Analysis of concrete structures by fracture mechanics: proceedings of the International RILEM workshop, Abisko, Sweden, June 28-30, 1989

Analysis of Concrete Structures by Fracture Mechanics

Other RILEM Proceedings available from Chapman and Hall 1 Adhesion between Polymers and Concrete. ISAP 86 Aix-en-Provence, France, 1986 Edited by H.R.Sasse 2 From Materials Science to Construction Materials Engineering Proceedings of the First International RILEM Congress Versailles, France, 1987 Edited by J.C.Maso 3 Durability of Geotextiles St Rémy-lès-Chevreuse, France, 1986 4 Demolition and Reuse of Concrete and Masonry Tokyo, Japan, 1988 Edited by Y.Kasai 5 Admixtures for Concrete Improvement of Properties Barcelona, Spain, 1990 Edited by E.Vázquez 6 Analysis of Concrete Structures by Fracture Mechanics Abisko, Sweden, 1989 Edited by L.Elfgren and S.P.Shah 7 Vegetable Plants and their Fibres as Building Materials Salvador, Bahia, Brazil, 1990 Edited by H.S.Sobral 8 Mechanical Tests for Bituminous Mixes Budapest, Hungary, 1990 Edited by H.W.Fritz and E.Eustacchio 9 Test Quality for Construction, Materials and Structures St Rémy-lès-Chevreuse, France, 1990 Edited by M.Fickelson Publisher’s Note This book has been produced from camera ready copy provided by the individual contributors, whose cooperation is gratefully acknowledged.

Analysis of Concrete Structures by Fracture Mechanics Proceedings of the International RILEM Workshop dedicated to Professor Arne Hillerborg, sponsored by RILEM (The International Union of Testing and Research Laboratories for Materials and Structures) and organized by RILEM Technical Committee 90—FM A Fracture Mechanics of Concrete Structures— Applications. Abisko, Sweden June 28–30, 1989

EDITED BY

L.Elfgren and S.P.Shah

CHAPMAN AND HALL LONDON • NEWYORK • TOKYO • MELBOURNE • MADRAS

UK Chapman and Hall, 2–6 Boundary Row, London SE1 8HN USA Van Nostrand Reinhold, 115 5th Avenue, New York NY10003 JAPAN Chapman and Hall Japan, Thomson Publishing Japan, Hirakawacho Nemoto Building, 7F, 1–7–11 Hirakawa-cho, Chiyoda-ku, Tokyo 102 AUSTRALIA Chapman and Hall Australia, Thomas Nelson Australia, 102 Dodds Street, South Melbourne, Victoria 3205 INDIA Chapman and Hall India, R.Seshadri, 32 Second Main Road, CIT East, Madras 600035 First edition 1991 This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” © 1991 RILEM ISBN 0-203-62676-1 Master e-book ISBN

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Dedication

This volume is dedicated to Professor Arne Hillerborg in recognition of his many outstanding contributions to the development of fracture mechanics for concrete structures.

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Participants in the RILEM Workshop on Fracture Mechanics of Concrete Structures dedicated to Professor Arne Hillerborg. The workshop took place in Abisko National Park in Northern Sweden in June 1989. From left to right: Lennart Elfgren (S), Herbert Linsbauer (A), Rune Sandström (S), Jan van Mier (NL), Ulf Ohlsson (S), Yu-Ting Zhu (PRC), Björn Täljsten (S). Surendra P.Shah (US), Ben Barr (UK), Anna Zolland (S), Carina Hannu (S), Herbert Duda (FRG), Jaime Planas (ES), Manuel Elices (ES), Ingegerd Hillerborg (S), Marianne Grauers (S), Per Anders Daerga (S), Crescentino Bosco (I) and Arne Hillerborg (S). Not present when the photo was taken were Mats Emborg (S), Jan-Erik Jonasson (S) and Gottfried Sawade (FRG).

Contents

1 PART ONE

Participants and contributing authors

ix

Preface

xi

Arne Hillerborg and fracture mechanics L.ELFGREN

1

BEHAVIOUR OF CONCRETE

15

2

Mode I behaviour of concrete: Influence of the rotational stiffness outside the crack-zone J.G.M. van MIER

16

3

Experimental analysis of mixed mode I and II behaviour of concrete J.G.M. van MIER, M.B.NOORU-MOHAMED, E.SCHLANGEN

26

4

Considerations regarding fracture zone response to simultaneous normal and shear displacement M.HASSANZADEH

36

5

Mixed mode fracture in compression A.K.MAJIS.P.SHAH

49

6

Thermal stresses in concrete at early ages M.EMBORG

63

7

Grain-model for the determination of the stress-crack-width-relation H.DUDA

79

PART TWO

STRUCTURAL MODELLING

89

8

Size effect and experimental validation of fracture models M.ELICES, J.PLANAS

9

General method for stability analysis of structures with growing interacting cracks Z.P.BAŽANT

117

Use of the brittleness number as a rational approach to minimum reinforcement design C.BOSCO, A.CARPINTERI, P.G.DEBERNARDI

121

10

90

viii

11

Fracture mechanics analyses using ABAQUS K.GYLLTOFT

138

12

Design and construction of concrete dams under consideration of fracture mechanics aspects H.N.LINSBAUER

144

PART THREE BENDING

152

13

Size dependency of the stress-strain curve in compression A.HILLERBORG

153

14

Influence of the beam depth on the rotational capacity of beams K.CEDERWALL, W.SOBKO, M.GRAUERS, M.PLOS

161

15

New failure criterion for concrete in the compression zone of a beam L.VANDEWALLE, F.MORTELMANS

166

PART FOUR SHEAR, BOND AND PUNCHING

178

16

Bond between new and old concrete YU-TINGZHU

179

17

Strengthening of existing concrete structures with glued steel plates B.TÄLJSTEN

187

18

Modelling, testing and strength analysis of adhesive bonds in pure shear P.J.GUSTAFSSON, H.WERNERSSON

197

19

Concrete surface loaded by a steel punch H.W.REINHARDT

211

PART FIVE

ANCHORAGE

220

20

Fracture mechanics based analyses of pull-out tests and anchor bolts R.BALLARINI, S.P.SHAH

221

21

Anchor bolts in concrete structures. Two dimensional modelling U.OHLSSON, L.ELFGREN

254

Index

272

Participants and contributing authors

Robert Ballarini, Case Western Reserve University, Department of Civil Engineering, Cleveland, Ohio 44106, USA. Ben I.G.Barr, University of Wales, Division of Civil Engineering, P.O. Box 917, Cardiff CF2 1XH, UK. Zdenek P.Bažant, Northwestern University, The Technological Institute, Evanston, Illinois 60208/3109, USA. Crescentino Bosco, Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degli Abruzzi 24, 1–10129 Torino, Italy. Alberto Carpinteri, Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degli Abruzzi 24, 1–10129 Torino, Italy. Krister Cederwall, Chalmers University of Technology, Division of Concrete Structures, S-412 96 Göteborg, Sweden. Per Anders Daerga, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. P.G.Debernardi, Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degli Abruzzi 24, 1–10129 Torino, Italy. Herbert Duda, Technische Hochschule Darmstadt, Institut für Massivbau, Alexanderstrasse 5, D-6100 Darmstadt, Germany. Lennart Elfgren, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. Manuel Elices, Universidad Politecnica de Madrid, Departamento de Ciencia de Materiales, Ciudad Universitaria, E-280 40 Madrid, Spain. Mats Emborg, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. Per Johan Gustafsson, Lund Institute of Technology, Department of Structural Mechanics, Box 118, S-22100 Lund, Sweden. Kent Gylltoft, National Swedish Testing Institute, Box 857, S-501 15 Borås, Sweden. Manoucheher Hassanzadeh, Lund Institute of Technology, Department of Building Materials, Box 118, S-221 00 Lund, Sweden.

x

Arne Hillerborg, Lund Institute of Technology, Department of Building Materials, Box 118, S-221 00 Lund, Sweden. Jan-Erik Jonasson, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. Herbert N.Linsbauer, Technische Universität Wien, Institut für Konstruktiven Wasserbau, Karlsplatz 13/ 222, A-1040 Wien, Austria. Arup K.Maji, The University of New Mexico, Department of Civil Engineering, Albuquerque, New Mexico 87131, USA. Jan G.M.van Mier, Delft University of Technology, Department of Civil Engineering, Stevin Laboratory, P.O. Box 5048, 2600 GA Delft, The Netherlands. Fernand Mortelmans, Katholieke Universiteit te Leuven, Departement Bouwkunde, Park van Arenberg de Croylaan 2, B-3030 Heverlee, Belgium. M.B.Nooru-Mohamed, Delft University of Technology, Department of Civil Engineering, Stevin Laboratory, P.O. Box 5048, 2600 GA Delft, The Netherlands. Ulf Ohlsson, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. Jaime Planas, Universidad Politecnica de Madrid, Departamento de Ciencia-de Materiales, Ciudad Universitaria, E-280 40 Madrid, Spain. Mario Plos, Chalmers University of Technology, Division of Concrete Structures, S-41296 Göteborg, Sweden. Hans W.Reinhardt, Stuttgart University, Pfaffenwaldring 4, D7000 Stuttgart 80, West Germany (formerly at Darmstadt University of Technology, Institut für Massivbau, Alexanderstrasse 5, D-6100 Darmstadt). Rune Sandström, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. Gottfried Sawade, Stuttgart University, Institut für Werkstoffe im Bauwesen, Pfaffenwaldring 4, D-7000 Stuttgart 80, Germany. Erik Schlangen, Delft University of Technology, Department of Civil Engineering, Stevin Laboratory, P.O. Box 5048, 2600 GA Delft, The Netherlands. Surendra P.Shah, Northwestern University, NSF Center for Advanced Cement-Based Materials (ACBM), The Technological Institute, Evanston, Illinois 602 08–3109, USA. Åke Skarendal, Swedish Cement and Concrete Institute (CBI), S-100 44 Stockholm, Sweden. Wanda Sobko, Chalmers University of Technolgy, Division of Concrete Structures, S-412 96 Göteborg, Sweden. Angelo di Tommaso, Università degli studi di Bologna, Facoltà di Ingegneria, Viale Risorgimento 2, I-40136 Bologna, Italy. Björn Täljsten, Luleå University of Technology, Department of Civil Engineering, S-951 87 Luleå, Sweden. Lucie Vandewalle, Katholieke Universiteit te Leuven, Department Bouwkunde, Park van Arenberg de Croylaan 2, B-3030 Heverlee, Belgium. Hans Wernersson, Lund Institute of Technology, Department of Structural Mechanics, Box 118, S-221 00 Lund, Sweden. Yu-Ting Zhu, Royal Institute of Technology, Department of Structural Mechanics, S-100 44 Stockholm, Sweden.

Preface

This volume contains the proceedings from an international workshop on the analysis of concrete structures by fracture mechanics. The workshop was dedicated to Professor Arne Hillerborg in recognition of his many outstanding contributions to this field. The workshop was organized by RILEM Technical Committee 90-FMA Fracture Mechanics of Concrete Structures—Applications. In addition to the presentations and discussions during the workshop, as summarized by the authors, the volume also contains some papers contributed by colleagues and friends of Arne Hillerborg. We would like to thank Arne Hillerborg and the other authors for their cooperation, the staff at the Division of Structural Engineering at Luleå University of Technology for the practical arrangements of the workshop, and the personnel at Chapman and Hall for the publication of this volume. L.Elfgren, S.P.Shah Luleå and Evanston February 1990

1 ARNE HILLERBORG AND FRACTURE MECHANICS L.ELFGREN Department of Civil Engineering, Luleå University of Technology, Luleå, Sweden

1 INTRODUCTION Arne Hillerborg has played an important part in the development of fracture mechanics for concrete structures. His aim has always been to analyze problems in such a way that conclusions can be drawn which are of value to practising desing engineers. In this paper a short outline is given of some of his contributions. 2 CAREER Arne Hillerborg was born in Stockholm, Sweden, on the 4th of January 1923. He graduated as a civil engineer (M.Sc.) from the Royal Institute of Technology in Stockholm in 1945. He subsequently worked as a research assistent in Structural Engineering at the Royal Institute of Technology, where he presented his PhD thesis “Dynamic influences of smoothly running loads on simply supported girders” in 1951. After a short period as design engineer he became involved in work for the Swedish concrete code, particularly regarding design of two-way slabs. During a five year period 1955–60 he was a lecturer at the Technical College in Stockholm, teaching structural engineering. Then, 1960–68 he was head of Siporex Central Laboratory, in charge of research and development for autoclaved aerated concrete. In 1968 he became associate professor in Structural Mechanics and in 1973 full professor in Building Materials at Lund Institute of Technology. Since January 1989 he is professor emeritus and in this function he is still very active. Arne Hillerborg married Ingegerd in 1944. They have one daughter. 3 THE STRIP METHOD The strip method for reinforced concrete slabs was first proposed by Hillerborg (1956). It is a design method based on the lower bound theory of plasticity. It was further developed (1959). These first papers were written in Swedish. A short English paper was published by Hillerborg (1960). The presentation to the

2

ANALYSIS OF CONCRETE STRUCTURES

English-speaking world is mainly due to Crawford (1962), Blakey (1964), Wood (1968) and Wood and Armer (1968). Later Hillerborg has summarized and extended his work, first in Swedish (1974), English

ARNE HILLERBORG AND FRACTURE MECHANICS

3

Fig. 1. Example of an analysis with the Hillerborg strip method. In (a) dimensions and support condition are given for a slab with a uniform load of 8 kN/m2 . The slab is supported by a column, two sides are built in and two sides are simply supported. In (b) and (c) chosen moment curves for the main strips are shown. In (d) resulting design moments are given. The load carried by the column is R=8×6.1×6.05=296 kN. From Hillerborg (1982a).

translation (1975), and In (1982a). An example of the simple and straight forward analysis is given in Figure 1.

