Fracture mechanics of concrete: Structural application and numerical calculation

ENGINEERING APPLICATION OF FRACTURE MECHANICS Editor-in-Chief' George C. Sih

G.C. Sih and L. Faria (eds.), Fracture mechanics methodology: Evalua­tion of structure components integrity. 1984. ISBN 90-247-2941-6.

E.E. Gdoutos, Problems of mixed mode crack propagation. 1984. ISBN 90-247-3055-4.

A. Carpinteri and A.R. Ingraffea (eds.), Fracture mechanics of concrete: Material characterization and testing. 1984. ISBN 90-247-2959-9.

G.C. Sih and A. DiTommaso (eds.), Fracture mechanics of concrete: Structural application and numerical calculation. 1984. ISBN 90-247-2960-2.

Fracture mechanics of concrete: Structural application and numerical calculation Edited by

George C. Sih Lehigh University, Institute of Fracture and Solid Mechanics, Bethlehem, PA 18015, USA

A. DiTommaso University of Bologna Bologna, Italy

1985 MARTIN US NIJHOFF PUBLISHERS a member of the KLUWER ACADEMIC PUBLISHERS GROUP DORDRECHT / BOSTON / LANCASTER

Distributors

for the United States and Canada: Kluwer Academic Publishers, 190 Old Derby Street, Hingham, MA 02043, USA for the UK and Ireland: Kluwer Academic Publishers, MTP Press Limited, Falcon House, Queen Square, Lancaster LAI lRN, England for all other countries: Kluwer Academic Publishers Group, Distribution Center, P.O. Box 322, 3300 AH Dordrecht, The Netherlands

Library of Congress Cataloging in Publication Data

Maln entry under tltle:

Fracture mechanics of concrete.

(Engineering application of fracture mechanics ; v. 4) Includes bibliographlcal references and lndexes. 1. Concrete--Fracture. 2. Fracture mechanics.

3. Structures, Theory of. 4. Numerlcal calculations. 1. Sih, G. C. (George C.) II. DiTommaso, A. III. Series. TA440.F735 1984 620.1'366 84-3996

ISBN-13: 978-94-009-6154-8 e-ISBN-13: 978-94-009-6152-4 DOl: 10.1007/978-94-009-6152-4

Copyright

© 1985 by Martinus Nijhoff Publishers, Dordrecht. Softcover reprint of the hardcover 1 st edition 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers, Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht, The Netherlands.

Contents

Series on engineering application of fracture mechanics ........ VII

Editors' preface .................................... IX

Contributing authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XII

1. Mechanics of fracture and progressive cracking in concrete struc-tures / Z.P. BaZant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction................................. 1 1.2 Blunt crack band theory ........................ 2 1.3 Finite element implementation . . . . . . . . . . . . . . . . . . .. 21 1.4 Energy considerations . . . . . . . . . . . . . . . . . . . . . . . . .. 34 1.5 Applications and practical analysis . . . . . . . . . . . . . . . .. 50 1.6 Crack development . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61 1.7 General model for progressive fracturing. . . . . . . . . . . .. 72 1.8 Conclusion.................................. 85 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85

2. Scale effects in fracture of plain and reinforced concrete struc-tures / A. Carpinteri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95 2.1 Introduction................................. 95 2.2 Dimensional analysis applied to plain and reinforced con-

crete structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100 2.3 Fracture stability in plain and reinforced concrete elements 114 2.4 Hysteretic behaviour of reinforced concrete elements .... 129 2.5 Appendix: Dimensional independence .............. 136 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 138

3. Numerical methods to simulate softening and fracture of concrete / A. Hillerborg .................................. 141 3.1 Introduction................................. 141 3.2 The behaviour of concrete in a tension test. . . . . . . . . . .. 141 3.3 A comparison between concrete and steel ............ 144

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3.4 Tensile fracture zones .......................... 146 3.5 A general model for the tensile fracture of concrete. . . . .. 149 3.6 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150 3.7 FEM analysis of a fracture zone: coincident with prede-

termined crack path ........................... 154 3.8 FEM analysis of a fracture zone: not coincident with

predetermined crack path. . . . . . . . . . . . . . . . . . . . . . .. 164 3.9 Some comparisons with test results . . . . . . . . . . . . . . . .. 166 3.10 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . .. 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170

4. Numerical modeling of discrete crack propagation in reinforced and plain concrete / A.R. Ingraffea and V. Saouma . . . . . . . .. 171 4.1 Introduction................................. 171 4.2 Discrete crack models for concrete ................. 182 4.3 The linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183 4.4 The nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . .. 206 4.5 Crack propagation modeling: the future ............. 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 222

