBN: 978-2-35158-120-9; e-ISBN: 978-2-35158-121-6; Eds. R.D. Toledo Filho, F.A. Silva, E.A.B. Koenders and E.M.R. Fairbairn PRO 82: 2nd International RILEM Conference on Progress of Recycling in the Built Environment (2011) 507 pp., e-ISBN: 978-2-35158-122-3; Eds. V.M. John, E. Vazquez, S.C. Angulo and C. Ulsen PRO 83: 2nd International Conference on Microstructural-related Durability of Cementitious Composites (2012) 250 pp., ISBN: 978-2-35158-129-2; e-ISBN: 978-2-35158-123-0; Eds. G. Ye, K. van Breugel, W. Sun and C. Miao PRO 84: CONSEC13—Seventh International Conference on Concrete under Severe Conditions—Environment and Loading (2013) 1930 pp., ISBN: 978-2-35158-124-7; e-ISBN: 978-2- 35158-134-6; Eds. Z.J. Li, W. Sun, C.W. Miao, K. Sakai, O.E. Gjorv and N. Banthia PRO 85: RILEM-JCI International Workshop on Crack Control of Mass Concrete and Related issues concerning Early-Age of Concrete Structures— ConCrack 3—Control of Cracking in Concrete Structures 3 (2012) 237 pp., ISBN: 978-2-35158-125-4; e-ISBN: 978-2-35158-126-1; Eds. F. Toutlemonde and J.-M. Torrenti PRO 86: International Symposium on Life Cycle Assessment and Construction (2012) 414 pp., ISBN: 978-2-35158-127-8, e-ISBN: 978-2-35158-128-5; Eds. A. Ventura and C. de la Roche

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PRO 87: UHPFRC 2013—RILEM-fib-AFGC International Symposium on Ultra-High Performance Fibre-Reinforced Concrete (2013), ISBN: 978-2-35158-130-8, e-ISBN: 978-2-35158-131-5; Eds. F. Toutlemonde PRO 88: 8th RILEM International Symposium on Fibre Reinforced Concrete (2012) 344 pp., ISBN: 978-2-35158-132-2; e-ISBN: 978-2-35158-133-9; Eds. Joaquim A.O. Barros PRO 89: RILEM International workshop on performance-based specification and control of concrete durability (2014) 678 pp., ISBN: 978-2-35158-135-3; e-ISBN: 978-2-35158-136-0; Eds. D. Bjegović, H. Beushausen and M. Serdar PRO 90: 7th RILEM International Conference on Self-Compacting Concrete and of the 1st RILEM International Conference on Rheology and Processing of Construction Materials (2013) 396 pp., ISBN: 978-2-35158-137-7; e-ISBN: 978-2-35158-138-4; Eds. Nicolas Roussel and Hela Bessaies-Bey PRO 91: CONMOD 2014—RILEM International Symposium on Concrete Modelling (2014), ISBN: 978-2-35158-139-1; e-ISBN: 978-2-35158-140-7; Eds. Kefei Li, Peiyu Yan and Rongwei Yang PRO 92: CAM 2014—2nd International Conference on advances in chemically-activated materials (2014) 392 pp., ISBN: 978-2-35158-141-4; e-ISBN: 978-2-35158-142-1; Eds. Caijun Shi and Xiadong Shen PRO 93: SCC 2014—3rd International Symposium on Design, Performance and Use of Self-Consolidating Concrete (2014) 438 pp., ISBN: 978-2-35158-143-8; e-ISBN: 978-2-35158-144-5; Eds. Caijun Shi, Zhihua Ou, Kamal H. Khayat PRO 94 (online version): HPFRCC-7—7th RILEM conference on High performance fiber reinforced cement composites (2015), e-ISBN: 978-2-35158-146-9; Eds. H.W. Reinhardt, G.J. Parra-Montesinos, H. Garrecht PRO 95: International RILEM Conference on Application of superabsorbent polymers and other new admixtures in concrete construction (2014), ISBN: 978-2-35158-147-6; e-ISBN: 978-2-35158-148-3; Eds. Viktor Mechtcherine, Christof Schroefl PRO 96 (online version): XIII DBMC: XIII International Conference on Durability of Building Materials and Components (2015), e-ISBN: 978-2-35158-149-0; Eds. M. Quattrone, V.M. John PRO 97: SHCC3—3rd International RILEM Conference on Strain Hardening Cementitious Composites (2014), ISBN: 978-2-35158-150-6; e-ISBN: 978-2-35158-151-3; Eds. E. Schlangen, M.G. Sierra Beltran, M. Lukovic, G. Ye PRO 98: FERRO-11—11th International Symposium on Ferrocement and 3rd ICTRC—International Conference on Textile Reinforced Concrete (2015), ISBN: 978-2-35158-152-0; e-ISBN: 978-2-35158-153-7; Ed. W. Brameshuber PRO 99 (online version): ICBBM 2015—1st International Conference on Bio-Based Building Materials (2015), e-ISBN: 978-2-35158-154-4; Eds. S. Amziane, M. Sonebi PRO 100: SCC16—RILEM Self-Consolidating Concrete Conference (2016), ISBN: 978-2-35158-156-8; e-ISBN: 978-2-35158-157-5; Ed. Kamal H. Kayat PRO 101 (online version): III Progress of Recycling in the Built Environment (2015), e-ISBN: 978-2-35158-158-2; Eds I. Martins, C. Ulsen and S. C. Angulo

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RILEM Publications

PRO 102 (online version): RILEM Conference on MicroorganismsCementitious Materials Interactions (2016), e-ISBN: 978-2-35158-160-5; Eds. Alexandra Bertron, Henk Jonkers, Virginie Wiktor PRO 103 (online version): ACESC’16—Advances in Civil Engineering and Sustainable Construction (2016), e-ISBN: 978-2-35158-161-2; Eds. T.Ch. Madhavi, G. Prabhakar, Santhosh Ram and P.M. Rameshwaran PRO 104 (online version): SSCS’2015—Numerical Modeling—Strategies for Sustainable Concrete Structures (2015), e-ISBN: 978-2-35158-162-9 PRO 105: 1st International Conference on UHPC Materials and Structures (2016), ISBN: 978-2-35158-164-3; e-ISBN: 978-2-35158-165-0 PRO 106: AFGC-ACI-fib-RILEM International Conference on Ultra-HighPerformance Fibre-Reinforced Concrete—UHPFRC 2017 (2017), ISBN: 978-2-35158-166-7; e-ISBN: 978-2-35158-167-4; Eds. François Toutlemonde & Jacques Resplendino PRO 107 (online version): XIV DBMC—14th International Conference on Durability of Building Materials and Components (2017), e-ISBN: 978-2-35158-159-9; Eds. Geert De Schutter, Nele De Belie, Arnold Janssens, Nathan Van Den Bossche PRO 108: MSSCE 2016—Innovation of Teaching in Materials and Structures (2016), ISBN: 978-2-35158-178-0; e-ISBN: 978-2-35158-179-7; Ed. Per Goltermann PRO 109 (2 volumes): MSSCE 2016—Service Life of Cement-Based Materials and Structures (2016), ISBN Vol. 1: 978-2-35158-170-4; Vol. 2: 978-2-35158-171-4; Set Vol. 1&2: 978-2-35158-172-8; e-ISBN : 978-2-35158-173-5; Eds. Miguel Azenha, Ivan Gabrijel, Dirk Schlicke, Terje Kanstad and Ole Mejlhede Jensen PRO 110: MSSCE 2016—Historical Masonry (2016), ISBN: 978-2-35158-178-0; e-ISBN: 978-2-35158-179-7; Eds. Inge Rörig-Dalgaard and Ioannis Ioannou PRO 111: MSSCE 2016—Electrochemistry in Civil Engineering (2016); ISBN: 978-2-35158-176-6; e-ISBN: 978-2-35158-177-3; Ed. Lisbeth M. Ottosen PRO 112: MSSCE 2016—Moisture in Materials and Structures (2016), ISBN: 978-2-35158-178-0; e-ISBN: 978-2-35158-179-7; Eds. Kurt Kielsgaard Hansen, Carsten Rode and Lars-Olof Nilsson PRO 113: MSSCE 2016—Concrete with Supplementary Cementitious Materials (2016), ISBN: 978-2-35158-178-0; e-ISBN: 978-2-35158-179-7; Eds. Ole Mejlhede Jensen, Konstantin Kovler and Nele De Belie PRO 114: MSSCE 2016—Frost Action in Concrete (2016), ISBN: 978-2-35158-182-7; e-ISBN: 978-2-35158-183-4; Eds. Marianne Tange Hasholt, Katja Fridh and R. Doug Hooton PRO 115: MSSCE 2016—Fresh Concrete (2016), ISBN: 978-2-35158-184-1; e-ISBN: 978-2-35158-185-8; Eds. Lars N. Thrane, Claus Pade, Oldrich Svec and Nicolas Roussel PRO 116: BEFIB 2016—9th RILEM International Symposium on Fiber Reinforced Concrete (2016), ISBN: 978-2-35158-187-2; e-ISBN: 978-2-35158-186-5; Eds. N. Banthia, M. di Prisco and S. Soleimani-Dashtaki PRO 117: 3rd International RILEM Conference on Microstructure Related Durability of Cementitious Composites (2016), ISBN: 978-2-35158-188-9; e-ISBN: 978-2-35158-189-6; Eds. Changwen Miao, Wei Sun, Jiaping Liu, Huisu Chen, Guang Ye and Klaas van Breugel

RILEM Publications

xxi

PRO 118 (4 volumes): International Conference on Advances in Construction Materials and Systems (2017), ISBN Set: 978-2-35158-190-2; Vol. 1: 978-2-35158-193-3; Vol. 2: 978-2-35158-194-0; Vol. 3: ISBN:978-2-35158-195-7; Vol. 4: ISBN:978-2-35158-196-4; e-ISBN: 978-2-35158-191-9; Ed. Manu Santhanam PRO 119 (online version): ICBBM 2017—Second International RILEM Conference on Bio-based Building Materials, (2017), e-ISBN: 978-2-35158-192-6; Ed. Sofiane Amziane PRO 120 (2 volumes): EAC-02—2nd International RILEM/COST Conference on Early Age Cracking and Serviceability in Cement-based Materials and Structures, (2017), Vol. 1: 978-2-35158-199-5, Vol. 2: 978-2-35158-200-8, Set: 978-2-35158-197-1, e-ISBN: 978-2-35158-198-8; Eds. Stéphanie Staquet and Dimitrios Aggelis PRO 121 (2 volumes): SynerCrete18: Interdisciplinary Approaches for Cement-based Materials and Structural Concrete: Synergizing Expertise and Bridging Scales of Space and Time, (2018), Set: 978-2-35158-202-2, Vol.1: 978-2-35158-211-4, Vol.2: 978-2-35158-212-1, e-ISBN: 978-2-35158-203-9; Eds. Miguel Azenha, Dirk Schlicke, Farid Benboudjema, Agnieszka Knoppik PRO 122: SCC’2018 China—Fourth International Symposium on Design, Performance and Use of Self-Consolidating Concrete, (2018), ISBN: 978-2-35158-204-6, e-ISBN: 978-2-35158-205-3; Eds. C. Shi, Z. Zhang, K. H. Khayat PRO 123: Final Conference of RILEM TC 253-MCI: MicroorganismsCementitious Materials Interactions (2018), Set: 978-2-35158-207-7, Vol.1: 978-2-35158-209-1, Vol.2: 978-2-35158-210-7, e-ISBN: 978-2-35158-206-0; Ed. Alexandra Bertron PRO 124 (online version): Fourth International Conference Progress of Recycling in the Built Environment (2018), e-ISBN: 978-2-35158-208-4; Eds. Isabel M. Martins, Carina Ulsen, Yury Villagran PRO 125 (online version): SLD4—4th International Conference on Service Life Design for Infrastructures (2018), e-ISBN: 978-2-35158-213-8; Eds. Guang Ye, Yong Yuan, Claudia Romero Rodriguez, Hongzhi Zhang, Branko Savija PRO 126: Workshop on Concrete Modelling and Material Behaviour in honor of Professor Klaas van Breugel (2018), ISBN: 978-2-35158-214-5, e-ISBN: 978-2-35158-215-2; Ed. Guang Ye PRO 127 (online version): CONMOD2018—Symposium on Concrete Modelling (2018), e-ISBN: 978-2-35158-216-9; Eds. Erik Schlangen, Geert de Schutter, Branko Savija, Hongzhi Zhang, Claudia Romero Rodriguez PRO 128: SMSS2019—International Conference on Sustainable Materials, Systems and Structures (2019), ISBN: 978-2-35158-217-6, e-ISBN: 978-2-35158-218-3 PRO 129: 2nd International Conference on UHPC Materials and Structures (UHPC2018-China), ISBN: 978-2-35158-219-0, e-ISBN: 978-2-35158-220-6 PRO 130: 5th Historic Mortars Conference (2019), ISBN: 978-2-35158-221-3, e-ISBN: 978-2-35158-222-0; Eds. José Ignacio Álvarez, José María Fernández, Íñigo Navarro, Adrián Durán, Rafael Sirera

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RILEM Publications

RILEM REPORTS (REP) Report 19: Considerations for Use in Managing the Aging of Nuclear Power Plant Concrete Structures (ISBN: 2-912143-07-1); Ed. D. J. Naus Report 20: Engineering and Transport Properties of the Interfacial Transition Zone in Cementitious Composites (ISBN: 2-912143-08-X); Eds. M. G. Alexander, G. Arliguie, G. Ballivy, A. Bentur and J. Marchand Report 21: Durability of Building Sealants (ISBN: 2-912143-12-8); Ed. A. T. Wolf Report 22: Sustainable Raw Materials—Construction and Demolition Waste (ISBN: 2-912143-17-9); Eds. C. F. Hendriks and H. S. Pietersen Report 23: Self-Compacting Concrete state-of-the-art report (ISBN: 2-912143-23-3); Eds. Å. Skarendahl and Ö. Petersson Report 24: Workability and Rheology of Fresh Concrete: Compendium of Tests (ISBN: 2-912143-32-2); Eds. P. J. M. Bartos, M. Sonebi and A. K. Tamimi Report 25: Early Age Cracking in Cementitious Systems (ISBN: 2-912143-33-0); Ed. A. Bentur Report 26: Towards Sustainable Roofing (Joint Committee CIB/RILEM) (CD 07) (e-ISBN 978-2-912143-65-5); Eds. Thomas W. Hutchinson and Keith Roberts Report 27: Condition Assessment of Roofs (Joint Committee CIB/RILEM) (CD 08) (e-ISBN 978-2-912143-66-2); Ed. CIB W 83/RILEM TC166-RMS Report 28: Final report of RILEM TC 167-COM ‘Characterisation of Old Mortars with Respect to Their Repair (ISBN: 978-2-912143-56-3); Eds. C. Groot, G. Ashall and J. Hughes Report 29: Pavement Performance Prediction and Evaluation (PPPE): Interlaboratory Tests (e-ISBN: 2-912143-68-3); Eds. M. Partl and H. Piber Report 30: Final Report of RILEM TC 198-URM ‘Use of Recycled Materials’ (ISBN: 2-912143-82-9; e-ISBN: 2-912143-69-1); Eds. Ch. F. Hendriks, G. M. T. Janssen and E. Vázquez Report 31: Final Report of RILEM TC 185-ATC ‘Advanced testing of cement-based materials during setting and hardening’ (ISBN: 2-912143-81-0; e-ISBN: 2-912143-70-5); Eds. H. W. Reinhardt and C. U. Grosse Report 32: Probabilistic Assessment of Existing Structures. A JCSS publication (ISBN 2-912143-24-1); Ed. D. Diamantidis Report 33: State-of-the-Art Report of RILEM Technical Committee TC 184-IFE ‘Industrial Floors’ (ISBN 2-35158-006-0); Ed. P. Seidler Report 34: Report of RILEM Technical Committee TC 147-FMB ‘Fracture mechanics applications to anchorage and bond’ Tension of Reinforced Concrete Prisms—Round Robin Analysis and Tests on Bond (e-ISBN 2-912143-91-8); Eds. L. Elfgren and K. Noghabai Report 35: Final Report of RILEM Technical Committee TC 188-CSC ‘Casting of Self Compacting Concrete’ (ISBN 2-35158-001-X; e-ISBN: 2-912143-98-5); Eds. Å. Skarendahl and P. Billberg

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Report 36: State-of-the-Art Report of RILEM Technical Committee TC 201-TRC ‘Textile Reinforced Concrete’ (ISBN 2-912143-99-3); Ed. W. Brameshuber Report 37: State-of-the-Art Report of RILEM Technical Committee TC 192-ECM ‘Environment-conscious construction materials and systems’ (ISBN: 978-2-35158-053-0); Eds. N. Kashino, D. Van Gemert and K. Imamoto Report 38: State-of-the-Art Report of RILEM Technical Committee TC 205-DSC ‘Durability of Self-Compacting Concrete’ (ISBN: 978-2-35158-048-6); Eds. G. De Schutter and K. Audenaert Report 39: Final Report of RILEM Technical Committee TC 187-SOC ‘Experimental determination of the stress-crack opening curve for concrete in tension’ (ISBN 978-2-35158-049-3); Ed. J. Planas Report 40: State-of-the-Art Report of RILEM Technical Committee TC 189-NEC ‘Non-Destructive Evaluation of the Penetrability and Thickness of the Concrete Cover’ (ISBN 978-2-35158-054-7); Eds. R. Torrent and L. Fernández Luco Report 41: State-of-the-Art Report of RILEM Technical Committee TC 196-ICC ‘Internal Curing of Concrete’ (ISBN 978-2-35158-009-7); Eds. K. Kovler and O. M. Jensen Report 42: ‘Acoustic Emission and Related Non-destructive Evaluation Techniques for Crack Detection and Damage Evaluation in Concrete’—Final Report of RILEM Technical Committee 212-ACD (e-ISBN: 978-2-35158-100-1); Ed. M. Ohtsu Report 45: Repair Mortars for Historic Masonry—State-of-the-Art Report of RILEM Technical Committee TC 203-RHM (e-ISBN: 978-2-35158-163-6); Eds. Paul Maurenbrecher and Caspar Groot Report 46: Surface delamination of concrete industrial ffioors and other durability related aspects guide—Report of RILEM Technical Committee TC 268-SIF (e-ISBN: 978-2-35158-201-5); Ed. Valerie Pollet

Contents

1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alain Millard and Pierre Pimienta

1

2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alain Millard

7

3 Engineering Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sven Huismann, Matthias Zeiml, Manfred Korzen and Alain Millard

15

4 Advanced Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fekri Meftah, Francesco Pesavento, Colin Davie, Stefano Dal Pont, Matthias Zeiml, Manfred Korzen and Alain Millard

27

5 Constitutive Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fekri Meftah, Colin Davie, Stefano Dal Pont and Alain Millard

67

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alain Millard and Pierre Pimienta

99

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Contributors

Colin Davie School of Engineering, Newcastle University, Newcastle upon Tyne, UK Sven Huismann Bundesanstanlt for Materialforschung Und Prüfung (BAM), Berlin, Germany Manfred Korzen Bundesanstanlt for Materialforschung Und Prüfung (BAM), Berlin, Germany Fekri Meftah Institut National des Sciences Appliquées, Rennes, France Alain Millard Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France Berenice Moreau Centre d’Etudes des Tunnels, Bron, France Francesco Pesavento Dipartimento di Ingegneria Civile, Edile e Ambientale, Universita degli Studi di Padova, Padua, Italy Pierre Pimienta Centre Scientifique et Technique du Bâtiment (CSTB), Université Paris-Est, Marne la Vallée, France Stefano Dal Pont Laboratoire 3SR, Université Grenoble-Alpes, Grenoble, France Fabienne Robert Centre d’Etudes et de Recherches de l’Industrie du Béton, Epernon (CERIB), Épernon, France Martin Schneider Carinthia University of Applied Sciences, Spittal a.d.D, Villach, Austria Benedikt Weber EMPA, Switzerland

Concrete/Construction

Chemistry,

Dübendorf,

Matthias Zeiml Material Technology, University of Innsbruck, Innsbruck, Austria

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Chapter 1

Scope Alain Millard and Pierre Pimienta

Abstract This book presents the work done by the RILEM Technical Committee 227-HPB (Physical properties and behaviour of High-Performance Concrete at high temperature). It contains the latest research results on the modelling of concrete behaviour at high temperature. Some monographs on the subject have been published already but generally they do not cover the whole range of possibilities which are encountered in the literature as well as in practice. Moreover, there has been a rapidly increasing development of computational models during the last twenty years, which deserves attention. Therefore, it is the aim of this report to compile and present most of the tools that are proposed in the literature and are nowadays available for practice in some commercial computational packages. This chapter presents the main literature produced during the last decades on the behaviour at high temperature of concrete. This literature mainly focus on 2 main fields of application: the structural response and safety analysis of buildings and structures under fire and the nuclear industry and in particular the nuclear reactor containment buildings generally made of pre-stressed concrete. A special attention is paid to high performance concrete and their specific properties.

1.1

General Comments

This report presents part of the work done by the RILEM Technical Committee 227-HPB (Physical properties and behaviour of High-Performance Concrete at high temperature). This Committee was created in 2007 and was adjourned in 2014. A. Millard (&) Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France e-mail: [email protected] P. Pimienta Centre Scientifique et Technique du Bâtiment (CSTB), Université Paris-Est, Marne la Vallée, France e-mail: [email protected] © RILEM 2019 A. Millard and P. Pimienta (eds.), Modelling of Concrete Behaviour at High Temperature, RILEM State-of-the-Art Reports 30, https://doi.org/10.1007/978-3-030-11995-9_1

1

2

A. Millard and P. Pimienta

This committee assured the continuity of the work that had been done by the RILEM Technical Committee 200-HTC (Mechanical concrete properties at high temperatures-modelling and applications) and previous ones on ordinary concretes. The main objectives of the TC HPB were to write the State-of-the-Art on two areas of research. The first, which is published in this book, focuses on the modelling of the behaviour of concrete at high temperatures. The second is published in a second book and gathers and analyses the latest experimental research results on the behaviour of high-performance concretes (HPC) at high-temperature (Pimienta et al. 2018). Because of their actual significant use and their specific properties at high temperature, special attention will be focus to HPC through the document.

1.2

Context and Objectives

As previously written, the RILEM Technical Committee 227-HPB has been established as a continuation of previous RILEM Committees working on the behaviour of concrete (mostly Ordinary Performance Concrete (OPC)) at high temperatures. These Committees have produced a State-of-the-Art report on the physical properties of OPC at high temperatures (RILEM Committee 44-PHT 1985 and Schneider and Horvath 2003), as well as recommendations for test procedures related to their measurements (RILEM TC 129-MHT 1995, 1997, 1998, 2000a, b, c, 2004 and RILEM TC 200-HTC 2005, 2007a, b, c). Although some analytical aspects have also been described in these reports, a detailed comprehensive State-of-the-Art report on concrete behaviour at high temperatures was still missing. Fundamental pioneering work has been performed on OPC by U. Schneider since the early 1970s, including material laws for strength, elasticity, creep, transient creep, thermal strain, decomposition and dehydration (Schneider 1973, 1979, 1982, 1986 and 1988). It has been synthesized in documents such as (Schneider 1982), opening the way for all the phenomena occurring in heated concrete and pointing out the fact that additional research was required to convert test data into constitutive equations that could be used for practical applications. In a recent State-of-the-Art report by CEB (CEB-FIP 2007) on HPC constitutive models, the temperatures are limited to 80 °C. In another State-of-the-Art report devoted to fire design of concrete structures (CEB-FIP 2008), numerical modelling is not covered, but judged as very promising. Some monographs on the subject have been published already some years ago, such as the one by Bažant and Kaplan (1996), but generally they do not cover the whole range of possibilities which are encountered in the literature as well as in practice. Moreover, there has been a rapidly increasing development of computational models during the last twenty years, which deserves attention. Therefore, it is the aim of this report to compile and present most of the tools that are proposed in the literature and are nowadays available for practice in some commercial computational packages.