4

ANALYSIS OF CONCRETE STRUCTURES

Fig. 2. The fictitious crack model as it was first proposed by Hillerborg et al (1976), and by Hillerborg (1978a).

4 FRACTURE MECHANICS Arne Hillerborg first got interested in fracture mechanics of concrete when he taught Building Materials at Lund Institue of Technology in the mid 70ies. He initiated a work for two of his students Petersson and Modeer (1976) and they together later in the same year published a paper on it, Hillerborg et al (1976). In that paper the model which has later become known as the fictitious crack model was introduced, see Figure 2. With the model it became obvious that linear elastic fracture mechanics (LEFM) could only be applied to very large concrete structures and not to concrete elements of normal size as was earlier done, Mindess (1983a, b). In 1979 and 1981 Matz Modéer and Per-Erik Petersson published their PhD theses on fracture mechanics of concrete. Their work was inspired by Hillerborg and separately or together they published many papers on related subjects. Some examples of results are given in Figures 3 to 5. In order to illustrate the size dependance in a simple and dimensionless way Arne Hillerborg early introduced the concept of a characteristic length 1ch of a material. The characteristic length 1ch is defined as

where E=the modulus of elasticity, GF=the fracture energy and ft= the tensile strength of the material.

ARNE HILLERBORG AND FRACTURE MECHANICS

5

Fig. 3. Theoretical relation between splitting strength fs and tensile strength ft for a concrete cube according to Modéer as function of the brittleness number w/1ch. (w is the height of the cube, 1ch is a characteristic length=EGF/ft2, E modulus of elasticity, and GF=fracture energy) From Hillerborg (1979b).

Size dependance can now be plotted as a function of d/1ch where d is 2 representative dimension of the studied structural element (e.g. the depth of a beam), see examples in Figures 3, 7, 8 and 9. The ratio d/ 1ch=dt2/EGF=(d3 ft2/E)/(d2 GF) can also be interpreted as a brittleness number giving the ratio of the stored elastic strain energy (d3 ft2 /E) to the fracture energy needed to break the specimen d2 GF), see e.g. DiTommaso and Bache (1989). When the first RILEM technical committee on fracture mechanics of concrete was formed in 1979 (TC 50FMC) with Professor Folker H Wittman as chairman, Arne Hillerborg was one of the members (other prominent members were H.K.Hilsdorf, M.Lorrain, H.Mihashi, S.Mindess, A. Rösli, R.N.Swamy, S.Ziegelsdorf and A.Di Tommaso). In the RILEM work Hillerborg had the main responsibility for a series of round robin tests on fracture energy of concrete according to a method proposed by Petersson. This resulted in a RILEM recommendation (1985) for a three-point beam method to determine the fracture energy of concrete, see Figures 6 and 7. This method is now used world-wide and has started much additional work on the testing methods of fracture mechanics properties. After the work of RILEM TC 50 FMC was finished in 1985, Wittmann (1983, 1986), two new RILEM technical committees were set up for the continuation of the work on fracture mechanics of concrete, one for further work on test methods (TC 89 FMT, chaired by S.P.Shah) and one for application (TC 90 FMA, chaired by L.Elfgren). The latter committee was proposed by Hillerborg, who has also taken an active part in its work, see Elfgren (1989a, b). In 1985 another of Hillerborgs students presented his PhD thesis, Gustafsson (1985). This thesis comprised among other things many comparisons between tests and analytical results by means of the

6

ANALYSIS OF CONCRETE STRUCTURES

Fig. 4. The fracture zone and the stress distribution in front of the notch tip at the maximum load for different beam depths. The figure is relevant for three-point bending with a ratio of notch depth a to beam depth d of a/d=0.25. The material properties are ft=3 MPa, GF=75 N/m, E=30 GPa, 1ch=0.25 m. From Petersson (1981).

fictitious crack model and application of the model to some practical design problems, e.g. shear fracture of reinforced beams. Some results from the thesis and related papers are given in Figures 8 and 9. Ongoing work by Hillerborg and his doctor students includes mixed mode properties of concrete and stability problems in fracture mechanics testning, see Figure 10 and 11. Recently the possibility of a formal application of the fictitious crack model to the failure of concrete in the compression zone of a bent beam has been studied by Hillerborg, and some preliminary conclusions of this work have been published (1988c, 1989a). A bibliography of the works presented by Hillerborg and his group is presented in the next section. 5

ARNE HILLERBORG AND FRACTURE MECHANICS

7

Fig. 5. Experimentally determined stress-deformation-curves for concrete in tension. From Petersson (1981).

Fig. 6. Proposed standard beam for test of fracture energy GF Hillerborg (1985c).

BIBLIOGRAPHY In this section a list of references are given to the works published by Arne Hillerborg and his group of students and collaborators. A few additional works of general interest are also cited. Avd för Byggnadsmateriallära (1988) Byggnadamateriallara LTH 1973– 1988. Tillägnad Arne Hillerborg vid hans avgång från professuren i december 1988 (Building Materials 1973–1988. A report dedicated to Arne Hillerborg). Lund Institute of Technology. Report TVBM-3038, 96 pp. Blakey, F.A. (1964) Strip method for slabs on columns, L-shaped plates etc. Translation of Hillerborg (1959). Melbourne, CSIRO, D.B.R. Translation No. 2. Crawford, R.E. (1962) Limit design of reinforced ete slabs. Thesis submitted to the University of Illinois for the degree of PhD. Urbana. DiTommaso, A. and Bache, H. (1989) Size effects and brittleness. Chapter 7 in Fracture Mechanics of Concrete Structures. From theory to applications (ed. L.Elfgren) Chapman & Hall, London, 191–207

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Fig. 7. Theoretical flexural strength of notched and unnotched concrete beams. Hillerborg (1985c). Elfgren, L. editor (1989a), Fracture Mechanics of Concrete Structures. From theory to applications. A RILEM Report by Technical Committee 90-FMA. Chapman & Hall, London, 407 pp (ISBN 0–412– 30680–8). Elfgren, L. (1989b) Applications of fracture mechanics to concrette structures. Fracture Toughness and Fracture Energy (eds. H Mishashi, H.Takahashi and F.H.Wittmann). Balkema, Rotterdam, pp 575–590. Gustafsson, P.J. (1983) Oarmerade betongrörs böjbrottlast. Teoretiska beräkningsmetoder. (Unreinforced concrete pipes.) Lund Institute of Technology, Rapport TVBM-3012. Gustafsson, P.J. (1985) Fracture mechanics studies of non-yielding materials like concrete: modelling of tensile fracture and applied strength. Doctor thesis. Lund Institute of Technology. Report TVBM-1007. 422 pp. Gustafsson, P.J. and Hillerborg, A. (1984) Improvements in concrete design achieved through the application of fracture mechanics. “Application of fracture mechanics to cementitious composites”, NATO Advanced Research Workshop, September 4–7, Northwestern University. Gustafsson, P.J. and Hillerborg, A. (1988) Sensitivity in shear strength of longitydinally reinforced concrete beams to fracture energy of concrete. ACI Structural Journal, Vol. 55, No 3, May-June, 286–294. Hassanzadeh, M. (1988) Determination of fracture zone properties in mixed mode I and II. Int. Conf. on Fracture of Concrete and Rock, Vienna July 4–6, 1988. Abstract published in Engineering Fracture Mechanics, Vol 35, No 1/2/3, 1990, p 614. Hassanzadeh, M. and Hillerborg, A. (1989a) Theoretical Analysis of test methods. Fracture of Concrete and Rock, (eds S.P.Shah and S.E.Swartz), Springer, New York, pp 388–395. Hassanzadeh, M. and Hillerborg, A. (1989b) Concrete properties in mixed mode fracture. Fracture Toughness and Fracture Energy Test Methods for Concrete and Rock, (eds H.Mihashi et al) Balkema, Rotterdam. pp 565–568. Hassanzadeh, M., Hillerborg, A. and Zhou, F.P. (1987) : Tests of material properties in mixed mode I and II. SEM— RILEM International Conference on Fracture of Concrete and Rock. (eds. S.P.Shah and S.E.Swartz). Society for Experimental Mechanics, Bethel, CT, pp 353–358 (ISBN 0–912053–13–5).

ARNE HILLERBORG AND FRACTURE MECHANICS

Fig. 8. Theoretical shear strength of reinforced concrete beams without shear reinforcement, forcement. Gustafsson & Hillerborg (1984).

9

is the percentage of rein-

Helmerson, H. (1978) Materialbrott för olika byggnadsmaterial. (Material Fracture for various building materials.) Diploma work Lund Institute of Technology, Division of Building Materials. Hillerborg, A. (1951) Dynamic influences of smoothly running loads on simply supported girders. Doctor Thesis, Department of Bridge Engineering, Royal Institute of Technology, Stockholm, 126 pp. Hillerborg, A. (1956) Jämviksteori för armerade betongplattor. (Equilibrium theory for concrete slabs). Betong (Stockholm). Vol. 41, No 4, 171–182. Hillerborg, A. (1959) Strimlemetoden (The strip method). Riksbyggen, Stockholm. Hillerborg, A. (1960) A plastic theory for the design of reinforced concrete slabs. IABSE Sixth Congress. Stockholm. Preliminary publication. pp 177–186. Hillerborg, A. (1974) Strimlemetoden (The strip method), Almqvist & Wiksell, Stockholm, 327 pp (ISBN 91–20– 03912–2). Hillerborg, A. (1975) Strip Method of design. English translation of Hillerborg (1974). Cement and Concrete Association, Wexham Springs, (Viewpoint publication), E &FN Spon, London, 256 pp, (ISBN 0–721–010121). Hillerborg, A. (1978a) A model for fracture analysis. Lund Institute of Technology. Report TVBM-3005, 8 pp. Hillerborg, A. (1978b) Brottmekanik tillämpad på betong (Fracture mechanics applied to concrete) Nordisk Betong (Stockholm) Nr 6–78, pp 5–12. Hillerborg, A. (1979a) The fictitious crack model and its use in numerical analysis. International Conference on Fracture Mechanics in Engineering Applications, Bangalore, March 26–30.