5. Fracture of steels for reinforcing and prestressing concrete / M. Elices ......................................... 226 5.1 Introduction................................. 226 5.2 Fracture.................................... 230 5.3 Fracture under extreme conditions ... . . . . . . . . . . . . .. 248 5.4 Fatigue .................................... 254 5.5 Environment sensitive cracking. . . . . . . . . . . . . . . . . . .. 262 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 268

Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 272

Author's index ..................................... 273

Series on engineering application of fracture mechanics

Fracture mechanics technology has received considerable attention in recent years and has advanced to the stage where it can be employed in engineering design to prevent against the brittle fracture of high-strength materials and highly constrained structures. While research continued in an attempt to extend the basic concept to the lower strength and higher toughness materials, the technology advanced rapidly to establish material specifications, design rules, quality control and inspection standards, code requirements, and regulations for safe operation. Among these are the fracture tough­ness testing procedures of the American Society of Testing Materials (ASTM), the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Codes for the design of nuclear reactor components, etc. Step-by-step fracture detection and prevention procedures are also being developed by the industry, government and university to guide and regulate the design of engineering products. This involves the interaction of individuals from the different sectors of the society that often presents a problem in communication. The transfer of new research findings to the users is now becoming a slow, tedious and costly process.

One of the practical objectives of this series on Engineering Application of Fracture Mechanics is to provide a vehicle for presenting the experience of real situations by those who have been involved in applying the basic knowledge of fracture mechanics in practice. It is time that the subject should be presented in a systematic way to the practicing engineers as well as to the students in universities at least to all those who are likely to bear a responsibility for safe and economic design. Even though the current theory of linear elastic fracture mechanics (LEFM) is limited to brittle fracture behavior, it has already provided a remarkable improvement over the conventional methods not accounting for initial defects that are inevitably present irr all materials and structures. The potential of the fracture mechanics technology, however, has not been fully recognized. There remains much to be done in constructing a quantitative theory of material damage that can reliably translate small specimen data to the design of large size structural components. The work of the physical metallurgists and the fracture mechanicians should also be brought together by reconciling the details of the material microstructure with the assumed continua of the computational methods. It is with the aim of developing a wider appreciation of the fracture mechanics technology applied to the design of engineering structures such as aircrafts, ships, bridges, pavements, pressure vessels, off-shore structures, pipelines, etc. that this series is being developed.

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Undoubtedly, the successful application of any technology must rely on the sound­ness of the underlying basic concepts and mathematical models and how they reconcile with each other. This goal has been accomplished to a large extent by the book series on Mechanics of Fracture started in 1972. The seven published volumes offer a wealth of information on the effects of defects or cracks in cylindrical bars, thin and thick plates, shells, composites and solids in three dimensions. Both static and dynamic loads are considered. Each volume contains an introductory chapter that illustrates how the strain energy criterion can be used to analyze the combined influence of defect size, component geometry and size, loading, material properties, etc. The criterion is particularly effective for treating mixed mode fracture where the crack propagates in a non-self similar fashion. One of the major difficulties that continuously perplex the practitioners in fracture mechanics is the selection of an appropriate frac­ture criterion without which no reliable prediction of failure could be made. This requires much discernment, judgement and experience. General conclusion based on the agreement of theory and experiment for a limited number of physical phenomena should be avoided.

Looking into the future the rapid advancement of modern technology will require more sophisticated concepts in design. The micro-chips used widely in electronics and advanced composites developed for aerospace applications are just some of the more well-known examples. The more efficient use of materials in previously unexperienced environments is no doubt needed. Fracture mechanics should be extended beyond the range of LEFM. To be better understood is the entire process of material damage that includes crack initiation, slow growth and eventual termination by fast crack propa­gation. Material behavior characterized from the uniaxial tensile tests must be related to more complicated stress states. These difficulties could be overcome by unifying metallurgical and fracture mechanics studies, particularly in assessing the results with consistency.

This series is therefore offered to emphasize the applications of fracture mechanics technology that could be employed to assure the safe behavior of engineering products and structures. Unexpected failures mayor may not be critical in themselves but they can often be annoying, time-wasting and discrediting of the technical community.