1 Scope

3

The field of application of such tools is constantly enlarging. First, the temperatures that are considered range from 20 °C up to the melting point of concrete, around 1350 °C. These temperatures can be obtained either in service or in fault conditions, depending on the nature and serviceability of the structure. Of course, one main field of application is related to the structural response and safety analysis of buildings and structures under fire. In particular, recent accidental fires in railway as well as roadway tunnels have highlighted the need to improve the safety of the structures under fire, as well as their design, in particular by improving the concrete performances under high temperature. This trend concerns not only concrete structures but, more generally, all types of cementitious-like materials (such as mortar, plaster, etc.) which are routinely used in buildings, and for which the same type of approaches can be contemplated. Another main field of applications concerns the nuclear industry. A first important issue is related to the safety analysis of nuclear reactor containment buildings generally made of pre-stressed concrete. Although the temperature increase up to about 210 °C in the case of a Loss of Coolant Accident (LOCA) in a Pressurized Water Reactor (PWR) is already a problem, advanced reactors such as liquid metal cooled reactors or high temperature gas cooled reactors are more critical, with temperatures reaching as high as 1150 °C (Schimmelpfennig and Altes 1982). In the case of a severe accident, several scenarios that can lead to high temperatures are studied, such as the spillage of melted core on to the containment basemat, or the thermal shock due to liquid metal on concrete walls. Another important issue is related to the intermediate or long-term surface storage of high-level nuclear wastes. In the case of a failure of the cooling system, temperatures up to 250 °C might be encountered. HPC is now often used because of its improved strength and durability compared to OPC. These increased performances are mainly obtained by means of reduced water-cement ratios and use of special additives such as silica-fume. As a consequence, the porosity and the permeability are reduced, thus improving the resistance of concrete to the ingress of chloride ions or carbon dioxide, which can in the long-term lead to the active corrosion of the reinforcement. A further consequence is that the behaviour of HPC under high temperatures may be rather different from that of OPC, especially in the case of rapid heating, as can be encountered during a fire. In particular, spalling may occur, leading to the detachment of concrete pieces or layers from the surface, and to the subsequent exposure of the remaining structure to the fire, thus accelerating the degradation process and finally the collapse of the structure. The needs for an improved design of HPC structures subjected to high temperatures, as well as their performance and safety assessments, have boosted the development of analytical and numerical tools during the last three decades. In particular, in view of its inherent complexity as outlined in the next sections, a satisfactory explanation of spalling requires a clear understanding not only of the fundamental mechanisms but also of their orders of magnitude. This can only be done by resorting to experimental tests complemented by modelling, because of the non-homogeneous response of the specimens that behave like true structures.

4

A. Millard and P. Pimienta

Moreover, the cost and time required by experimental testing can be significantly reduced by appropriate modelling investigations, which allow a better understanding of what is happening, what the dominating physical mechanisms are and the most influential parameters. In the next section, a description of the main phenomena and processes taking place in concrete structures exposed to high temperature, is given for a better understanding of the difficulties faced in the formulation of models that can be considered as really predictive of the structure behaviour.

References Bažant, Z.P., Kaplan, M.F.: Concrete at High Temperatures: Material Properties and Mathematical Models. Longman, Harlow (1996) CEB-FIP: Fire design of concrete structures – materials, structures and modelling, State-of-the-art report, Bulletin 38. International Federation for Structural Concrete (fib) (2007) CEB-FIP: Constitutive modelling of high strength/high performance concrete, State-of-the-art report, Bulletin 42. International Federation for Structural Concrete (fib) (2008) Pimienta P., Jansson McNamee R. and Mindeguia J-C. (Eds.) “Physical Properties and Behaviour of High-Performance Concrete at High Temperature - State-of-the-Art Report of the RILEM Technical Committee 227-HPB” Book published by Springer (2018) RILEM Committee 44-PHT: Properties of materials at high temperature – Concrete. U. Schneider (eds), Dept. of Civil Engineering, Kassel University (1985) RILEM Committee HTC: Behaviour of ordinary concrete at high temperatures, U. Schneider and J. Howarth (eds), Institute of Building Materials, Vienna University of Technology (2002) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures: Part 3: Compressive strength for service and accident conditions, Materials and Structures 28 (7), 410–414 (1995) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures, Recommendations Part 6: Thermal strain. Materials and Structures, Supplement March 1997, 17–21 (1997) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures, Recommendations, Part 7: Transient creep for service and accident condition. Materials and Structures 31, 290–295 (1998) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures, Part 4: Tensile Strength for service and accident conditions. Mater. Struct. 33(228), 219–223 (2000a) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures, Recommendations, Part 8: Steady-state creep and creep recovery for service and accident conditions. Materials and Structures 33, 6–13 (2000b) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures, Recommendations, Part 9: Shrinkage for service and accident conditions.Materials and Structures 33, 224–228 (2000c) RILEM TC 129-MHT: Test methods for mechanical properties of concrete at high temperatures: Part 5: Modulus of elasticity for service and accident conditions, Materials and Structures 37 (2) 139–144 (2004) RILEM TC 200-HTC: Test methods for mechanical properties of concrete at high temperatures, Recommendations, Part 10: Restraint stress. Materials and Structures 38, 913–919 (2005)

1 Scope

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RILEM TC 200-HTC: Test methods for mechanical properties of concrete at high temperatures, Recommendation, Part 1: Introduction-General presentation. Materials and Structures 40(9), 841–853 (2007a) RILEM TC 200-HTC: Mechanical concrete properties at high temperatures - modelling and applications, Part 2: Stress-strain relation, Materials and Structures 40(9) 855–864 (2007b) RILEM TC 200-HTC: Mechanical concrete properties at high temperatures-modelling and applications, Part 11: Relaxation. Materials and Structures 40, 449–458 (2007c) Schimmelpfennig, K., Altes, J.: Special aspects on the behaviour of PCRV under extremely high core temperature loading. Nuclear Engineering and Design. 75, 291–302 (1982) Schneider U.: Kinetic of strength reducing reactions of concrete under high temperatures. PhD Dissertation (1973) Schneider U.: A contribution to creep and relaxation of concrete at elevated temperatures, Habilitation Thesis (1979) Schneider U.: Behaviour of concrete at high temperatures. Deutscher Ausschuss für Stahlbeton, Heft 337 (1982) Schneider, U.: Modelling of concrete behaviour at high temperatures. In: Anchor, R.D, Malhotra, H.L. and Purkiss, J.A. (eds.) Design of structures against fire, pp. 53–69. Elsevier Applied Science Publishers, London (1986) Schneider, U.: Concrete at high temperatures – A general review. Fire Safety Journal 13(1), 55–68 (1988) Schneider, U., Horvath, J., Behaviour of ordinary concrete at high temperature, distributed by, Institute of Building Materials, Vienna University of Technology, Austria, (2003)

Chapter 2

Introduction Alain Millard

Abstract As an introduction, this chapter summarizes in a first step the physical aspects of heated concrete. Interactions between moisture transport and physical changes, heat flux, pore pressure changes, microstructure and associated volumetric changes, mechanical properties and spalling phenomena are described. Main types of models are introduced in a second step. Concerning design purposes, codes and standards generally apply to classical situations, i.e. typical structural members, such as beams, columns and plates. Prescriptive methods and performance-based approaches are introduced. The possibilities offered by the routinely made calculations and their limitations are discussed. More comprehensive ThermoHydro-Chemo-Mechanical (THCM) models are needed for special applications. Refined mechanical constitutive equations are then required (as for example description of elastic, plastic, damage, thermal, creep and shrinkage strains changes during heating).

2.1

Physical Aspects of Heated Concrete

In general, moisture transport in concrete structures during fire may include air-vapour mixture flow due to forced convection, free convection, and infiltration through cracks and pores, vapour transport by diffusion, flow of liquid water due to diffusion, capillary action, or gravity, and further complications associated with phase changes due to condensation/evaporation, solid condensation/sublimation, and adsorption/desorption. Movement of air and water through the concrete is accompanied by significant energy transfer, associated with the latent heat of water and the heats of hydration and dehydration. At temperature largely above water critical point important chemical transformations of components of concrete take place. A. Millard (&) Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France e-mail: [email protected] © RILEM 2019 A. Millard and P. Pimienta (eds.), Modelling of Concrete Behaviour at High Temperature, RILEM State-of-the-Art Reports 30, https://doi.org/10.1007/978-3-030-11995-9_2

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At ambient temperature, HPC presents much better features than a normal concrete because of their lower permeability, lower porosity and higher compactness. In fact, at normal ambient temperatures, the cement matrix increases the strength of high performance concrete, because of its higher density and homogeneity, involving a better distribution of the stresses among aggregates compared to traditional concrete. At higher temperature this matrix becomes the weak point of the materials showing low mechanical strength: With increasing temperature, the aggregates progressively expand as long as they are not chemically altered, while the cement matrix, after an initial expansion, is subject (over 150 °C) to a progressive shrinkage. These two opposite phenomena involve a micro-cracking process that causes damaging of the material microstructure. Further, low permeability inhibits water mass transfer causing high gas pressure values, crack-opening and then an increase of intrinsic permeability. During fire the surface of a concrete element is heated both by a convective heat flux from the surrounding air of higher temperature and by a radiation heat flux, which can be direct (from the flames) or mutual (from other heated surfaces). The heating results in a gradual increase of the element temperature, starting from the surface zone. Due to this process the temperature gradients in this zone are high because almost all moisture must evaporate in the temperature range of 100–200 °C before a further temperature increase, which requires considerable amounts of heat. For this reason, even after several tens of minutes of fire duration the temperature of the inner part of wall remains almost unchanged (Gawin et al. 2003, 2006). Due to moisture evaporation, the water content in the surface zone decreases to a very low value and a sharp front, separating the moist and dry material, moves slowly inwards, see Fig. 2.1. At this front intensive evaporation takes place, increasing considerably the vapour pressure. The maximum values of vapour and gas pressures increase initially, (Kalifa et al. 2000), then they remain constant as the surface temperature increases and the front moves inwards (Gawin et al. 2003, 2006). The maximum value of gas pressure usually occurs at the position where the temperature equals approximately to 170–280 °C. In the region with lower temperatures, below about 130 °C, the gas pressure increase is caused mainly by a growth of the dry air pressure due to heating. In the regions with higher temperature the effects of a rapid increase of vapour pressure due to heating and temperature-dependence of the saturation vapour pressure predominate. At temperatures 160–350 °C the gas in the material pores consists mainly of the water vapour (Gawin et al. 2006). The gradients of vapour pressure cause the vapour flow both towards the surface and inwards. The latter mass flow results in vapour condensation when the hot vapour inflows the colder, internal layers of concrete, and in an increase of the pore saturation with liquid moisture above the initial value, usually referred as the so called “moisture clog” phenomenon, (England and Khoylou 1995). An additional increase of the liquid water volume in the pores is due to the release of chemically bound water (dehydration) and to the liquid thermal

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Fig. 2.1 The four observed zones and the process for the build-up of pressure

expansion, which is particularly important above the temperature of about 160 °C, (Gawin et al. 2002a). These effects cause a significant decrease of the gas permeability, because the space available for the gas is decreased. Increasing temperature causes the material expansion which in part is due to concrete dehydration (products of the thermal dissolution of concrete components have greater volume than their initial volume), in part due to the material cracking and progressive cracks opening (Schneider 1988), and finally due to ‘normal’ thermal expansion of the material skeleton. The concrete cracking during heating is caused by an incompatibility of thermal expansion of the aggregate and the cement paste, resulting in high traction stresses and development of local micro-cracks. Due to these cracks and chemical transformations of concrete (generally called dehydration), the concrete strength properties degrade gradually (Gawin et al. 2003, 2005).

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A thermal expansion of the external layers of a heated element is constrained by the core material which has significantly lower temperature. This causes a considerable macro-stress in the external layers of the element (traction in the direction perpendicular to the surface) and accumulation of elastic strain energy. Development of the cracks, both of thermo-chemical origin and the macro-stress induced ones, causes a considerable increase of the material intrinsic permeability (Gawin et al. 2002b, 2003), and thereupon a decrease in gas pressure in the external layers where high temperatures are observed (Dal Pont and Ehrlacher 2004). The highest values of gas pressure usually correspond to the temperatures 170–280 °C (Kalifa et al. 2000), and this is also the range where thermal spalling of concrete occurs. Physical causes of the phenomenon, and in particular a role played by elastic strain energy of a constrained thermal accumulated in the surface layer and by a high value of gas pressure, are discussed in detail in (Gawin et al. 2006). Hence, for concrete, particularly at high temperature, using advanced models, one cannot predict heat transfer only from the traditional thermal properties: thermal conductivity and volumetric specific heat. In the next section, a review of the approaches used in the last three decades for modelling the behaviour of concrete at high temperature is presented.

2.2

Modelling

Concerning design purposes, codes and standards generally apply to classical situations, i.e. typical structural members, such as beams, columns and plates. In case of non-conventional projects, it is necessary to turn to specific modelling tools to demonstrate, for example, the fire resistance of a building. Moreover, modelling tools can nowadays be contemplated as possible tools to optimize the HPC formulation itself, in order to improve its behaviour under high temperature. Obviously, there are a variety of modelling approaches, which on one hand reflect the complexity of the phenomena, and on the other hand correspond to the expected level of accuracy in the description of these phenomena. The most straightforward approach is based on the so-called prescriptive methods. It consists in assessing the load bearing capacity of a given structure (which can be either a structural member or a substructure) subjected to high temperatures, by applying codes and standards, such as Eurocode 2 (EN 1992-1-2 2004) for structural fire design. For example, in Eurocode 2, geometrical tabulated data are available to allow a design which guarantees the resistance of the structure under high temperature for a given time. Although very simple, this approach can be too severe and does not cover all the structural configurations (heating scenarios, geometries, loadings, support conditions, etc.), which can be encountered in practice. Therefore, there is an increasing trend to have recourse to performance-based approaches in order to optimize

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structure design. It consists in studying the response of part of the structure or the entire structure during the whole real heating process. When fire safety engineering approach is applied, the cooling phase is also taken into account. For this purpose, computer simulations are generally required, which can be more or less sophisticated, the differences coming from the scale and the level of description of the physico-chemical mechanisms. Nevertheless, whatever the level of accuracy or complexity, a thermal analysis is required to determine the evolution of the temperature distribution in the structure. A common practice consists in adopting a thermo-mechanical framework, together with phenomenological models. In this context, the simplest model is based on a simplified limit state analysis, known as the “isotherm method”. In this method, a temperature threshold is defined (around 400 °C for HPC), above which the concrete is supposed to be totally damaged, while below, the concrete is not affected by the temperature increase. Even though this simple method gives good results for example in case of normalized fires, it may give non-conservative results in case of slow heating rates and does not take into account indirect fire actions. Moreover, it cannot be used for the cooling phases since it does not capture the irreversible changes of the concrete strength. Most of the routinely made calculations resort to thermo-mechanical analysis where, in general, the temperature evolution is coupled to the mechanical analysis through the thermal strains and the variation of the mechanical properties (Young’s modulus, thermal expansion coefficient, compressive strength, etc.), whereas the mechanical evolution has no significant influence on the temperature. Therefore, the transient thermal and mechanical analysis are performed sequentially. Despite the fact that major phenomena like moisture transfer and chemical reactions, are not explicitly modeled, the thermo-mechanical approach can predict rather accurate results provided that ad-hoc phenomenological equations are established to account indirectly for their effect. For example, the variation of the concrete specific heat with respect to the temperature accounts for different hydro-chemical phenomena such as the vaporization of the free water around 100 °C, the decomposition of calcium carbonate above 400 °C, etc. A great effort must be put on the constitutive models in order to be able to describe the various strains which develop during heating, such as drying shrinkage, transient creep and transitional thermal creep strains, etc. These strain components can be carefully identified from experimental measurements, following for example the recommendations for test methods issued by the RILEM Committee 129-MHT (see references in Chap. 1). However, one main underlying assumption in these models is to consider the concrete as behaving like a sealed or an unsealed specimen, with respect to moisture transfer. It is obvious that, in reality, the situation is much more complex and generally lies between these two extreme cases, depending on the time-scale considered. Moreover, these models cannot account for the changes in pore pressure due to heating, and in particular to the vapour pressure increase which can partly explain spalling.

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Therefore, more comprehensive models are nowadays proposed in the literature, within a Thermo-Hydro-Chemo-Mechanical (THCM) framework. In these approaches, the concrete is considered as a porous medium, the pores of which are partly filled by gas (a mixture of air and vapour) and water. Mass, momentum and energy conservation equations are solved in a coupled way. Physico-chemical transformations such as evaporation-condensation, hydration-dehydration (chemically bound water is released from the solid skeleton upon heating), calcium carbonate decomposition, transformation of quartz, etc. are taken into account. The changes in the micro-structure, such as micro-cracking, and their consequences on macroscopic properties like porosity and permeability, can also be described. For this purpose, refined mechanical constitutive equations are required, which include the description of elastic, plastic, damage, thermal, creep and shrinkage strains. Moreover, many material properties depend upon the state variables evolution during heating. Multi-scale approaches are also possible (Meftah 2009), which enable a deep understanding of the different behaviours of the cement paste and of the aggregates and their interactions, and allow describing the various mechanisms at the relevant scale, thus requiring more intrinsic and physically based material data. Nevertheless, such sophisticated tools are not yet ready for everyday use.

References Dal Pont S., Ehrlacher A.: Numerical and experimental analysis of chemical dehydration, heat and mass transfer in a concrete hollow cylinder submitted to high temperatures. International Journal of Heat and Mass transfer 1(47):135–147 (2004) England, G.L., Khoylou, N.: Moisture flow in concrete under steady state non-uniform temperature states: experimental observations and theoretical modelling. Nuclear Engineering Design 156, 83–107 (1995) EN 1992-1-2, Eurocode 2 Design of concrete structures, Part 1.2 General rules – Structural fire design, (2004) Gawin, D., Alonso, C., Andrade, C., Majorana, C.E., Pesavento, F.: Effect of damage on permeability and hygro-thermal behaviour of HPCs at elevated temperatures, Part 1. Experimental results. Computers and Concrete 2(3), 189–202 (2005) Gawin, D., Pesavento, F., Schrefler, B.A.: Modelling of hygro-thermal behaviour and damage of concrete at temperature above critical point of water. International Journal of Numerical and Analytical Methods in Geomechanics 26(6), 537–562 (2002a) Gawin, D., Pesavento, F., Schrefler, B.A.: Modelling of thermo-chemical and mechanical damage of concrete at high temperature. Computer Methods in Applied Mechanics and Engineering 192, 1731–1771 (2003) Gawin, D., Pesavento, F., Schrefler, B.A.: Simulation of damage–permeability coupling in hygro-thermo-mechanical analysis of concrete at high temperature. Communications in Numerical Methods in Engineering 18(2), 113–119 (2002b) Gawin, D., Pesavento, F., Schrefler, B.A.: Towards prediction of the thermal spalling risk through a multi-phase porous media model of concrete. Computer Methods in Applied Mechanics and Engineering 195(41–43), 5707-5729 (2006) Kalifa, P., Menneteau, F.D., Quenard, D.: Spalling and pore pressure in HPC at high temperatures. Cement and Concrete Research 30, 1915–1927 (2000)

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Meftah, F.: A meso-scale thermo-hygro-mechanical analysis of micro-structure effects on heated concrete behaviour. In: F. Dehn, E. Koenders (eds.), 1st International Workshop on Concrete Spalling due to Fire Exposure, Leipzig (2009) Schneider, U.: Concrete at high temperatures – A general review. Fire Safety Journal 13(1), 55–68 (1988)

Chapter 3

Engineering Modelling Sven Huismann, Matthias Zeiml, Manfred Korzen and Alain Millard

Abstract This chapter presents the most commonly used approach to analyse the thermo-mechanical behaviour of concrete structures subjected to high temperatures as in the case of fire loading. Prescriptions of the Eurocode are detailed for the thermal as well as the mechanical analysis. Finally, recommendations from two national (Austrian and German) guidelines give some improvements for underground infrastructure.

3.1

General Approach

Engineering models generally serve the purpose of producing numerical results of sufficient accuracy for engineering purposes, with acceptable computational costs. Hence, simplifications are applied whenever suitable, provided that they only introduce deviations from reality with acceptable magnitude. This means that the level of simplification is chosen with regard to the complexity of the problem under consideration. Most engineering models can be classified as thermo-mechanical models. During the analysis two consecutive steps are applied. First, the thermal field is determined during a thermal analysis. Based on the obtained information on the S. Huismann (&)
M. Korzen Bundesanstanlt for Materialforschung Und Prüfung (BAM), Berlin, Germany e-mail: [email protected] M. Korzen e-mail: [email protected] M. Zeiml University of Innsbruck, Material Technology, Innsbruck, Austria e-mail: [email protected] A. Millard Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France e-mail: [email protected] © RILEM 2019 A. Millard and P. Pimienta (eds.), Modelling of Concrete Behaviour at High Temperature, RILEM State-of-the-Art Reports 30, https://doi.org/10.1007/978-3-030-11995-9_3

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distribution of temperature and, hence, mechanical material properties and resulting strains and stresses, the mechanical behaviour of the structure and/or structural component is simulated. At the moment, a variety of national and international standards and guidelines are available. In the following section, the Eurocode is presented as the most prominent in Europe. Most of the regulations and recommendations included in the Eurocode are derived and developed with respect to fire loading corresponding to the standard temperature-time fire curve according to EN 1991-1-2, 2010 (which also corresponds to the ISO 834 standard). Therefore, selected national guidelines focusing on more severe fire-loading scenarios such as might occur in tunnel fires are presented in addition to the Eurocode.

3.2

Eurocode

According to Eurocode 2 (EN 1992-1-2, 2004), a structural member or a structure is considered to fulfil the design requirements in the load case fire, provided that Ed;fi  Rd;t;fi

ð3:1Þ

where, Ed,fi is the design effect of actions for the fire situation, determined in accordance with Eurocode 1 (EN 1991-1-2, 2010), including effects of thermal expansions and deformations, and Rd,t,fi is the corresponding design resistance in the fire situation. In general, the fire design of concrete structures or structural components is performed in two consecutive steps. First, a thermal analysis gives access to the temperature distribution within the member. Within the subsequent mechanical analysis, the previously-determined temperature field serves as input for the simulation of the mechanical behaviour of the structural member and/or structure.

3.3

Thermal Analysis

During the thermal analysis, the energy-conservation equation is solved considering the appropriate fire-loading curve (temperature-time curve) and thermal boundary conditions. For the thermal material properties of concrete, empirical temperature-dependent functions are given in Eurocode 2 (see Fig. 3.1 and (EN 1992-1-2, 2004) for details on the corresponding empirical functions). These phenomenological functions indirectly account for effects related to mass transport. The function for the specific heat (see Fig. 3.1b) accounts for vapourisation of pore water by a peak starting at 100 °C the magnitude of which depends on the moisture content.

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Fig. 3.1 Thermal material properties of concrete according to Eurocode 2 (EN 1992-1-2, 2004): a thermal conductivity and b specific heat for different values of the moisture content by weight

For fire loading following the standard temperature-time fire curve according to EN 1991-1-2, 2010, the corresponding temperature field can be determined directly from Eurocode 2 where pre-calculated temperature distributions are available for typical types of members (e.g. slabs, beams, columns) and selected time durations (i.e., 30, 60, 90, 120, 180, 240 min).

3.4

Mechanical Analysis

Within the design process, the following design methods can be used in order to satisfy condition (3.1):

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– detailing according to recognised design solutions (tabulated data or testing) – simplified calculation methods for specific types of members – advanced calculation methods for simulating the behaviour of structural members, parts of the structure or the entire structure. The design method shall be chosen based on the design task under consideration: – tabulated data according to Eurocode 2: Tabulated data given in Eurocode 2 up to a fire duration of 240 min are based on experimental results obtained with ISO-fire loading. For typical design situations and structural members (e.g., slabs, beams, columns), minimum design requirements are given for, e.g. member thickness (dimensions) and axis distance, depending on the required fire resistance (e.g., REI 90). For temperature loading outside the scope of the standard temperature-time fire curve or structural design situations other than those listed in the code, this design method cannot be applied. – simplified calculation methods according to Eurocode 2: Simplified calculation methods apply to the fire resistance of structural components or parts of fire-exposed structures. The underlying temperature profiles can be determined either from tests or by calculation (see also the temperature distributions in the code for typical types of members such as slabs, beams or columns and selected time durations). Subsequently, the structural resistance of a structural member or the critical section of a substructure is in particular determined by reducing its cross-section. Hereby, all concrete regions with temperatures exceeding 500 °C are disregarded in the design process and all other concrete regions are considered with their original strength at room temperature. The strength of the reinforcement is determined based on the actual temperature in the respective region. Finally, the load-bearing capacity of the member is determined with conventional calculation methods that are also applicable for room temperature. The so-called zone method, which is applicable for the standard temperature-time curve only, applies a similar approach. Hereby, the member is subdivided into a certain number of zones of equal thickness. For each zone, the corresponding values for temperature and relevant mechanical parameters for concrete are determined. Then, the thickness for a theoretical damaged region is determined based on the calculated values and this region is disregarded within the subsequent determination of the load-bearing capacity. Calculation of the latter again follows conventional calculation methods which are also applicable for room temperature. – advanced calculation methods according to Eurocode 2: According to the code, the aim of advanced calculation methods is to provide a realistic analysis of structures exposed to fire. They shall be based on fundamental physical behaviour leading to a reliable approximation of the expected behaviour of the relevant structural component under fire conditions. Advanced

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calculation methods apply to the analysis of the fire resistance of structural components, parts of fire-exposed structures or the entire structure. The temperature field can be determined in a similar manner as for simplified calculation methods. According to the code, the influence of moisture content and of migration of moisture may be conservatively neglected within a thermal analysis. Within the mechanical analysis, the code requires that the applied model shall be based on the acknowledged principles and assumptions of the theory of structural mechanics, taking into account the changes of mechanical properties with temperature. Furthermore, the effects of thermally induced strains and stresses both due to temperature rise and due to temperature differentials, shall be considered. Additionally, deformations shall be checked with respect to compatibility of all structural components. Moreover, a comment on spalling (and falling-off) is included in the code, with special attention to the compressive zone of concrete. Simulation tools, generally known as nonlinear thermo-mechanical simulation tools, which fulfil all above-mentioned prerequisites are hence able to simulate the realistic build-up of thermal strains depending on the governing boundary conditions. The corresponding mechanical parameters for concrete are given for normal-weight concrete with siliceous or calcareous aggregates (see Fig. 3.2 for the compressive strength and refer to EN 1992-1-2, 2004 for the tensile strength and for details on the corresponding empirical functions). The relevant material properties are the compressive strength fc(T) and the corresponding strain ec1(T). The specified phenomenological stress-strain relationship implicitly takes into account all relevant mechanical strain components (i.e. stress-related strain, creep strain, transient strain). Originally, the above values were developed for normal-strength concrete. In the latest version of Eurocode 2, the range of the compressive strength fc(T) was extended to cover high-strength concretes also. The thermal strain is given as an empirical temperature-dependent function depending on the type of aggregate (see Fig. 3.3 and refer to Eurocode 2 for the corresponding empirical functions) since the latter governs the thermal strain of concrete.