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Fig. 9. Theoretical strength of unreinforced concrete pipes, Gustafsson & Hillerborg (1984). Hillerborg, A. (1979b) Some practical conclusions from the application of fracture mechanics to concrete. Studies on concrete “technology. Dedicated to Professor Sven G.Bergström on his 60th anniversary, Swedish Cement and Concrete Institute, Stockholm, December 14, 43–54. Hillerborg, A. (1980a) Brott i betong (Fracture of concrete). CBI:s informationsdag, Swedish Cement and Concrete Institute, Stockholm 1980. Hillerborg, A. (1980b) Analysis of fracture by means of the fictitious crack model, particularly for fibre reinforced concrete. International Journal of Cement Composites, November, 177–184. Hillerborg, A. (1981) The application of fracture mechanics to concrete. Contemporary European Concrete Research, Stockholm June 9–11. Hillerborg, A. (1982a) The advanced strip method—a simple design tool. Magazine of Concrete Research, Vol 34, No 121, December 1982, pp 175–181. Hillerborg, A. (1982b) The influence of the tensile toughness of concrete on the behaviour of reinforced concrete structures. The Ninth International Congress of the FIP, Stockholm June 6–10. Hillerborg, A. (1983a) Theoretical analysis of the double torsion test. Cement and Concrete Research, Vol 13, 69–80. Hillerborg, A. (1983b) Analysis of one single crack. Fracture mechanics of Concrete. Editor F.H.Wittmann. Elsevier. 223– 250. Hillerborg, A. (1983c) Examples of practical results achieved by means of the fictitious crack model. William Prager Symposium on Machanics of Geomaterials: Rocks, Concrete, Soils. Northwestem University, September 11–15. Hillerborg, A. (1983d) Concrete fracture energy tests performed by 9 laboratories according to a draft RILEM Recommendation. Report to RILEM TC 50-FMC. Lund Institute of Technology, Report TVBM-3015. Hillerborg, A. (1983e) The fracture energy GF as a material property and its significance in structural engineering. An outline of an introductory chapter of a report from RILEM TC 50-FMC. Hillerborg, A. (1984a) Additional concrete fracture energy tests performed by 6 laboratories according to a draft RILEM Recommendation. Report to RILEM TC 50-FMC. Lund Institute of Technology, Report TVBM-3017.

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Fig. 10. Displacements and stresses in a cohesive zone where unstable conditions arize. From Hillerborg (1989c). Hillerborg, A. (1984b) Numerical methods to simulate softening and fracture of concrete. Fracture Mechanics of Concrete: Structural Application and numerical calculation. (Eds. G.Sih and A.DiTommaso). Martinus Nijhoff. 141–170. Hillerborg, A. (1985a) Predictions of nonlinear fracture process zone in concrete. J. of Engineering Mechanics, January. Discussion of paper by Wecharatana and Shah, October 1983. Hillerborg, A. (1985b) Influence of beam size on concrete fracture energy determined according to a draft RILEM Recommendation. Report to RILEM TC 50-FMC. Lund Institute of Technology, Report TVBM-3021. Hillerborg, A. (1985c) The theoretical basis of a method to determine the fracture energy GF of concrete. Materials and Structures. No 106, 291–296. Hillerborg, A. (1985d) Results of three comparative test series for determining the fracture energy GF of concrete. Materials and Structures, No 107, 407–413. Hillerborg, A. (1985e) Determination and significance of fracture toughness of steel fibre concrete. “Steel fiber concrete”, US-Sweden joint seminar, Stockholm June 3–5. Hillerborg, A. (1985f) A comparison between the size effect law and the fictitious crack model. A Festschrift for the seventieth birthday of professor Sandro Dei Poli, Milano. Hillerborg, A. (1986a) Dimensionless presentation and sensitivity analysis in fracture mechanics. Fracture Toughness and Fracture Energy of Concrete, (ed. F.H.Wittmann), Elsevier, Amsterdam, pp 413–421. Hillerborg, A. (1986b) Fracture aspects of concrete. The 6th European Conference on Fracture, June 15–20. Hillerborg, A. (1988a) Application of fracture mechanics to concrete. Summary of a series of lectures 1988. Lund Institute of Technology, Report TVBM-3030. Hillerborg, A. (1988b) Fracture mechanics and the concrete codes. Fracture mechanics: Application to Concrete, SP-118, American Concrete Institute, Dec 1989, pp 157–170.

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Fig. 11. Theoretical stress-deformation curves compared to correct curves (dashed lines) for different values of rotational stiffness k, compare Figure 10. From Hillerborg (1989c). Hillerborg, A. (1988c) Rotational capacity of reinforced concrete beams, Nordic Concrete Research (Oslo) No 7, pp 121–134. Hillerborg, A. (1989a) Compression stress-strain curve for design of reinforced concrete beams. Fracture Mechanics: Application to Concrete, SP-118, American Concrete Institute, Dec 1989, pp 281–294.

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Hillerborg, A. (1989b) Mixed mode fracture In concrete. Seventh international Conference on Fracture, Houston, Texas, March 20–24, 1989. Hillerborg, A. (1989c) Stability problems in fracture mechanics testing. Fracture of concrete and rock. Recent Developments (eds. S.P.Shah, S.E.Swartz and B.Barr). Elsevier, Amsterdam, pp 369– 378. Hillerborg, A. (1989d) Existing methods to determine and evaluate fracture toughness of aggregative materials— RILEM recommendation on concrete. Fracture Toughness and Fracture Energy—Test Methods for Concrete and Rock, (eds. H.Mihashi, H.Takahashi & F.H.Wittmann), Balkema, Rotterdam, pp 145–151. Hillerborg, A. (1990) Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams. Engineering Fracture Mechanics, Vol 35, No 1/2/3. pp 233–240. Hillerborg, A. Modéer, M. and Petersson, P-E. (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and concrete research, Vol 6, 773–782. Hillerborg, A. Petersson, P-E. (1981) Fracture mechanical calculations, test methods and results for concrete and similar materials. Fifth international conference on fracture, Cannes March 29– April 3. Horvath, R. and Persson, T. (1984) The influence of the size of the specimen on the fracture energy of concrete. Lund Institute of Technology. Report TVBM-5005. Kaplan, F.M. (1961) Crack propagation and the fracture of concrete. J. American Concrete Institute, 58, 591–610. Mindess, S. (1983a) The application of fracture mechanics to cement and concrete: A historical review, in Fracture Mechanics of Concrete (ed. F.H.Wittman), Elsevier, Amsterdam, 1–30. Mindess, S. (1983b) The cracking and fracture of concrete: an annotated bibliography 1928–1981, in Fracture Mechanics of Concrete, (ed. F.H. Wittman), Elsevier, Amsterdam, 539–680. Modéer, M. (1979a) Brottmekaniska analysmetoder för betong (Fracture mechanics methods for concrete). Nordisk Betong (Stockholm), Nr 1–79, pp 24–29. Modéer, M. (1979b) A fracture mechanics approach to failure analyses of concrete materials. Doctor thesis. Lund Institute of Technology. Report TVBM-1001. 102+44 pp. Petersson, P-E. (1979) Betongs brottmekaniska egenskaper (Fracture mechanical properties of concrete). Nordisk Betong (Stockholm) Nr 5–79, pp 31–38. Petersson, P-E. (1980a) Fracture energy of concrete: Method of determination. Cement and Concrete Research, vol 10, 1980, pp 78– 89. Petersson, P-E. (1980b) Fracture energy of concrete: Practical performance and experimental results. Cement and Concrete Research, vol 10, 1980, pp 91–101. Petersson, P-E. (1980c) Fracture mechanical calculations and tests for Fibre reinforced cementitious material. Advances in Cementmatrix Composites. Materials Research Society, Annual meeting, Boston November 17–18, 1980, Proceeding, Symposium L, pp 95–106. Petersson, P-E. (1981) Crack growth and development of fracture zones in plain concrete and similar materials. Doctor Thesis. Lund Institute of Technology. Report TVBM-1006. 174+10 pp. Petersson, P-E. (1982a) Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Proposed RILEM Recommendation, 29th January 1982. Petersson, P-E. (1982b) Comments on the method of determining the fracture energy of concrete by means of three-point bend tests on notched beams. Lund Institute of Technology. Report TVBM3011. Petersson, P-E. and Gustafsson, P.J. (1980) A model for calculation of crack growth in concrete-like materials. Numerical Methods in Fracture Mechanics. (Proceedings of the Second International Conference held at University College, Swansea, July 1980) Pineridge press, Swansea, pp 707–719. Petersson, P-E. and Modéer, M. (1976) Brottmekanisk modell för beräkning av sprickutbredning i betong. (A fracture mechanics model for the calculation of crack development In concrete.) Lund Institute of Technology, Division of Building Materials, Report No 70, 47 pp. RILEM Draft Recommendation (1985) Determination of the fracture energy of mortar and concrete by means of threepoint bend tests on notched beams. Materials and Structures, vol 18, No 106, pp 285–290. Wittman, F.H. editor (1983) Fracture mechanics of concrete. Elsevier, Amsterdam, 8+680, (ISBN 0–444–42199–8).

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Wittman, F.H. editor (1986) Fracture toughness and fracture energy of concrete. Elsevier, Amsterdam, 15+699, (ISBN 0–444–42733– 3). Wood, R.H. (1968) The reinforcement of slabs in accordance with a predetermined field of moments. Concrete. vol 2, No 2. February 1968. pp 69–76. Wood, R.H and Armer, G.S.T. (1968) The theory of the strip method for design of slabs. Proceedings of the Institution of Civil Engineers. Vol. 41, No. 10. October 1968. pp 285–311. Zhou, F. (1988) Some aspects of tensile fracture behaviour and structural response of cementitious materials. Lund Institute of Technology. Report TVBM-1008. 76 pp.

PART ONE BEHAVIOUR OF CONCRETE

2 MODE I BEHAVIOUR OF CONCRETE: INFLUENCE OF THE ROTATIONAL STIFFNESS OUTSIDE THE CRACKZONE J.G.M. van MIER Delft University of Technology, Department of Civil Engineering, Stevin Laboratory, Delft, The Netherlands

ABSTRACT In the paper, the influence of the rotational stiffness of the specimen outside the crack-zone in an uniaxial tensile test is discussed. Both the allowable rotations of the specimens loading platens as well as the flexural stiffness of the specimen itself determine the shape of the descending branch in a displacement controlled experiment. An LEFM based model is proposed, and it is shown that hardened cement paste fulfils the assumptions made in the model. In contrast, mortar and concrete show different behaviour, which may be explained by considering fracture in tension as a growth process in three dimensions. INTRODUCTION Since a number of years, displacement controlled uniaxial tensile tests are carried out for determining the fracture mechanics parameters for concrete and other cement-based materials under mode I loading. This specific experiment was recomended with the introduction of the fictitious crack model [4]. Since then it has become clear that some of the assumptions of this model are not fulfilled in an uniaxial tensile test, more specifically the development of a “uniform” process zone [14]. The rotational stiffness of the specimen outside the crack-zone determines the shape of the descending branch in an uniaxial tensile test. Testing a specimen between rotating end-platens will result in a gradual descending branch as shown in [8] (see Fig. 1), whereas between fixed end-platens a typical bump arises in the softening branch (see Fig. 2). The bump depends largely on stress redistributions during crack growth within the entire machine-specimen system. The stiffness of the specimen is crucial in this respect: the length of a specimen [5], but also its shape [15] will determine if possible stress redistributions will occur within the system. Utilising a simplified LEFM based analysis, the shape of the descending branch can be calculated for different boundary conditions. The fracture process in hardened cement paste seems to be in agreement with some of the peculiarities predicted with such an LEFM approach. Yet for concrete and mortar different response is measured, which can be explained by considering the fracture process as a growth process in three dimensions.

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Fig. 1 Uniaxial tensile test between rotating end platens, after [8]. The numbers between brackets along the descending branch are the optically measured crack lengths at the specimens surface.

Fig.2 Results of uniaxial tensile tests between non-rotating end-platens, Single-edge-notched specimens, after [15].

SIMPLIFIED LEFM ANALYSIS In [14] a physical model was presented for explaining the fracture process in concrete. Let us assume that the assumptions made are valid, and that at peak (maximum load), a critical flaw has developed in the specimen. The growth of this flaw results in a descending branch and can only be studied experimentally in a stable controlled testing machine (provided that the elastic energy release during crack growth is limited and that no snap-backs occur). In Fig. 3, the crack growth beyond peak is shown using reflection photoelasticity. In this particular experiment, the crack nucleated at the left notch of a Double-Edge-Notched (DEN) tensile specimen, and propagated gradually towards the other side while the load-displacement diagram (displacement measured in the centre of the specimen, with a gauge length of 65 mm) described a descending branch. Note that In a DEN tensile specimen the crack will nucleate always from one of the

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Fig. 3a Crack growth in an uniaxial tensile test on a DEN specimen between fixed end-platens, detected with reflection photo-elasticity: load-displacement diagram, after [15].

