Table Of ContentFracture 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: Evalua
tion 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
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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 1s t 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
v
VI
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 tough
ness 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.
VII
VIII
Undoubtedly, the successful application of any technology must rely on the sound
ness 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 frac
ture 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 propa
gation. 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 G.c. Sih
1984 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 environmen
tal 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
IX
x
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 proper
ties 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 signifi
cantly alter the structure behavior. The break down of the interfacial