3.5

Spalling

Only limited information and recommendations are included in Eurocode 2 concerning spalling of fire-loaded concrete, showing the demand for further research in this field. According to the code, spalling is unlikely to occur when the moisture content of the concrete is less than a certain moisture content k [% by weight]. This value is

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Fig. 3.2 Stress-strain relationship for concrete under compression at elevated temperatures (EN 1992-1-2, 2004)

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Fig. 3.3 Stress-strain relationship for concrete under compression at elevated temperatures (EN 1992-1-2, 2004)

specified in the national annex to Eurocode 2 for each country. The recommended value is 3%. For moisture contents greater than k, a more accurate assessment of moisture content, type of aggregate, permeability of concrete and heating rate should be considered. From a scientific point of view, this simplistic recommendation is to be questioned, since it is commonly accepted that spalling is caused by the interaction of numerous thermo-hydro-chemo-mechanical processes with the moisture content being only one of numerous influencing parameters.

3.6

Examples of National Guidelines

Since most of the recommendations of the Eurocode are valid only for fire-loading scenarios up to the standard temperature-time fire, national guidelines were established for more severe fire-loading scenarios. Such fire-loading scenarios are to be considered, e.g. in tunnels and other underground structures. Austrian ÖBV-guideline “Improved structural fire protection with concrete for underground infrastructure” (ÖBV 2013) This Austrian guideline was established in order to specify the details and typical areas of application of numerical models with different levels of complexity within the model category defined in the Eurocode 2 as advanced calculation methods. In this guideline it is suggested that determination of the thermal loading and the resulting temperature distributions within the structural members should either be based on a thermal analysis according to Eurocode 2 or on empirical temperature distributions determined from fire experiments (Kusterle et al. 2005) with

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fire-loading conditions typical for tunnel fires (see Fig. 3.4 and (ÖBV 2013) for details as well as the numerical values corresponding to the displayed curves). For the subsequent mechanical analysis following the recommendations given in Eurocode 2, (ÖBV 2013) distinguishes between linear and nonlinear model assumptions: – linear model assumptions: In order to be used in standard linear-elastic beam-spring models, the nonlinear temperature distribution resulting from the thermal analysis is transformed into a so-called equivalent temperature (see Kusterle et al. 2005; Ring et al. 2013 for details). The internal forces caused by the governing load combination for the design case fire are determined considering linear-elastic material behaviour of both concrete and reinforcing steel. – nonlinear model assumptions: In the case that nonlinear model assumptions are applied, the nonlinear temperature distributions are incorporated directly into the mechanical analysis model. The internal forces are determined considering elasto-plastic material behaviour and the mechanical material parameters according to Eurocode 2 (see Sect. 3.5 for details). Based on the above statements of linear and nonlinear model assumptions, recommendations are given on when linear or nonlinear model assumptions are to be applied. Based on numerical simulations of fire experiments and benchmark examples (see Ring 2012; Ring et al. 2013a, b for details), the main influencing parameters determining the choice of the model assumptions are the duration of the

Fig. 3.4 Empirical temperature distributions for structural fire design according to (ÖBV 2013) (curves are valid for PP-fibre reinforced concrete with a moisture content comparable to (Kusterle et al. 2004) and a dry density between 2000 and 2600 kg/m3; curves cover the following fire-loading scenarios: RWS, HCinc, HC, EUREKA, ISO 834)

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fire (i.e. the required fire resistance) and the complexity of the structure (e.g. rectangular, frame-like structures, arched or circular cross sections). German DB-guideline 853 “Design, construction and maintenance of railway tunnels” (DB-guideline 853 2011) This German guideline deals with, among numerous other topics, the structural design of tubbing tunnels and recommends a specific type of advanced calculation method (as defined in Eurocode 2). Hereby, the special model feature of tubbing tunnels, namely nonlinear hinges representing the joints between tubbing segments, is incorporated into the model by the following design steps: – determination of time-dependent internal forces due to thermal restraint: The structural model for determining the internal forces due to thermal restraint represents a tubbing tunnel without nonlinear hinges representing the joints. The overall stiffness of the numerical model is determined at the time instant where the restraint forces reach their maximum values. – determination of the restraint internal forces for a unit temperature loading: With the same structural model as in the previous step, the internal forces due to thermal restraint are determined with a unit temperature loading (i.e. a fixed value for the components of the so-called equivalent temperature). – determination of unit temperatures for the given structural model: The unit temperatures for the structural model under consideration are determined by the quotient of the internal stress resultants of the two above-mentioned analysis steps. – fire simulation of final tunnel structure: With the temperature values determined in the previous analysis step, the final structural model (i.e. tubbing tunnel including the nonlinear hinges representing the joints between tubbing segments) is analysed. Hereby, the previously determined stiffness is considered. The reason for recommending this rather laborious analysis scheme is that common engineering software tools can consider nonlinear material behaviour together with nonlinear temperature distributions (that typically develop in concrete with its low heat diffusivity), but are not able to consider these features together with nonlinear hinges at nodes, which are necessary to take into account all relevant nonlinearities present in common tubbing tunnels. Nonlinear hinges are currently incorporated only in standard beam-spring models which in turn cannot incorporate nonlinear temperature distributions. Since software tools will evolve in the future, eventually becoming capable of incorporating all above-mentioned nonlinearities in one numerical model, the described analysis scheme will become obsolete and a direct non-linear analysis (in the sense of an advanced calculation method according to Eurocode 2) will be possible.

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Discussion, Limits

Engineering models are mainly developed for a straight-forward structural design under fire loading, i.e. the verification that a specific structure is able to withstand the fire for a specific period of time, expressed as a fire resistance (e.g. REI90). Engineering models in general do not consider the cooling phase since numerous nonlinear and irreversible processes require to take into account hysteretic effects, which is a quite challenging task. For standard problems, which are the majority in engineering practice, there are no significant differences between the numerical results obtained from engineering models and those obtained from advanced models (Franssen 2004) (see the following section for details on advanced models). Some of the main differences between engineering models and advanced models are – Engineering models perform a thermo-mechanical analysis implicitly taking into account hydro-chemical phenomena via empirical material functions; advanced models perform a fully-coupled thermo-hydro-chemo-mechanical analysis taking into account in detail the relevant interactions. – Engineering models consider strain components implicitly by empirical stress-strain relationships (e.g. according to Eurocode 2); advanced models consider the different strain components explicitly (see following section for details). The aim of engineering models is to simplify the analysis in order to optimize the computational cost and necessary manpower. The simplification has to be limited, however, to an extent at which the quality of prediction is still within the boundaries of reasonable accuracy. Since these boundaries as well as the quality of prediction of a certain model depends on the problem under consideration, the level of idealization and simplification has to be chosen with regard to the complexity of the design task under consideration. Concerning spalling, only limited information has been implemented in standards and guidelines. At the most, recommendations on investigating/considering spalling from an empirical point of view are included. In some guidelines, it is only mentioned that, or under which circumstances, spalling needs to be taken into account within structural simulations. This clearly shows the need for the establishment of a common standard spalling assessment, whether experimental, based on a risk analysis or through modelling.

References EN 1992-1-2:2004: Eurocode 2: Design of concrete structures – Part 1–2: General rules – Structural fire design (2004) EN 1991-1-2:2010: Eurocode 1: Actions on structures – Part 1–2: General actions – Actions on structures exposed to fire (2010)

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Franssen, J.-M.: Plastic analysis of concrete structures subjected to fire. Fire design of concrete structures: What now? What next?, Milan, 133–145 (2004) ÖBV-guideline “Improved structural fire protection with concrete for underground infrastructure” (“Erhöhter baulicher Brandschutz mit Beton für unterirdische Verkehrsbauwerke”). Austrian Society for Construction Technology, in German (2013) Kusterle, W., Lindlbauer, W., Hampejs, G., Heel, A., Donauer, P.-F., Zeiml, M., Brunnsteiner, W., Dietze, R., Hermann, W., Viechtbauer, H., Schreiner, M., Vierthaler, R., Stadlober, H., Winter, H., Lemmerer, J., Kammeringer, E.: Brandbeständigkeit von Faser-, Stahl- und Spannbeton [Fire resistance of fiber-reinforced, reinforced, and prestressed concrete]. Tech. Rep. 544, Bundesministerium für Verkehr, Innovation und Technologie, Vienna, in German (2004) Ring, T., Zeiml, M., Lackner, R.: Underground concrete frame structures subjected to fire loading: Part II – re-analysis of large-scale fire tests. Engineering Structures, in print (2013a) Ring, T., Wikete, C., Kari, H., Zeiml, M., Lackner, R.: Der Einfluss des Rechen- und Materialmodells auf die Strukturantwort bei der Simulation von Tunnel unter Brandbelastung [The influence of numerical and material model on the structural response of tunnels subjected to fire loading], Bauingenieur 88, p. 35–44, in German (2013b) Ring, T.: Experimental characterization and modelling of concrete at high temperatures – structural safety assessment of different tunnel cross-sections subjected to fire loading. PhD thesis, Vienna University of Technology, Vienna, Austria (2012) DB-guideline 853 “Design, construction and maintenance of railway tunnels” (“Eisenbahntunnel planen, bauen und instandhalten”). Deutsche Bahn AG, in German (2011)

Chapter 4

Advanced Modelling Fekri Meftah, Francesco Pesavento, Colin Davie, Stefano Dal Pont, Matthias Zeiml, Manfred Korzen and Alain Millard

Abstract In this chapter, advanced models based on a general ThermoHydro-Chemo-Mechanical (THCM) framework are considered. In these approaches, the concrete is considered as a porous medium, the pores of which are partly filled by gas (a mixture of air and vapour) and water. First, the main physicochemical phenomena that take place in heated concrete are explained. Then, mass, momentum and energy conservation equations as well as the constitutive model are detailed for the general model considering three fluid pressure fields. Indications are also given for the numerical solution of the obtained final equations. Finally, a simplified model based on a single fluid pressure field is presented.

F. Meftah (&) Institut National des Sciences Appliquées, Rennes, France e-mail: [email protected] F. Pesavento Dipartimento di Ingegneria Civile, Edile e Ambientale, Universita degli Studi di Padova, Padua, Italy e-mail: [email protected] C. Davie School of Engineering, Newcastle University, Newcastle upon Tyne, UK e-mail: [email protected] S. D. Pont Laboratoire 3SR, Université Grenoble-Alpes, Grenoble, France e-mail: [email protected] M. Zeiml Material Technology, University of Innsbruck, Innsbruck, Austria e-mail: [email protected] M. Korzen Bundesanstanlt for Materialforschung und prüfung (BAM), Berlin, Germany e-mail: [email protected] A. Millard Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France e-mail: [email protected] © RILEM 2019 A. Millard and P. Pimienta (eds.), Modelling of Concrete Behaviour at High Temperature, RILEM State-of-the-Art Reports 30, https://doi.org/10.1007/978-3-030-11995-9_4

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28

4.1

F. Meftah et al.

Introduction

The advanced models can be classified into two main categories: models essentially focused on the thermo-mechanical behaviour of concrete, and the models considering also the fluid phases (one or more) inside the pores of the material. Firstly, we will discuss about the models considering the fluids inside the pore network of concrete. Then, we will illustrate the more commonly used, purely thermo-mechanical models (Sect. 4.6). The first class of models aims at the simulation of the overall thermo-hygral and mechanical behaviour of the material, considering sometimes also some chemical aspects (e.g. the reactions of hydration and dehydration). To analyse hygro-thermal phenomena in porous media, two different approaches are currently used: phenomenological and mechanistic ones. In phenomenological approaches (Bažant and Thonguthai 1978, 1979; England and Khoylou 1995; Abdel-Rahman and Ahmed 1996) moisture and heat transport are described by diffusive type differential equations with temperature- and moisture content-dependent coefficients. The model coefficients are usually determined by inverse problem solution, starting from experimental tests results, in order to obtain the best agreement between theoretical prediction and experimental evidence. A characterisitc of such models is that they are very accurate for interpolation, but rather poor for extrapolation of the known experimental results. Moreover, various physical phenomena are lumped together and important processes, such as phase changes, especially at high temperatures where they play a relevant role, are neglected. Mechanistic models (Bažant and Kaplan 1996; Consolazio et al. 1998; Ulm et al. 1999a, b; Gawin et al. 1999, 2002a) are more complicated from a mathematical point of view and, contrary to phenomenological ones, their coefficients have a clear physical meaning. This kind of model is often obtained from microscopic balance equations written for each constituent of the medium, which are then averaged in the space applying special averaging operators. Several mathematical and numerical models, usually based on extensive laboratory tests, have been developed for the analysis of heat and mass transfer in concrete at high temperature, e.g. Gawin et al. (2003, 2004, 2006), Tenchev et al. (2001, 2005), Ichikawa and England (2004), Davie et al. (2006), Chung et al. (2006), Dwaikat and Kodur (2009). However, one should remember that every interpretation possesses all limitations and simplifications that are assumed in the mathematical model on which the simulations are based. Thus, for this purpose one should use models considering possibly the whole complexity and mutual interactions of the analysed physical processes. Hence, in the case of concrete at high temperature, a proper choice may be models based on mechanics of multiphase porous media, taking into account chemical reactions (dehydration), phase changes, cracking and thermo-chemical material degradation, as well as their mutual couplings and influence on the hygral, thermal, chemical and mechanical properties of concrete. These physical aspects of the behaviour of concrete subjected to high temperature are described below.

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4.2

29

Concrete as Multiphase Porous Material

Concrete is essentially a mixture of two components: aggregates and paste. The aggregate component is usually sand, gravel or crushed stone. It is mainly responsible for the unit weight, the elastic modulus, and the dimensional stability of concrete. Therefore, it has a very important effect on the volume changes of concrete exposed to high temperatures. The paste component is typically used to bind aggregates in concrete and mortar. It is a porous medium, which is composed of a solid skeleton, produced from the hydration of Portland cement, and pores, which are filled by different fluid phases. The principal solid phases generally present in the hydrated cement paste (hcp), that can be resolved by an electron microscope, are the calcium silicate hydrates C-S-H, which are very important in determining the properties of the paste such as strength and permeability, and calcium hydroxide Ca(OH)2 (also called portlandite). The pore structure is relevant for concrete. It contributes to the mechanical strength of concrete, but it also allows interaction with the external environment, which takes place through the connected pores. Besides, it is the container of the liquid water and gas phases (vapour and dry air). Figure 4.1 gives a schematic representation of cement paste seen as a partially saturated medium. For a given Representative Elementary Volume (REV) X, the volume of pores Xp allows the porosity to be defined u = Xp/X. Furthermore, we introduce the degree of saturation of pores with liquid water as Sl = Xl/Xp. It is an experimentally determined function of the relative humidity RH (obtained through the sorption isotherms). The remaining volume of pores is filled with a gaseous mixture with a gas saturation: Sg = (1 − Sl) = Xg/Xp (Xl and Xg are the volumes of the liquid and gas phases respectively). The interfacial surface tension of water in a capillary pore leads to a concave meniscus between liquid water and gas phase, see Fig. 4.2. This may give rise to a discontinuity in fluid pressure. The difference between the liquid water pressure pl

Fig. 4.1 Schematic representation of a non-saturated porous material

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F. Meftah et al.

Fig. 4.2 Schematic representation of a capillary pore

and the gas pressure pg = pv + pa (pv being the pressure of water vapour and pa the pressure of the dry air) is called the capillary pressure pc and is a function of the liquid water saturation Sl: pc ðSl Þ ¼ pg  pl

ð4:1Þ

Assuming the contact angle c between the liquid phase and the solid matrix to be zero, the capillary pressure of water pc can be related to the pore radius r with the Laplace equation: pc ¼

2rðT Þ ¼ 2rðT Þv r

ð4:2Þ

where r(T) is the surface tension of water which depends upon temperature, and v is the curvature of the capillary meniscus. Any change in the curvature of meniscus will change the equilibrium between liquid and vapour phases. A relationship between the liquid water and the vapour can be obtained by means of Kelvin’s equation considering that liquid is incompressible and the vapour is a perfect gas. Water can exist in the hydrated cement paste in many forms: • capillary and physically adsorbed water; their loss is mainly responsible for the shrinkage of the material while drying • interlayer water; it can be lost only during strong drying, which leads to considerably shrinkage of the C-S-H structure All these types of water can evaporate at a temperature of 105 °C (at atmospheric pressure and slow rate of heating). Another type of water, which is non-evaporable, is chemically bound water. It is considered to be an integral part of the structure of various cement hydration products and it is released when the

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x- interlayer water o - adsorbed water A - inter particles links B - CSH layers

Fig. 4.3 Probable structure of hydrated silicates (Feldman and Sereda 1968)

hydrates decompose on heating. Based on the Feldman-Sereda model (Feldman and Sereda 1968), the different types of water associated with the C-S-H are illustrated in Fig. 4.3. In fact, there is a third phase in the concrete microstructure, known as the transition zone, which represents the interfacial region between the particles of coarse aggregates and the hcp. Concrete has micro-cracks in the transition zone even before a structure is loaded. This has a great influence on the stiffness of the concrete. In the composite material, the transition zone serves as a bridge between the two components: the mortar matrix and the coarse aggregate particles. Therefore, the broken bridges (i.e. voids and micro-cracks in this zone) would not permit the transfer of stresses (Mehta and Monterio 2014). For what has been described above, concrete can be treated as a multiphase/ multicomponent porous material. Because of this, in the last fifteen years, mechanics of multi-phase porous media proved to be a theory that enables an efficient modelling of cement-based materials at high temperature (Gawin et al. 1999, 2003, 2006; Davie et al. 2006). It allows for consideration of the porous and multiphase nature of the materials, their chemical transformations, water phase changes and different behaviour of moisture below and above the critical temperature of water, mutual interactions between the thermal, hygric and degradation processes, as well as several material nonlinearities, especially those due to temperature changes, material cracking and thermo-chemical degradation. In modelling, it is usually assumed that the material phases are in thermodynamic equilibrium state locally. In this way their thermodynamic state is described by one common set of state variables and not by separate sets for every component of the material, which allows the number of unknowns in a mathematical model to be reduced. Thus we have, for example, one common temperature for the multi-phase material, and not different temperature values for the skeleton, liquid water, vapour and dry air, etc. When fast hygro-thermal phenomena in a material at high temperature are analysed, the assumption is debatable, but it is almost always

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used in modelling, giving reasonable results from a physical point of view and in part confirmed by the available experimental results. Concrete is here considered to be a multiphase medium where the voids of the solid skeleton could be filled with various combinations of liquid- and gas-phases (Fig. 4.2). In the specific case the fluids filling the pore space are the moist air (mixture of dry air and vapour), capillary water and physically adsorbed water. The chemically bound water is considered to be part of the solid skeleton until it is released on heating. Below the critical temperature of water Tcr, the liquid phase consists of physically adsorbed water, which is present in the whole range of moisture content, and capillary water, which appears when the degree of water saturation Sl exceeds the upper limit of the hygroscopic region, Sssp (i.e. below Sssp there is only physically adsorbed water). Above the temperature Tcr the liquid phase consists of the adsorbed water only. In the whole temperature range the gas phase is a mixture of dry air and water vapour. Thanks to the mechanics of porous media and the schematisation of the material described above, it is possible to analyse three different classes of phenomena which characterise the behaviour of concrete at high temperature: • Hygral phenomena • Thermal phenomena • Mechanical phenomena

4.2.1

Hygral Phenomena

When temperature increases in a concrete structure (e.g. in a wall), water vapour pressure continuously increases in a zone close to the heated surface. This derives principally from the evaporation of water inside the wall, when the temperature reaches and passes the boiling point of water. Vapour pressure is also due to the water that is liberated during the dehydration of cement paste. This increase of the water vapour pressure in the hot region will create a thermodynamic imbalance between the hot and the cold regions. This will entail a diffusion process of the water vapour and of the dry air through the wall and towards the external atmosphere in order to maintain the equilibrium between liquid and vapour (see Fig. 4.4). For an appropriate prediction of the moisture distribution in a concrete structure subjected to high temperatures one needs to know the material properties that control the movement of the fluids inside the porous medium. Permeability and diffusivity are the most important properties of the cementitious materials. These are very sensitive to porosity changes or micro-cracking phenomena. In fact, the increases in permeability and porosity of such materials are currently accepted as providing a reliable indication of their degradation whether it has thermal,

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33

Fig. 4.4 Mass transport mechanism in a non-saturated porous medium subjected to heating (Gens and Olivella 2001)

mechanical or physico-chemical origins. Therefore, in this document these properties and their evolutions when the concrete is subjected to high temperatures will be studied.

4.2.2

Thermal Phenomena

Concerning thermal phenomena, it can be stated that in most cases the main mechanism for heat transport is heat conduction. Heat conduction responds to gradients of temperature T. However, additional heat transfer will also be accomplished by advection due to the movement of the three phases: solid, liquid and gas. The latent heat inherent to phase changes may also have significant thermal effects (Gens and Olivella 2001) (Fig. 4.5). The evolution of the temperature distribution in any structure is governed by the thermal properties of the material, particularly heat capacity and thermal conductivity. In the case of concrete, it is difficult to determine these properties because of the numerous phenomena that occur simultaneously within the microstructure of concrete and cannot be separated easily. These phenomena are affected in particular

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F. Meftah et al.

Fig. 4.5 Heat transport mechanism in a porous medium (Gens and Olivella 2001)

by the evolution of the porosity, by the moisture content, by the type and amount of aggregate, by changes in the chemical composition and by the latent heat consumption generated by certain chemical phenomena. Because of these effects, a unique relationship cannot rigorously describe the dependence of concrete properties on temperature (Harmathy 1965, 1970).

4.2.3

Mechanical Phenomena

When mechanically loaded and simultaneously heated, the overall measured strain of concrete is assumed as an additive combination of different components. These components can be conventionally classified into three families according to the origin of the driving mechanisms: – Mechanical strains that occur due to an applied mechanical load only. Elastic strain, cracking strain and basic creep strain are the main mechanical components

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– Thermo-hygral strains related to the occurrence of physico-chemical processes within the material such as drying, temperature increase and dehydration. Thermal expansion and shrinkage are the most important components. They are carried out in (mechanically) load-free configurations. Interaction strains are additional components generated when the abovementioned physico-chemical processes occur with a concomitant applied load. In this case, the overall measured strain differs from the sum of all strains induced by each single mechanism. For instance, additional elastic strain, due to the evolution of Young’s modulus with temperature, is an illustration of interaction strains. Also, drying and dehydration creep strains belong to this category.

4.3

Heat and Mass Transfer

The general approach to heat and mass transfer processes in a partially saturated open porous medium is to start from a set of balance equations governing the time evolution of mass and heat of the solid matrix and the fluids filling the porous network, taking into account the exchange between the phases and with the surrounding medium. These balance equations are supplemented with an appropriate set of constitutive relationships, which permit a reduction in the number of independent state variables that control the physical process under investigation (Hassanizadeh and Gray 1979a, b, 1980; Lewis and Schrefler 1998; Pesavento 2000; Schrefler 2002). In the following, the full set of balance equations will be presented. The governing equations of the model are given in terms of the chosen state variables: the capillary pressure pc the gas pressure pg and the temperature T. This choice is of particular importance: the chosen quantities must describe a well-posed initial-boundary value problem, should guarantee a good numerical performance of the solution algorithm, and should make their experimental identification simple. A detailed discussion about the choice of state (i.e. primary) variables can be found in Sect. 4.3.