Fig. 3b Crack growth at δ=13.5 µm, after [15].

notches as a direct result of the heterogeneity of the material under consideration. See also Fig. 1, where the optically measured crack lengths at different stages along the descending branch are indicated. Now assume that, due to pre-peak microcracking, at peak stress a critical flaw of size a0/W has developed in a Single-Edge-Notched (SEN) tensile specimen as indicated in Fig. 4. It is assumed that the crack front remains straight during crack propagation. The stress intensity factor KI is equal to (1) where ∞ is the nominal externally applied stress, and Y is a geometrical factor, which depends on the specimen geometry and boundary conditions. When a0 reaches the critical size, KI reaches its critical value KI, crit. As mentioned before,it is assumed that a0 has reached the critical size at peak stress σp: (2) When the macro-crack grows, KI=KI, crit=constant. Thus when the crack has extended to a length a1, the stress σ1 that can be carried by the cracked specimen is equal to

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Fig. 4 Single-edge-notched tensile specimen.

(3) Thus the ratio σ1/σp can be calculated following

(4)

Eq. (4) determines the shape of the descending branch as function of the relative crack length a1/W. When the geometrical factor Y is known, the exact shape of the softening branch can be calculated. The present model is in fact similar to an R-curve type model, in which a gradual increase of KI is allowed for small crack lengths (in the current model when a1
(5)

Using equations (4) and (5), the shape of the descending branch of a SEN specimen loaded between free rotating end platens can be determined. The result is given in Fig. 5 as a σ1/σp—a1/W relation for different initial notch sizes a0/W. When the initial notch size is small, a steeper descending branch is obtained, whereas a more shallow softening curve is calculated for specimens with a larger initial notch. It might be argued that for concretes with coarse gravel, a larger initial flaw is needed for triggering macro-crack growth. Experimentally a more shallow softening curve was observed for large size aggregates [10].

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Fig. 5 Calculated shape of the descending branch for a SEN tensile specimen loaded between free rotating end-platens.

Fixed end-platens A similar result can be obtained for a SEN tensile specimen loaded between non-rotating end-platens (θedge=0). A solution for the geometrical factor of the problem was recently published by Marchand et al. [9]. The solution is based on Tada’s geometrical factors for bending and normal load respectively, and the expression for the geometrical factor is:

(6)

where F1 and F2 are Tada’s functions for normal load (eq.5) and pure bending respectively, and ξ=a/W. The Cij are the dimensionless crack compliances that contain all information regarding the M, N, δ, θ relations. The details of the analysis can be found in [9]. Inherent to the solution is the development of a closingbending moment as soon as the crack starts to grow. This phenomenon is shown in Fig. 6, where a plot of dimensionless stress σ1/σp versus relative crack length a1/W is given. The initial flaw was equal to a0/ W=0.05. After an initial steep stress drop for crack lengths a1/W<0.2, the growing crack is arrested due to the development of the closing bending moment. The analysis is valid for specimens with a sufficiently long length, such that Saint Venant’s principle is not affected. For very short cracks, the decay length is proportional to the crack length, and the solution is valid for L/W>1. For longer cracks, the decay length is proportional to the specimen width, and a valid solution is obtained only when L/W is sufficiently large. The descending branch has some interesting characteristics. The plateau level increases with decreasing L/W ratio, which demonstrates the influence of the flexural stiffness of the specimen outside the crackzone. Furthermore in a truly LEFM material, the external stress has to be increased near relative crack lengths of approximately 0.8–0.9, which implies that the load-excentricity and the associated closing bending moment have become so large that the axial load must be increased to facilitate further crack growth. DISCUSSION AND COMPARISON WITH EXPERIMENTAL RESULTS In tensile tests on concrete or mortar specimens between fixed end-platens, an increase of applied stress has never been observed in the descending branch, see for example [5], [15] (see Fig. 2), and [17]. For DEN

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Fig. 6 Calculated shape of the descending branch for a SEN tensile specimen loaded between fixed end-platens.

Fig. 7 a P-δ diagrams for single-edge-notched hcp specimens loaded in tension between fixed end-platens. The numbers between brackets indicate the number of days of under water curing. Between removal from the water basin and testing at an age of 35–40 days, the specimens were kept at 50 % RH.

tensile specimens also a plateau is observed which depends on the actual specimen dimensions and notch depths. However, the length of the plateau is smaller in terms of deformations ([15], see Fig.3). In contrast to this, for hardened cement paste (hcp), the external stress must be increased during some interval of the descending branch as shown in Fig. 7. This may indicate that LEFM-concepts are applicable to hcp, and that the model assumptions in the preceding paragraph are correct. Note however that experiments on hcp specimens are extremely difficult

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Fig. 7b Fracture surfaces for the two specimens of Fig. 7a.

mainly because of the sensitivity to shrinkage cracking of the material. The results of Fig. 7a were obtained on specimens that were cured under water for different periods, viz. 14 and 28 days as indicated. Shrinkage cracking was extreme in the specimen which was cured for 14 days only. As shown in Fig. 7b, the crack plane is in this case largely determined by the presence of shrinkage cracks which will extend primarily from the specimens surface into the interior of the specimen [19]. It must be concluded that concrete and mortar do not follow the assumptions of the model, at least at this size level. Yet the descending branches of both mortar and concrete specimens have the typical plateau in tests with fixed end-platens, and the different behaviour can be explained by considering the fracture of a specimen as a three dimensional growth process. In Fig. 8, the hypothesized process is shown schematically. Due to non-uniform drying, the surfaces of a specimen will always be subjected to tensile eigen-stresses. Therefore it is quite likely that crack nucleation is from the outer parts of the specimen towards the centre of it. The stress redistributions in a SEN specimen loaded between fixed end-platens are limited in the case of concrete and mortar as demonstrated by the fact that no stress increase is measured during some interval of the descending branch. This can occur when the crack front is not straight, but rather curved as indicated in Fig. 8 (see cross-section A-A). The crack has extended further in parts of the specimen near the surfaces as compared to the specimens centre. Because the core is still intact, the developing load-excentricity during macro-crack propagation is considerable less as compared to a growing crack in a truly LEFM material with a straight crack front. Consequently, the bending moment is relatively small and no increase of external stress is needed for facilitating further crack growth. Note that curved crack fronts are observed also in three-point bend tests by means of impregnation techniques (see for example [1], [12]). When the crack front reaches the other side of the specimen (which will first occur near the surfaces), the specimen will unload rapidly as soon as the last part of the crosssection cracks. However, a small part of the specimens core may still be intact as will be discussed next. Several reasons can be given for the “development” of an intact core in a specimen. First of all, the crack branches which develop from the surfaces do not necessarily grow in the same cross-sectional plane, and may avoid each other, [15]. In this way, flexural ligaments develop internally. Flexural failure of the ligaments may be responsible for the long tail that is normally observed in the load-deformation curve in tension (see Fig. 8, inset (a.)). This follows also from theoretical considerations, [2]. The second reason may be that compressive eigen-stresses which develop in the specimens core due to non-uniform drying-out, will prevent the formation of a straight crack front (see Fig. 8, inset (b.)). In this

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23

Fig. 8 Three-dimensional fracture mechanism for concrete and mortar specimens loaded in tension.

case, the long tail in the load-deformation diagram is explained by the fact that the increased tensile resistance of the core must be overcome. The third reason may be that the final separation of the crack planes is prevented by friction due to the pull-out of aggregates. This might also explain the long tail of the load-deformation curve (see Fig. 8, inset (c.)). Currently It is not clear which of the mechanisms is responsible for the observed long tail in the descending branch. Most likely a combination of the three factors occurs. The end of the plateau results normally in a sudden unloading, which might be explained from the second mechanism. The explosive unloading is facilitated by the presence of tensile eigen-stresses near the specimens edges. One last remark should be made. The non-uniform opening and the associated plateau in the descending branch can be simulated numerically with a computational softening diagram, [14], [11], [18]. The computational softening diagram is a simplified two-dimensional representation of the three dimensional fracture process for plane-stress analysis. Crucial is that the first part of the softening curve is extremely steep, and corresponds to almost purely brittle behaviour (see Fig. 9). If the first steep part of the bi-linear diagram is exceeding the upper bound for non-uniform opening as indicated in Fig. 9, the non-uniformity of the fracture process cannot be simulated in a numerical analysis [14]. This is in agreement with the above reasoning. Yet, if the shallow tail of the computational softening diagram is dependent of the specimen dimensions (thickness) and possibly also of non-uniform drying-out, the size dependency of the Gfapproach might be explained [3]. Note that the largest contribution to the fracture energy Gf is due to the shallow second branch in the computational softening diagram. CONCLUSIONS The influence of the boundary conditions on the load-deformation response of Single-Edge-Notched tensile specimens can be explained from a simple analysis based on linear fracture mechanics. Assumption in the model is that a critical notch has developed in the specimen before the peak load is reached. Subsequently

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Fig. 9 Computational bi-linear softening diagrams for plane-stress analysis, after [16].

this critical flaw grows under constant stress intensity factor. When a specimen is loaded between free rotating end-platens, a smooth descending branch will occur, whereas the analysis of a specimen loaded between fixed end-platens results in a typical plateau in the descending branch. The plateau is associated with severe stress redistributions in the specimen-machine system, which depend on the specimens geometry (shape and size) and boundary conditions, and the machine stiffness. A material fulfils the assumptions made in the analysis if the external load must be increased during some interval of the descending branch when a SEN specimen is loaded under fixed end displacement. Hardened cement paste specimens fulfil this requirement, whereas concrete and mortar specimens give different results. Note that the model gives insight in the post-peak behaviour only. The maximum load depends on the micro-crack processes that take place before the maximum is reached [14]. The behaviour of concrete and mortar specimens can be explained by considering the fracture of a specimen as a three-dimensional growth process. Primarily, the critical flaw will propagate with a curved crack front as observed also in three point bend tests [1], [12]: the crack has extended further along the edges, and is delayed in the centre of a specimen. When the macro-crack grows towards the free (unnotched) edge of the specimen, sudden unloading will occur, which marks the end of the plateau. The residual carrying capacity of a specimen, i.e. the long tail of the load-deformation diagram, can be explained from three different (possibly interacting) mechanisms: the development of internal flexural ligaments, the compressive eigen-stresses in the specimens core due to non-uniform drying-out and frictional pull-out of aggregates. Future research should clarify the mechanism. REFERENCES 1

[] Bascoul, A., Kharchi, F. and Maso, J.C., Concerning the Measurement of the Fracture Energy of a Micro-Concrete According to the Crack Growth in a Three Points Bending Test on Notched Beams, in Fracture of Concrete and Rock, Proceedings of the SEM-RILEM International Conference, Houston, June 17–19, 1987, (S.P.Shah and S.E.Swartz, eds.), Springer Verlag 1989, pp.396–408.

