Solid Mechanics and Its Applications

Zdeněk P. BažantMilan Jirásek

Creep and Hygrothermal Effects in Concrete Structures

Zdeněk P. Bažant • Milan Jirásek

Creep and HygrothermalEffects in Concrete Structures

123

Zdeněk P. BažantDepartment of Civil and EnvironmentalEngineering

Northwestern UniversityEvanston, ILUSA

Milan JirásekDepartment of Mechanics, Faculty of CivilEngineering

Czech Technical University in PraguePragueCzech Republic

ISSN 0925-0042 ISSN 2214-7764 (electronic)Solid Mechanics and Its ApplicationsISBN 978-94-024-1136-2 ISBN 978-94-024-1138-6 (eBook)https://doi.org/10.1007/978-94-024-1138-6

Library of Congress Control Number: 2017952930

© Springer Science+Business Media B.V. 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer Science+Business Media B.V.The registered company address is: Van Godewijckstraat 30, 3311 GX Dordrecht, The Netherlands

To the memory ofTreval Clifford Powers

1900–1997

widely regarded as the leading cement and concrete physicist of the middle partof the twentieth century, who made major lasting contributions to the understandingof the structure of fresh and hardened cement paste and achieved fundamentalresults on concrete rheology, workability, consistency, durability, shrinkage andswelling, creep, and resistance of concrete to frost, sulfates, and abrasion.

Born on February 8, 1900, in Palouse, WA, Powers studied chemistry at WillametteUniversity in Salem, Oregon, a small private college founded in 1846. From 1930until his retirement in 1965, he conducted research at the famous Portland CementAssociation (PCA) Laboratories, located in Chicago and later in Skokie, Illinois.For many years until his retirement in 1965, he served as the PCA Director. Hisstellar research achievement earned him top honors. The American ConcreteInstitute (ACI) bestowed on Powers its highest award, the Wason Medal forMaterials Research; in fact, it did so three times, in 1933, 1940, and 1948, which isa singular case in ACI history testifying to his fundamental achievements. In 1957,Powers received the S.E. Thompson Award from the American Society for Testingand Materials (ASTM) and, in 1976, the Arthur R. Anderson Award from the ACI.In 1961, he became an Honorary Member of ACI and, also in 1961, he wasawarded an honorary Doctor of Science degree by the University of Toledo. During1967–68, he lectured as Visiting Professor at the University of Toronto. He died onJune 30, 1997, in Green Valley, AZ. The founding conference of IA-ConCreep atM.I.T. in 2001 was dedicated to Powers, to honor his memory.

vii

Guest Preface

Concrete, the solid that forms at room temperature from mixing Portland cementwith water, sand, and aggregates, suffers from time-dependent deformation underload. This creep occurs at a rate that degrades the durability and truncates the lifespan of concrete structures, and it can lead in some (fortunately rare) cases tocatastrophic failures unless a rational approach is put in place that blends theunderlying mechanics and physics of concrete creep and shrinkage with engi-neering ingenuity and mathematical eloquence to gain predictive impact at the scaleof engineering operations.

It is precisely this challenging mission that Zdeněk Bažant and Milan Jirásekembraced and accomplished in this generational masterpiece; the definite book oncreep and shrinkage of concrete the community of engineers and scientists has beenwaiting for.

For generations of scientists and engineers, concrete creep has been a dauntingtask: Creep rates are intrinsically low, thus requiring typically long time scales forlaboratory experimentation under highly controlled hygrothermal conditions. Theload-induced deformation must be separated from other sources of deformationrelated to an ever evolving microstructure and other chemo-physical aging andout-of-equilibrium phenomena that define the very nature of concrete’s life cycle.Moreover, creep of concrete is by its very nature dissipative. This means that thework provided to the material or structural system in form of load is not recovered,but irreversibly lost in the creation of deformation in excess of elastic reversibledeformation. An engineer in charge of a structural design will thus aim at moni-toring via modeling and simulations a controlled energy dissipation, so as to avoidthat this energy would be dissipated in an uncontrolled way in, e.g., fracture cre-ation. With limited experimental creep data thus available, engineers must rely onmodels to predict, over extended periods of time, the impact of creep deformationon structural functionality, integrity and stability, force and moment distribution (instatically indeterminate structures), to ultimately minimize, by inverse design ofmaterials and structures, the impact of structural creep on performance.

With a life worth of experience in Structural Engineering and Design, the coreproposition of Zdeněk Bažant and Milan Jirásek is the need for reliable engineering

ix

models of concrete creep that permit engineers to meet the tasks ahead of designinghigh-performance concrete structures with high confidence levels. These models arecalibrated against a large database of creep and shrinkage painstakingly collectedand developed by Zdeněk Bažant since the early 1960s. In their most advancedversion, these models permit a rapid recalibration from short-term tests. Enabled bythe physics of the phenomena at stake, these models become an integral part of theinnovation pathway for sustainable concrete and concrete structures.