4.3.1

Balance Equations

The balance equations can be obtained by using the procedure of space averaging of the microscopic balance equations, written for the individual constituents of the medium (i.e. the governing equations at local level). The theoretical framework is based on the works of Hassanizadeh and Gray (1979a, b, 1980), Gray and Schrefler (2007) and Lewis and Schrefler (1998). A detailed description of the procedure can be found also in (Pesavento 2000, Gawin et al. (2003).

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For the sake of brevity, only the form of the conservation equations at macroscopic level is given (Gawin et al. 2003; Dal Pont et al. 2001). Hence, mass balance equations read, for the solid matrix: @ms þ r  ðms vs Þ ¼ m_ dehyd @t

ð4:3Þ

@ml þ r  ðml vl Þ ¼ m_ vap  m_ dehyd @t

ð4:4Þ

@mv þ r  ðmv vv Þ ¼ m_ vap @t

ð4:5Þ

@ma þ r  ð m a va Þ ¼ 0 @t

ð4:6Þ

for the liquid water:

for the vapour:

and for the dry air:

where mp is the mass per unit volume of porous medium of each constituent: ms ¼ ð1  uÞqs ; ml ¼ ql Sl u mv ¼ qv ð1  Sl Þu; ma ¼ qa ð1  Sl Þu

ð4:7Þ

in which qp is the corresponding density, u the porosity and Sl the degree of saturation of pores with liquid water, the complementary part of the pore volume being filled with the gas mixture. These equations take into account phase changes due to dehydration, evaporationevaporation and condensation phenomena (Gawin et al. 2003; Dal Pont and Ehrlacher 2004; Sabeur et al.2008; Sabeur and Meftah 2008). Source terms, corresponding to dehydration mass rate m_ dehyd and to evaporationevaporation/condensation mass rate m_ vap , are therefore considered. The dehydration mass rate m_ dehyd is the mass loss rate due to the release of chemically-bound water from the solid phase when temperature increases and exceeds a threshold value of (conventionally) 105 °C. This threshold is hardened to fit continuously the greatest temperature attained by the porous medium. After cooling and heating again, dehydration restarts only when the temperature exceeds the actual (hardened) threshold, which is the medium’s memory with regard to temperature history. Hence, the occurrence of dehydration may be described by a first order kinetics model (Feraille 2000; Dal Pont et al. 2004; Sabeur et al. 2008):

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37

m_ dehyd ¼ 

D E mdehyd  meq dehyd

þ

sdehyd

ð4:8Þ

where the symbol hi þ stands for the positive value of the argument (otherwise the value is set to zero), sdehyd is the characteristic time and meq dehyd ðT Þ is the dehydration mass at equilibrium, that is, the mass loss when the temperature rate T_ is slow enough; i.e. when the time interval of variation of temperature is great enough with regard to sdehyd . Alternatively, a time impendent evolution law may be adopted (Gawin et al. 2003) by considering that mass loss attains systematically its equilibrium value irrespective of the temperature rate: m_ dehyd ¼

@meq dehyd  _  T þ @T

ð4:9Þ

Note that meq dehyd ðT Þ is a negative function of temperature since it corresponds to mass loss. In most applications, the term r  ðms vs Þ  ms r  vs may be neglected since it corresponds to the effect of solid deformability (volume variation) on transfer mechanisms. Under this simplified assumption, the solid mass balance equation (4.3) may be used only to determine the evolution of porosity appearing in the mass balance equations of the fluids according to: @u m_ dehyd ¼ @t qs

ð4:10Þ

where the solid phase density qs is assumed here to be constant, otherwise the variation of qs with time should appear in Eq. (4.10). It is worth noting that Eq. (4.10) is strictly local since the effect of the deformation of the solid skeleton on the porosity has been neglected in the adopted context of small-strain deformations. In this case, the variation of porosity of the medium is only due to the temperature induced dehydration: porosity increases with a rate that is proportional to dehydration rate. A more general presentation including deformability of the solid skeleton is discussed in Sect. 4.2.3. Moreover, fluid velocities vp in conservation equations (4.4)–(4.6) are split into relative components in order to describe mass transport within the porous network, by both permeation and diffusion phenomena due to pressure and concentration gradients, respectively. They read: vl ¼ vs þ vls  vls vp ¼ vs þ vgs þ vpg  vgs þ vpg

with p ¼ v; a

ð4:11Þ

where vs (which can be neglected) is the velocity of the solid phase, vp−s is the velocity of the liquid water (p = l) and gas mixture (p = g) with respect to the solid

38

F. Meftah et al.

phase and vp−g is the velocity of the vapour (p = v) and dry air (p = a) with respect to the gas mixture (moist air). Having assumed here that all phases of the material are locally in thermodynamic equilibrium, i.e. their temperatures are the same, Tp = T (p = s, w, g), with regard to the energy conservation equation, the space averaging procedure gives (Gawin et al. 2003): qCp

 @T  þ ml Cl vls þ mg Cg vgs  rT þ r  q ¼ rHdehyd m_ dehyd Þ  DHvap m_ vap @t ð4:12Þ

with the heat capacity of the whole porous medium given by: qCp ¼

X

mp Cp

with p ¼ s; l; v; a

ð4:13Þ

p

where q is the density of the whole porous medium, Cp (p = s, w, g) is the specific heat capacity of each constituent, DHvap is the enthalpy of evaporation, DHdehyd the enthalpy of dehydration and q the flux vector of heat conduction. Note that the second term of the left-hand side of Eq. (4.12) gives the heat convection process which encompasses both advective and diffusive heat transfer within the porous network. Moreover, the right-hand side corresponds to the latent heat due to phase changes (evaporation, condensation, dehydration). The governing equations in the form described above are now complemented by the constitutive equations.

4.3.2

Constitutive Equations

Fluid state equations The liquid water is considered to be incompressible such that its density depends on the temperature only. The vapour, dry air and the gas mixture are considered to behave as ideal gases, which gives (p = v, a, g): qp ¼

Mp pp RT

ð4:14Þ

where pp is the pressure, Mp the molar mass and R the universal gas constant. Furthermore, the pressure and density of the gas mixture can be related to the partial pressures and densities of the constituents by:

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qg ¼ qv þ qa ;

pg ¼ pv þ pa

which gives: M g ¼ M a þ ðM v  M a Þ

pv pg

ð4:15Þ

ð4:16Þ

Liquid–Vapour equilibrium By assuming that the evaporation process occurs without energy dissipation, that is, liquid and vapour water have equal free enthalpies, one can derive the generalized Clausius–Clapeyron equation, which is a relationship between liquid and vapour pressure: 

pv ¼ pvs

  Mv  pg  pc  pvs exp ql RT

ð4:17Þ

where the capillary pressure: pc ¼ pg  pl

ð4:18Þ

has been introduced for the porous medium and pvs is the saturation vapour pressure, which only depends on temperature. Mass fluxes According to the velocity decomposition (Eq. 4.11), the mass fluxes can be made explicit: J ls ¼ ml vls ¼ K

ð4:19Þ

  qv krg Mv Ma pv rpg  Dqg r lg Mg2 pg

ð4:20Þ

  qa krg Mv Ma pa ¼ K rpg  Dqg r lg Mg2 pg

ð4:21Þ

J vs ¼ mv vgs þ mv vvg ¼ K J as ¼ ma vgs þ ma vag

ql krl rpl ll

In the above equations, where Darcy’s and Fick’s laws are introduced, K is the intrinsic permeability (assumed isotropic), krp is the relative permeability, lp is the dynamic viscosity, D is the diffusivity and Mp is the molar mass. For the gas mixture, the Darcy’s part of the mass flux J ps with p = v, a is controlled  by the barycentric velocity of the gas vps while the Fick’s part J pg ¼ mp vpg , which is controlled by the concentration gradient, gives the diffusion of each component within the gas mixture. Furthermore, the concentration gradient can be written as:

40

F. Meftah et al.

       pv 1 qv pv qv pa r  ¼ rpp  rpc ¼ r pg pg ql pg ql pg

ð4:22Þ

Thus, the vapour and dry air fluxes can be recast in the following generic form: J ps

  qp krg M v M a q v pv Mv Ma qv ¼ K rpg  D  rpc rpg  D lg RTMg ql pg RTMg ql

ð4:23Þ

where the upper signs correspond to vapour case (p = v) while the lower signs give the dry air flux (p = a). This form depends directly on the state variables pg and pc. Conductive heat flux In the energy balance equation (4.12), the heat conduction process in the porous medium can be described by Fourier’s law which relates the temperature to the heat flux as follows: q ¼ kðSl ; T Þ rT

ð4:24Þ

where k(Sl, T) is the effective thermal conductivity, which is a function of the temperature and of the degree of saturation of the pores with liquid water (see Chap. 6). Sorption–desorption isotherm Solving the equations presented above works towards determining the spatial and temporal distribution of temperature, masses of constituents and corresponding pressures within the porous medium. By introducing the previous constitutive equations, the problem involves, at this stage, as main unknowns (Sl, pc, pg, T, m_ vap ) or alternatively (Sl, pv, pa, T, m_ vap ). This set is reduced by introducing an additional explicit relationship (van Genuchten 1980; Baroghel-Bouny et al. 1999; Pesavento 2000; Gawin et al. 2003): Sl ¼ Sl ð pc ; T Þ

ð4:25Þ

which is the sorption–desorption isotherm, characterizing phenomenologically the microstructure of the porous medium (pore size distribution). Using this additional relationship, the reduced set of unknowns is then ðpc ; pg ; T; m_ vap Þ where a choice is made here to retain the state variables (pc, pg) instead of (pv, pa) or (pl, pa). At this point it is necessary to decide how to gather the mass conservation equations of the phases. For instance, it is possible to sum up the mass balance equation of the water vapour with the one of the dry air in order to obtain the mass balance equation of the gas phase. In such a case the mass balance equation of the water reduces to the equation for the mass conservation of the liquid phase. Alternatively, a classical approach is described in Schrefler (2002), Gawin et al.

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41

(2003), in which the equation of vapour mass conservation is summed up with the liquid water mass balance equation. This has the effect of reducing the set of unknowns further as the evaporation source term m_ vap is eliminated, thus obtaining the balance equation for total water mass mw:

with:

@mw þ r  J ws ¼ m_ dehyd @t

ð4:26Þ

mw ¼ ml þ mv

ð4:27Þ

J ws ¼ J ls þ J vs

ð4:28Þ

In this way, at the end of the mathematical developments, we have a model with two mass balance equations: one for the dry air and another one for the water species (liquid plus water vapour). The final form of the governing equations of heat and mass transfer will be described in the next subsection, taking into account the set of primary variables chosen at the beginning of the formulation. Some additional discussions about the choice of the state variables and about some key points in modelling concrete exposed to high temperature can be found in Sect. 4.3.

4.3.3

Final Form of the Heat and Mass Balance Equations

The hygro-chemo-thermal state of cement based materials at high temperature is described by three primary state variables, i.e. gas pressure, pg, capillary pressure, pc, and temperature, T, as well as one internal variable describing advancement of the dehydration processes, i.e. degree of dehydration, Cdehyd . The mathematical model describing the material performance consists of three equations: two mass balances (continuity equations of water and dry air), enthalpy (energy) balance and one evolution equation (degree of dehydration). For convenience of the reader, the final form of the model equations, expressed in terms of the primary state variables, are listed below. The full development of the equations is presented in Gawin et al. (2003). The Mass balance equation of dry air takes into account both the diffusive (described by the term L.27.7) and advective air flow (L.27.8), the variations of the saturation degree of water (L.27.1) and air density (L.27.4), as well as the variations of porosity caused by: dehydration process (L.27.5), temperature variation (L.27.2),

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skeleton density changes due to dehydration (L.27.6) and by skeleton deformations (L.27.3). The mass balance equation (without neglecting the solid skeleton volume variation and assuming a non-constant solid density) has the following form:

L.27.1

L.27.2

L.27.3

L.27.4

L.27.5 ð4:29Þ

L.27.6

L.27.7

L.27.8

_ dehyd and to where C_ dehyd is a normalized dehydration rate that may be related to m the apparent solid density (Gawin et al. 2003). The Mass balance equation of gaseous and liquid water considers the diffusive (L.28.6) and advective flows of water vapour (L.28.7) and water (L.28.8), the mass sources related to phase changes of vapour (evaporation-condensation, physical adsorption–desorption) (the sum of those mass sources equals to zero) and dehydration (R.28.1), and the changes of porosity caused by variation of: temperature (L.28.3), dehydration process (L.28.10), variation of skeleton density due to dehydration (L.28.9) and deformations of the skeleton (L.28.2), as well as the variations of: water saturation degree (L.28.1) and the densities of vapour (L.28.4) and liquid water (L.28.5). This gives the following equation (Gawin et al. 2003):

L.28.1

L.28.6

L.28.2

L.28.7

L.28.9

L.28.3

L.28.4

L.28.5

L.28.8

L.28.10

R.28.1 ð4:30Þ

with bslg the thermal expansion coefficient of the multiphase porous medium, defined by:   bslg ¼ bs ð1  uÞ ql Sl þ qv Sg þ ubl ql Sl ð4:31Þ in which bp is the thermal expansion coefficient of each phase (p = s, w, g). The Enthalpy balance equation of the multi-phase medium, accounting for the conductive (L.30.3) and convective (L.30.2) heat flows, the heat effects of phase

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43

changes (R.30.1) and dehydration process (R.30.2), and heat accumulation by a material (L.30.1), can be written as follows:

L.30.1

L.30.2

L.30.3

R.30.1

R.30.2 ð4:32Þ

where q is the conductive thermal flux and the enthalpies of vaporisation and dehydration are: DHvap ¼ Hv  Hl ð4:33Þ DHdehyd ¼ Hl  Hls the vapour mass source is given by Gawin et al. (2003):

L.32.1 R.32.1

R.32.2

R.32.3

R.32.4 ð4:34Þ

R.32.5

R.32.6

R.32.7

where bsl is the thermal expansion coefficient of the solid-liquid phase, defined as: bsl ¼ ql Sl ½ð1  uÞbs þ ubl 

ð4:35Þ

Equation (4.34) is obtained from the water mass conservation equation, considering the advective flow of water (R.32.4), the mass sources related to dehydration (R.32.7) and the changes of porosity caused by: variation of temperature (R.32.3), dehydration process (R.32.6), variation of skeleton density due to dehydration (R.32.5) and deformations of the skeleton (R.32.2), as well as the variations of water saturation degree (R.32.1).

4.3.4

Boundary Conditions

The hygro-thermal problem consists of determining the temperature field T and the pressure fields pg and pc, satisfying the conservation equations (4.29), (4.30) and (4.32) within the domain X with the following boundary conditions at the domain boundary R: pg ¼ pg on Rp ð4:36Þ pc ¼ pc

on Rp

ð4:37Þ

44

F. Meftah et al.

T ¼ T

ð4:38Þ

on RT

  J vs  n ¼ qv  hg qv  q1 v

on Rp

ð4:39Þ

  p J as  n ¼ qa  hg qa  q1 on R a

ð4:40Þ

J ls  n ¼ ql

p on R

    4  q  DHvap J ls  n ¼ qT  hT ðT  T1 Þ  rB T 4  T1

ð4:41Þ T on R

ð4:42Þ

where R is part of the boundary at which the pressures and the temperature are  is the complementary part known (Dirichlet type boundary conditions), while R (with unit outward normal n) at which the mass fluxes of fluids and heat flux are imposed. The terms qp (p = T, a, v, l) are the imposed fluxes, while q1 p with p = v, a, are the densities of vapour and dry air, respectively. T∞ is the temperature in the far field surrounding gas, while hg, and hT, are the convective mass and energy, exchange coefficients, respectively. Finally  is the emissivity and rB is the Stefan– Boltzmann constant. Note that the convective term on the right-hand-side of Eq. (4.42) corresponds to Newton’s law of cooling and describes the conditions occurring in most practical situations at the interface between a porous medium and the surrounding fluid (moist air in this case). The second left-hand-side term gives the energy exchange at the external surface due to the vaporisation process. Particular attention has to be given to the mass convective boundary conditions (4.39) and (4.40) if the staggered approach discussed below is used for time integration. The mass densities qp (p = v, a) have to be expressed in terms of the retained variables of the problem (pc, pg, T). However, the relationships relating the densities to the state variables are strongly nonlinear, which does not permit a straightforward algebraic form of these boundary conditions suitable for the dis p should be cretised problem to be obtained. Indeed, the variables at the boundary R factorised such that additional terms, associated to these boundary conditions, arise in the operators of the discretized formulation in order to make them well conditioned. In order to overcome this difficulty, it is proposed here to linearize, for a given time stage t, the relationships relating the densities to the state variables in the neighbourhood of a reference time stage tref: qp  qref p þ

@q   @qp   @qp  þ T  T ref pg  pref þ p pc  pref g c @pg @pc @T

ð4:43Þ

where (pc, pg, T)ref are the values of the variables at tref. The choice of the reference time will depend on the adopted iterative algorithm presented in the following. It is worth noting that this peculiar problem does not exist in the case where the monolithic approach algorithm is adopted for the solution of the numerical model (Gawin et al. 2003, 2006). In such a case all the variables related to the calculation

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of the boundary conditions are evaluated at the same time instant and at the same iteration.

4.4

Key Points for Modelling Cement-Based Materials at High Temperature

Dealing with cement-based materials at high temperature introduces some physical and theoretical difficulties, which should be overcome to be able to formulate a mathematical model giving reliable results for the material. These key points are discussed in the following subsections. Choice of state variables A proper choice of state variables for the description of materials at high temperature is of particular importance. From a practical point of view, the physical quantities used should be possibly easy to measure during experiments, and from a theoretical point of view, they should uniquely describe the thermodynamic state of the medium. They should also assure a good numerical performance of the computer code based on the resulting mathematical model. As mentioned previously, the necessary number of state variables may be significantly reduced if the existence of local thermodynamic equilibrium at each point of the medium is assumed. In such a case, the physical state of different phases of water can be described by the same variable. Having in mind all the aforementioned remarks, we will briefly discuss now the state variables chosen for the presented model. Use of temperature, which is the same for all constituents of the medium because of the assumption about the local thermodynamic equilibrium state, and use of solid skeleton displacement vector is rather obvious; thus it needs no further explanation. As a hygrometric state variable, various physical quantities which are thermodynamically equivalent may be used, e.g. moisture content by volume or by mass, liquid water saturation degree, vapour pressure, relative humidity, or capillary pressure. Analysing materials at high temperature, one must remember that at temperatures higher than the critical point of water (i.e. 374.15 °C or 647.3 K) there is no capillary (or free) water present in the material pores, and there exists only the gas phase of water, i.e. vapour. Then, very different moisture contents may be encountered at the same moment in a heated cement-based material, ranging from full saturation with liquid water (e.g. in some nuclear vessels or in the so called “moisture clog” zone in a heated concrete— see England and Khoylou 1995) up to almost completely dry material. Moreover, some quantities (e.g. saturation or moisture content), which can be chosen as primary variable are not continuous at interfaces between different materials. For these reasons it is apparently not possible to use, in a direct way, one single variable for the whole range of moisture contents. However, it is possible to use a single variable that has a different meaning depending on the state. The moisture

46

F. Meftah et al.

state variable selected in the model is capillary pressure (Gawin et al. 2002b), which was shown to be a thermodynamic potential of the physically adsorbed water and, with an appropriate interpretation, can be also used to describe water at pressures higher than atmospheric. The capillary pressure has been shown to assure good numerical performance of the computer code (Gawin et al. 2002b, 2003, 2006) and is very convenient for analysis of the stress state in concrete, because there is a clear relationship between pressures and stresses (Gray et al. 2009). Therefore, as already shown, the chosen primary state variables of the presented model are the volume averaged values of: gas pressure, pg, capillary pressure, pc, temperature, T, and displacement vector of the solid matrix, u. For temperatures lower than the critical point of water, T < Tcr, and for capillary saturation range, Sl > Sssp(T) (Sssp is not only the upper limit of the hygroscopic moisture range as mentioned earlier, but also the lower limit of the capillary one), the capillary pressure is defined according to Eq. (4.18). For all other situations, and in particular for T Tcr where condition Sl < Sssp is always fulfilled (there is no capillary water in the pores), the capillary pressure is substituted—only formally—by the water potential Wc, defined as: Wc ¼

  RT pv ln vs Mw f

ð4:44Þ

where Mw is the molar mass of water, R the universal gas constant and f vs the fugacity of water vapour in thermodynamic equilibrium with a saturated film of physically adsorbed water (Gawin et al. 2002b). For physically adsorbed water at lower temperatures (S < Sssp and T < Tcr) the fugacity f vs should be substituted in the definition of the potential Wc, by the saturated vapour pressure pv. Having in mind Kelvin’s equation, valid for the equilibrium state of capillary water with water vapour above the curved interface (meniscus):   pv pc M w ln ¼ pvs ql RT

ð4:45Þ

We can note that in the situations where (4.44) is valid, the capillary pressure may be treated formally as the water potential multiplied by the density of the liquid water, ql , according to the relation: pc ¼ Wc ql

ð4:46Þ

Thanks to this similarity, it is possible to “formally” use the capillary pressure during computations even in the low moisture content range, where the capillary water is not present in the pores and capillary pressure has no physical meaning. Passing the critical point of water In a concrete structure exposed to fire conditions, temperatures higher than the critical point of water can be encountered in a part of the structure after some time.

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47

Above this temperature, the liquid water and gaseous water phase (water vapour) cannot be distinguished and only the gas phase exists. As a result, there are no phase changes of the pore water (condensation–evaporation) and capillary pressure has no physical meaning. Hence, after reaching, in a region of moist porous material, temperatures higher than the critical point of water, we deal with a kind of Stefan’s problem, where two regions, with the temperatures below and above the critical point of water, are separated by the moving interface boundary. In Gawin et al. (2002b) a method, which formally allows direct tracking of the boundary position in the space to be avoided, is presented. It consists of giving formally different physical meaning to the capillary pressure (as discussed above) and using it still for description of the hygrometric state of concrete in the zone where the temperature exceeds the critical point of water. Then, a special ‘switching’ procedure is applied for a finite element where this temperature is encountered, from the below- to the above- critical temperature description of the medium, still using the governing equations of the same form, but with different physical meaning.

4.5

Numerical Approach

In the following, some numerical procedures that are adopted for the discretisation and the solution of the thermo-hydraulic model are presented. The set of non-linear partial differential equations that controls the processes of mass transport (see 4.2.3) is usually discretised in the space domain by means of two methods, Finite Elements or Finite Volumes. The latter has the advantage of being flux conservative but the former is usually adopted in the literature and will be retained for this presentation. According to the FEM procedure, the governing equations are re-written using the weighted residual method, and the standard Galerkin procedure is usually adopted. Time discretisation is achieved by means of the standard Finite Difference Method. A non-symmetric, non-linear system is finally obtained and linearization, usually by means of the Newton-Raphson method, is required (Zienkiewicz and Taylor 2000). The final set of discretised, linearized governing equations is solved as a monolithic system or, alternatively, by means of a staggered coupled procedure (see Gawin et al. 2003 and Meftah and Dal Pont 2010 or Meftah et al. 2012).

4.5.1

Variational Formulation

Starting from the previous initial-boundary value problem, the variational formulation can be obtained as below, respectively, for total water, dry air and energy:

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F. Meftah et al.

Z

@mw dX  p @t 

X

Z

Z



  p ðql þ  q v  h g q v  q1 v ÞdR

rp J ws dX ¼ Rp

X

Z



p m_ dehydr dX

ð4:47Þ

  p ðqa  hg qa  q1 a ÞdR

ð4:48Þ

X

Z

p

X

Z X

@ma dX  @t

Z

rp J as dX ¼

Z Rp

X

  Z  @T  @ml þ ml Cl vls þ mg Cg vgs rT dX  T Hvap dX T  qCp @t @t X Z Z þ T rHvap ml vls dX þ rT Hvap ml vls dX ZX  X

rT qdX  Z

¼

Z

X

  T Hvap þ Hdehydr m_ dehydr dX

ð4:49Þ

X

  4 T ðqT  hT ðT  T1 Þ  er T 4  T1 ÞdR 

Rp

where p* and T* are weighting functions that respectively vanish at the boundaries Rp and RT and in which essential boundary conditions are used.

4.5.2

Spatial Discretisation

The standard Galerkin discretisation, replacing the weighting functions with the shape functions, is adopted. For the state variables we can write: pg ¼ N p pg ; pc ¼ N p pc ;

T ¼ Nt T

ð4:50Þ

After some manipulations, the weak formulation previously stated can be written in the following form: Cij ðY Þ where:

@Y þ K ij ðY ÞY ¼ f i ðY Þ @t

ð4:51Þ

4 Advanced Modelling

0

Cgg Cij ¼ @ 0 0

Cgc Ccc Ctc

49

1 0 K gg Cgt Cct A; K ij ¼ @ K cg K tg Ctt

K gc K cc K tc

1 0 1 K gt fg K ct A; f i ¼ @ f c A ð4:52Þ K tt ft

For the sake of brevity, the matrix C and K are not given (see e.g. Gawin et al. 2003; Meftah and Dal Pont 2012 for further details).