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2 3 4 5

6 7

8 9 10 11 12

13 14 15

16

17

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[] Bazant, Z.P., Snapback Instability at Crack Ligament Tearing and its implications for Fracture Micromechanics, Cement & Concrete Research, 17(6) (1987), 951–967. [] Brameshuber, W., Bruchmechanische Eigenschaften von jungem Beton, Schriftenreihe des Instituts fur Massivbau und Baustofftechnologie, Karlsruhe, 5 (1988), pp.233. [] Hillerborg, A., Petersson, P.E. and Modeer, M., Analysis of Crack Formation and Crack Growth in Concrete by means of Fracture Mechanics and Finite Elements, Cement & Concrete Research, 6(6) (1976), 773–782. [] Hordijk, D.A., Reinhardt, H.W. and Cornelissen, H.A.W., Fracture Mechanics Parameters from Uniaxial Tensile Tests as influenced by Specimen Length, in Preprints SEM-RILEM Intern. Conf. on Fracture of Concrete and Rock, Houston, June 17–19, 1987 (S.P.Shah and S.E.Swartz eds.), pp.138–149. [] Jenq, Y.S. and Shah, S.P., Two Parameter Fracture Model for Concrete , Journal of Eng. Mech., ASCE, 111(10) (1985), pp.1227–1241 [] Karihaloo, B.L., Do Plain and Fibre-Reinforced Concretes have an R-curve Behaviour?, in Fracture of Concrete and Rock, Proceedings of the SEM-RILEM International Conference, Houston, June 17–19, 1987 (S.P.Shah and S.E.Swartz eds.), Springer Verlag, 1989, pp.96–105. [] Labuz, J.F., Shah, S.P. and Dowding, C.H., Experimental Analysis of Crack Propagation in Granite, Int.J.Rock Mech. Min. Sci. & Geomech. Abstr., 22(2), (1985), 85–98. [] Marchand, N., Parks, D.M., and Pelloux, R.M., KI-Solutions for Single Edge Notch Specimens under Fixed End Displacements, Int J. Fracture, 31 (1986), 53–65. [] Peterson, P.E., Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials, Report no. TVBM 1006, Lund Institute of Technology, Sweden, (1981), pp.174. [] Rots, J.G., Computational Modelling of Concrete Fracture, PhD Dissertation, Delft University of Technology (1988). [] Swartz, S.E. and Refai, T., Cracked Surface Revealed by Dye and its Utility in Determining Fracture Parameters, in Preprints Int’I. Workshop on Fracture Toughness and Fracture Energy—Test Methods for Concrete and Rock, Tohoku University, Sendai, Japan, Oct. 12–14, 1988, pp.393–405. [] Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, USA (1973). [] Van Mier, J.G.M., Fracture of Concrete under Complex Stress, HERON, 31 (3) (1986), pp.1–90. [] Van Mier, J.G.M. and Nooru-Mohamed, M.B., Geometrical and Structural Aspects of Concrete Fracture, International Conference on Fracture and Damage of Concrete and Rock, Vienna, July 4–6, 1988 To appear in Eng. Fract Mech. (1989). [] Van Mier, J.G.M., Mixed-Mode Fracture of Concrete under Different Boundary Conditions, in Proceedings SEM Spring Conference on Experimental Mechanics, Cambridge (Ma), May 28-June 1 (1989), SEM, Bethel, pp.51–58. [] Willam, K., Hurlbut, B. and Sture, S. (1985), Experimental and Constitutive Aspects of Concrete Failure, in Proceedings US-Japan Seminar on Finite Element Analysis of Reinforced Concrete Structures, May 21–24, (1985), ASCE Special Publ.(Ch.Meyer and H.Okamura eds.), pp.226–254. [] Wittmann, F.H., Roelfstra, P.E., Mihashi, H., Huang, Y.-Y., Zhang, X.-H. and Nomura, N., Influence of Age of Loading, Water-Cement Ratio and Rate of Loading on Fracture Energy of Concrete, Materials and Structures, RILEM, 20 (1987), 103–110. [] Wittmann, F.H. and Roelfstra, P.E., Constitutive Relations for Transient Conditions, In Proceedings IABSE Colloquium on Computational Mechanics of Concrete Structures—Advances and Applications, Delft, 54 (1987), pp.239–259.

3 EXPERIMENTAL ANALYSIS OF MIXED MODE I AND II BEHAVIOUR OF CONCRETE J.G.M. van MIER, M.B.NOORU-MOHAMED, E.SCHLANGEN Delft University of Technology, Department of Civil Engineering, Stevin Laboratory, Delft, The Netherlands

ABSTRACT Results are presented of combined tensile and shear tests on concrete, mortar and Lytag lightweight concrete plates in the recently developed biaxial test rig of the Stevin Laboratory. The experimental procedure is explained in detail, and some preliminary results are presented. The behaviour of pre-cracked concrete specimens subjected to lateral (tensile or compressive) shear is, at small crack openings (smaller than 250 µm), governed by diagonal cracking (tensile shear) or strut splitting (compressive shear) due to the rotation of the principal stresses. Sliding failure was observed only for larger crack openings. The results suggest that assumptions made in conventional aggregate interlock theories are not valid for small crack openings. Rather, the assumptions made in recently developed so-called smeared rotating crack models are confirmed. INTRODUCTION Mixed mode I and II fracture of concrete and other brittle disordered materials is currently studied extensively worldwide. The efforts can be subdivided between two basic approaches: the conventional fracture mechanics approach, and an approach in which the constitutive relations of a fracture zone are studied. In the first approach the primary goal is to study the growth of cracks under combined tensile and shear loading. Of main interest are the derivation of a crack growth mechanism and the evaluation of the mixed mode fracture toughness of materials (e.g. [1], [2]). Because concrete and mortar do not obey the assumptions of linear fracture mechanics, it is assumed that a fracture process zone preceeds the development of a macroscopic traction free crack. In the second approach it is tried to isolate a fracture process zone and to study its behaviour under combined tensile and shear loadings (e.g. [3], [7], [10]). As a result constitutive equations should be derived (normal and shear stress versus crack opening and crack sliding displacement, see pp. 94–96 in [6]) for implementation in finite element codes. This last approach is in fact an extension of the fictitious crack model [4] and related models for mode I loading. The input parameters for the fictitious crack models must be derived by loading small specimens in a very stiff loading apparatus. However, recently researchers became aware of the fact that besides the stiffness of the machine, the stiffness of the specimen itself has a considerable influence on the measured response in tension.

EXPERIMENTAL ANALYSIS OF MIXED MODE

27

In [13], It was argued that the fracture in uniaxial tension can also be regarded as a growth process in three dimensions. Macroscopic cracks grow through the specimens cross-section with a curved crack front. The surfaces of a specimen are cracked first, and the propagation of the surface cracks complies to assumptions from linear fracture mechanics. Yet because different mechanisms take part in the fracturing of the central core of the specimen, a deviation from linear fracture mechanics occurs. These mechanisms must be borne in mind when fracture processes under mixed mode I and II conditions are studied. Currently, experiments are carried out in the Stevin Laboratory, where the fracture process of concrete specimens subjected to combined tension and shear is studied. The experiments should lead to the derivation of a mixed mode crack growth criterion for brittle disordered materials. In this paper, the experimental method and some preliminary results are discussed. MIXED MODE I AND II EXPERIMENTS Description of biaxial test rig The experiments were carried out in the biaxial test rig of the Stevin Laboratory [10]. Basically, the apparatus consists of two independent rigid frames that are fixed in an overall frame by means of plate springs. An exploded view of the apparatus is shown in Fig. 1. One of the two rigid frames consists of two coupled frames, and the second frame can slide in between the coupled frames. The coupled frames can move in the horizontal direction and are fixed to the overall frame in the vertical direction. The middle frame can move vertically, and is fixed in the horizontal direction. To both the middle frame and the coupled outer frames a load-cell and a hydraulic actuator, with a capacity of 100 kN in tension or compression, are connected as indicated. The maximum load is however restricted to 50 kN. Square concrete plates of size 200×200×50 mm are loaded in the apparatus. The loading procedure is clarified in Fig. 2. In uniaxial tension, a concrete plate is glued between the upper side of the middle frame and the lower side of the coupled outer frames (Fig. 2a). When the glue has set, the specimen is loaded by moving the middle frame upward. Because the coupled outer frames are fixed in the vertical direction, a tensile stress develops in the concrete specimen, which will eventually fracture the plate (Fig. 2b). The concrete plates are double or single-edge-notched at half height (as indicated) in order to generate crack growth at a known location. This facilitates deformation controlled testing. The same procedure is used for applying a shear load to a pre-cracked specimen, as shown in Fig. 3. In these biaxial experiments not only the upper and loweredges of a specimen are glued to the frames, but also parts of the left and right edges. After pre-cracking a specimen as described before, a lateral in-plane shear load can be applied over the cracked area. This can be done either in displacement-control (when the differential shear deformation between the two specimen halves is used as control variable), or in loadcontrol. Specimens and materials Specimens were double-edge-notched square concrete plates of size 200*200 *50 mm. The notches were 25 mm deep saw-cuts. Up to now, experiments are carried out using three materials: a 2 mm mortar, a 16 mm concrete and Lytag high strength lightweight concrete with a maximum aggregate size of 12 mm. The details of the various mixes as well as the average compressive strength and splitting tensile strength of the three materials are given in [8] and [12]. The plates were cast in a vertical position in a battery mould (containing six plates of size 300*300*50 mm) . Two days after casting, the plates were demoulded and

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Fig. 1 Exploded view of the biaxial test rig

placed in a fresh water basin. After 14 days, the specimens were sawn to the required size and were placed in a fresh water basin again. At 28 days the specimens were removed from the basin, and were allowed to dry in the laboratory (17–18°C and 50% R.H.). The age at loading varied a little for the different test-series, but was always larger than 35 days. The differences in treatment are reported in the previous publications [8], [12]. Load-paths investigated Three different load-paths were studied so far: (1) shear at constant axial crack opening, (2) shear at zero normal confinement and (3) shear at small constant compressive normal confinement. Always a specimen was pre-cracked in tension (in displacement-control in the vertical direction of the test rig) to a prescribed axial crack opening. Displacement control in the vertical direction was done by using the average deformation (meas. length 65 mm) measured with either two [12] or four [8] LVDTs as a feedback signal in the servo-controlled system. Subsequently the axial load was decreased to zero or to the prescribed normal confinement (-1000 N, corresponding to an average normal stress of -0.13 N/mm2). The axial opening was in most experiments larger than 50 µm, at which stage the crack process was uniform again (see [12]). Only a small number of tests was carried out at smaller crack openings. In load-paths (1) and (2) the shear load was either compressive or tensile, whereas in load-path (3) compressive shear was applied. In the initial experiments [12], the shear was applied in load-control, whereas in the more recent tests [8], shear loads were applied in displacement-control via two LVDTs measuring the relative displacements of the two specimen halves (that are separated partially by the axial crack). The loading rates were as follows: 1.65–2.0 µm/min in the vertical (tensile) direction and 1.60 µm/min (when a test was carried out in displacement-control) or 20 N/s (in load-controlled testing) in the horizontal (shear) direction. When the prescribed axial crack opening was reached, the specimen was unloaded in the

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29

Fig. 2 Sectional view of the biaxial apparatus: at the beginning of an experiment (a), and after cracking (b). Note the bending of the plate springs of the middle frame in Fig. b as a result of the upward movement of this frame with respect to the coupled frames.

Fig. 3 Double-edge-notched square plate for the mixed-mode tests.

vertical (tensile) direction by an immediate change of polarity of the axial loading. Therefore, the influence of creep-effects can be neglected. EXPERIMENTAL RESULTS Depending on the actual boundary conditions (load-paths) in the experiments, rather distinct responses were measured. The fracturing of the specimens was recorded in detail, in some cases even using reflection photoelasticity [11]. For all details, the reader is refered to [8], [11] and [12]. Here only some of the results will be discussed.

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Fig. 4 Tensile shear at constant crack opening: (a) P-δ diagram, (b) load-path P-Ps and (c) photo-elastic fringes at stage 27.

Tensile shear at constant crack-opening (load-path 1) A number of tests was carried out using reflection photo-elasticity as a crack detection technique. A very thin coating (0.25 mm) of photo-elastic material was glued to the concrete specimen. In Fig. 4 some of the results are shown. After axial pre-cracking, in this case up to 11.6 and 23.4 µm respectively, atensile shear load was applied in the horizontal direction. The axial P-δ diagram and load-path P- Ps are shown in Fig. 4a and b. The growth of macro-cracks in the specimen could be followed quite accurately by means of the photo-elastic reflection technique. In Fig. 4c, the cracking at step 27 is shown. The corresponding point in the P-δ diagram is indicated in Fig. 4a. At the moment of application of tensile shear, a diagonal crack nucleated from the left saw-cut, and propagated towardsthe upper edge of the specimen. This was observed already during the first shear cycle at δ=11.2 µm. Redistributions in the specimen could be followed closely, and at some moment even the closure of the axial crack was observed. In fact, the axial crack was shielded by the propagating diagonal “shear” crack. Final failure of the specimen was through the development of a

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Fig. 5 P-δ and Ps-δ diagrams for tensile shear tests at constant zero axial load.

second diagonal crack from the small crack branch near the lower right notch. For this boundary condition (viz. δ=const. during shear) the in-plane “shear resistance” of the axial crack was higher than the crack resistance of the intact specimen parts. Tensile shear at zero axial load (load-path 2) In addition to the constant crack opening experiments (load-path 1), several tests were carried out in which tensile shear was applied to a specimen containing a partially developed axial crack, while maintaining a zero load normal to the crack plane (viz. P=0=const. during shearing). In Fig. 5, the results are shown as P-δ and Ps-δ plots for a number of experiments with various axial crack openings. A sudden decrease of shear capacity, or rather maximum applicable shear-load PS, was measured near crack openings of 250 µm. Beyond this point, a more or less constant shear load (Ps=2 kN) was measured. When the post-mortem crack patterns were compared, it was found that failure was through sliding for crack openings larger than 250 µm, and that diagonal cracks developed when the initial crack width at shearing was smaller than 250 µm (see Fig. 6). A parallel might be drawn with shear in beams, when after stress redistributions a new set of diagonal cracks grows through a previously developed crack pattern. Compressive shear with normal confinement (load-path 3) Examples of load-path 3 experiments are shown in Fig. 7. In Fig. 7a, the P-δ diagrams for three tests with different axial crack opening are given, in Fig. 7b the corresponding Ps-δs diagrams. The replicability of the experiments was good [8], so only single results are compared. Basically the fracture mechanism of the experiments is of interest, and not (yet) the quantitative evaluation of the results. With increasing crack