This is science-enabled engineering at its best! Decoded by two engineeringscientists of eminent status and encyclopedic knowledge of the mechanics andphysics of concrete creep, this book is a must-read for any structural engineer andengineering scientist in search of concrete innovation.

Franz-Josef UlmMassachusetts Institute of Technology,

Former Vice-President of the EngineeringMechanics Institute of the

American Society of Civil Engineers,Director, Concrete Sustainability Hub at MIT

x Guest Preface

Preface

The literature on concrete creep and shrinkage is vast and scattered over manymedia. A host of books have already been devoted to this subject. Some arevaluable compilations of diverse properties and experimental results, but lack acoherent system. A few champion formalistic beauty of sophisticated mathematicaltreatment based, however, on oversimplified unrealistic hypotheses. Others arefocused on simple methods of analysis for practical design (nothing is, of course,wrong with simple methods except when they are simplistic).

In this book, we present a different kind of exposition of the subject. We attemptto balance a sound, theoretically justified, mathematical modeling at the level ofcurrent knowledge with careful attention to laboratory test results, measurements onstructures, structural design applications, analysis of design standards or recom-mendations, and numerical algorithms.

To make our book useful to different kinds of readers, we divide it in two parts:Part I deals with the essentials required for designing structures, and Part II dealswith advanced subjects concerned with the effects of moisture, solidification, aging,cracking, and temperature, the consideration of which is necessary for moreaccurate predictions. We mark by asterisks the titles of the sections that elaborate onvarious highly theoretical aspects and can be skipped by a reader interested mainlyin practical design. Throughout the book, we emphasize the randomness of creepand shrinkage effects and their probabilistic treatment.

Both parts together give an all-encompassing presentation, although with someexceptions. We must admit that our exposition of the nanoscale mechanism of creepand shrinkage may soon be regarded incomplete because the current research,driven by increased concern with sustainability of infrastructure and facilitated byadvanced micro- and nanoscale measurements as well as computer simulations, is,at the time of writing, advancing rapidly (especially in the research group ofFranz-Josef Ulm at M.I.T., with collaborators worldwide). We must also admit thatour coverage of the consequences of the large autogenous shrinkage andself-desiccation in modern concretes is only superficial because the long-term dataand the understanding of nanomechanics are still too limited. We must admit, too,that our discussions of time-dependent growth of cracking damage and fracture, of

xi

deformation rate effects under impact and explosions, and of high-temperatureeffects in fire or hypothetical nuclear accidents, could be more thorough if the scopeof the book permitted.

Our book is intended for university researchers and educators, for members ofcommittees formulating design recommendations, and for practicing engineersdesigning or evaluating creep sensitive structures such as large-span prestressedbridges, supertall buildings, large roof shells and nuclear containments, airportpavements, underground excavation linings, and ocean oil platforms. Parts of thebook can be used for teaching graduate level courses. Supplementary materials willbe made available at http://mech.fsv.cvut.cz/ConcreteCreep.

The book grew out of various courses taught by each of the authors, including:sections of the first author's short intensive courses on Material Modeling ofConcrete (including creep and shrinkage) taught at Swedish Cement & ConcreteInstitute in Stockholm in 1976, Chalmers University in 1977, University of Mexicoin 1977 and École nationale des ponts et chausées in Paris in 1978, his short courseon Concrete Creep and Shrinkage at Politecnico di Milano in 1982, sections of hisshort courses on Inelastic Materials and Structures at EPF de Lausanne in 1983,1988, and 1991 and at Luleå University in 1994, and sections of his course onMaterial Modeling taught at Northwestern University in the 1980s; and sectionsof the second author's course on Deformation and Failure of Materials taught atCTU Prague. The first author also deeply values the three-year experience in creepanalysis of large bridges that he gained as a bridge engineer in Dopravoprojekt,Prague (1961–63). Valuable was also his experience as a staff consultant at Sargent& Lundy Engineers, Chicago, during 1974. He also benefited from the experiencewith creep and hygro-thermal effect that he gained while serving during 1974–94 asa staff consultant to the Reactor Analysis Division of Argonne National Laboratory.He feels particularly grateful for 48 years of almost continuous funding ofnumerous research projects, concerned fully or partly with creep, shrinkage, anddurability, by the US National Science Foundation, Department of Transportation,Department of Energy, Electric Power Research Institute and W.R. Grace Co.