4.5.3

Time Discretisation

Time discretisation is obtained by means of the standard #-method (see Zienkiewicz and Taylor 2000), thus giving: ~ cc Dpn þ 1 þ K cc pn ¼ f n þ #  K ~ cg Dpn þ 1 þ K cg pn  K ~ cT DT n þ 1 þ K cT T n ð4:53Þ K c c c g g ~ gg Dpn þ 1 þ K gg pn ¼ f n þ #  K ~ gc Dpn þ 1 þ K gc pn  K ~ gT DT n þ 1 þ K gT T n ð4:54Þ K g g g c c ~ TT DT n þ 1 þ K TT T n ¼ f n þ #  K ~ Tc Dpn þ 1 þ K Tc pn  K ~ Tg Dpn þ 1 þ K Tg pn ð4:55Þ K T c c g g with: Dpnc þ 1 ¼ pnc þ 1  pnc ;

Dpng þ 1 ¼ png þ 1  png ; DT n þ 1 ¼ T n þ 1  T n

ð4:56Þ

and: ~  ¼ C þ #K  K Dt

ð4:57Þ

where Dt is the time step, the index n makes reference to the time station tn and tn þ # to the time station tn þ #Dtn .

4.5.4

Solution Strategy

A monolithic approach is often adopted for the solution of the linearized system. In this approach all the equations are solved at the same time and all the state variables are updated simultaneously during the iterative procedure. This approach guarantees the optimal convergence of the system especially when dealing with strong non-linearities (Pesavento 2000; Gawin et al. 2003). Alternatively, the system can be solved using a staggered procedure that permits the equations to be solved sequentially (Dal Pont et al. 2007, Meftah and Dal Pont 2010). A specific iterative algorithm to account for the interactions between all of

50

F. Meftah et al.

the transfer processes to conserve their full coupling is required. Staggered procedures show a superior flexibility compared to monolithic ones if successive fields have to be introduced in the problem. Moreover, an appropriate partitioning reduces the size of the discretized problem to be solved at each time step. For the sake of clarity, some details are provided on this latter procedure. The set of nonlinear equations (4.53)–(4.55) is solved using a staggered iterative scheme with two nested levels of iterations. The first level (local iteration k) concerns the convergence process for each of the three equations when solved for one variable while the other variables are kept constant. The second level (global iteration j) concerns the convergence of the interaction among the three equations when considering simultaneously the three updated variables. Moreover, a quasi-Newton iterative algorithm is adopted; each equation is solved for the total increment Dxn þ 1 (with Dxc ¼ Dpc , Dxg ¼ Dpg ; Dxt ¼ DT) by keeping fixed the linearization point at the previous converged step (at time stage tn ) and by cumulating the residuals during the iterations. Accordingly, the set of Eqs. (4.53)–(4.55) can be put in the following generic form: ~ n Dxn þ 1;j þ 1;k þ 1 ¼ f n þ #  K n xn  K ~ n Dxn þ 1;j K a aa a aa a ab b ~ n Dxn þ 1;j  K n xn þ R  n þ #;j þ 1;k  K nab xnb  K ac c ac c a

ð4:58Þ

 n þ #;j þ 1;k is the sum of the residuals of with a ¼ c; g; T; b; c 6¼ a; b 6¼ c and where R a the k previous iterations. Solving the above equation gives: xna þ 1;j þ 1;k þ 1 ¼ xna þ Dxna þ 1;j þ 1;k þ 1 xna þ #;j þ 1;k þ 1 ¼ xna þ #Dxna þ 1;j þ 1;k þ 1

ð4:59Þ

 n þ #;j þ 1;k to be determined: which allows the updated residual R a  na þ #;j þ 1;k ¼ fan þ #  fan þ #;j þ 1;k þ 1 R

ð4:60Þ

Hence, after some manipulations, the final form of the set of Eqs. (4.53)–(4.55) to be solved at each local iteration k + 1 is given by: ~ n Dxn þ 1;j þ 1;k þ 1 Dxn þ 1;j þ 1;k þ 1 ¼ Y n þ #;j þ 1;k K aa a a a

ð4:61Þ

with:  n  ~ K ~ n þ #;j þ 1;k Dxn þ 1;j þ 1;k  K n þ #;j þ 1;k xn Y na þ #;j þ 1;k ¼ f na þ # þ K aa aa a aa a n þ #;j þ 1;k n þ 1;j n þ #;j þ 1;k n ~ K Dx K x ab

b

ab

b

~ n þ #;j þ 1;k Dxn þ 1;j  K n þ #;j þ 1;k xn K ac c ac c The procedure is summarized in Fig. 4.6.

ð4:62Þ

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4.6

51

Simplified One-Fluid Model

In this section, we describe the Bažant’s model as the most representative one in the class of the phenomenological models. Bažant’s model has been, probably, the most popular model for analysing the behaviour of concrete at high temperature between the 1980s and 1990s. Bažant and Thonguthai (1978, 1979), on the basis of Luikov’s theory, proposed field equations for the coupled heat and moisture transfer in concrete at high temperatures. Making use of experimental data, they simplified the expression of the mass flux (a function only of the gradient of the pressure) and presented some semi-empirical relationships for the desorption isotherms, for the dependence of permeability on temperature, for the change of porosity because of dehydration, and so on. Here, only a brief description will be given and the reader is referred to Bažant and Thonguthai (1978, 1979), and Bažant and Kaplan (1996) for further details. This model is based on a typical phenomenological approach and considers a single moisture flux, i.e. a single fluid phase is present, which includes both liquid water and moist air. Thus, the flux of mass in concrete J consists of a part due to the gradient of moisture concentration, i.e. is a diffusive flux (Fick’s law), and a part due to the gradient of temperature, i.e. convective flux (Soret flux). Similarly, the heat flux q, consists of a flux due to the gradient of temperature, i.e. conductive flux (Fourier’s law) as well as the flux due to the gradient of moisture concentration, i.e. convective flux (Dufour flux). We can write (Bažant and Kaplan 1996): J ¼ Dww rmw  DwT rT q ¼ DTw rmw  DTT rT

ð4:63Þ

where mw is the water-content (Eq. 4.27), i.e., the total mass of free water (not chemically bound) per m3 of concrete at a certain temperature. Note that generally the diffusion coefficients Dij (i, j = w, T) in Eq. (4.63) are not symmetric and that the relationship is written assuming that the mass and heat fluxes are not affected by solid (concrete) deformability. From Eq. (4.63)1, Bažant gives the following relationship for moisture flux by neglecting the convective flux due to thermal gradient (Soret effect): J ¼ Dh rh

ð4:64Þ

where Dh is the diffusion coefficient of the moisture (without distinction between liquid and vapour phase) transfer into the partially saturated porous network due to the gradient of relative humidity h with: h¼

pv pvs

ð4:65Þ

52

Fig. 4.6 Staggered iterative algorithm—two nested levels of iterations

F. Meftah et al.

4 Advanced Modelling

53

Using Eq. (4.64) and Kelvin’s equation (4.45), the moisture flux in Eq. (4.64) can be alternatively expressed in terms of gradient of capillary pressure: J ¼ Dc rpc

ð4:66Þ

Likewise, it is possible to write down the relationship for heat flux, neglecting the convective part of the flow (Dufour): q ¼ brT

ð4:67Þ

where b is the thermal conduction coefficient. In such a way a partial uncoupling of the Eqs. (4.63) is possible, obtaining a final system of conservation equations (mass plus heat) only partly coupled. The uncoupling is only partial because Eq. (4.64) also gives a nonzero thermal moisture flux. Thus, the final form of the conservation equations for moisture and energy in concrete exposed to high temperature will be: @w @wd  ¼ r  J @t @t qC

@T @w  Ca  Cw JrT ¼ r  q @t @t

ð4:68Þ ð4:69Þ

in which wd is the mass of chemically bound (hydrate) water that has been released into the pores as a result of dehydration of cement paste (i.e. part of the solid phase) caused by heating, q and C are the density and specific heat of concrete including chemically bound water, but excluding free water, and finally, Ca and Cw are the heat of sorption of free water and the isobaric heat capacity of bulk (liquid) water, respectively. The term Cw JrT represents the rate of heat supply due to heat convection by moving water. Usually this term is neglected in Bažant’s model, but in rapid heating, such as during a fire or a nuclear accident, it can be of importance. The most important shortcoming of this model is that water vapour and liquid water content are treated as one variable (moisture). The formulation of the desorption isotherms proposed by Bažant is analytically derived from the thermodynamics of water at high temperature, but it is empirically modified to match the experimental data. In such a way the equilibrium between the liquid phase and its vapour phase is no longer assured and it is possible to obtain values of relative humidity greater than 100%, which is physically meaningless. Moreover, part of the description of the material microstructure, i.e. the porosity, is automatically encompassed in the isotherms definition. Furthermore, in the original formulation of the model the mechanical behaviour is not considered, so it is limited to the heat and mass transfer simulation.

54

4.7

F. Meftah et al.

Simplified Thermo-mechanical Approach

Most of the routinely made calculations are based on simplified thermo-mechanical equations where the temperature evolution is coupled to the mechanical analysis through the thermal strains and the variation of the mechanical properties with temperature, without explicitly describing important phenomena like moisture transfer and chemical reactions. Ad hoc phenomenological equations are established to account indirectly for their effect.

4.7.1

Heat Balance Equation

In this approach, only the heat transfer process is considered, with the temperature field as the unknown. It is determined by solving the linearized energy balance equation, which can be put in the following form when neglecting mechanical interaction (which has negligible contribution when high temperatures are considered) and local source terms: qCp

@T þr  q ¼ 0 @t

ð4:70Þ

where q is the material density, Cp is the specific heat and q is the heat flux. It is worth noting that, in a simplified thermo-mechanical approach, the rate of energy release or energy consumption due to phase changes (as evaporation, condensation and dehydration) at any material point depends only on the temperature at this point. In other words, the supply (or loss) of water to this point by the surrounding medium cannot be considered during the heating process. Thus, phase changes are taken into account by adopting suitable phenomenological evolution laws of the specific heat capacity as functions of temperature. For instance, an increase in specific heat capacity can be considered in the range of temperature where water evaporation is assumed to intensively occur. This simplification is a limitation in the thermo-mechanical approach since the heat capacity of the material point is constrained by the adopted temperature-dependent evolution law disregarding the transient hygral state of the point. Another limitation in the simplified thermo-mechanical approach is that advective heat transfer (which may affect significantly the temperature field due to mass fluxes of fluids) is ignored.

4 Advanced Modelling

4.7.2

55

Heat Flux

The heat flux is classically given by Fourier’s constitutive equation for heat transfer: q ¼ kðT ÞrT

ð4:71Þ

where kðT Þ is the thermal conductivity. Assuming that the thermal conductivity depends on the temperature only restricts the model in a way that there is no possibility of accounting for the effect of the transient hygral state on the heat transfer mechanism.

4.7.3

Boundary Conditions

The heat transfer problem consists of determining the temperature field T ðx; tÞ satisfying Eq. (4.70) (supplemented by constitutive Eq. (4.71)) within a given domain X. The boundary conditions can be of various types:  on a part @XT of the domain boundary: – prescribed temperatures T T ¼ T

8 x 2 @XT

ð4:72Þ

– prescribed heat flux q(T) on a part @Xq of the boundary: n  q ¼ q 8 x 2 @XT

ð4:73Þ

where n is the outward normal to the boundary and q is the heat flux across the boundary, accounting for conduction, convection and radiation where appropriate:   4 q ¼ hT ðT  T1 Þ þ rB T 4  T1

ð4:74Þ

where, T is the temperature of the concrete, T1 is the temperature surrounding of the environment, hT is the coefficient of convective heat transfer,  is the emissivity and rB is the Stefan-Boltzman constant.

4.8 4.8.1

Mechanical Modelling Momentum Balance

For completeness, the momentum balance equations are recalled here while limiting the analysis to quasi-static mechanical loadings. Introducing the Cauchy stress

56

F. Meftah et al.

second-order tensor rðx; tÞ at time stage t and spatial position x of any material point of the studied domain X, the momentum balance equations can be written, in the quasi-static case, i.e. neglecting the inertia effects, as: x2X:

div r þ f ¼ 0

ð4:75Þ

where f is the vector of applied forces per unit volume. These equations are complemented by the boundary conditions, which are either  on part @Xu of the boundary @X of the studied domain X, prescribed displacements u  x 2 @Xu : u ¼ u

ð4:76Þ

or prescribed surface forces on the complementary part @Xr of the boundary, x 2 @Xr : r  n ¼ f

ð4:77Þ

where n denotes the outward normal vector to the boundary @Xr and stands for the scalar product.

4.8.2

Strain Decomposition

Different strains can be identified in a concrete which is mechanically loaded and simultaneously heated. Accordingly, the overall measured strain tensor e may conventionally be split as follows (Schneider 1988; Khoury et al. 2002): e ¼ ee þ ef þ eth þ ec

ð4:78Þ

where ee is the temperature dependent elastic strain, ef is the cracking induced strain, eth is the free thermal strain and ec is the transient creep strain. The free thermal strain eth is the apparent strain experimentally measured during a standard free (unconstrained) thermal dilatation test. It results from the superposition of the effective thermal strain et , the shrinkage strain es and the micro-cracking strain emf which is induced by the incompatibility between shrinking cement paste and expanding aggregates. Hence, the thermo-hygral strain tensor can be written as: eth ¼ et þ es þ emf

ð4:79Þ

Note that the identification of the component emf requires analysis of the behaviour of the material at the meso-scale, i.e., the scale at which concrete is seen as a set of granular inclusions embedded in a cementitious matrix.

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57

With regard to the shrinkage strain es two main driving mechanisms can be discriminated: – Drying shrinkage which involves the mechanical action of the capillary pressure which increases due the loss of water by desiccation. This component appears to be less significant than the second mechanism since the liquid saturation of heated concrete quickly drops to zero in addition to the counteracting effect of dilatation of initially available capillary water. – Dehydration shrinkage which is due to a reduction of the material volume induced by the departure of the initially chemically bounded water. The progressive reorganization, at the microstructure scale, of the formed anhydrous phases lead to this shrinkage component, the magnitude of which appears to be more important than the previous one. Furthermore, the transient creep strain ec in Eq. (4.78) encompasses both the basic creep strain ebc and the transient thermal creep strain etc : ec ¼ ebc þ etc

ð4:80Þ

This strain is obtained from an experimental test during which a non-sealed (mass exchange allowed) specimen is continuously heated under a constant load. It is obtained by removing from the overall measured strain the other strain components (obtained from an ad hoc companion test) according to Eq. (4.78). Therefore, the transient thermal creep component can be obtained from Eq. (4.80) when the basic creep component is measured on a companion specimen subjected to a constant load while sealed and kept at a constant temperature. It is generally assumed that basic creep of heated concrete is negligible which means that transient creep strain ec mainly reduces to the transient thermal creep strain etc . The latter appears (Thelanderson et al. 1988; Torrenti et al. 1997; Benboudjema 2002; Feraille 2000; Pasquero 2004; Anderberg and Thelandersson 1973; Khoury et al. 1985; Bažant and Kaplan 1996; Gawin et al. 2004) to be the additional strain which comes from the evolution of the microstructure due to the material drying and dehydration when under a sustained stress state. Moreover, some works (Khoury et al. 1985; Schneider 1988; Khoury 2003) refer to the so-called load-induced transient strain etm (LITS) which is the sum of the elastic ee , cracking ef and transient thermal creep etc strains: etm ¼ ee þ ef þ etc

ð4:81Þ

From experimental tests, LITS is obtained by removing the free thermal strain eth from the strain measured during a test at which the specimen is firstly loaded and then heated while the load is maintained. It is worth noting that the identification of strain components of heated concrete according to Eqs. (4.78)–(4.81) is not straightforward since it requires a proper quantification of the contribution of cracking processes. Cracking strain needs to be evaluated by ad hoc cracking models since it cannot be measured experimentally.

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A back analysis based on a numerical approach should accompany the experiment in order to assess the effective contribution of each strain component. In the next section, constitutive models are presented for each strain component according to additive decomposition given by Eqs. (4.78)–(4.81). It is worth mention that the following constitutive equations are given for illustrative purposes; that is, other laws can be adopted instead. Note also that intermediate decomposition of total strain may be adopted with constitutive equations written for these intermediate strains each of them gathering some elementary components of the decomposition presented above. For instance a constitutive law may be elaborated for the intermediate strain em ¼ ee þ ef .

4.8.3

Constitutive Equations

Free thermal strain Usually, free thermal strain is considered by adopting the following phenomenological constitutive relationship for the strain rate tensor e_ th : _ e_ th ¼ ath ðT ÞTd

ð4:82Þ

where d is the second-order unit tensor, T_ is the temperature rate and T_ is the tangent temperature-dependent thermal expansion coefficient. Constitutive equation (4.82) is the usual adopted approach for modelling thermal expansion. However, the thermal expansion coefficient is identified experimentally from the standard free thermal expansion test during which mass and heat transfer occur within the specimen with some characteristic times that depend on the specimen size and the transfer properties (conductivity, permeability, dehydration kinetics…) of the material. Since the identified thermal expansion coefficient depends on the test conditions it cannot be considered as an objective constitutive (material) parameter. To overcome the inherent drawback in the standard experimental identification procedure, the free thermal strain eth should be split according to Eq. (4.79) with an appropriate constitutive law for each strain. Then, experiment and computation should be used in conjunction to identify all involved physical parameters. Effective thermal strain Similarly to the approach of apparent thermal strain, an effective thermal expansion coefficient at ðT Þ may be considered for expressing the effective thermal strain rate e_ t : _ e_ t ¼ at ðT ÞTd

ð4:83Þ

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The identification of the effective coefficient at ðT Þ from the apparent thermal strain eth needs the knowledge of appropriate constitutive equations for the two other components in the right-hand-side of Eq. (4.79). The shrinkage component es is made explicit hereafter while the meso-scale cracking strain emf comes from modelling of the cracking process at this scale (due to incompatibility between aggregate and cement paste) with an appropriate cracking, plasticity, damage… based model. Shrinkage strain A simple model for this strain component considers it as proportional to water loss without distinguishing previously described mechanisms of shrinkage. Accordingly, the shrinkage strain rate tensor can be written as (Benboudjema 2002, Al Najim et al. 2003; Benboudjema et al. 2007; Sabeur et al. 2008; Sabeur and Meftah 2008): e_ s ¼ as hm_ l i d

ð4:84Þ

where as is the shrinkage coefficient, m_ l is the mass loss of water that comes from mass and heat transfer modelling (cf. Sect. 4.2) and the symbol hi stands for the negative part, that is, only a water loss (negative m_ l ) will lead to shrinkage. An alternative constitutive equation for the drying shrinkage component comes from making the two driving mechanisms previously described (cf. Sect. 4.7.2) explicit in the constitutive law. Distinction is made between shrinkage induced by drying and shrinkage induced by dehydration. The former is assumed to be the deformation of the solid skeleton that is induced mechanically by capillary effects: capillary forces give rise to an increase of suction within the pore space leading to a contraction of the porous medium. The second mechanism is rather related to a volume decrease of the solid phase (skeleton) due to loss of chemically bound water by dehydration. This is conventionally assumed to start when the temperature exceeds a threshold of about 105–115 °C. Hence, the shrinkage strain tensor in this alternative approach reads (Benboudjema et al. 2007; Sabeur et al. 2008; Sabeur and Meftah 2008): e_ s ¼ e_ drys þ e_ hyds ¼ e_ drys þ ahs m_ dehyd d

ð4:85Þ

While an explicit constitutive law is needed for strain rate tensor of dehydration shrinkage e_ hyds , the drying (capillarity) induced shrinkage may be implicitly embedded in the total strain by the use of a poro-mechanical constitutive behaviour (cf. section on effective stress). Indeed, the capillary pressure (Eq. 4.18) as a part of pore pressure will generate an effective stress r0 (Eq. 4.89) that leads to a deformation of the porous medium. For instance, a time dependent constitutive law (creep) may be adopted for the porous medium leading to a drying shrinkage strain that is the time dependent response of the medium due to capillary pressure (suction) acting on the creeping skeleton (Benboudjema et al. 2007; Sabeur et al. 2008; Sabeur and Meftah 2008).

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Transient Creep During first-time heating of a concrete specimen under sustained load, a significant increase in strains associated with the increase in temperature occurs. This phenomenon cannot be solely explained by the thermo-mechanical degradation of the elastic properties nor by the occurrence of micro-cracking. It has rather been linked to an additional strain component, which takes place in a very short period of time. This strain component, called transient thermo-mechanical interaction strain or transient creep strain, is considered to be a function of the temperature as well as of the stress. Even though there is no consensus on the physical mechanisms driving this phenomenon, a federating twofold assumption is made: whereby between 100 and 200 °C drying of the cement matrix is the most important factor (drying creep), while for higher temperatures a change of the chemical structure of the hydrates is primarily assumed to be responsible. Based on experimental investigation, Anderberg and Thelandersson (1976) proposed a uni-axial constitutive law for transient creep strain rate that has been extended to multi-axial behaviour by Thelandersson (1987): _ e_ tc ¼ TQ:r

ð4:86Þ

where Q is the forth order tensor of transient creep compliance given by: Qijkl ¼

    ath btc 1 ð1 þ ctc Þ dik djl þ dil djk  ctc dij dkl 2 fc

ð4:87Þ

in which fc is the compressive strength at ambient temperature and ðbtc ; ctc Þ are model parameters identified from transient creep tests: btc comes from a uni-axial test while ctc refers to the “Poisson’s” ratio of transient creep. The constitutive relationship (4.86) has been successfully incorporated into a thermo-elasto-plastic modelling approach for heated concrete by Khennane and Baker (1992) and by Heinfling (1998). The approach was also extended to a damage-based framework by Nechenech et al. (2002): _ r e_ tc ¼ TQ:~

ð4:88Þ

~ is the effective stress (cf. section Stress-strain relationship) according to where r damage mechanics (Kachanov 1986). More recently, Sabeur et al. (2008) have proposed a constitutive model for transient creep that relates the strain rate to two assumed driving mechanism: drying creep and dehydration creep. Effective stress Within the framework of poro-mechanics, the effective stress is the stress acting on the solid skeleton of the porous medium due to the combination of the apparent stress acting on the porous medium and the stress induced by fluids filling the pore space. Accordingly, a common expression of the apparent stress tensor r is written as:

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r ¼ r0 þ aps d

ð4:89Þ

where r0 is the effective (Bishop’s) stress tensor, a is the Biot coefficient, ps is the pore pressure transferred to the solid skeleton and d is the identity tensor (Gray et al. 2009). Pore pressure definition Different expressions for the pore pressure that transfers to the solid concrete material phase as stress can be found in the literature. A first definition is (Tenchev et al. 2001): ps ¼ pg

ð4:90Þ

where pg is the gas pressure. A second definition is (Gawin et al. 1999):

ps ¼



Sl 

Sbl

 pg patm  pl þ 1  Sl  Sbl pg  patm

for Sl  Sbl for Sl [ Sbl

ð4:91Þ

where pl is the liquid water pressure, patm is the atmospheric pressure and Sbl is the saturation threshold beyond which the liquid phase present in the pore cannot exert a significant pressure on the solid skeleton. Based on the same key idea (Gawin et al. 2003), an alternative expression to the previous reads: ( ps ¼

Sl Sbl 1Sbl

pg  patm pl þ

1Sl 1Sbl

pg  patm

for Sl  Sbl for Sl [ Sbl

ð4:92Þ

Finally, the pore pressure can be related to gas pressure and capillary pressure as follows: ps ¼ pg  v l pc

ð4:93Þ

where vl is the solid surface fraction in contact with the wetting film (Al Najim et al. 2003; Benboujema et al. 2007; Sabeur and Meftah 2008; Gray et al. 2009). Stress-strain relationship Typically when modelling concrete exposed to elevated temperatures, it is considered most appropriate and convenient to assume elastic behaviour of the solid component and account for non-linear behaviour via a damage formulation (e.g. Gawin et al. 1999, 2004; Tenchev and Purnell 2005; Davie et al. 2010; de Sa and Benboudjema 2011). That said, some authors have recently started to combine damage and plasticity models in order to more comprehensively capture observed behaviour, especially in multi-axial stress conditions (e.g. Nechnech et al. 2002; Al Najim et al. 2003; Gernay et al. 2013).