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Fig. 6 Post-mortem crack patterns of the experiments of Fig. 5.

opening at shearing, a decreasing shear stiffness was measured in the experiments. Also, the tests at larger crack opening showed a lower maximum shear-load. The typical transition as was observed in the tensile shear tests at zero axial load was not found. The failure pattern was completely different in the compressive shear tests. Failure was not through the development of diagonal “shear-cracks”, but rather by tensile splitting in the direction of the compressive strut that would develop in a specimen. This is shown in Fig. 8. Crack A-B-C developed due to the axial tensile loading. Under subsequent shear, the diagonal splitting crack C-D developed parallel to the compressive strut direction. The residual carrying capacity could be associated with the length of crack B-C, which was more or less perpendicular to the compressive loading direction. Clearly, the shear strength of the crack, in combination with the axial confinement was still so large that no shear failure could occur. This was observed only In the tests without axial confinement. Shear stiffness reduction In Fig. 9, the initial shear stiffness Ps/δs, which is defined as the secant modulus between 10 % and 30 % of the maximum shear load Ps, is shown as function of the axial crack opening δa* (which is the true crack opening at the beginning of shearing, and is smaller than the unloading deformation in the axial direction, see Fig. 7a). The results of 2 mm mortar, 16 mm concrete and Lytag concrete are included in the same figure. As expected, a decreasing shear stiffness was measured with increasing crack width. For crack openings larger than 200 mm, there seemed to be a slight influence of the confining stress normal to the crack plane. In computational models, the decrease of shear stiffness of cracked concrete is expressed by means of a shear retention factor β (see for example in [6], p.95). For normal-weight concrete with a maximum aggregate size of 16 mm, values of β between 0.12 and 0.40 were calculated based on the new experimental results [8]. This is far in excess of values determined from conventional aggregate interlock experiments

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33

Fig. 7 P-δ (a) and Ps-δs (b) diagrams for three compressive shear tests with compressive confinement normal to the crack plane. The tests were carried out at different values for the axial crack opening.

Fig. 8 Crack pattern for one of the experiments of Fig. 7 (δ=150 µm crack opening at shearing).

(e.g. [9]), viz. 0.04 -0.09 for the same crack opening rangeas tested (200–400 µm for normal-weight concrete, see [8]). As argued in [6], the differences might be explained from neglecting the coupling between shear stiffness and shear displacements. Furthermore, an assumption in conventional aggregate interlock theories is that the principal stress axes are fixed and that shear is transfered in the crack plane. Rotation of principal stresses occurred in the experiments reported in this paper (Figs. 4–7), and it is believed that the current tests are suitable for testing so-called smeared rotating crack concepts (see in [5], pp.138–146). The assumptions made in this type of modelling are in agreement with the current experimental results. CONCLUSIONS Results are presented of mixed mode I and II tests of concrete. A shear-like load was applied to a specimen after it was pre-cracked to a prescribed axial crack opening. The results indicate that the response of the specimens upon shearing is governed by the development of a rotated principal stress state, especially for very small axial crack openings. Due to this rotation of principal stresses, diagonal cracking (under tensile shear) or splitting of compressive struts (under application of compressive shear) was observed in the

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Fig. 9 Decrease of shear stiffness with increasing axial crack opening.

experiments. Sliding in the crack plane has been found for very large crack openings only (larger than 250 µm). This suggests that aggregate interlock models are not applicable for small crack openings (smaller than 250 µm). The experimental results seem to be in agreement with assumptions made in so-called smeared rotating crack models [5]. REFERENCES 1 2 3

4 5 6 7 8

9

[] Arrea, M. and Ingraffea, A.R., Mixed Mode Crack Propagation in Mortar and Concrete, Report No. 81–13, Dept. Struct.Eng., Cornell University, Ithaca, N.Y., (1982), pp. 143. [] Bazant, Z.P. and Pfeiffer, P.A., Shear Fracture Tests of Concrete, Materials and Structures, RILEM, 19(110) (1986), 111–121. [] Hassanzadeh, M., Hillerborg, A. and Zhou Fan Ping, Tests of Material Properties in Mixed Mode I and II, in Preprints SEM-RILEM Intern. Conf. on Fracture of Concrete and Rock, Houston, June 17–19, (1987), (S.P.Shah and S.E.Swartz eds.), pp. 353–358. [] Hillerborg, A., Peterson, P.E. and Modeer, M., Analysis of Crack Formation and Crack Growth in Concrete by means of Fracture Mechanics and Finite Elements, Cement and Concrete Research, 6(6) (1976), 773–782. [] Hillerborg, A. and Rots, J.G., Crack Concepts and Numerical Modelling, Chapter 5 in Fracture Mechanics of Concrete Structures—From Theory to Applications, (L.Elfgren ed.), Chapman & Hall (1989), pp.128–146. [] Hordijk, D.A., Van Mier, J.G.M. and Reinhardt, H.W., Material Properties, Chapter 4 in Fracture Mechanics of Concrete Structures—From Theory to Applications, (L.EIfgren ed.), Chapman & Hall (1989), pp. 67–127. [] Keuser, W., Kornverzahnung bei Zugbeanspruchung, Forschunngskolloquium DAfStb, Darmstadt (1986), pp. 13–18. [] Nooru-Mohamed, M.B. and Van Mier, J.G.M., Fracture of Concrete under Mixed-Mode Loading, in Fracture of Concrete and Rock-Recent Developments, (S.P.Shah, S.E.Swartz, B.Barr, eds.), Elsevier Science Publishers, London/New York, 1989, pp.458–467. [] Paulay, T. and Loeber, P.J., Shear Transfer by Aggregate Interlock, ACISpecial Publication 42, Shear in Reinforced Concrete, (1974), Vol. I, pp. 1–16.

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10

11

12

13

35

[] Reinhardt, H.W., Cornelissen, H.A.W., and Hordijk, D.A., Mixed Mode Fracture Tests of Concrete, in Fracture of Concrete and Rock, Proceedings of the SEM-RILEM International Conference, Houston, June 17–19, 1987 (S.P.Shah & S.E.Swartz eds.), Springer Verlag (1989), pp.117–130. [] Van Mier, J.G.M., Fracture Study of Concrete Specimens Subjected to Combined Tensile and Shear Loading, in Proceedings GAMAC Int’I. Conf. on Measurement and Testing in Civil Engineering, (J.F.Jullien ed.), LyonVilleurbanne, Sept. 13–16 (1988), Vol. 1, pp. 337–347. [] Van Mier, J.G.M. and Nooru-Mohamed, M.B., Fracture of Concrete under Tensile and Shear-like Loadings, in Preprints Int’I. Workshop on Fracture Toughness and Fracture Energy—Test Methods for Concrete and Rock, Tohoku University, Sendai, Japan (H.Mihashi ed.), Oct. 12–14, (1988), pp. 433–447. [] Van Mier, J.G.M., Model Behaviour of Concrete: Influence of the Rotational Stiffness outside the Crack-Zone, in Analysis of Concrete Structures by Fracture Mechanics, Proceedings of the RILEM workshop dedicated to Prof. A.Hillerborg, Abisko, Sweden, June 1989 (L.Elfgren ed.), Chapman & Hall, (1990).

4 CONSIDERATIONS REGARDING FRACTURE ZONE RESPONSE TO SIMULTANEOUS NORMAL AND SHEAR DISPLACEMENT M.HASSANZADEH Lund Institute of Technology, Sweden

Abstract A testing arrangement has been developed in order to determine the fracture process zone properties, affected by simultaneously applied normal and shear displacement. The testing arrangement makes it possible to perform stable displacement-controlled mixed-mode tests according to any arbitrary displacement path. The paper presents the testing arrangement and discusses a few aspects of the determination of the fracture zone properties. 1 Introduction The present project is part of a major fracture-mechanics project which has been going on since 1974 . The project has resulted in the development of the Fictitious Crack Model (FCM), Hillerborg (1989-a), which has become the principal model regarding fracture-mechanics analyses at the Division of Building Materials, Lund Institute of Technology. The goal of this particular project has been to determine material properties which are necessary for mixed mode I and II fracture-mechanics applications of FCM. 2 Mixed mode according to the fictitious crack model According to the FCM, three phases may be distinguished in mixed-mode crack propagation (see figure 1). The first one is the initiation phase characterized by the formation of the fracture process zone or the formation of a zone in which the material starts to soften. In homogeneous material, the condition for initiation is fulfilled when the first principal tensile stress reaches the tensile strength of the material. At the instant the fracture zone is formed, it also assumes the final orientation which is perpendicular to the first principal tensile stress direction. The second phase is the continuous softening of the fracture zone, which is characterized by the gradual weakening of the material inside the fracture zone. The subsequent state of stress within the fracture zone depends on alterations in the first principal stress direction in the subsequent stages. If the direction remains unchanged during the complete softening process, the state of stress will be mode I, i.e. only normal stresses will occur inside the fracture zone despite global mixed-mode loading.

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Figure 1. Mixed-mode crack propagation according to the fictitious crack model.

However, if the direction of the first principal tensile stress is changed, the state of stress inside the fracture zone will be mixed-mode, i.e. both normal and shear stresses will occur within the zone. The third phase is the formation of a real crack characterized by stress-free crack surfaces. The fracture zone properties which are needed for incremental mixed-mode applications of the FCM can be formulated as follows:

∆σ and ∆wn are stress and displacement increments normal to the fracture plane. ∆r and ∆Ws are shear stress and shear displacement increments inside the fracture process zone. The diagonal terms in the stiffness matrix (K11 and K22) are normal and shear crack stiffnesses, and the off-diagonal terms (K12 and K21) are the coupling functions. The terms in the stiffness matrix are not constants but are functions of both total normal and total shear displacements. The FCM differs from the linear elastic mixed-mode models in two aspects. FCM utilizes the strength criterion for the initiation of crack propagation, while the linear elastic models utilize the stress intensity criterion (Hillerborg 1989). Furthermore, the linear elastic models neglect the effects of the fracture process zone which, leads to a completely different stress field in front of a propagating real crack. 3 Testing arrangement Figure 2 shows test equipment which, together with a closed-loop testing machine, forms an arrangement for mixedmode I and II tests. The details, performance and design of the equipment are discussed in references

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Figure 2. Test equipment.

[2, 3, 4, 6, 7]. Prior to the test, a prismatic notched concrete specimen is glued in between the beams. After hardening of the adhesive, the equipment is put in position in the closed-loop machine. The tests are displacement-controlled and are run in a semi-automatic manner, which means that the closedloop machine continuously runs a programmed normal dis placement scheme and that the shear displacement is manually imposed by rotating the crank indicated in figure 2. By means of continuous adjustment of the shear displacement, it is possible to perform mixed-mode tests according to any arbitrary displacement path, or relation between normal and shear displacements. The mixed mode procedures are such that they fulfil the criterion outlined by the FCM, i.e. normal displacement is first imposed so that a tensile fracture zone starts to form. Shortly after the normal stress starts to decrease, shear displacement is also imposed. In the subsequent stages, the test is continued in such a way that the normal and shear displacements follow a predefined displacement path. 4 Mixed mode tests A predefined displacement path is needed to conduct the mixed-mode tests by means of the test arrangement presented here. At present there are no established sets of displacement paths which represent real structural behaviour. Therefore, arbitrary displacement paths have been used so far. Two types of displacement path have been tested, a linear path and a parabolic path which are defined by the following formulae.