We wish to express our deep thanks to many respected colleagues for valuablediscussions on the subject. The first author was introduced to the subject in 1959 byhis undergraduate advisor Jan Klimeš at CTU Prague. During 1967–69, the firstauthor was lucky to have visionary mentors in Robert L'Hermite at CEBTP Paris,Boris Bresler at UC Berkeley, and especially Treval C. Powers of PCA Skokie, agiant of cement physics who inspired the first author's research direction while bothheld visiting appointments at the University of Toronto during 1967–68. The firstauthor wishes to acknowledge the stimulating interactions and collaborations withGianluca Cusatis at Northwestern University, Qiang Yu of Pittsburgh University,Franz-Josef Ulm and Roland Pelenq of M.I.T., Jialiang Le at University ofMinnesota, Kaspar Willam, Yunping Xi, and Mija Hubler at UC Boulder, RomanWendner at BOKU Vienna, Folker H. Wittmann, Christian Huet, and ThomasZimmermann at EPF Lausanne, Ignacio Carol at UPC Barcelona, MatthieuVandamme of Université Paris Est, Vladimír Křístek, Vít Šmilauer, Zdeněk Bittnar,and Petr Havlásek at CTU Prague, and Joško Ožbolt at Stuttgart University. Thanks

xii Preface

for valuable collaborations on the subject are also due to many of the first author'sformer doctoral students at Northwestern University,1 as well as postdoctoralassociates and visiting scholars.2

The second author is grateful for stimulating discussions with his colleagues atthe Czech Technical University in Prague, in particular with Jan Zeman, VítŠmilauer, Zdeněk Bittnar, Bořek Patzák, Radek Štefan, Vladimír Křístek, Jan Vítek,Lukáš Vráblík, Tomáš Vogel, and Pavel Demo, as well as with many internationalexperts, including Christian Huet, Gilles Pijaudier-Cabot, Ignacio Carol, JoškoOžbolt, Peter Grassl, Franz-Josef Ulm, Mija Hubler, Dariusz Gawin, FrancescoPesavento, Luca Sorelli, and Jean-Michel Torrenti. Special thanks are due to PetrHavlásek, who collaborated with the second author on the development of mod-eling techniques and numerical algorithms for creep, shrinkage, moisture transport,and heat transfer; this research was funded by the Czech Science Foundation(projects 103/09/H078 and P105/10/2400) and by the European Social Fund(project CZ.1.07/2.3.00/30.0034). Assistance with the preparation of some of thefigures was provided by students of the Czech Technical University.3

Last but not least, as expressed in our dedication, we wish to thank our belovedwives Iva and Vlasta for their sustained support of our professional activities, whichmade the arduous work on this book possible.

Evanston, USA Zdeněk P. BažantPrague, Czech Republic Milan JirásekJuly 2017

1 They included Leonard J. Najjar, Spencer T. Wu, Ali A. Asghari, Elmamoun Abdalla Osman,Werapol Thonguthai, Sang-Sik Kim, Liisa Panula, Tatsuya Tsubaki, Jenn-Chuan Chern, SantoshPrasannan, Joong-Koo Kim, Yunping Xi, Ravindra Gettu, Sandeep Baweja, Goangseup Zi, QiangYu, Guang-Hua Li, and Mija Helena Hubler.2 They included Laurent Granger, Anders Boe Hauggaard, Franz-Josef Ulm, Alexander Steffens,Geir Horrigmoe, Henrik O. Madsen, Tong-Sheng Wang, Joško Ožbolt, M. Elisabeth Karr, JaroslavNavrátil, Larissa Molina, Zhishen Wu, Milan Holický, Daniele Ferretti, Vít Šmilauer, GoangseupZi, Abdullah Dönmez, Enrico Masoero, and Mohammad J.A. Qomi.3 They included Marek Vinkler, Hana Hasníková, Pavel Fišar, Dominika Majerová, AnetaBulíčková, and Michal Šmejkal.

Preface xiii

Contents

Part I Fundamentals

1 Introduction: How the Theory Evolved and How It ImpactsPractice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Fundamentals of Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . 92.1 Characterization of Creep by Compliance Function . . . . . . . . . . 92.2 Integral Stress–Strain Relation . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Relaxation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Viscoelasticity Under Multiaxial Stress . . . . . . . . . . . . . . . . . . 252.5 Operator Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Basic Properties of Concrete Creep, Shrinkage, and Drying . . . . . . 293.1 Sources and Characterization of Time-Dependent

Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Asymptotic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Basic Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Creep Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Mean Cross-Sectional Shrinkage . . . . . . . . . . . . . . . . . . . . . . . 433.6 Mean Drying Creep in the Cross Section . . . . . . . . . . . . . . . . . 483.7 Common Misconceptions in Measuring and Defining Creep . . . 51

3.7.1 Incompatible Initial Strain . . . . . . . . . . . . . . . . . . . . . . 513.7.2 Plotting Creep Curves in Actual, Rather than

Logarithmic, Time Scale . . . . . . . . . . . . . . . . . . . . . . . 523.7.3 Creep “Inflation” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.7.4 Is Tensile Creep Different from Compression