Bethlehem. Pennsylvania 1984

G.c. Sih Editor-in-Chief

Editors' preface

Concrete has traditionally been known as a material used widely in the construction of roads, bridges and buildings. Since cost effectiveness has always been one of the more important aspects of design, concrete, when reinforced and/or prestressed, is finding more use in other areas of application such as floating marine structures, storage tanks, nuclear vessel containments and a host of other structures. Because of the demand for concrete to operate under different loading and environmen­tal conditions, increasing attention has been paid to study concrete specimens and structure behavior. A subject of major concern is how the localized segregation of the constituents in concrete would affect its global behavior. The degree of nonhomogeneity due to material property and damage.by yielding and/or cracking depends on the size scale and loading rate under consideration. Segregation or clustering of aggregates at the macroscopic level will affect specimen behavior to a larger degree than it would to a large structure such as a dam. Hence, a knowledge of concrete behavior over a wide range of scale is desired. The parameters governing micro- and macro-cracking and the techniques for evaluating and observing the damage in concrete need to be better understood. This volume is intended to be an attempt in this direction.

The application of Linear Elastic Fracture Mechanics to concrete is discussed in several of the chapters. Depending on the specimen size, loading rate and type, concrete can behave linearly up to fracture or nonlinearly to failure by plastic collapse and/or fracture. Such behavior can be generally observed in other materials as well. This indicates the importance of identifying damage in concrete with load history as a path dependent process.

Chapter 1 considers modeling crack extension in concrete by the finite element method. An approach assuming that fracture is simulated by a smeared crack band in concrete is presented. The model aims to reflect the densely distributed cracks and to offer computational convenience. Energy consumed in the crack band and the remaining portion of the specimen is calculated. Strain-softening is included in the analysis such that the results exhibit the gradual transition from failure predicted by

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strength at one extreme and by linear fracture mechanics at the other. In most situations, failure occurs in this transition range depending on the structural size. The stability aspect of fracture is also analyzed by focusing attention on strain localization instability and crack spacing. Furthermore a study is proposed of triaxial strain-softening and triaxial constitutive relations for the fracture process zone. An exhaustive amount of references is given at the end of this chapter.

Chapter 2 applies the principle of dimensional analysis to investigate crack behavior in concrete. Fracture or damage patterns in the large and smail structural components are recognized to be different. The large structure can fail by brittle fracture, while the smaller structure can fail by plastic collapse. A brittleness number is defined which is directly dependent on the critical stress intensity factor and inversely on the beam height and yield strength. Bounds on this number are presented for failure by plastic collapse and brittle fracture. The idea is also extended to treat fracture stability in plain and reinforced concrete structural members.

Chapter 3 emphasizes the need to include softening due to the damage in the fracture zone for describing concrete behavior. Such data can be collected from a simple tension test. Application of the single crack model in Linear Elastic Fracture Mechanics may not lead to realistic results because of the inability to identify a real crack in practice. The concept of fictitious crack model is discussed. Data on material proper­ties are still lacking for making reliable theoretical predictions.

The application of finite element to treat crack extension in concrete is presented in Chapter 4. A revival of interest considering the discrete character of cracking is discussed in contrast to the smeared crack approach. This is mainly due to the advent of interactive graphics providing capability in remeshing of elements that were not available previously. The tensile strength criterion is compared with the critical stress intensity factor approach in Linear Elastic Fracture Mechanics. Crack growth direction can be predicted from the stationary values of the strain energy density criterion. The results are compared with those obtained from the maximum circumferential stress and maximum energy release rate criteria. Discussed is an example on the cracking of a full scale dam. This involved stress and failure analysis in three dimensions.

Chapter 5 deals with the fracture of steels used for reinforcing and prestressing concrete. Such information is essential for understanding the load transfer character between the steel and concrete which can signifi­cantly alter the structure behavior. The break down of the interfacial

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bond between steel and concrete is emphasized. Analysis of defects can be involved as it may require the three-dimensional stress analysis with elastoplastic constitutive relations. Initiation and growth of fatigue cracks are also discussed in connection with aggressive environments.

The valuable time spent by the authors to complete this work is acknowledged. The contribution belongs solely to those who have made the publication of this volume possible.

Lehigh University University of Bologna April 1983

G.C. Sib A. DiTommaso

Contributing authors

Z.P. BaZant Northwestern University, Evanston, Illinois

A. Carpinteri University of Bologna, Bologna, Italy

M. Elices Universidad Politecnica de Madrid, Madrid-3, Spain

A. Hillerborg Lund Institute of Technology, Lund 7, Sweden

A.R. Ingraffea Cornell University, Ithaca, New York

v. Saouma University of Pittsburgh, Pittsburgh, Pennsylvania

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