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The use of a damage formulation has the added advantage that, for thermohydro-chemo-mechanical models, the change in some other material properties (particularly for example permeability (see Sect. 5.1.5) can be directly associated with the damage formulation (e.g. Gawin et al. 2002b; Al Najim et al. 2003; Davie et al. 2010). In the general case, the damage formulation is based on the concept of the ~ in the sense of damage mechanics (Kachanov 1986), which may effective stress r be related to the effective stress r0 in the sense of poro-mechanics as: r ¼ ð1  xÞC0 :ee r0 ¼ ð1  xÞ~

ð4:94Þ

Whereby the damage parameter, x.,ontrols the reduction of the initial elasticity tensor C0 due to the effective loss of load carrying cross-section as micro-cracks develop in the solid skeleton of the concrete. Classic damage models consider the effects of mechanical stress in the development of damage, but when considering concrete at elevated temperatures it is necessary to consider also the fact that damage can be caused by thermal degradation of the material (i.e. the breakdown of the solid and the development of micro-cracks due to an increase in temperature, even without load). This thermal damage xT is typically accounted for through the degradation of the elastic modulus (as described in Sect. 5.3.1.3) (e.g. Baker and Stabler 1999; Nechnech et al. 2002; Al Najim et al. 2003; Gawin et al. 2003; Davie et al. 2010) and is treated multiplicatively with the mechanical damage xM such that: r0 ¼ ð1  xT Þð1  xM ÞC0 : ee

ð4:95Þ

However, some authors account for thermal effects through the temperature dependent evolution of the damage and/or yield surfaces (e.g. Gernay, Millard et al. 2013). The definition of the mechanical damage can take many forms and the most appropriate form is dependent on the required application. Generally, thermal dependence of the parameters of the model is required in order to capture the change in mechanical behaviour at elevated temperatures. Examples of various approaches to thermally dependent mechanical damage formulations can be found in Gawin et al. (1999, 2003), Nechnech et al. (2002), Al Najim et al. (2003), Luccioni et al. (2003), Tenchev and Purnell (2005), Davie et al. (2010), de Sa and Benboudjema (2011), Gernay et al. (2013).

References Abdel-Rahman, A.K., Ahmed, G.N.: Computational heat and mass transport in concrete walls exposed to fire, Numerical Heat Transfer, Part A: Applications. 29(4), 373–395 (1996)

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Al Najim, A., Meftah, F., et al.: A Non-Saturated Porous Medium Approach for the Modelling of Concrete Behaviour Submitted to High Temperatures. Computational Modelling of Concrete Structures, Proceedings of the EURO-C Conference, Austria, Balkema (2003) Anderberg Y. and Thelandersson S. (1973), Stress and deformation characteristics of concrete at high temperature, Lund Institute of technology (Sweden): Division of Structural Mechanics and Concrete Construction, 84p. Internal rep. no Alba15/04-01 Anderberg, Y., Thelandersson, S.: Stress and deformation characteristics of concrete at high temperature:2. Bulletin 54, Lund Institute of technology, Lund. (1976) Baker, G., Stabler, J.: Computational modelling of thermally induced fracture in concrete. Proc. Euro-C, 530–545 (1999) Baroghel-Bouny, V., et al.: Characterization and Identification of Equilibrium and Transfer Moisture Properties for Ordinary and High-Performance Cementitious Materials. Cement and Concrete Research. 29, 1225–1238 (1999) Bažant, Z.P., Kaplan, M.F.: Concrete at High Temperatures: Material Properties and Mathematical Models. Longman, Harlow (1996) Bažant, Z.P., Thonguthai, W.: Pore pressure and drying of concrete at high temperature. J. Engng. Mech. Div. ASCE. 104, 1059–1079 (1978) Bažant, Z.P., Thonguthai, W.: Pore pressure in heated concrete walls: theoretical prediction. Mag. of Concr. Res. 31(107), 67–76 (1979) Benboudjema F. (2002), Modélisation des déformations différées du béton sous sollicitations biaxials. Application aux enceintes de confinement de bâtiments réacteurs des centrals nucléaires, Thèse de doctorat, UMLV, France, 258 p (In French) Benboudjema, F., Meftah, F., Torrenti, J.M.: A viscoelastic approach for the assessment of the drying shrinkage behaviour of concrete, Materials and Structures, 40(2), 163–174 (2007). Chung, J.H., Consolazio, G.R., McVay, M.C.: Finite element stress analysis of a reinforced high-strength concrete column in severe fires. Computers and Structures. 84(21), 1338–1352 (2006) Consolazio, G.R., McVay, M.C., Rish, J.W.: Measurement and prediction of pore pressures in saturated cement mortar subjected to radiant heating. ACI Mater. J. 95(5), 526–536 (1998) Dal Pont, S., Ehrlacher A.: Numerical and experimental analysis of chemical dehydration, heat and mass transfer in a concrete hollow cylinder submitted to high temperatures. International Journal of Heat and Mass Transfer. 1(47), 135–147 (2004) Dal Pont, S., Durand, S., Schrefler, B.A. A multiphase thermo-hydro-mechanical model for concrete at high temperatures - finite element implementation and validation under LOCA load. Nuclear Eng. and Design, 237/20: 2137–2150 (2007) Davie, C.T., Pearce, C.J., Bicanic, N.: Coupled Heat and Moisture Transport in Concrete at Elevated Temperatures-Effects of Capillary Pressure and Adsorbed Water. Numerical Heat Transfer, Part A. 49(8), 733–763 (2006) Davie, C.T., Pearce, C.J., et al.: A Fully Generalised, Coupled, Multi-Phase, Hygro-Thermo-Mechanical Model for Concrete, Materials and Structures 43 (Sup. 1) (2010) de Sa, C., Benboudjema, F.: Modeling of concrete nonlinear mechanical behavior at high temperatures with different damage-based approaches, Materials and Structures 44(8): 1411 – 1429 (2011) Dwaikat, M.B., Kodur, V.K.R.: Hydrothermal model for predicting fire-induced spalling in concrete structural systems. Fire Safety Journal. 44, 425–434 (2009) England, G.L., Khoylou, N.: Moisture flow in concrete under steady state non-uniform temperature states: experimental observations and theoretical modeling. Nucl. Eng. Des. 156, 83–107 (1995) Feldman, R.F., Sereda, P.J. :The model for hydrated Portland cement as de-duced from sorption-length change and mechanical properties. Materials and Con-struction, 1, 509–520 (1968) Feraille A.: Le rôle de l’eau dans le comportement à haute température des bétons. Dissertation, ENPC, Marne la Vallée (2000)

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Gawin, D., Majorana, C.E., Schrefler, B.A.: Numerical analysis of hygro-thermic behaviour and damage of concrete at high temperature. Mechanics of Cohesive-Frictional Materials. 4, 37–74 (1999) Gawin, D., Pesavento, F., Schrefler, B.A.: Simulation of damage–permeability coupling in hygro-thermo-mechanical analysis of concrete at high temperature. Communications in Numerical Methods in Engineering. 18(2), 113–119 (2002a) Gawin, D., Pesavento, F., Schrefler, B.A.: Modelling of hygro-thermal behaviour and damage of concrete at temperature above critical point of water. International Journal of Numerical and Analytical Methods in Geomechanics. 26(6), 537–562 (2002b) Gawin, D., Pesavento, F., Schrefler, B.A.: Modelling of thermo-chemical and mechanical damage of concrete at high temperature. Computer Methods in Applied Mechanics and Engineering. 192, 1731–1771 (2003) Gawin, D., Pesavento, F., Schrefler, B.A.: Modelling of deformations of high strength concrete at elevated temperatures. Materials and Structures. 37(268), 218–236 (2004) Gawin, D., Pesavento, F., Schrefler, B.A.: Towards prediction of the thermal spalling risk through a multi-phase porous media model of concrete. Computer Methods in Applied Mechanics and Engineering. 195(41–43), 5707–5729 (2006) Gens, A. and Olivella, S.: THM phenomena in saturated and unsaturated porous media, RFGC. Environmental Geomechanics. 5(6), 693–717 (2001) Gernay, T., Millard, A., et al.: A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation, International Journal of Solids and Structures 50(22–23): 3659-3673 (2013) Gray, W.G., Schrefler, B.A.: Analysis of the solid phase stress tensor in multiphase porous media. Int. J. Numer. Anal. Meth. Geomech., 31(4), 541–581 (2007) Gray, W.G., Schrefler, B.A., Pesavento, F.: The solid phase stress tensor in porous media mechanics and the Hill-Mandel condition. J. Mech Phys Solids. 57, 539–554 (2009) Harmathy, T.Z.: Effect of moisture on the fire endurance of building elements. In ASTM special technical publication 385. Philadelphia (1965) Harmathy, T.Z.,. Thermal properties of concrete at elevated temperatures. J. Mech. JMLSA. 5(1), 47–74 (1970) Hassanizadeh, S.M., Gray W.G.: General Conservation Equations for Multi-Phase Systems: 1. Averaging Procedure. Adv. Water Resources. 2, 131–144 (1979a) Hassanizadeh, S.M., Gray, W.G.: General Conservation Equations for Multi-Phase Systems: 2. Mass, Momenta, Energy and Entropy Equations. Adv. Water Resources. 2, 191–203 (1979b) Hassanizadeh, S.M., Gray, W.G.: General Conservation Equations for Multi-Phase Systems: 3. Constitutive Theory for Porous Media Flow. Adv. Water Resources, 3, 25–40 (1980) Heinfling G. (1998), Contribution à la modélisation numérique du comportement du béton et des structures en béton armé sous sollicitations thermomécaniques à hautes températures, Thèse de doctorat, INSA de Lyon, 227 p (In French) Ichikawa, Y, England, G.L.: Prediction of moisture migration and pore pressure build-up in concrete at high temperatures. Nucl. Eng. Des. 228(1–3), 245-259 (2004) Kachanov, L. M.: Introduction to continum damage mechanics, Martinus Nijhoff Publishers, Dordrecht, The Netherlands (1986) Khennane, A., Baker, G.: Thermo-plasticity models for concrete under varying temperature and biaxial stress. Proc. Royal Soc. Lond. A. 439, 59–80 (1992) Khoury G.A. (2003), Creep & Shrinkage, Course of Heat on Concrete, International Center for Mechanical Sciences (CISM), 9-13 June 2003, Udine, Italy Khoury G. A. ., Grainger B. N. and Sullivan, P. I. E. (1985), Grainger B. N. and Sullivan P. J. E., Transient thermal strain of concrete: literature review, conditions within specimen and behaviour of individual constituents. Magazine of Concrete Research, 37 (132), 131–144 Khoury G.A, Majorana C.E, Pesavento F. and Schrefler B.A. (2002), Modelling of heated concrete, Magazine of concrete Research, 2002, 54, No.2, April, 77–101 Lewis, R.W., Schrefler, B.A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. second ed. John Wiley & Sons, Chichester (1998)

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Meftah, F., Dal Pont, S.: Staggered finite volume modeling of transport phenomena in porous materials with convective boundary conditions. Transport Porous Media. 82(2), 275–298 (2010) Meftah, F., Dal Pont, S., Schrefler, B.A.: A three-dimensional staggered finite element approach for random parametric modeling of thermo-hygral coupled phenomena in porous media. Int. J. Num and Anal. Meth. Geomech. 36(5), 574–596 (2012) Mehta PK, Monteiro PJM (2014) Concrete: Structure, Properties, and Materials (4th Ed). Prentice-Hall, Englewood Cliffs, NJ, USA Nechnech, W., Meftah, F., et al.: An elasto-plastic damage model for plain concrete subjected to high temperatures, Engineering Structures 24(5): 597–611 (2002) Pasquero D. (2004), Contribution à l'étude de la déshydratation dans les pâtes de ciment soumises à haute température, Thèse de doctorat, ENPC, France, 191 p. (In French) Pesavento, F.: Non-linear modeling of concrete as multiphase porous material in high temperature conditions. Dissertation, University of Padova (2000) Sabeur H, Meftah F. Dehydration creep of concrete at high temperatures. Materials and Structures 41(1), 17–30 (2008) Sabeur H, Meftah F, Colina H, Plateret G. Correlation between transient creep of concrete and its dehydration. Magazine of Concrete Research, 60(3):157–163 (2008) Schneider U. (1988), Concrete at high temperatures: A general review, Fire safety Journal, vol. 13, p 55–68 Schrefler, B.A.: Mechanics and Thermodynamics of Saturated-Unsaturated Porous Materials and Quantitative Solutions. Applied Mechanics Review. 55(4): 351–388 (2002) Tenchev, R., Purnell, P.: An application of a damage constitutive model to concrete at high temperature and prediction of spalling. Int. J. Solids Struct. 42(26), 6550–6565 (2005) Tenchev, R.T., Li, L.Y., Purkiss, J.A.: Finite element analysis of coupled heat and moisture transfer in concrete subjected to fire. Num. Heat Transfer, Part A: Appl. 39(7), 685–710 (2001) Thelandresson, S.: Modeling of combined thermal and mechanical action in concrete. J. Engrg. Mech., ASCE, 113(6), 893–906 (1987) Thelandersson S., Martensson A. and Dahlblom O. (1988), Tension softening and cracking in drying concrete, Materials and Structures, 21, p. 416–424 Torrenti J.M., Granger L., Diruy M. and Genin P. (1997), Modélisation du retrait du béton en ambiance variable, Revue Française de Génie Civil, 1 (4), p. 687–698 (In French) Ulm, F.-J., Coussy, O., Bažant, Z.P.: The “Chunnel” fire. I. Chemoplastic softening in rapidly heated concrete. J. Eng. Mech. ASCE 125(3), 272–282 (1999a) Ulm, F.-J., Acker, P., Levy, M.: The “Chunnel” fire. II. Analysis of concrete damage. J. Eng. Mech. ASCE 125(3), 283–289 (1999b) van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980) Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, vol. 1: The Basis, Butterworth-Heinemann, Oxford (2000)

Chapter 5

Constitutive Parameters Fekri Meftah, Colin Davie, Stefano Dal Pont and Alain Millard

Abstract The numerous parameters which appear in the advanced models described in the previous chapter are in general dependent of some of the state variables, such as temperature. In this chapter, many expressions used in practice for the evolution of the hygral, thermal and mechanical parameters are presented.

5.1 5.1.1

Hygral Parameters Evolution of Pore Structure During Heating

The loss of stability of solid phases in concrete, paste and aggregates, when exposed to high temperatures, affects its pore structure. The physical and chemical changes occurring in solid phases induce changes in pore size distribution and total porosity. Noumowe (1995) has given the variation of porosity with different levels of temperatures for an ordinary concrete and a high performance concrete heated up to 600 °C. The total pore volume of a heated concrete increases non-linearly with the increase of temperature. On heating up to 300 °C, a relatively small increase in pore volume is observed in comparison to the weight losses. This was considered to F. Meftah (&) Institut National des Sciences Appliquées, Rennes, France e-mail: [email protected] C. Davie School of Engineering, Newcastle University, Newcastle upon Tyne, UK e-mail: [email protected] S. D. Pont Laboratoire 3SR, Université Grenoble-Alpes, Grenoble, France e-mail: [email protected] A. Millard Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France e-mail: [email protected] © RILEM 2019 A. Millard and P. Pimienta (eds.), Modelling of Concrete Behaviour at High Temperature, RILEM State-of-the-Art Reports 30, https://doi.org/10.1007/978-3-030-11995-9_5

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be due to structural changes resulting from the desiccation and dehydration processes occurring mainly in pores of radii less than 40 Å (Andrade et al. 2003). Heating up to 600 °C will increase the total pore volume. This increase is higher than would be expected in comparison with weight losses (Fig. 5.1). On the other hand Bažant and Thonguthai (1978, 1979) observed an upward jump in concrete permeability of two orders of magnitude. This was related to the fact that the pore volume available for capillary water must increase significantly when temperature and pore pressure increase. Then, it may therefore be suspected that either an expansion of pores has occurred by break down of partition walls, or micro-cracks are formed (Andrade et al. 2003). Many authors have investigated micro-cracking (Dougill 1968; Blundell et al. 1976). They have indicated that the differential strain between the aggregate (which expands) and the cement paste (which shrinks) at different temperatures will initially induced a small compressive stress in the paste. As the temperature increases, the compressive stress is reduced and changes to a larger tensile stress. The changes of porosity with the increase of temperature were measured for several types of concrete by Schneider and Herbst (1989). Their results showed that the dependence of porosity on temperature can be approximated for concrete by a linear relationship:

Fig. 5.1 Weight variations (Noumowe 1995)

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69

Fig. 5.2 Porosity from experimental data (Schneider and Herbst 1989) and approximations using Eq. 5.1

uT ¼ u0 þ Au ðT  T0 Þ

ð5:1Þ

where Au is a constant dependent on the type of concrete, and T0 corresponds to room temperature. For the experimental data for three types of the B35-concrete, the following coefficients in (5.1) have been found by using the least-squares method (Gawin et al. 1999) (Fig. 5.2):

u0 Au ðK1 Þ

Silicate concrete

Limestone concrete

Basalt concrete

0.060 0.000195

0.087 0.000163

0.0802 0.00017

Another relationship for the evolution of porosity due to the dehydration processes has been proposed by Feraille (2000) as follows: uT ¼ u0 þ 0:72  103 mdehyd

ð5:2Þ

where u0 is the initial porosity and mdehyd is the mass of water which is to be experimentally determined from the dehydration processes, as will be seen in detail in the following section. It should be noted also that the above equation takes into account the Le Chatelier contraction.

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Dehydration Processes—Chemical Reaction

The combined reaction in cement paste and aggregate in concrete, which are initiated when concrete is subjected to elevated temperature, have been examined by many authors (Harmathy and Allen 1973; Schneider 1982) with the aid of differential thermal analysis investigations. Dehydration (i.e. release of chemically bound water) from the C-S-H becomes significant above 105 °C, while the dehydration of the calcium hydroxide Ca(OH)2 takes place at about 500 °C producing CaO and H2O. Both will produce a reduction in the mass of the cement paste of the solid skeleton. However, the strength of concrete is more related to the C-S-H than the Ca (OH)2. The amount of water that is released upon heating due to dehydration into the pores of concrete depends on the amount of hydrated water mhydr (consumed by the hydration process) before heating. Therefore, it is a function of the degree of hydration. The degree of hydration before heating depends on the age of concrete as well as its temperature and humidity history. The degree of hydration may conveniently be referred to the equivalent hydration period (maturity) te. It represents the period of hydration, in water and at 25 °C, needed to give the same degree of hydration as the actual time period gives at variable RH and T. By fitting the results of Powers and Brownyard (1948) at 25 °C, it has been found (Bažant and Kaplan 1996) that:


te mhyd ðte Þ  0:21c te þ se

 ð5:3Þ

where the characteristic period se ¼ 23 days for a concrete with w/c = 0.43. The amount of water that is released by the dehydration mdehydr (kg per m3 of concrete), when reaching various temperatures is experimentally obtained by weight loss measurements on heated specimens and can be expressed according to Bažant and Kaplan (1996) as: 

C mdehyd ðT Þ ¼ m105 hydr f ðT Þ

ð5:4Þ



105 C is the hydrated water content at 105 °C and f ðT Þ is a function of in which mhydr dehydration. Gawin et al. (2001) have calculated the dehydrated water content mdehyd ðT Þ, when temperature increases, by using a step function showing a sharp change between 200 and 500 °C. The following law can represent this kind of behaviour:

mdehyd ðT Þ ¼ c  faging  fstechio  f ðT Þ

ð5:5Þ

where c is the cement content, faging is the aging degree of the concrete (between 0 and 1), fstechio is stechiometric factor and f ðT Þ is a function of dehydration (Gawin et al. 2001).

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71

Based on an experimental work on cement paste, Feraille (2000) has noted the evolution of dehydration and observed thatit is not an instantaneous process but needs some time to take place. The author has proposed the following formula, which takes into account the asymptotic evolution of dehydration m_ dehyd through the characteristic time sdehyd of mass loss: m_ dehyd ¼ 

D E mdehyd  meq dehyd sdehyd

þ

ð5:6Þ

where meq dehyd is the water mass at the equilibrium measured during the thermo-gravimetric tests. It is expressed as: meq dehyd


 7:5 105  C T  105 2 105  C mhydr 1  exp mhydr ¼ H ðT  105Þ þ 100 200 100 
 
 T  400 1:5 105  C T  540 m 1  exp 1  exp H ðT  400Þ þ 10 100 hydr 5 H ðT  540Þ ð5:7Þ

  105  C where H is the Heaviside function and mhydr ¼ c 1 þ 0:2faging in (kg/m3). Equation 5.7 is used to calculate the dehydrated water content mdehyd ðT Þ, and the evolution of dehydration as follows: m_ dehyd ¼

5.1.3

 @T @  mdehyd ðT Þ @T @t

ð5:8Þ

Liquid-Vapour Equilibrium

It is assumed here that no dissipation occurs during phase change. This in turn requires the assumption that the liquid water (both capillary and adsorbed) and the vapour remain in a local thermodynamic equilibrium. Therefore, this local equilibrium gives rise to a set of restrictions in the equilibrium configuration that must be satisfied at all times. However, it can be considered that the specific Gibbs functions of the two phases, considered at the same temperature, are equal at all times. In the work of Dal Pont (2004) the following steps are followed to establish a relationship between the pure liquid water and vapour phases; starting with the consideration that the enthalpies of both liquid water gl ðpl ; T Þ and vapour gv ðpv ; T Þ are functions of temperature T and pressure p, the partial derivatives of these two functions with respect to their pressures can be written as follows:

72

F. Meftah et al.

@ 1 gl ð pl ; T Þ ¼ @pl ql ðpl ; T Þ

ð5:9Þ

@ 1 gv ð pv ; T Þ ¼ @pv q v ð pv ; T Þ

ð5:10Þ

and considering that the liquid water is incompressible (ql is independent of pl ) and   v Ml . After integrating the two equations above, we the vapour is an ideal gas qv ¼ pRT obtain: pl þ C1 ðT Þ q l ðT Þ

ð5:11Þ

RT lnðpv Þ þ C2 ðT Þ Ml

ð5:12Þ

gl ð pl ; T Þ ¼ gv ð pv ; T Þ ¼

from the equality of enthalpies of the two phase we can get: pl RT þ lnðpv Þ þ C1 ðT Þ þ C2 ðT Þ ¼ 0 Ml q l ðT Þ

ð5:13Þ

To determine C1 ðT Þ þ C2 ðT Þ Dal Pont (2004) considered the case in which the liquid water and the vapour coexist in a big and closed container. The separation surface between the two phases is flat (no capillary effect). In this case and at equilibrium conditions, the pressure of both liquid and vapour is the same (no body force is considered) and will be equal to the saturation pressure of pure vapour pvs ðT Þ. This state can be taken as a reference state and should verify (5.13). So we can find C1 ðT Þ þ C2 ðT Þ. Finally, the relationship between the liquid and the vapour, which is known as the generalised Clapeyron relationship, would be expressed as follows (Dal Pont 2004): pl ¼ pvs ðT Þ þ


 ql ðT ÞRT pv ln Ml pvs ðT Þ

ð5:14Þ

Now if we consider the case in which three phases (i.e. liquid water, vapour and dry air) can coexist at the same time, like in the case of a partially saturated concrete, then the condition of equilibrium over the separation surface must change so that (pv ¼ pvs and pl ¼ pvs þ pa ) and we can get the following relationship: pl ¼ pvs ðT Þ þ pa þ


 ql ðT ÞRT pv ln Ml pvs ðT Þ

ð5:15Þ

If we know that at 20 °C the saturation vapour pressure is pvs ¼ 2350 Pa and the dry air pressure is pa ¼ 99;000 Pa, then pg ¼ patm ¼ 101;325 Pa. This can be a

5 Constitutive Parameters

73

particular case of (5.15) which can give the known relationship of Kelvin as follows: pl ¼ patm þ


 ql ðT ÞRT pv ln Ml pvs ðT Þ

ð5:16Þ

The saturation pressure of pure vapour pvs ðT Þ, which depends only upon temperature, could be obtained from empirical correlation. Tabulated results (Raznjevic 1970), which link the water vapour saturation pressure pvs with temperature T could be approximated by the following formula (Monteith and Unsworth 1990): 7:5T

pvs ðT Þ ¼ 610:7  10T þ 237:3

ð5:17Þ

It should be noted that this equation is taken until the critical point of water 647.15 K, after which it is not possible to distinguish between the liquid and vapour. Another equation can be taken for pvs after the critical temperatures following the L-function given in Ju and Zhang (1998) for the superheated steam: "


2 # T T þ L2 pvs ðT Þ ¼ pvs ð647:15Þ L0 þ L1 647:15 647:15

ð5:18Þ

where L0 ¼ 15:8568, L1 ¼ 34:1706 and L2 ¼ 15:7437. In (5.15), the saturation vapour pressure pvs should be the pressure that takes into account the coexistence of air. In fact, it is different from that one given in (5.14), which is the saturation pressure of pure vapour. However, Feraille (2000) showed that below the critical temperature of water, the two saturation vapour pressures coincide. A boiling-point temperature can be introduced which, when reached, gives an additional evaporation in the pore until the full drying of material. This boiling temperature is the temperature for which the saturation vapour pressure is equal to the gas pore pressure. When this situation is met, the temperature remains constant while the water saturation degree decreases. Note that beyond the boiling-point temperature, no liquid water exists. Therefore, there is no need to define specific liquid water properties (density, viscosity, heat capacity…) after the critical temperature. The state equation of liquid water is rather complex, and usually creates numerical problems. As a sufficient approximation of the dependence of the bulk density of liquid water upon its temperature and pressure, the following formula may be used (Gawin et al. 1999),  ql ¼ q0l 1  bl ðT  T0 Þ þ al ðpl  p0l

ð5:19Þ

where q0l ¼ 999:84 kg m3 is the water density at the reference temperature T0   l and pressure p0l , bl ¼ m1l @m @T pl is volume thermal expansion coefficient of the

74

water,

F. Meftah et al.

(bl ¼ 4:3  10



@ml @pl T 1 9

al ¼ m1l

is

the

isothermal

compression

modulus

of water

Pa ), and ml ¼ q1 is the water specific volume. The coefficient bl l

changes in a non-linear manner with temperature (e.g. bl ¼ 0:68  104 K1 at T = 273.15 K and bl ¼ 10:1  104 K1 at T = 420 K), thus an average value for the temperature range of interest should be used in calculations. The state Eq. (5.19), basically valid for the bulk (free) liquid water, is often used also for the description of the capillary and bound water, e.g. (Gawin et al. 1999). However, due to a very complex nature of interaction between the water and skeleton, its applicability is questionable, especially as far as pressure dependence is concerned (although capillary water is in traction, its density is not expected to be lower than that for bulk water). Nevertheless, because of a lack of any reliable data, Eq. (5.19) may be used, but assuming water incompressibility, i.e. al ¼ 0 (Gawin et al. 1999). We can use the experimental results given by Raznjevic (1970) for the density of liquid water as a function of temperature, which could be approximated by the following formula given in Deseur (1999): "




T  273:15 ql ¼ 314:4 þ 685:6 1  374:14

#0:55 1 0:55

ð5:20Þ

Gas Properties Ideal Gases The following expressions of density of gases can be found in the literature: qp ¼

Mp pp RT

ð5:21Þ

where, pp is the density of gas phase, R is the ideal gas constant, T is the temperature, Mp is the molar mass of gas and pp is the partial pressure of the gas phase (e.g. Gawin et al. 1999; Tenchev et al. 2001; Davie et al. 2006) qv ¼ pv

Mv þ Dqv RT

ð5:22Þ

where, qv is the density of the vapour phase, Mv is the molar mass of vapour, pv is the partial pressure of the vapour phase and Dpv is a correction factor to bring the ideal gas behaviour in line with measured values of vapour density with temperature (Gawin et al. 2002).