Creep? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.7.5 Autogenous Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . 55

xv

3.8 Updating Long-Time Creep and Shrinkage Predictions fromShort-Time Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8.1 Updating Creep Predictions . . . . . . . . . . . . . . . . . . . . . 563.8.2 Difficulties in Updating Shrinkage Predictions . . . . . . . 60

4 Structural Effects of Creep and Age-Adjusted Effective ModulusMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Homogeneous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Elastic–Viscoelastic Analogy . . . . . . . . . . . . . . . . . . . . 644.1.2 Change of Structural System . . . . . . . . . . . . . . . . . . . . 70

4.2 Age-Adjusted Effective Modulus Method . . . . . . . . . . . . . . . . . 774.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.2 Fundamental Equation of AAEM . . . . . . . . . . . . . . . . 804.2.3 Alternative Derivation of AAEM� . . . . . . . . . . . . . . . . 824.2.4 Ramifications of AAEM . . . . . . . . . . . . . . . . . . . . . . . 834.2.5 Approximation of Relaxation Function . . . . . . . . . . . . 854.2.6 Simple Applications of AAEM . . . . . . . . . . . . . . . . . . 89

4.3 Nonhomogeneous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.1 Stress Redistributions Due to Differences

in Age of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3.2 Stress Redistributions in Beams of Composite

Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.3.3 Effects of Nonuniform Drying . . . . . . . . . . . . . . . . . . . 1044.3.4 Stress Relaxation in Prestressed Members . . . . . . . . . . 1184.3.5 Creep Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.3.6 Reduction of Flexural Creep Due to Cracking

in Unprestressed Reinforced Concrete . . . . . . . . . . . . . 139

5 Numerical Analysis of Creep Problems . . . . . . . . . . . . . . . . . . . . . . 1415.1 Numerical Analysis of Structural Creep Problems Based

on History Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2 Efficient Rate-Type Creep Analysis . . . . . . . . . . . . . . . . . . . . . 153

5.2.1 Generalized Trapezoidal Rule� . . . . . . . . . . . . . . . . . . 1545.2.2 First-Order Exponential Algorithm� . . . . . . . . . . . . . . . 1595.2.3 Second-Order Exponential Algorithm . . . . . . . . . . . . . 1625.2.4 Nonaging Kelvin Chain . . . . . . . . . . . . . . . . . . . . . . . 1675.2.5 Solidifying Kelvin Unit . . . . . . . . . . . . . . . . . . . . . . . 1705.2.6 Solidifying Kelvin Chain . . . . . . . . . . . . . . . . . . . . . . 1715.2.7 Aging Kelvin Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.2.8 Aging Kelvin Chain . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6 Uncertainty Due to Parameter Randomness via Samplingof Deterministic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.1 Random Parameters in Creep and Shrinkage Model . . . . . . . . . 1786.2 Latin Hypercube Sampling of Parameters of Creep

and Shrinkage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

xvi Contents

6.3 Histograms and Statistics of Response, and ConfidenceLimits for Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.4 Bayesian Improvement of Statistical Prediction of Creepand Shrinkage Effects� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.4.1 Background on Bayesian Statistics� . . . . . . . . . . . . . . . 1976.4.2 Method of Bayesian Analysis� . . . . . . . . . . . . . . . . . . 198

7 Paradigms of Application, Phenomena Affecting CreepDeformations, and Comparisons to Measurementson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057.1 Drying Effects in Viaduct La Lutrive . . . . . . . . . . . . . . . . . . . . 2067.2 Description of the KB Bridge in Palau and Input Data . . . . . . . 2077.3 Creep Structural Analysis Utilizing Commercial

General-Purpose Finite Element Program . . . . . . . . . . . . . . . . . 2117.4 Numerical Implementation and Algorithmic Aspects . . . . . . . . . 2137.5 Effects of Slab Thickness, Temperature, and Cracking . . . . . . . 2167.6 Determination of Model Parameters . . . . . . . . . . . . . . . . . . . . . 2187.7 Results of Simulations and Comparisons to Measurements . . . . 221

7.7.1 Calculated Deflections . . . . . . . . . . . . . . . . . . . . . . . . 2217.7.2 Calculation of Prestress Losses Due to Creep,

Shrinkage, Cyclic Creep, and Steel Relaxation . . . . . . . 2257.8 Excessive Long-Term Deflections of Other Box Girders . . . . . . 2287.9 Approximate Multidecade Extrapolation of Medium-Term

Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.10 Uncertainty of Deflection Predictions and Calculation of

Confidence Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2367.11 Precautionary Deflection-Minimizing Design and Tendon

Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.12 Deflection-Mitigating Layout of Tendons . . . . . . . . . . . . . . . . . 2407.13 Effect of Cyclic Stress Variations on Creep Compliance . . . . . . 242