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39

Wn and Ws are normal and shear displacements within the fracture process zone. Low a and β values describe situations where the normal displacement is somehow suppressed, for instance due to reinforcement. On the contrary, high a and β values describe situations where the normal displacement is not considerably suppressed. The results of such tests are presented and discussed in reference [4]. Some additional tests will be presented in this paper. Four additional tests were performed on ordinary concrete specimens with 50 MPa compressive strength and 8 mm maximum aggregate size. The specimen geometry is demonstrated in figure 3. The first and the second tests were conducted on three-month-old specimens. The first test was conducted on a watersaturated specimen. The second test was conducted on a specimen which was water-saturated during the first month and conditioned for two months in a laboratory climate. The displacement path for both tests was parabola with β=0.5 mm1/2 (see figure 4a) . The results are presented in figures 4b and 4c. The figures also show the mean value of three similar tests conducted on 28-day-old specimens. The third and the fourth tests were conducted on specimens which were treated in the same way as above, but the displacement path was different. The displacement path, a mixed-path, is shown in figure 5a, in which four parts may be distinguished. Part (a) is a straight line with α=60°, part (b) is a parabola with β=0.4 mm1/2 , part (c) is a straight line with slope 76° and part (d) is a parabola with β=0.6 mm . The test results are demonstrated in figures 5b and 5c. Figure 6 shows the results of two parabola paths (β=0.4 and β=0.6) and a straight line path (α=60°) conducted on 28-day-old specimens (Hassanzadeh 1989). The results of the parabola paths are the mean values of three tests. 5 Discussion Due to the low number of samples, the statistic significance of the results presented in the previous section is dubious. Nevertheless the results are useful, to a certain extent, in studying the degree of influence of some parameters. Figures 4 and 5 confirm that the moisture content has a striking influence on the tensile strength but almost no influence on the mixed-mode behaviour of concrete. In figure 4, the influence of the age of the concrete can be observed. It is low due to the fact that most of the hardening occurs during the first month. In figures 5 and 6, the influence of the history of the displacement is studied. It is interesting to compare the shear stress—shear displacement curves at: points where the first straight line (α=60°) intersects the parabolas. The first intersection is with parabola β=0.4 mm1/2 (point ws=0.05 mm wn=0.09 mm) where the shear stresses pertaining to the straight line displacement path are slightly higher than the corresponding stress pertaining to the parabola path. The second intersection is with parabola β=0.6 mm1/2 (point ws=0.12 mm wn=0.21 mm) where the shear stresses pertaining to the straight line displacement path are much higher than the corresponding stress pertaining to the parabola path. Further, the linear path stresses descend while the parabola path stress rises. The same tendencies can be observed with the normal stress—normal displacement curves. It can be concluded that there are two, as in this case, or more stress states for one total displacement point. As was mentioned above, the displacement paths have been chosen arbitrarily and it is not known to what extent they cover the real displacement paths occurring in practice. The displacement paths which occur in practice depend on geometry, loading and boundary conditions. Therefore, it is necessary that this type of

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Figure 3. Specimen geometry and measuring points, in mm, Normal displacements are measured at points 1 1 and 2. Shear displacements are measured at points 3 and 4.

mixed-mode tests be combined with numerical applications in order to check the reasonableness of the assumed displacement path.

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Figure 4a. Displacement path, parabola β=0.5 mm1/2.

6 Conclusion Some degree of path dependency has been observed, which may cause difficulties in the derivation of a simple, general expression which describes the mixed-mode behaviour of a fracture process zone. Since the tests and the conclusions drown from the test results have been based on arbitrary deformation paths, further research is needed to verify the phenomena which have been observed. Such research should combine both laboratory and theoretical experiments. REFERENCES 1 2 3 4 5 6 7

. Gustafsson, P.J. (1985), “Fracture mechanics studies of non-yielding materials like concrete”, report TVBM-1007, thesis, Div. of Building Materials, Univ. of Lund, Sweden. . Hassanzadeh, M., Hillerborg, A., Zhou, F.P. (1987), “Tests of material properties in mixed mode I and II”. proc of the International Conference on Fracture of Concrete and Rock, pp 353–358. Houston, U.S.A. . Hassanzadeh, M., Hillerborg, A. (1989), “Concrete prope rties in mixed mode fracture”, Fracture toughness and fracture energy, Ed Mihashi, H., Takahashi, H., Wittmann, F.H., A.A. Balkema Publishers. . Hassanzadeh, M. (1989), “Determination of fracture zone properties in mode I and II”. proc of the 1989 SEM Spring Conference on Experimental Mechanics, pp 521–527. Cambridge, MA, U.S.A. . Hillerborg, A. (1989-a), “Discrete crack approach”, chapter 5 in Fracture mechanics of concrete structures, RILEM report, Ed L. Elfgren, Chapman and Hall. . Hillerborg, A. (1989-b), “Mixed mode fracture in concrete”, Seventh International Conference on Fracture, Houston, Texas, March 20–24. . Zhou, F.P. (1988), “Some aspects of tensile fracture behaviour and structural response of cementitious materials”, report TVBM-1008, thesis, Div. of Building Materials, Univ. of Lund, Sweden.

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Figure 4b. Normal stress—normal displacement curves pertaining to the parabola path β=0.5 mm1/2.

CONSIDERATIONS REGARDING FRACTURE ZONE

Figure 4c. Shear stress—shear displacement curves pertaining to the parabola path β=0.5 mm1/2

43

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Figure 5a. Different displacement paths.

CONSIDERATIONS REGARDING FRACTURE ZONE

Figure 5b. Normal stress—normal displacement curves pertaining to the mixed-path.

45

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Figure 5c. Shear stress—shear displacement curves pertaining to the mixed-path.

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Figure 6a. Normal stress—normal displacement curves pertaining to the parabola path β=0.4 mm1/2 β=0.6 mm1/2 and α=60°.

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Figure 6b. Shear stress—shear displacement curves pertaining to the parabola path β=0.4 mm β=0.6 mm1/2 and α=60°.

5 MIXED MODE FRACTURE IN COMPRESSION A.K.MAJI Department of Civil Engineering, University of New Mexico, Albuquerque, USA S.P.SHAH Center for Advanced Cement-Based Materials, Northwestern University, USA

SUMMARY Mechanical response of concrete under compressive loading is largely dependent on the development of microfracture in the material. It is generally accepted that interface debonding and matrix cracking is responsible for nonlinear behavior and eventual failure of concrete. A study of these phenomena is therefore critical in understanding the material and for developing adequate constitutive models. In order to study the influence of compressive loading, a phenomenological study was first undertaken using model concrete specimen with prefabricated microstructure. Holographic Interferometry was used as the technique for observing formation and growth of cracking in real-time under different stages of loading. Loading was applied by a closed-loop testing machine under axial displacement control. A fracture mechanics based study was then conducted to model the initiation of aggregate-mortar interface cracks and their subsequent propagation into the mortar. Experiments were conducted on model concrete specimens with limestone inclusions placed at different orientations to the direction of applied load to study the effect of loading geometry. Finite element method was used to model the actual test specimen and crack propagation geometry to evaluate the experimental results. Development of a fracture mechanics criterion to predict interface debonding and matrix crack propagation has been discussed and recommendations for future research directions have been made. INTRODUCTION Crack initiation and propagation are dominant mechanisms responsible for nonlinear response of concrete subjected to uniaxial compressive loading. Cracks may initiate in matrix, aggregates or at matrix-aggregate interface. Studies conducted using microscopic analysis (Hsu, et. al. [1], Shah and Chandra [2] and Shah and Sankar [3]) have revealed that cracks frequently initiate at the interface and then propagate into matrix where mortar cracks join to form a continuous crack path prior to ultimate load. To analyze this phenomenon of interfacial cracking (bond cracking) and subsequent matrix cracking (mortar cracking), researchers have examined bond strength of an isolated interface between stone and portland cement mortar matrix subjected to tension-shear (Kao and Slate [4]), as well as compression-shear (Taylor and Broms [5], and Shah and Slate [6]). Criteria based on Coulomb-Mohr theory have been suggested to predict bond crack

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initiation. However, adequate attention has not been given to determine how an interfacial crack will propagate into the matrix. The aim of the work presented here was to obtain an understanding of crack initiation and propagation from the aggregate-matrix interface under uniaxial compression using fracture mechanics approach, finite element modeling and holographic interferometry techniques. Laser holographic interferometry (HI) was applied to study the whole field deformation pattern during loading. This technique uses the interference of coherent laser light to form a three-dimensional image of the specimen. Interference of the holographic images at two different loading stages create fringes. These fringes correspond to the displacement of the object between the two stages of loading (Ranson, Sutton and Peters, [7]). This technique is called ‘Double Exposure HI’. Small cracks (about 0.2 microns wide) could be detected as discontinuities in the fringe pattern. The loads at which the interface and matrix cracks initiate and grow can be noted. It is necessary to conduct the experiment in a vibration-free environment for interference of the two beams to take place. The loading device therefore has to be mounted on a vibration isolation table or secured firmly to a rigid support [8]. EXPERIMENTAL PROCEDURE Model Concrete with Cylindrical Aggregates Model concrete specimens with cylindrical limestone aggregates were tested first. Specimens were 3. 75”×6”×1” (9.4×15.2×2.5 cm) blocks made with a mortar having mix proportions by weight of 1:2:.5 (cement, sand and water respectively). Inclusions were placed as shown in Figure 1a. Specimens were tested in uniaxial compression in a 120 kip (534 KN) capacity closed loop testing machine. The signal from the two LVDTs (Linear Variable Displacement Transducer) which monitored axial displacement was averaged and used as the feedback control. It is necessary to have a vibration isolated environment in order to make holograms. This was obtained by building an extension to the existing loading platten. Details of the test have been reported by Maji and Shah [8]. Fringe patterns from holographic interferometry are shown in Figures 1c and 1d corresponding to stages shown in the load-deformation diagrams. The two exposures to make each double exposure hologram were made at loads slightly less and greater than the points shown. The holographic fringes observed at two different stages of loading have been presented. Corresponding crack patterns are shown below the photographs. Bond cracking initiates at loads at about 30% of peak load (Figure 1c). Nonlinearity in the load deformation diagram also starts beyond that load. As the loading increases the bond cracks increase. Vertical matrix cracks originate predominantly from these bond cracks and propagate vertically with increased load (Figure 1d). Cracks also originate in the matrix under increased loads. Cracks continue to propagate vertically in the matrix until close to peak load. Similar experiments were conducted on specimens with smaller size (1/2”, 1.25 cm) diameter aggregates and on specimens having cylindrical voids in place of aggregates [8]. Smaller aggregate specimens showed lesser interfacial cracking and less nonlinear behavior prior to the peak load. Apparently the fracture patterns and nonlinearity are a function of the size of aggregates or voids. Fracture Mechanics Based Study In order to determine the criterion for crack initiation and propagation, a simpler specimen geometry was adopted. Thin rectangular slabs were cast in mortar matrix at different orientations to study the effect of interface inclination to the fracture phenomena. Specimens shown in Figure 2 were made of mortar with a 1:

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Fig. 1a. Model Concrete Specimen

Fig. 1b. Load-Displacement, Diagram

3.0:0.6 mix-ratio (cement:sand:water by weight) using Type I cement. A 0.3” (0.8 cm) wide and 2” (5.1 cm) long piece of limestone was placed vertically through the thickness of the specimen during casting (Figure 2). The specimens were cast in a plexiglass mold. The orientation of the limestone inclusion was varied to study the effect of different loading situations to be discussed later. Four different stone orientation angles (β=18°, 36°, 54° and 72°) with respect to the loading direction were used. Specimens were 6” (15.3 cm) wide, 8” (30.3 cm) high, and 3” (7.6 cm) thick. Two separate end blocks (6”×1.5”×3”) (15.3×3.8× 7.6 cm) were cast simultaneously by using plexiglass separators to reduce end restraint during testing (Shah and Sankar [3]). Three specimens were prepared for each inclination. The tests were carried out on a MTS 120 Kip (534 kN) capacity closed loop compression testing machine under axial displacement control similar to the tests described in the previous section. The faces between the specimen and the end blocks were greased to reduce friction and ensure smooth seating. Two specially

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Fig. 1c, 1d. Crack Initiation and Propagation (Holographic Fringes)

designed clip gages were mounted on the.specimen by super glue and epoxy. The two gages were placed on the top and bottom of the aggregate such that they measured the sliding and opening displacements at the mouth of the propagating kink (Figure 2). The gage output was passed through a bridge circuit and amplified by an Accudata 218 Bridge Amplifier. The load and displacement gage data were stored in digital format on a Nicolet digital oscilloscope and later read into an IBM-AT computer. FINITE ELEMENT ANALYSIS There are theoretical solutions available for determination of the Stress Intensity Factors (SIF) at the tip of a kink emanating from a preexisting inclined crack (Horii and Nemat Nasser [9], Steif [10]). However, these solutions make assumptions of a straight kink and an infinite geometry. To account for curved kinks, and for finite dimensions of the specimens, finite element analysis was performed. The software used, FRacture ANalysis Code (FRANC), is an interactive workstation based program. A two-dimensional elastic analysis was performed using six node triangular and eight node quadrilateral elements. The program features automated remeshing after propagation of a crack and used eight quarterpoint triangular singularity elements placed around the crack tip to simulate the singular stress field (Wawrzynek [11]) Results of FEM