5 Constitutive Parameters

75

Partial Pressures and Densities

pg ¼ pv þ pa

ð5:23Þ

qg ¼ qv þ qa

ð5:24Þ

where pg is the pressure of the combined gas phase, pv is the partial vapour pressure and pa is the partial pressure of dry air. pg is the density of the combined gas phase, pv is the density of the vapour phase and pa is the density of the dry air (Gawin et al. 1999; Tenchev et al. 2001; Davie et al. 2006).

5.1.4

Sorption and Desorption Isotherms

The sorption and desorption isotherms are useful for calculation of pores water vapour pressures. These isotherms are the equilibrium curves of pore relative vapour pressure versus specific water content at constant temperatures. In another words, relative humidity RH is controlled while the moisture content within the sample is measured. Since the degree of saturation Sl is fundamental for realistic modelling of thermo-hygral behaviour of the specific concrete, it must be determined during sorption tests at several temperatures (Bažant and Thonguthai 1978). However, isotherms are only available for moderate temperature. Theoretically, it should be possible to determine these isotherms also above 100 °C, but this would require measurements at very high ambient pressure. Such data are not available (Bažant and Kaplan 1996). At elevated temperature Bažant and Thonguthai (1978) assumed that a local thermodynamic equilibrium always exists between the phases of pore water (vapour and liquid) within a very small element of concrete. An approximate, but rational, constitutive relation, which relates pore pressure pv, water content w and temperature T is presented as follows: im1 w h ws0  ¼  RH c c

with RH ¼

pv pvs ðT Þ

ð5:25Þ

where c = mass of (anhydrous) cement per m3 of concrete; ws0 = saturation water content at 25 °C. The value of ws0/c can be determined if the concrete mix is specified (Neville 1973; Powers and Brownyard 1948). The same value for concrete that was mentioned in Bažant and Thonguthai (1978), ws0/c = 100/300 has been taken. The function m(T) in Eq. (5.21) has been empirically corrected by fitting with test data given by England and Ross (1970). It has been found that the following semi-empirical expression (5.22) is acceptable (Bažant and Thonguthai 1978):

76

F. Meftah et al.

mðT Þ ¼ 1:04 

T0 22:34 þ T 0

with T 0 ¼




T þ 10 T0 þ 10

2 ð5:26Þ

where T is the temperature in °C; T0 ¼ 25  C. From (5.21) we can write the degree of saturation Sl ¼ Sl ðRH; T Þ as a function of relative humidity and temperature as follows: Sl ðRH; T Þ ¼

ðw=cÞRH;T ðw=cÞRH0 ;T0

ð5:27Þ

where ðw=cÞRH0 ;T0 is calculated at the initial state. According to this equation it can be seen that at a higher temperature the saturated state is reached with lower water content. The degree of liquid saturation Sl can also be related to the capillary pressure pc. Since a direct experimental determination is difficult, an indirect determination of the Sl − pc relationship by using the desorption isotherm is generally employed. Applying Kelvin’s law, which relates the relative humidity to the capillary pressure, yields an expression for the missing capillary pressure curve and the degree of saturation Sl ¼ Sl ðpc ; T Þ as follows: Sl ð pc ; T Þ ¼

 
1=mðT Þ 1 ws0 Ml pc  exp ðw=cÞRH0 ;T0 c ql RT

ð5:28Þ

The following equation between capillary pressure, temperature and saturation can be alternatively developed based on the work of Bendar (2002): S l ð pc ; T Þ ¼ Sl ð pc Þ

rðT Þ rð T 0 Þ

ð5:29Þ

where r(T) is the surface tension of water which depends upon temperature, and can be given by Le Neindre (1993) as follows:
 T n rðT Þ ¼ r0 1  Tcr

ð5:30Þ

where r0 = 0.155 [N/m2] and n = 1.26. Based upon the pore network model of Mualem (1976), a relationship between the capillary pressure and the liquid saturation is proposed in van Genuchten (1980) in the form: "


1=ð1AÞ #A j pc j Sl ðpc Þ ¼ 1 þ B

ð5:31Þ

where A, B are parameters of the material, evaluated with a least-square fit in comparison with the experimental results of the retention curve. Figure 5.3 shows

5 Constitutive Parameters

77

Fig. 5.3 Desorption isotherms Sl(pc, T)

the evolution of Sl with capillary pressure pc using (5.31). We can relate the capillary pressure pc to the saturation degree Sl by the following relationship Baggio et al. (1993):  11=b pc ðSl Þ ¼ a Sb l 1

ð5:32Þ

where a and b are material parameters identified for each studied case.

5.1.5

Permeability

Darcy’s Law Pressure driven fluid flow can be written as: vi ¼

K  Kp rpp li

ð5:33Þ

78

F. Meftah et al.

where vp is the velocity of the fluid p, K is the intrinsic permeability of the concrete, Kp is the relative permeability of the phase, lp is the dynamic viscosity of the phase and pp is the driving pressure. pp is typically the pressure of the phase pl or pg but may be calculated from capillary pressure and may include pressures due to gravity (e.g. Gawin et al. 1999; Tenchev et al. 2001; Davie et al. 2006; Mounajed and Obeid 2004; Baroghel-Bouny et al. 1999; Chung and Consolazio 2005). In a partially saturated porous medium like concrete the liquid and gas transport will be governed by the absolute permeability of the material, which can be defined as: K0 ¼

K  krp lp

ð5:34Þ

where, K′ in [m3/kg s] is the conventional permeability (apparent permeability) of the material, K in [m2] is defined as the intrinsic permeability, which is a material property independent of the fluid, krp [−] is the relative permeability of the fluid and depends on the degree of saturation of the liquid phase and lp in [Pa s] is the dynamic viscosity of fluid and p = l, g. In the following sections we will carry out a detailed study of each one of the above properties and their evolutions with high temperatures. Intrinsic Permeability Intrinsic permeability is a characteristic of the geometrical connectivity of the porous network, irrespective of the filling fluid. Therefore, the relative permeability is the additional variable that must take into consideration the penetration of gas and liquid and their coexistence within the porous network. In their investigations on pore pressures in heated concrete (Bažant and Thonguthai 1978, 1979) found an upward jump in the permeability of concrete by two orders of magnitude (about 200 times) as the temperature rises above 100 °C. It was concluded that the pore volume available to capillary water must increase significantly as temperature and pore pressure increase. That means porosity changes from a system of closed, isolated pores to an open interconnected network. Kalifa et al. (1998) studied the influence of high temperature on the intrinsic permeability K [m2] of concrete. The main features of this work can be seen in Fig. 5.4. At 105 °C, the permeability of high performance concrete (HPC) (1017 m2 ) is 10 times inferior to the one of a normal concrete (NC) (1016 m2 ). Between 105 and 400 °C, the permeability of the HPC increases more quickly than the one of the NC; At 400 °C, the permeability of the HPC (3  1015 m2 ) is slightly superior to that one of the NC (1015 m2 ). According to the authors, below 300 °C the augmentation of permeability is due to an increase in the capillary pores, while above 300 °C micro-cracking plays a very important role in permeability augmentation. Besides the temperature effect, the mechanical load also influences the evolution of intrinsic permeability. Indeed, a load applied up to the ultimate strength is assumed to cause extensive damage in the specimen generating macroscopic

5 Constitutive Parameters

79

Fig. 5.4 The intrinsic permeability versus temperature for an ordinary concrete (NC) and for high performance concrete (HPC) (Kalifa et al. 1998)

cracking and, therefore, a sharp increase in permeability (Gérard et al. 1996; Wang et al. 1997; Torrenti et al. 1999). An experimental study was conducted by Picandet et al. (2001) to characterize the effect of external load-induced cracking on the permeability of concrete after unloading. Compared to the undamaged sample, a uniaxial compressive load at 90% of the ultimate strength can increase the axial permeability by about one order of magnitude after unloading. This increase in permeability is directly related to the maximum applied strain during loading and is due to the formation of a connected network of micro-cracks, which does not close down completely after the samples are unloaded (see Fig. 5.5). An expression has been proposed by the author in order to reproduce the experimental results as follows: K ðuM Þ ¼ K0  expðA  uM ÞB

ð5:35Þ

A generic relationship by Gawin et al. (1999) gives the intrinsic permeability as a function of temperature and gas pressure: 



K T; pg ¼ K0  10

AT ðTT0 Þ




pg  pg0

Ap ð5:36Þ

where K0 is the intrinsic permeability at reference temperature T0 and gas pressure pg0.

80

F. Meftah et al.

Fig. 5.5 Relationship between increase in permeability and damage value from ‘Grindosonic’ evaluation and the curve fitted according to (5.35) (Picandet et al. 2001)

An approach was developed by Meschke and Grasberger (2003) based on an additive decomposition of the permeability into two portions: the first related to the moisture transport through the partially saturated pore space ku multiplied by the initial intrinsic permeability K0 and the second one related to the moisture flux within a crack Kd. K ¼ K0 ku ðuÞ þ Kd

ð5:37Þ

The relationship between permeability and pore structure of hardening cement paste at early ages was investigated experimentally by Nyame and Illston (1981). According to this work a relationship between permeability and the pore structure can be expressed as follows Meschke and Grasberger (2003): ku ðuÞ ¼ 10d

with d ¼

6ðu  u0 Þ 0:3  0:4u0

ð5:38Þ

where u0 denotes the initial porosity and u  u0 is the change of porosity. When it is necessary to consider cracking of concrete, the increase of permeability due to the change of porosity is insufficient. The effect of cracks on moisture transport is significantly larger compared to the effect of elastic and inelastic change of porosity. In a smeared crack approach, isotropic damage permeability Kd is introduced as a function of the width wc [µm] of a discrete crack smeared over an equivalent strain localisation zone with a characteristic length lc (Meschke and Grasberger 2003). The authors proposed the following relationship:

5 Constitutive Parameters

81

Kd ¼

w3h 12lc

with wh ¼

Wc2 R2:5

for wc  wh

ð5:39Þ

where wh in [µm] is the equivalent hydraulic width and R is a parameter which describes the crack roughness. More recently (Rastiello et al. 2014) developed a phenomenological correction coefficient for the latter relationship. Relative Permeability of Fluids Few experimental results exist concerning the relative permeability of cement-based materials to gas and even less concerning the relative permeability to liquid. To overcome this difficulty, analytical expressions have been derived by van Genuchten (1980), based on the model of Mualem, which predicts the hydraulic conductivity from the statistical pore size distribution: krl ðSl Þ ¼




A 2 pffiffiffiffi 1=A Sl 1  1  Sl

ð5:40Þ

In addition, the author proposed a methodology to estimate the parameters of this equation, by using the fitted expression. Thus the parameter A is the exponent appearing in the capillary pressure curve of Eq. (5.31). In the particular case of cement-based materials (Savage and Janssen 1997) have shown the relevance of (5.40) in considering the liquid-water movement in a non-saturated Portland cement concrete. According to Couture et al. (1996) and Nasrallah and Perre (1988), the water relative permeability for concrete depends upon the saturation with liquid phase and can be usually expressed as the following: krl ¼


 S  Sir Al 1  Sir

for S [ Sir

ð5:41Þ

where Sir is the irreducible saturation (sometimes assumed as Sir = 0), and Al is a constant, with the value from the range 〈1, 3〉. When the relative humidity RH reaches a value higher than 75%, a rapid increase of the capillary water flux is observed (Bažant and Najjar 1972). Such behaviour can be described by the following relationship (Gawin et al. 1999). "


 #1 1  RH Bl krl ¼ 1 þ  SA l 0:25

ð5:42Þ

where Al, Bl are constants with values in the range 〈1, 3〉. It should be noted that the term in square brackets has a form originally proposed in Bažant and Najjar (1972). According to Gawin et al. (1999) this equation has good numerical properties and allows the use of the irreducible saturation concept to be avoide; a concept that creates serious theoretical and numerical problems (Couture et al. 1996).

82

F. Meftah et al.

The gas relative permeability of concrete, similarly to most capillary porous materials, may be described from the analytic expressions given by Luckner et al. (1989), based on the model of Mualem (1976): krg ðSÞ ¼

2A pffiffiffiffiffiffiffiffiffiffiffi

1  S 1  S1=A

ð5:43Þ

Moreover it can also be described by the formula given by Couture et al. (1996) and Nasrallah and Perre (1988):


S krg ¼ 1  Scr

Ag for S\Scr

ð5:44Þ

where Scr is the critical saturation value, above which there is no gas flow in the medium, Ag is a constant, which usually has value from the range 〈1, 3〉. Viscosity of Fluids The dynamic viscosity of liquid water µl [Pa s] depends strongly upon temperature and can be evaluated, with sufficient accuracy, in a wide temperature range, using the approximate formula (Thomas and Sansom 1995): ll ¼ 0:6612ðT  229Þ1:562

ð5:45Þ

The dynamic viscosity of moist air µg [Pa s], which depends upon the temperature and the ratio of the vapour and gas pressure, using the data from Mason and Monchik (1965), can be approximated using the following formula (Forsyth and Simpson 1991): lg ¼ lv þ ðla  lv Þ


a 0:608 p pg

ð5:46Þ

where the ratio pa/pg is the mole fraction of dry air in the gas phase, and lv is the water vapour dynamic viscosity: lv ¼ lv0 þ av ðT  T0 Þ

ð5:47Þ

 with lv0 ¼ 8:85  106 ½Pa s, av ¼ 3:53  108 Pa s K1 . Further, the dry air dynamic viscosity la is given by: la ¼ la0 þ aa ðT  T0 Þ þ ba ðT  T0 Þ2

ð5:48Þ

 with la0 ¼ 17:17  106 ½Pa s, aa ¼ 4:73  108 Pa s K1 , ba ¼ 2:22  1011  Pa s K2 . Another formula exists in the literature (Pezzani 1988), which is simpler since it is function of temperature only:

5 Constitutive Parameters

83

lg ¼ 3:85  108 T

5.1.6

ð5:49Þ

Diffusivity

The diffusivity of vapour in the air at temperature T and pressure pg is given in Daian (1989): 

Dva T; pg




Av T pg0 ¼ Dv0 T0 pg

ð5:50Þ

where Dv0 ¼ 2:58  105 ½m2 s1  is the diffusion coefficient of vapour species in the air at the reference temperature T0 ¼ 273:15 K and pressure pg0 ¼ 101;325 Pa (Forsyth and Simpson 1991). Av is a constant, and for the value Av ¼ 1:667 there is a good correlation with the experimental data concerning vapour diffusion at different temperatures (Mason and Monchik 1965). The pore space in concrete has a very complex inner structure, that influences the vapour diffusion process. Therefore, and because of the lack of experimental results, the simplest way of taking it into account is the introduction of a tortuosity factor, as it has been suggested by Bažant and Najjar (1972) and Perre (1987), which considers the tortuous nature of the pathway in the porous media. The tortuosity factor is usually assumed to be constant in the whole range of moisture content although (Millington 1959) showed that it can be approximated by the following formula: sð/; Sl Þ ¼ /1=3 ð1  Sl Þ7=3

ð5:51Þ

For concrete the range is typically s ffi 0:40:6, used by several authors (e.g. Daian 1989). Several authors (e.g. Perre 1987) have defined the effective diffusion coefficient, taking into account the reduction of space offered to the diffusion of gaseous constituents and the tortuosity, as: Deff ðSl Þ ¼ u ð1  Sl ÞBv sDva ðT; pl Þ

ð5:52Þ

where Bv is a constant, usually in the range 〈1, 3〉 (Daïan 1988). As an alternative, the resistance factor (structure factor) fs can be introduced, which considers both the tortuosity effect s and the reduction of space offered to the diffusion of gaseous constituents.

84

F. Meftah et al.

  Deff ðSl Þ ¼ fs ðu; Sl ÞDva T; pg

ð5:53Þ

where ug ¼ ð1  Sl Þu. This structure factor fs could be also approached through the expression derived by Millington (1959) for variably saturated porous media and used by several authors (e.g. Adenekan et al. 1993; Sleep and Sykes 1993; Mainguy et al. 2001), 2 4=3 fs ðu; Sl Þ ¼ u4=3 ð1  Sl Þ10=3 g ð1  Sl Þ ¼ u

ð5:54Þ

During diffusion of vapour in a porous material with very narrow pores (e.g. in concrete), the number of collisions of water molecules with the solid matrix is non-negligible compared with the number of collisions with air molecules. This is known as the Knudsen effect, but it is generally ignored.

5.2 5.2.1

Thermal Parameters Thermal Conductivity

For an ordinary concrete thermal conductivity will decrease when the temperature rises. Thermal conductivity as a function of high temperature is very difficult to measure due to the influence of many parameters such as porosity, moisture content, type and amount of aggregates. The reduction in thermal conductivity due to the temperature is marked for a concrete with siliceous aggregates, weak for calcareous aggregates, and very weak for lightweight aggregates [Fig. 5.6 (Collet 1977)]. The thermal conductivity, k, of siliceous concrete presented in Anderberg (2003) and used in Swedish fire engineering design is shown in Fig. 5.7. In Eurocode 2 (2004) and Eurocode 4 (2004) a compromise has been made in order to permit the thermal conductivity to be chosen between an upper and a lower limit as shown in Fig. 5.7. These limits, for normal weight concrete, are described by Eq. 5.55 in the range of temperature 20  C T 1200  C: k ¼ 2  0:2451ðT=100Þ þ 0:0107ðT=100Þ2 k ¼ 1:36  0:136ðT=100Þ þ 0:0057ðT=100Þ

ðupper limitÞ 2

ðlower limitÞ

ð5:55Þ

The effective thermal conductivity of a partially saturated concrete depends upon the degree of saturation and the temperature (Gawin et al. 1999).
k ¼ kd 1 þ 4

Sl uql ð1  uÞqs

 ð5:56Þ

5 Constitutive Parameters

Fig. 5.6 Thermal conductivity of different structural concretes (Collet 1977)

Fig. 5.7 Thermal conductivity of siliceous aggregate concrete (Anderberg 2003)

85

86 Table 5.1 Thermal conductivity parameters

F. Meftah et al. kdo Ak qs ql T0

1.39 (W/(m K)) 0.0006 (K−1) 2590 (kg/m3) 999.84 (kg/m3) 293.15 (K)

Fig. 5.8 Dependence of thermal conductivity from temperature according to (5.56)

where ql is the liquid water density, u is the porosity, ql is the density of liquid water, S is the saturation with liquid water, qs is the density of the solid phase, and kd(T) is thermal conductivity of the dry material, whose temperature dependence can be expressed by means of the following relationship (Harmathy 1970): kd ¼ kd0 ½1  Ak ðT  T0 Þ

ð5:57Þ

kd0 being the dry conductivity at the reference temperature T0 and Ak being a coefficient which takes into account the effect of the thermal damage on the dry thermal conductivity. An example of the thermal conductivity for concrete using (5.56), with the following data (Table 5.1) is presented in Fig. 5.8.

5 Constitutive Parameters

5.2.2

87

Heat Capacity

Heat capacity is the amount of heat per unit mass required to change the temperature of the material by one degree. At a constant pressure p, the heat capacity, Cp, may be expressed as follows (Harmathy 1970; Harmathy and Allen 1973):
Cp ¼

@H @Tp

 ð5:58Þ

where H is enthalpy and T is temperature. At elevated temperatures the changes in the heat capacity for different concrete types may be caused by the latent heat of the different reactions involved during heating (water release, dehydration, decarbonisation, a ! b quartz inversion). From test results, Schneider (1988) has concluded that: • The type of aggregate has little influence on the heat capacity if temperatures below 800 °C are considered. • The water content is important for temperatures below 200 °C. Wet concrete shows an apparent heat capacity nearly twice that of oven-dried concretes. Also according to Franssen (1987), wet concretes present an obvious calorific capacity that is nearly two times higher than that of dry concretes. For concretes that are initially wet, heating up to 90 °C can cause a rapid but temporary increase in heat capacity between two and three times the magnitude of the initial value (Blundell et al. 1976; Ohigishi et al. 1972). This is due to a rapid vaporisation of free or evaporable water. At about 150 °C, the heat capacity is the same as the initially pre-dried concrete and it increases linearly with temperature (Blundell et al. 1976). Harmathy (1970) estimated the heat capacity of idealised Portland cement pastes from theoretical considerations coupled with experimental data. The most important reactions were considered to be the dehydration of both C-S-H gel and calcium hydroxide Ca(OH)2. Note that the latter gives the most conspicuous peak of the curve at about 500 °C (see Fig. 5.9). The two reactions indicate that at the temperatures for which dehydration occurs, i.e. from 100 to 850 °C, the latent heat contribution to the heat capacity Cp is very significant. Its value may be several  p ” due to the absorption of heat in the times higher than the “sensible heat capacity C dehydration reaction. If heating is accompanied by chemical reactions or phase transitions that take place at a given temperature, the enthalpy is a function of both the temperature T and the degree of conversion n of the reactants into the products. According to Harmathy (1970) and Harmathy and Allen (1973) the Cp is usually referred to “apparent heat capacity” at a constant pressure and it may be expressed as follows:

88

F. Meftah et al.

Fig. 5.9 Heat capacity of a cement paste (Harmathy 1970)



 @H @H @n  p þ DHp @n ¼C Cp ¼ þ @T p; n @n p; T @T @T

ð5:59Þ

 p is the sensible heat contribution to the heat capacity at a given degree of where C conversion n, and DHp is the enthalpy of the reaction that occurs (evaporation, dehydration).

5.2.3

Enthalpy of Evaporation and Enthalpy of Dehydration

The enthalpy of evaporation, also called the latent heat of evaporation, depends on the temperature and may be approximated by the Watson formula (Forsyth and Simpson 1991): DHvap ¼ 2:672  105 ðTcr  T Þ0;38

ð5:60Þ

where Tcr ¼ 647:15 K is the critical temperature of water. Concerning the enthalpy of dehydration, the non-evaporable water must be first discussed. The non-evaporable water is the water driven from the cement paste when it is heated to temperatures above 105 °C and generally it contains nearly all

5 Constitutive Parameters

89

chemically combined water in the C-S-H, all the water in the Ca(OH)2, and also some water not held by chemical bonds. It takes 1670 J/kg to establish a bond of non-evaporable water in the C-S-H while the energy of the water of crystallisation of Ca(OH)2 is 3560 J/kg (Khoury and Majorana 2003). The enthalpy of dehydration is supposed constant by Tenchev et al. (2001) and Davie et al. (2006).