7.13.1 History of Cyclic Creep Models . . . . . . . . . . . . . . . . . 2427.13.2 Macroscopic Strain Due to Small Growth of

Microcracks� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2437.13.3 Strain According to Paris Law for Subcritical

Microcrack Growth� . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.13.4 Compressive Cycles via Dimensional Analysis

and Similitude� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.13.5 Cyclic Creep Compliance and Multiaxial

Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.13.6 Calibration by Existing Test Data . . . . . . . . . . . . . . . . 250

7.14 Effects of Cyclic Creep on Bridge Deflectionsand Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.14.1 Stress Distribution in a Prestressed Cross

Section Under Variable Loading . . . . . . . . . . . . . . . . . 252

Contents xvii

7.14.2 Curvature and Residual Stresses Due to CyclicCreep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.14.3 Appraisal of the Magnitude of Cyclic Creep Effectsin Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

7.14.4 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.15 Conclusions for Method of Analysis and Design . . . . . . . . . . . 266

Part II Advanced Topics

8 Moisture Transport in Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718.1 Water in Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2728.2 Pore Fluids at Thermodynamic Equilibrium . . . . . . . . . . . . . . . 276

8.2.1 Multiphase Porous Medium . . . . . . . . . . . . . . . . . . . . 2768.2.2 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2788.2.3 Capillary Pressure and Relative Humidity . . . . . . . . . . 2808.2.4 Kelvin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2848.2.5 Sorption Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2908.2.6 Free and Hindered Surface Adsorption, Disjoining

Pressure, and Its Continuum Thermodynamics� . . . . . . 2948.3 Moisture Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

8.3.1 Transport Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 3028.3.2 Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3038.3.3 Mass Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . 3098.3.4 Differential Equations for Moisture Transport . . . . . . . . 3128.3.5 Scaling Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3208.3.6 Effect of Distributed Cracking on the Rate

of Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3218.4 One-Dimensional Moisture Transport . . . . . . . . . . . . . . . . . . . . 323

8.4.1 One-Dimensional Diffusion Equation . . . . . . . . . . . . . . 3238.4.2 Numerical Solution by Finite Differences . . . . . . . . . . . 3258.4.3 Drying of a Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3458.4.4 Initial Drying and Analysis of Infinite Half-Space . . . . 3538.4.5 Evolution of Total Water Loss from a Specimen . . . . . 3618.4.6 Effects of Variable Environmental Humidity . . . . . . . . 373

8.5 Spreading of Hydraulic Pressure Front Into UnsaturatedConcrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

8.6 Shrinkage and Stresses Due to Nonuniform Drying . . . . . . . . . 3868.7 Effects of Self-Desiccation and Autogenous Shrinkage in

Drying or Swelling Specimens—A Problem Requiring FurtherResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028.7.1 Recent Paradigm-Changing Observations . . . . . . . . . . . 4028.7.2 Improved Aging Characterization via a Model

for Hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4048.8 Creep and Diffusion as Processes Controlling Alkali–Silica

Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

xviii Contents

9 Solidification Theory for Aging Effect on Stiffness and BasicCreep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4099.1 Growth of Volume Fraction of Calcium Silicate Hydrates

and Polymerization Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 4109.2 Basic Creep Model for Concrete . . . . . . . . . . . . . . . . . . . . . . . 4139.3 Basic Creep Compliance Function of Model B3 . . . . . . . . . . . . 4159.4 Absence of a Characteristic Time as the Reason for Using

Power Functions� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4179.5 Asymptotic Matching Properties of Solidification Theory

and Insufficiency of Log-Double-Power Law . . . . . . . . . . . . . . 4199.6 Nondivergence of Compliance Curves . . . . . . . . . . . . . . . . . . . 4249.7 Change of Sign of Relaxation Function . . . . . . . . . . . . . . . . . . 4399.8 Thermodynamically Admissible Rheological Chains� . . . . . . . . 442

9.8.1 General Properties� . . . . . . . . . . . . . . . . . . . . . . . . . . . 4429.8.2 Relation to Retardation Spectrum� . . . . . . . . . . . . . . . . 445

10 Microprestress-Solidification Theory and Creep at VariableHumidity and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45510.1 Overview of Physical Mechanisms . . . . . . . . . . . . . . . . . . . . . . 45610.2 Relevant Aspects of Pore Structure and Water Adsorption in

Hardened Cement Gel� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45810.3 The Concept of Microprestress and Its Relaxation . . . . . . . . . . 46010.4 Generation and Relaxation of Microprestress . . . . . . . . . . . . . . 46310.5 Unification of Microprestress and Solidification Models . . . . . . 46710.6 Temperature and Humidity Effects . . . . . . . . . . . . . . . . . . . . . . 468

10.6.1 Effects on Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46810.6.2 Hygrometric and Thermal Strains . . . . . . . . . . . . . . . . 473

10.7 Alternative Computational Approach: Viscosity EvolutionEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

10.8 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47710.8.1 Evaluation of Flow Viscosity . . . . . . . . . . . . . . . . . . . 47710.8.2 Evaluation of Flow Strain Increment . . . . . . . . . . . . . . 47910.8.3 Incorporation of Transformed Times . . . . . . . . . . . . . . 48110.8.4 Incremental Stress Evaluation Algorithm . . . . . . . . . . . 482

10.9 Analysis of Experimental Data on Temperature Effect . . . . . . . . 48610.9.1 Basic Creep at Constant Elevated Temperature . . . . . . . 48710.9.2 Transitional Thermal Creep . . . . . . . . . . . . . . . . . . . . . 493

10.10 Comment on Applications and Review of Main Points . . . . . . . 496

11 Physical and Statistical Justifications of Models B3 and B4and Comparisons to Other Models . . . . . . . . . . . . . . . . . . . . . . . . . 49911.1 Main Criteria of Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50011.2 Theoretically Based Physical Justifications of Model B3� . . . . . 501

11.2.1 Overview of Mechanisms and Phenomena� . . . . . . . . . 502

Contents xix

11.2.2 Thermodynamic Restrictions� . . . . . . . . . . . . . . . . . . . 50311.2.3 Microprestress Relaxation and the Question of

Characterizing Creep Aging by Strength Gain� . . . . . . 50411.2.4 Activation Energy, Power Laws, and Lack of

Bounds� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50411.2.5 Diffusion Theory for Pore Water� . . . . . . . . . . . . . . . . 50611.2.6 Effect of Cracking� . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

11.3 Statistical Aspects of Model Calibration and Validation . . . . . . 50811.3.1 Unbiased Statistical Verification of Model . . . . . . . . . . 50811.3.2 Importance of Validating Model Form by Individual

Tests on Many Different Concretes� . . . . . . . . . . . . . . 50911.3.3 Need for Short-Time Data Extrapolation by Linear

Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51211.4 Statistical Methods Applied to Model Evaluation . . . . . . . . . . . 514

11.4.1 Suppressing Database Bias Due to NonuniformSampling of Parameter Ranges . . . . . . . . . . . . . . . . . . 519

11.4.2 Reducing Anti-High-Strength Bias . . . . . . . . . . . . . . . 52111.4.3 Standard Least-Square Regression Statistics of the

Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52211.4.4 Bias Due to Different Density of Readings . . . . . . . . . 523

11.5 Statistical Comparison of Creep and Shrinkage Models . . . . . . . 52311.5.1 Model Evaluation by Standard Regression Statistics . . . 52311.5.2 Statistical Justification of Model B3 . . . . . . . . . . . . . . 524

11.6 Statistical Justification of RILEM Model B4 . . . . . . . . . . . . . . . 52811.6.1 Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52811.6.2 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

11.7 Analytical Methods for Predicting Concrete Creep from ItsComposition� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55111.7.1 Predicting Creep and Shrinkage from Heterogenous

Microstructure Using Homogenization Theory� . . . . . . 55211.7.2 Extracting Creep Properties of C-S-H via Cement

Paste Homogenization� . . . . . . . . . . . . . . . . . . . . . . . . 553

12 Effect of Cracking and Fracture Mechanics Aspects of Creepand Shrinkage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55512.1 Limitations of Simplistic Nonlinear Models for Concrete

Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55612.2 Fracture Mechanics Aspects and Crack Band Model . . . . . . . . . 55712.3 Role of Cracking and Irreversibility in Shrinkage . . . . . . . . . . . 56412.4 Role of Cracking in Drying Creep (Pickett Effect) . . . . . . . . . . 56612.5 Role of Creep in Cohesive Fracture and Size Effect on

Structural Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

xx Contents

12.6 Derivation of Crack Opening Rate Effect from FractureKinetics at Atomic Scale� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

12.7 Models Combining Damage and Creep . . . . . . . . . . . . . . . . . . 58212.8 Microplane Modeling of Cracking Damage with Creep . . . . . . . 594

12.8.1 Basic Ideas of Microplane Modeling . . . . . . . . . . . . . . 59412.8.2 Incorporation of Creep with Aging and Shrinkage

into Microplane Constitutive Laws . . . . . . . . . . . . . . . 60012.8.3 The Lattice Discrete Particle Model Generalized

for Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60612.9 Why Creep Rate at Low Stress Depends on Stress Linearly . . . 606

13 Temperature Effect on Water Diffusion, Hydration Rate,Creep and Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60713.1 Heat Transfer in Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