5.2.4

Density

According to Anderberg (2003) the density of concrete changes with temperature. Due to a first loss of free water up to about 200 °C and then to a second loss of chemically bound water, the density decreases with increasing temperature. The relative decrease of density up to 1000 °C is between 11 and 13%. These effects of water loss are reflected in the definition of concrete density q in Eurocode 2 and are presented in Table 5.2 (Anderberg 2003).

5.2.5

Volumetric Heat Capacity

In order to assess the volumetric heat capacity qCp which is normally used in the energy conservation equation, taking into account the effect of latent heat, Eq. (5.61) and Table (5.2) can be used. In a partially saturated concrete the volumetric heat capacity qCp [J/(m3 K)] can be expressed as a combination of the volumetric capacities of its constituents, as well as the effect of the evaporation and dehydration processes (Gawin et al. 1999). qCp

 @T @T  ¼ qCp eff þ ðDH m_ Þvap þ ðDH m_ Þdehydr @t @t

ð5:61Þ

where (qCp)eff is the effective volumetric heat capacity, DHvap and DHdehydr are the enthalpies of evaporation and dehydration respectively and m_ vap and m_ dehydr are the rates of evaporation and dehydration respectively. The effective volumetric heat capacity in Eq. (5.61) reads:

Table 5.2 Variation of density with temperature influenced by water loss (Anderberg 2003) q ¼ q20  C q ¼ q20  C (1–0.02(T − 115)/85) q ¼ q20  C (0.98–0.03(T − 200)/200) q ¼ q20  C (0.95–0.07(T − 400)/800)

For For For For

20 °C T 115 °C 115 °C T 200 °C 200 °C T 400 °C 400 °C T 1200 °C

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F. Meftah et al.



qCp

 eff

   ¼ ð1  uÞqs Cps þ u Sql Cpl þ ð1  Sl Þ qa Cpa þ qv Cpv

ð5:62Þ

where the thermal capacities Cpi [J/(kg K)] depend upon the temperature of the material and qi is a density with i ¼ s; l; v; a:S: The dependence of temperature on the volumetric heat capacity for the solid skeleton may be approximated by a linear relationship (Harmathy and Allen 1973): ð1  uÞqs Cps ¼ ð1  u0 Þqs0 Cps0 ½1 þ Ac ðT  T0 Þ

ð5:63Þ

where ð1  u0 Þqs0 Cps0 is the volumetric heat capacity of the solid skeleton at the reference temperature T0, and Ac is a coefficient which considers the effect of temperature on the dry volumetric heat capacity. The thermal capacity for the gaseous constituents, in the case of a perfect gas and at a constant pressure, would be chosen independent from temperature changes for both air and vapour. Their heat capacities will be Cpa ¼ 1003:5 and Cpv ¼ 1880:0 (Feraille 2000). Based on the results for the thermal capacity of the liquid water given by Raznjevic (1970), an approximated formula has been given by Feraille (2000) as follows:
 T  273 2 Cpl ¼ 4180 þ 300  T  715

5.3

ð5:64Þ

Mechanical Parameters

5.3.1

Material Properties

The solid phase skeleton of concrete generally undergoes degradation upon heating such that its mechanical properties change (reduce) with increasing temperature. When considering numerically the mechanical performance of concrete exposed to elevated temperatures it is necessary to take into account these evolutions. Primarily, the changes in strength and stiffness need to be considered but other properties such as fracture energy may also be of interest.

5.3.1.1

Tensile Strength

While the strength of concrete may be shown to increase slightly in the early stages of heating it is generally accepted that there is an overall loss of tensile strength and various simplified relationships have been employed for numerical models in the literature.

5 Constitutive Parameters

91

For example (Gernay et al. 2013) employ the function presented in Eurocode 2 (see also Sect. 4.2.2) fck;t ðT Þ ¼ kc;t ðT Þ  fck;t

ð5:65Þ

where, kc;t ðT Þ ¼ 1:0 kc;t ðT Þ ¼ 1:0  ðT  100Þ=500

for 20  C T 100  C for 100  C\T 600  C

ð5:66Þ

while (Davie et al. 2012) employed a similarly shaped quadratic curve: f t ðT Þ ¼

ft0



2 T  T0 1  0:1 100

ð5:67Þ

Tenchev and Purnell (2005) present an exponential curve, for a specific concrete, that predicts a much slower reduction in strength with increasing temperature:   ft ¼ 6:5  14 exp 430T 0:9

ð5:68Þ

Nielsen et al. (2002) proposed an alternative, convex curve: ft ðT Þ ¼

ft0


 ! T  T0 2 1  0:016 100

ð5:69Þ

and (Nechnech et al. 2002) present a relationship derived from experimental work, which reduces more steeply even than the Eurocode curve. Luccioni et al. (2003) also present relationships derived from experimental work and these lie in close proximity to the curve employed by Davie et al. (2012). These functions are all presented in Fig. 5.10.

5.3.1.2

Compressive Strength

Similarly, the compressive strength is seen to reduce with increasing temperature. However, functions describing this numerically are more rarely presented in the literature. Once again (Gernay et al. 2013) employ the tabulated functions presented in Eurocode 2 (although it should be noted that the Eurocode states that these values are for normal weight concrete). The Eurocode presents two sets of values, for concrete with siliceous and calcareous aggregates. Nechnech et al. (2002) again use a function derived from experimental work, which is very similar to the Eurocode curve for concrete with siliceous aggregate and (Gawin et al. 2003) use a linearly reducing function, also derived from

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1

Normalised Tensile Strength (-)

0.9 0.8 0.7

EC2

0.6

Davie et al.

0.5

Nielsen et al.

0.4

Nechnech et al.

0.3

Luccioni et al. 1

0.2

Luccioni et al. 2

0.1

Tenchev and Prunell

0

0

200

400

600 800 Temperature (C)

1000

1200

Fig. 5.10 Dependence of tensile strength on temperature according to various authors

experimental data, but reducing more steeply than the Eurocode curves. Luccioni et al. (2003) present two experimentally derived curves that both suggest significantly lower strengths than other authors in the early stages of heating but meet the upper curves at higher temperatures. Davie et al. (2012) do not explicitly employ compressive strength in their work but implicitly the compressive strength follows the same evolution as the tensile strength (5.67). These functions are all presented in Fig. 5.11.

fc

Normalised Compressive Strength ( -)

1.0 0.9 0.8 0.7

EC2 (Siliceous)

0.6

EC2 (Calcareous)

0.5

Davie et al.

0.4

Nechnech et al.

0.3

Gawin et al.

0.2

Luccioni et al. 1

0.1

Luccioni et al. 2

0.0

0

200

400

600

800

1000

1200

Temperature (C)

Fig. 5.11 Dependence of compressive strength on temperature according to various authors

5 Constitutive Parameters

5.3.1.3

93

Elastic Modulus

The elastic modulus of concrete is also known to generally decrease with increasing temperature due to the breakdown of the solid skeleton of the concrete and the development of micro-cracking. Again, several authors have presented relationships to describe this although, again, a numerical description is rarely found. Davie et al. (2010) use a quadratically reducing curve:

2 T  T0 EðT Þ ¼ E0 1  0:1 100

ð5:70Þ

while (Tenchev and Purnell 2005) proposed an exponential curve, for a specific concrete: E¼

32:3

ð5:71Þ

1 þ expð0:0068T  1:5Þ1:13

and (Gawin et al. 2003; Luccioni et al. 2003) and (Nechnech et al. 2002) employ functions derived from experimental data. When plotted (see Fig. 5.12) these functions are all broadly similar.

1 0.9

Normalised Elasc Modulus (-)

0.8 0.7 Davie et al. 0.6

Gawin et al. Nechnech et al. 1

0.5

Nechnech et al. 2

0.4

Luccioni et al. 1 0.3

Luccioni et al. 2

0.2

Tenchev and Prunell

0.1 0

0

200

400

600

800

1000

1200

Temperature (C)

Fig. 5.12 Dependence of elastic modulus on temperature according to various authors

94

F. Meftah et al.

5.3.1.4

Poisson’s Ratio

The variation of Poisson’s ratio with temperature is much less often considered and most numerical work has considered it to remain constant with increasing temperature (e.g. Gawin et al. 1999; Nechnech et al. 2002; Davie et al. 2010). However (Gernay et al. 2013) employed the following function derived from the work of Luccioni et al. (2003)

 T mðT Þ ¼ m20 0:2 þ 0:8 TTvv20 mðT Þ ¼ 0:2m20

for T Tv for T [ Tv

ð5:72Þ

This function is plotted in Fig. 5.13, along with experimental data from Luccioni et al. (2003).

5.3.1.5

Fracture Energy

The dependence of fracture energy release rate in temperature is not often considered explicitly when modelling concrete at elevated temperatures. However (Davie et al. 2010) do employ a function as below:

 0 TT0 2  Gf ¼ G0f 1 þ 0:39 TT 100  0:07 100

for 0 T 500  C

ð5:73Þ

and (Luccioni et al. 2003) do report experimentally derived values in their work. These are plotted in Fig. 5.14. Interestingly the function presented by Davie et al.

1

Normalised Poisson's rao (-)

0.9 0.8 0.7 0.6 0.5

Luccioni et al. 1

0.4

Luccioni et al. 2

0.3

Gernay et al.

0.2 0.1 0

0

200

400

600 800 Temperature (C)

1000

1200

Fig. 5.13 Dependence of Poisson’s ratio on temperature according to various authors

5 Constitutive Parameters

95 Gf

1.8

Normalised Fracture Energy ( -)

1.6 1.4 1.2 1 Davie et al. 0.8

Luccioni et al. 1

0.6

Luccioni et al. 2

0.4 0.2 0

0

100

200

300

400

500

600

700

800

Temperature (C)

Fig. 5.14 Dependence of fracture energy on temperature according to various authors

(2010) indicates that the fracture energy initially increases with temperature before decreasing, while the work of Luccioni et al. (2003) shows the fracture energy simply to decrease with increasing temperature.

References Adenekan A.E., Patzek T. & Pruess K. (1993) Modelling of multiphase transport of multicomponent organic contaminants and heat in the subsurface: Numerical model formulation, Water Resour. Res., 29(11), 3727–3740. Anderberg Y. (2003) Course on effect of heat on concrete, International Center for Mechanical Sciences (CISM), 9–13 June 2003, Udine, Italy. Andrade C., Alonso C. & Khoury G.A. (2003) Course on effect of heat on concrete, International Center for Mechanical Sciences (CISM), 9–13 June 2003, Udine, Italy. Baggio P., Bonacina C. & Strada M. (1993) Trasporto di calore e di massa nel calcestruzzo cellulare. La Termotecnica, 47(12):53–60. Baroghel-Bouny V. & al. (1999) Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials, Cement and Concrete Research, 29, pp. 1225–1238. Bažant Z.P. & Najjar L.J. (1972) Non linear water diffusion in non saturated concrete, Matériaux Constructions, Paris, 5(25):3–20. Bažant Z. P. & Thonguthai W. (1978), Pore pressure and drying of concrete at high temperature, In J. Eng. Mech. Div. ASCE. 104: 1059–1079. Bažant Z. P. & Thonguthai W. (1979) Pore pressure in heated concrete walls: theoretical prediction, In Mag. of Concr. Res. 31(107): 67–76. Bažant Z.P. & Kaplan M.F. (1996) Concrete at High Temperatures: Material Properties and Mathematical Models, Harlow: Longman.

96

F. Meftah et al.

Bendar T. (2002) Approximation of liquid moisture transport coefficient of porous building materials by suction and drying experiment. Demands on determination of drying curve. Trondheim, Norway, 17–19 June 2002. 6th Symposium on Building Physics in the Nordic Countries. Blundell R., Diamond C. & Browne R.G. (1976) The properties of concrete subjected to elevated temperatures, Technical Note No. 9, June, CIRIA Underwater Engineering Group, London. Chung JH, Consolazio GR. (2005) Numerical modelling of transport phenomena in reinforced concrete exposed to elevated temperatures. Cem Concr Res, 35(3), 597–608. Collet Y. (1977) Etude des propriétés du béton soumis a des températures élevées entre 200 et 900 °C, Annales des Travaux Publics Beiges, no 4, pp. 332–338. Couture F., Jomaa W. & Puiggali J.-R. (1996) Relative permeability relations: a key factor for a drying model, Transp. Porous Media, 23, 303–335. Daïan J.F. (1988) Condensation and isothermal water transfer in cement mortar. Part 1—Pore size distribution water condensation and imbibition, Transport Porous Med. 3(6), 563–589. Daian J.F. (1989) Condensation and isothermal water transfer in cement mortar, Part II–transient condensation of water vapour, Transp. Porous Media, 44, 1–16. Dal Pont S. (2004) Lien entre la perméabilité et l’endommagement dans les bétons à haute température, Thèse de doctorat, ENPC, France. Davie C.T., Pearce C.J. & Bićanić N. (2010) Mater Struct. 43(Suppl 1), 13. Davie, C.T., Pearce, C.J., Bicanic, N. (2006) Coupled heat and moisture transport in concrete at elevated temperatures: effects of capillary pressure and adsorbed water. Numer Heat Transf, A 49(8), 733–763. Davie C.T., Zhang H.L., & Gibson A. (2012) Investigation of a continuum damage model as an indicator for the prediction of spalling in fire exposed concrete. Comput. Struct. 94–95, 54–69. Deseur B. (1999) Modélisation du comportement du béton à hautes températures, Mémoire de DEA, ENPC, France. Dougill J.W. (1968) Some effects of thermal volume changes on the properties and behaviour of concrete, The Structure of Concrete. Cement and Concrete Association, London, pp. 499–513. England G.L. & Ross A.D., “Shrinkage, Moisture and Pore Pressures in Heated Concrete” A.C.I. International Seminar on Concrete for Nuclear Reactors, Berlin (1970). European Committee for Standardisation (CEN) (2004) Eurocode 2: Design of concrete structures. European Standard EN 1992, CEN, Brussels. European Committee for Standardisation (CEN) (2004) Eurocode 4: Design of composite steel and concrete structures. European Standard EN 1994, CEN, Brussels. Feraille A. (2000) Le rôle de l’eau dans le comportement a haute température des bétons, Thèse de doctorat, ENPC, France, 186p. Forsyth P.A. & Simspon R.B. (1991) A two phase, two component model for natural convection in a porous medium. International Journal for Numerical Methods in Fluids, 12(7):655–682. Franssen J.M. (1987) Etude du comportement au feu des structures mixtes acier-béton. Thèse de Doctorat, Université de liège, Belgique, 276p. Gawin D., Majorana C.E. & Schrefler B.A. (1999) Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature. Mech. Cohes. Frict. Mater., 4:37–74. Gawin D., Majorana C.E., Pesavento F. & Schrefler B.A. (2001) Modelling thermo-hygromechanical behaviour of high performance concrete in high temperature environments, Proc. of FraMCoS-4 Fourth International Conference on Fracture Mechanics of Concrete and Concrete Structures, R. de Borst, J. Mazars, G. Pijaudier-Cabot, J.G.M. van Mier (eds) Balkema Publishers, Cachan, pp 199–206. Gawin D., Schrefler B.A. & Pesavento F. (2002) Modelling of hygro-thermal behaviour and damage of concrete at temperature above the critical point of water. International Journal for Numerical and Analytical Methods in Geomechanics, 26(6):537–562. Gawin D., Pesavento F. & Schrefler B.A. (2003) Modelling of hygro-thermal behaviour of concrete at high temperature with thermo-chemical and mechanical material degradation. Computer Methods in Applied Mechanics and Engineering, 192(13–14):1731–1771.

5 Constitutive Parameters

97

Gérard B., et al. (1996) Cracking and permeability of concrete under tension, Mater. Struct. 29, pp. 141–151. Gernay, T., Millard, A., & Franssen, J.-M. (2013). A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation. International Journal of Solids and Structures, 50 (22–23), 3659–3673. Harmathy T.Z. (1970) Thermal properties of concrete at elevated temperatures. ASTM Journal of Materials, 5(1):47–74. Harmathy T.Z & Allen L.W. (1973) Thermal properties of selected masonry unit concretes, Journal of American Concrete Institute, vol. 70, no 2, pp. 132–142. Ju J.W. & Zhang Y. (1998) Axisymmetric thermo-mechanical constitutive and damage modelling for air field concrete pavement under transient high temperature, Mechanics of Materials, 29, 307–323. Kalifa P., Tsimbrovska M. & Baroghel-Bouny V. (1998) High performance concrete at elevated temperature – An extensive experimental investigation on thermal, hydral and microstructure properties. Proc. of Int. Symp. On high-performance and reactive powder concrete. Aug. 16–20, Sherbrooke Canada. Khoury G. & Majorana C. (2003) Course on effect of heat on concrete, International Center for Mechanical Sciences (CISM), 9–13 June 2003, Udine, Italy. Le Neindre B. (1993) Tensions superficielles des composés inorganiques et des mélanges, volume K476 of Traite Constantes Physico-Chimiques. Techniques de l’Ingenieur. Luccioni B M, Figueroa M I, Danesi R F. (2003) Thermo-mechanic model for concrete exposed to elevated temperatures. Eng Struct, 25 (6): 729–742. Luckner L., van Genuchten M.T. & Nielsen D.R. (1989) A consistent set of parametric models for the two-phase-flow of immiscible fluids in the subsurface, Water Res., 25(10), 2225–2245. Mainguy M., Coussy O. & Baroghel-Bouny V. (2001) Role of air pressure in drying of weakly permeable materials, J. Eng. Mech., 127(6), 582–592. Mason E.A. and Monchik L. (1965) Survey of the equation of state and transport properties of moist gases, Humidity Moisture Measurement Control Science, 3, 257–272. Meschke G. & Grasberger S. (2003) Numerical modelling of coupled hygro-mechanical degradation of cementitious materials, J. Eng. Mech., 129(4), 383–392. Millington R.J. (1959) Gas diffusion in porous media, Science, 130, 100–102. Monteith J.L. & Unsworth M.H. (1990) Principles of environmental physics. Edward Arnold, London. Mounajed G. & Obeid W. (2004) A new coupling F.E. model for the simulation of thermalhydro-mechanical behaviour of concretes at high temperatures, Mater. Struct. 37, 422–432. Mualem Y. (1976) A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., 12, 513–522. Nasrallah S. B. & Perre P. (1988) Detailed study of a model of heat and mass transfer during convective drying of porous media, Int. J. Heat Mass Transfer, 31(5), 957–967. Neville A.M. (1973) Properties of Concretes. A Halsted Press Book, J. Wiley & Sons, New York, N.Y. Nechnech W., Meftah F., Reynouard J. M. 2002) An elasto-plastic damage model for plain concrete subjected to high temperatures, Engineering Structures, 24, pp. 597–611. Nielsen, C. V., C. J. Pearce, et al. (2002). Theoretical model of high temperature effects on uniaxial concrete member under elastic restraint. Magazine of Concrete Research 54(4): 239–249. Noumowé A. (1995) Effet de hautes températures (20–600 °C) sur le béton - Cas particulier du béton à hautes performances. Thèse de Génie Civil: Institut National des Sciences Appliquées de Lyon et Univ. Lyon I, 1995. Nyame B.K. & Illston J.M. (1981) Relationships between permeability and pore structure of hardened cement paste, Magazine of Concrete Research, 33(116):139–146. Ohigishi S., Miyasaka S. & Chida J. (1972) On properties of magnetite and serpentine concrete at elevated temperatures for nuclear reactor shielding, In International Seminar on Concrete for Nuclear Reactors, AC1 Special Publication No. 34, Vol. 3, Paper PS34–57, American Concrete Institute, Detroit, pp. 1243–53.

98

F. Meftah et al.

Perre P. (1987) Measurements of softwoods’ permeability to air: importance upon the drying model, Int. Comm. Heat Mass Transfer, 14, 519–529. Pezzani P. (1988) Propriétés thermodynamiques de l’eau (K585), Techniques de l’ingénieur, traité constantes phisico-chimiques. Picandet V., Khelidj A. & Bastian G. (2001) Effect of axial compressive damage on gas permeability of ordinary and high-performance concrete, Cement and Concrete Research, 31, 1525–1532. Powers T.C. & Brownyard T.L. (1948) Studies of the physical properties of hardened cement paste. Research Department Bulletin No. 22. Portland Cement Association, Chicago. Rastiello G., Boulay C., Dal Pont S., Tailhan J., Rossi P., Real-time water permeability evolution of a localized crack in concrete under loading. Cement and Concrete Research, Vol.56/2, pp. 20–26: 2014. Raznjevic K. (1970) Tables et diagrammes thermodynamiques, Editions Eyrolles. Savage B.M. & Janssen D.J. (1997) Soil physics principles validated for use in predicting unsaturated moisture movement in Portland cement concrete, ACI Mat J., 94(1), 63–70. Schneider U. (1982) Behaviour of concrete at high temperatures. Paris: RILEM, 72p. Report to Committee no 44-PHT. Schneider U. (1988) Concrete at high temperatures: a general review, Fire safety J. 13, 55–68. Schneider U. & Herbst H.J. (1989) Permeabilitaet und Porositaet von Beton bei hohen Temperaturen (in German), Deutscher Ausschuss Stahlbeton, 403, 23–52. Sleep B.E. & Sykes J.F. (1993) Compositional simulation of groundwater contamination by organic compounds, 1. Model development and verification, Water Resour. Res., 29(6), 1697– 1708. Tenchev R.T., Li L.Y. & Purkiss J.A. (2001) Finite element analysis of coupled heat and moisture transfer in concrete subjected to fire, Numer. Heat Transfer, 2001, 39(7), 685–710. Tenchev R.T. & Purnell P. (2005) An application of a damage constitutive model to concrete at high temperature and prediction of spalling, Int. J. Solids Struct. 42(26), 6550–6565. Thomas H.R. & Sansom M.R. (1995) Fully coupled analysis of heat, moisture and air transfer in unsaturated soil, J. Eng. Mech., 121(3), 392–405. Torrenti J.-M. et al. (1999) La dégradation des bétons, Hermès (Eds.), Paris. Van Genuchten M. Th. (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America, 44, 892–898. Wang K., Jansen D.C. & Shah S.P. (1997) Permeability study of cracked concrete, Cem. Concr. Res. 27 3, pp. 381–393.

Chapter 6

Conclusion Alain Millard and Pierre Pimienta

Classical engineering approaches as well as recent sophisticated multi-physics approaches have been presented in this report, as relevant tools to study and appraise the behavior of reinforced concrete structures when subjected to high temperatures. From an engineering point of view, concrete is generally considered as an homogeneous material, at a macroscopic scale, but such an approach relies mostly on empirical behavior laws, and might not be sufficient to evaluate, in a predictive way, the risk of spalling of a given structure. For this purpose, it is necessary to take into account the real complex nature of the material in order to describe internal phenomena which occur under thermal loading. Therefore, as an introduction, Chap. 2 has presented the detailed physical phenomena which take place in a concrete volume subjected to high temperature. A mesoscopic description of the concrete has been given, where the pore are partly filled with liquid water and a mixture of vapor and dry air. When a concrete block is subjected to a thermal load, the temperature increase leads to a local increase of gas pressure in the pores. The resulting vapor gradient drives the vapor through the concrete, towards the free surface and the inner part of the block, where the vapor condenses and finally may create a ‘water clog’. The thermal expansion of the concrete as well as the pore pressure increase are the two main causes of the spalling event. Chapter 3 has been devoted to the engineering modelling approach, which is based on prescriptions of regulatory codes such as the Eurocode, and which genA. Millard (&) Service d’Études Mécaniques et Thermiques, Commissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Saclay, France e-mail: [email protected] P. Pimienta Centre Scientifique et Technique du Bâtiment (CSTB), Université Paris-Est, Marne La Vallée, France e-mail: [email protected] © RILEM 2019 A. Millard and P. Pimienta (eds.), Modelling of Concrete Behaviour at High Temperature, RILEM State-of-the-Art Reports 30, https://doi.org/10.1007/978-3-030-11995-9_6

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erally requires a thermo-mechanical approach, without consideration of the fluids in the concrete. Nevertheless, this approach might be sufficient for many practical situations. The more advanced models have been described in details in Chap. 4. They are based on a multiphysics approach which involves the temperature, the liquid water and dry air pressures, as well as the mechanical displacements. These quantities obey the classical balance equations (mass, momentum and energy) as well as specific constitutive equations. Among these equations, the sorptions isotherms, which relate the capillary pressure to the pore water content, and the concrete dehydration law play an important role in the heat and mass transfers. Concerning the mechanical aspects, it is necessary to account for the occurrence of damage in the concrete because of its significant influence on the mass transfer properties. The numerical aspects including the discretisation aspects and the solution strategy have also been covered in Chap. 4. Finally, for more practical applications, some simplifications of the general advanced approach have been proposed. Both engineering and advanced modelling approaches require a large number of parameters, which may depend on state variables such as temperature or water content. Most of these parameters have been reviewed in Chap. 5, with indication of their currently used values and expressions, for ordinary as well as high performance concretes.