13.1.1 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60813.1.2 Characteristic Times of Heating and Drying . . . . . . . . . 61113.1.3 Boundary Conditions for Heat Transfer . . . . . . . . . . . . 61313.1.4 Role of Heat Convection� . . . . . . . . . . . . . . . . . . . . . . 61413.1.5 Hydration Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61713.1.6 Temperature Increase Induced by Hydration . . . . . . . . 622

13.2 Heat and Moisture Transfer, and Hygrothermal Effectsin Heated Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62813.2.1 Structure of Bažant–Thonguthai Model . . . . . . . . . . . . 62913.2.2 Distributed Source of Water . . . . . . . . . . . . . . . . . . . . 63313.2.3 Isotherms at High Temperatures . . . . . . . . . . . . . . . . . 63613.2.4 Permeability at High Temperatures . . . . . . . . . . . . . . . 64113.2.5 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 64513.2.6 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64613.2.7 Latent Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

13.3 Strains and Stresses at High Temperature . . . . . . . . . . . . . . . . . 64913.3.1 Thermal and Hygral Volume Changes . . . . . . . . . . . . . 64913.3.2 Mechanical Properties at High Temperature . . . . . . . . . 65113.3.3 Extension of Creep Models to High Temperature . . . . . 65313.3.4 Application Example: Explosive Thermal Spalling

Due to Microwave Heating . . . . . . . . . . . . . . . . . . . . . 65813.4 Finite Volume Method for Problems with Moving

Interfaces� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66113.5 Mass, Momentum and Energy Balance Laws� . . . . . . . . . . . . . 663

13.5.1 Mass Conservation� . . . . . . . . . . . . . . . . . . . . . . . . . . 66313.5.2 Momentum Balance� . . . . . . . . . . . . . . . . . . . . . . . . . 66613.5.3 Energy Balance� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66813.5.4 Entropy Balance� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67013.5.5 Heat Equation� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

Contents xxi

13.5.6 Balance Laws for Multiphase Media� . . . . . . . . . . . . . 67913.6 Comments on Multiphase Modeling of Hygrothermal

Processes and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684

Appendix A: Viscoelastic Rheologic Models . . . . . . . . . . . . . . . . . . . . . . 687A.1 Maxwell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

A.1.1 Compliance Function . . . . . . . . . . . . . . . . . . . 688A.1.2 Relaxation Function . . . . . . . . . . . . . . . . . . . . 690

A.2 Kelvin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691A.2.1 Compliance Function . . . . . . . . . . . . . . . . . . . 691A.2.2 Relaxation Function . . . . . . . . . . . . . . . . . . . . 692

A.3 Rheologic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693A.3.1 Kelvin Chain . . . . . . . . . . . . . . . . . . . . . . . . . 694A.3.2 Maxwell Chain. . . . . . . . . . . . . . . . . . . . . . . . 696

A.4 Aging Rheologic Chains . . . . . . . . . . . . . . . . . . . . . . . 697A.4.1 Aging Maxwell Chain . . . . . . . . . . . . . . . . . . 697A.4.2 Aging Kelvin Chain . . . . . . . . . . . . . . . . . . . . 699

A.5 Solidifying Rheologic Chains . . . . . . . . . . . . . . . . . . . 700A.5.1 Solidifying Kelvin Chain . . . . . . . . . . . . . . . . 700A.5.2 Solidifying Maxwell Chain. . . . . . . . . . . . . . . 703

Appendix B: Historical Note on Old Creep Models . . . . . . . . . . . . . . . . 705

Appendix C: Estimates of Parameters Used By RILEM Model B3 . . . 709C.1 Sectional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 709C.2 Prediction of Model Parameters . . . . . . . . . . . . . . . . . 711C.3 Material Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

Appendix D: Estimates of Parameters Used By RILEM Model B4 . . . 717D.1 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

D.1.1 Basic Creep Compliance . . . . . . . . . . . . . . . . 718D.1.2 Additional Compliance Due to Drying . . . . . . 719

D.2 Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721D.2.1 Drying Shrinkage . . . . . . . . . . . . . . . . . . . . . . 722D.2.2 Autogenous Shrinkage . . . . . . . . . . . . . . . . . . 722

D.3 Effect of Admixtures. . . . . . . . . . . . . . . . . . . . . . . . . . 723D.4 Simplified Strength-Based Model B4s . . . . . . . . . . . . 723D.5 Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 725D.6 Examples of Compliance Curves . . . . . . . . . . . . . . . . 727D.7 Aging Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

D.7.1 Aging of Elastic Modulus . . . . . . . . . . . . . . . 731D.7.2 Effect of Aging on Creep . . . . . . . . . . . . . . . . 738

xxii Contents

D.8 Improvements of Model B4 . . . . . . . . . . . . . . . . . . . . 743D.8.1 Better Prediction of Drying Creep . . . . . . . . . 743D.8.2 Anticipated Future Improvements

of Model B4. . . . . . . . . . . . . . . . . . . . . . . . . . 749

Appendix E: Creep Models Recommended by Design Codes . . . . . . . . 751E.1 General Structure of Creep Design Formulae . . . . . . . 751E.2 CEB and fib Model Codes . . . . . . . . . . . . . . . . . . . . . 754

E.2.1 CEB Model . . . . . . . . . . . . . . . . . . . . . . . . . . 754E.2.2 fib Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 756

E.3 ACI Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757E.4 GL2000 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759E.5 JSCE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761E.6 Comparison of Compliance Functions. . . . . . . . . . . . . 762

Appendix F: Continuous Retardation Spectrum . . . . . . . . . . . . . . . . . . 767F.1 Relation Between Compliance Function

and Retardation Spectrum . . . . . . . . . . . . . . . . . . . . . . 767F.2 Spectrum of Log-Power Law. . . . . . . . . . . . . . . . . . . . 771

F.2.1 Straightforward Application of Post-WidderFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772

F.2.2 Improved Technique Based on ShiftedRetardation Times . . . . . . . . . . . . . . . . . . . . . . 777

F.3 Spectra of ACI and CEB Models . . . . . . . . . . . . . . . . 779F.4 Spectrum of Drying Creep Compliance Function

of B3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784F.5 Spectrum of JSCE Model . . . . . . . . . . . . . . . . . . . . . . 788F.6 Spectrum of fib Model . . . . . . . . . . . . . . . . . . . . . . . . . 790F.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792

Appendix G: Free-Energy Potentials for Aging LinearViscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Appendix H: Updating Long-Time Shrinkage Predictions fromShort-Time Measurements. . . . . . . . . . . . . . . . . . . . . . . . . 807H.1 Measuring Water Loss to Update Shrinkage

Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807H.2 Procedure of Updating Shrinkage Prediction from

Short-Time Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810H.3 Example of Shrinkage Updating . . . . . . . . . . . . . . . . . 812

Appendix I: Moisture Transport Characteristics . . . . . . . . . . . . . . . . . . 817I.1 Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817I.2 Sorption Hysteresis Due to Nonuniqueness

of Menisci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

Contents xxiii

I.3 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827I.4 Moisture Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

I.4.1 Dependence of Diffusivity on Humidity. . . . . . 828I.4.2 Aging of Diffusivity. . . . . . . . . . . . . . . . . . . . . 828

Appendix J: Moisture Transport in Porous Materials . . . . . . . . . . . . . . 831J.1 Fick Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831J.2 Darcy–Buckingham Law . . . . . . . . . . . . . . . . . . . . . . . 833J.3 Richards Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835J.4 Coussy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836J.5 Relation to the Bažant–Najjar Model . . . . . . . . . . . . . . 841J.6 Künzel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842J.7 Heat and Moisture Transport—Model of Beneš

and Štefan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Appendix K: Nonstandard Statistics Used in Support of SomeCreep and Shrinkage Models . . . . . . . . . . . . . . . . . . . . . . 857K.1 Linear Coefficient of Variation (L.C.o.V.) . . . . . . . . . 857K.2 CEB Coefficient of Variation . . . . . . . . . . . . . . . . . . . 860K.3 CEB Mean-Square Relative Error . . . . . . . . . . . . . . . . 860K.4 CEB Mean Relative Deviation . . . . . . . . . . . . . . . . . . 861K.5 Coefficient of Variation of the Data/Prediction

Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861K.6 Why Is the Method of Least Squares the Only

Correct Approach to Central Range Statistics? . . . . . . 862K.7 Comparison of Models by Standard and

Nonstandard Statistical Indicators . . . . . . . . . . . . . . . . 863K.8 Stochastic Process for Extrapolating

Concrete Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865

Appendix L: Method of Measurement of Creep and Shrinkage . . . . . . 867L.1 Testing Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867L.2 Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868

L.2.1 Form and Dimensions. . . . . . . . . . . . . . . . . . . 868L.2.2 Specimen Production . . . . . . . . . . . . . . . . . . . 869L.2.3 Curing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869L.2.4 Environmental Conditions. . . . . . . . . . . . . . . . 869L.2.5 Companion (or Control) Specimens . . . . . . . . 870

L.3 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870L.3.1 Preparation of Specimens . . . . . . . . . . . . . . . . 870L.3.2 Measurements Prior to Loading . . . . . . . . . . . 871L.3.3 Measurements of Total Strain

Under Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 871

xxiv Contents

L.3.4 Measurements of Water Loss . . . . . . . . . . . . . 872L.3.5 Recommended Test Parameters. . . . . . . . . . . . 872L.3.6 Reporting of Results . . . . . . . . . . . . . . . . . . . . 873

L.4 Ring Test of Restrained Shrinkage and Crackingand Its Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 873

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915

Contents xxv

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