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Fig. 2. Fracture Specimen With Different Stone Orientation

analysis were validated against available theoretical solutions and have been presented elsewhere (Tasdemir, Maji and Shah [12]). A program was implemented to generate the input mesh for different values of β. The automated remeshing feature of the program was used to generate various cracks as was observed experimentally. The stress intensity factors at the tip of the kink, and the opening and sliding displacements at the mouth of the kink were obtained for the various experimentally observed crack paths at the corresponding experimentally observed loads. The initial mesh is shown in Figure 3a. The nodes on the top and bottom faces of the crack were fixed in the vertical Y direction. This was necessary to simulate absence of closure of the rectangular inclusion under applied load. In the real life situation of the experiment it was assumed that the limestone inclusion was rigid enough for the thickness of the rectangle to remain unchanged. One middle node at the upper side of the crack was fixed in the X direction to prevent rigid body translation. Hence, the crack faces were allowed to slide but not allowed to close and open. The Finite Element (FEM) Analysis was done on all the four types of specimens (4 different inclusion angles) using the experimentally observed crack path. The cracks were gradually propagated in the FEM mesh for each type of specimen along the experimentally observed direction. The loads were applied at the specimen ends to simulate a uniaxial compression situation. A second set of runs were made where the top and bottom crack faces were loaded with shear traction to simulate the aggregate-matrix friction. The applied shear stress used was of the magnitude “µσ sin 2β” where µ=0.25, determined experimentally and σ was the applied uniaxial compressive stress. These stresses were lumped at the interfacial nodes. Superposition of these two runs yielded the solutions

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 3a. Initial FEM Mesh

discussed later. A typical mesh with kink extension of about 0.50” (1.27 cm) and P=36° showing the singular elements at the kink tip is shown in Figure 3b. EXPERIMENTAL RESULTS Photographs of HI fringe patterns for a specimen with P=36° are shown in Figure 4 along with sketches of corresponding crack patterns. From such fringe patterns it was observed that the debonding at the stonematrix interface always preceded the initiation of the matrix crack (kink) (compare Figure 4a and Figure 4b). The far-field compressive stresses at which debonding occurred ( σb) and at which the initial matrix cracking ( σi) was observed for different values of β are recorded in Table 1. The values reported are the average of three sets of observations. A plot of resolved shear stress (σs) component and the corresponding normal stress component (σn) at debonding is shown in Figure 5. Table I. Experimentally Observed Values of Stresses Stresses (σ) , psi. (1 psi=6.89 k Pa) β

Debonding

First Crack

At 1=0.5”

At 1=1.0”

Compressive

(σb)

(1=0, σi)

(1 .27 cm)

(2.54 cm)

Strength

573 550

1057 983

1478 1200

4657 5380

*1=crack propagation length 18° 36°

293 327

MIXED MODE FRACTURE IN COMPRESSION

55

Stresses (σ) , psi. (1 psi=6.89 k Pa) β

Debonding

First Crack

At 1=0.5”

At 1=1.0”

Compressive

(σb)

(1=0, σi)

(1 .27 cm)

(2.54 cm)

Strength

851 2986

1187 3560

1524 –

– –

*1=crack propagation length 54° 72°

366 1450

Fig. 3b. FEM Mesh With Singular Elements.

Assuming that the relationship between these two components is a straight line, the coefficient of friction was calculated at 0.25. This is the value µ used henceforth. Crack Propagation Criterion The critical values of stress intensity factors for matrix crack propagation were calculated using the finite element analysis (Figure 3). For these calculations, an experimentally observed crack path was used. The actual crack path was simulated as a series of straight line 0.25” (0.63 cm) long. For each kink extension the values of KI and KII were calculated based on the experimentally observed load using the value of µ=.25 for the stone-matrix interface and the value of E=3.9×10 psi, ν=0.21 (Poisson’s ratio) for the matrix. The calculated value of the stress intensity factors at which the cracks propagated (Figure 6) show that the value of KQ are not constant for different crack extensions and they vary significantly for different values of β. This indicates that the crack propagation criteria KI=KIc does not appear to be valid for crack propagation in matrix.

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 4. Mixed Mode Crack Propagation Under Compression.

A comparison of crack mouth opening and sliding displacement as calculated by numerical analysis above with those measured using clip pages (Figure 2) for β=36° is shown in Figure 7. It can be seen that there is large discrepancy between the measured and the calculated values. Such large discrepancy was also observed for other values of β The matrix crack in the FEM analysis above was so far considered to be traction-free. One way of explaining the differences between calculated values and the measured ones is the

MIXED MODE FRACTURE IN COMPRESSION

57

Fig. 5. Normal and Shear Stresses at Interface.

Fig. 6. Stress Intensity Factors at Crack Propagation Stages.

possible influence of normal and shear tractions along the cracks in matrix. This observation motivated additional analysis. Studies of Mode I tensile crack growth in concrete have indicated that traction exists behind the crack tips partly as a result of unbroken ligament, so-called aggregate interlock, grain-boundary sliding, microcracking and tortuosity (Maji and Shah [14], Gopalaratnam and Shah [15], Miller. Shah and Bjelkhagen [16]). Since this zone of disturbance (also termed fracture process zone) can be quite large, it must be included in a fracture mechanics analysis. Cohesive crack type models have been suggested (Hillerborg et al. [17], Bazant and Oh [18], Jenq and Shah [19]). In such models, crack closing pressure

58

ANALYSIS OF CONCRETE STRUCTURES

Fig. 7. Clip-Gage vs. FEM Measurements.

whose value depends inversely on crack opening displacement is incorporated in addition to far-field stresses in predicting crack propagation. Effect of Crack-Face Tractions In the present study, both opening and sliding displacement occur and as a result both normal and shear tractions should be included in the analysis. The relation between shear stress and slip depends on crack opening. Sliding of the crack faces is also accompanied by dilatancy (Bazant and Tsubaki [20], Van Mier [21]). Ingraffea and Gerstle [22] have used in their finite element analysis of mixed-mode crack propagation a shear stress versus slip relation suggested by Fenwick and Paulay [23]. Several researchers are studying the relationship between shear stresses, normal stresses and the sliding and opening displacement (Van Mier [21], Divakar et al. [24]). To incorporate shear and normal tractions in the present study, uniformly distributed traction forces were applied along the kink length (Figure 8). Their magnitudes were determined such that the calculated opening and sliding displacements matched the measured value. To be able to find the amount of tractions the following equations were employed: (1) (2) where in L=[L1 L2 L3 L4], L1 and L2 are the known externally applied load and resulting aggregate-mortar friction loads, respectively (see section on Finite Element Analysis). Matrices L3 and L4 represent the unknown uniformly distributed normal and shear traction loads. Matrices [O] and [S] are flexibility matrices obtained from FEM analysis. These matrices [O] and [S] contain the CMOD (Crack Mouth Opening Displacement) and CMSD (Crack Mouth Sliding Displacement) caused by unit loads of L1, L2, L3, L4 respectively applied separately to the FEM mesh for different crack lengths. The right hand side of the equation represent the measured values of crack mouth opening and sliding displacement. The values of L3 and L4 are obtained from the solution of the above equations. These four different loading configurations

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59

Fig 8. Four Different Loads Considered in the FEM Analysis.

are shown in Figure 8. The values of assumed uniformly distributed normal traction forces were calculated for each β and are tabulated in Table 2. The resultant stress intensity factors which include normal and shear traction are shown in Figure 9b. By comparing Figure 9b with Figure 6 it can be seen that the values of (KQ) now lie within a reasonable band. This implies if traction forces are included then one can use KI=KIC criteria for mixed-mode crack propagation. A similar conclusion was reached for mode I crack propagation by Jenq and Shah [19]. It should be noted that for simplicity, uniformly distributed traction forces were observed. The resulting value of KI is always numerically larger than KII and the ratio tends to increase with increasing kink extension (Figure 9a). These results support the mixed-mode criteria proposed by Steif [10] and Horii and Nemat-Nasser [9]. It is expected that the uniform tractions on the crack face assumed in this analysis is inadequate. More realistic models of interface traction is required to improve accuracy of such an analysis. One possible way of improving the model is to study the crack profiles all along the propagating crack rather than at the crack mouth only as reported in this research. A technique based on holographic interferometry to perform such measurements have been reported by Maji and Shah [25]. Table II. Calculated Magnitudes of Normal Traction Crack

Normal Stresses Applied on the Propagating Crack, psi.

Extension (1)

β=18°

β=36°

0.25” 598 –108 0.5” 455 –101 526 –120 0.75” + Pulling out normal traction – Pushing in normal traction (1 psi=6.89 kPa, 1”=2.54 cm)

β=54°

β=72°

–507 –336 –325

–245 –150 –

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 9a. Comparison of the Two Stress Intensity Factors at Crack Propagation.

Fig. 9b. Stress Intensity Factors Incorporating Traction Effects.

CONCLUSIONS Initiation and propagation of bond and matrix cracks were observed in real-time at different loading stages. The crack patterns were correlated with observed mechanical response of concrete under uniaxial compression. A model was developed to study the formation of interface debonding and subsequent matrix cracking based on a fracture mechanics approach. A finite element based analysis was necessary to study the fracture parameters of a propagating crack. The effect of interface friction had to be explicitly modeled in this analysis based on the simplified assumption of the Coulomb type friction law.

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61

It was found that a fracture mechanics based model shows promise only if the effect of crack-face tractions can be incorporated in the analysis. It is felt that the results obtained by assuming uniform traction along the crack faces can be improved by incorporating more realistic models of traction. Acknowledgement This research was supported by a grant from the Air Force Office of Scientific Research; (AFOSR-88C-0188) under a program managed by Dr. Spencer Wu. The authors also appreciate the support from the National Science Foundation Grant DMR 880–8432 (Program Manager: Dr. Lance Haworth). References 1 2 3 4 5 6

7 8 9 10 11 12 13 14 15 16 17

) Hsu T.T.C, Slate F.O. Sturman G.M. and Winter G., “Microcracking of Plain Concrete and the Shape of the Stress-Strain Curve”, ACI J., Proceedings V 60, No. 2, 1963, pp. 209–224. ) Shah S.P. and Chandra S. “Fracture of Concrete Subjected to Cyclic and Sustained Loading”, ACI J., Proceedings V 67, No. 10, 1970, pp. 816–825. ) Shah S.P. and Sankar R. “Internal Cracking and Strain Softening Response of Concrete Under Uniaxial Compression”, ACI Mat. J., V84, No.3, 1987, pp. 200. ) Kao C.C. and Slate F.O. “Tensile-Shear Bond Strength and Failure Between Aggregate and Mortar”, J. of Testing and Evaluation, JTEVA, V 4, No. 2, 1976. ) Taylor M.A. and Broms B.B. “Shear Bond Strength Between Coarse Aggregate and Cement Paste or Mortar”, ACI J., Proc. V61, No. 8, 1964, pp. 937–957. ) Shah S.P. and Slate F.O. “Internal Microcracking, Mortar-Aggregate Bond, and the Stress-Strain Curve of Concrete”, Proceedings of Int. Conf. on the Structure of Concrete, Editor : A.E. Brooks and K. Newman, London, 1965, pp. 82–91. ) Ranson W.F., Sutton M.A. and Peters W.H. “Holographic and Laser Speckle Interferometry” in SEM Handbook on Experimental Mechanics, Editor: A.S. Kobayashi, Prentice-Hall, 1987, pp. 388–429. ) Maji A.K. and Shah S.P. “Application of Acoustic Emission and Laser Holography to Study Microfracture of Concrete”, ACI-SP 112, Nondestructive Testing, 1989, pp. 83–109. ) Horii H. and Nemat Nasser S. “Brittle Failure in Compression: Splitting, Faulting and Brittle-Ductile Transition”, Phil. Transactions Royal Soc. London, A319, 1986, pp. 337–374. ) Steif P.S. “Crack Extension Under Compressive Loading”, Engineering Fracture Mechanics, V 20, No. 3, 1984, pp. 463–473. ) Wawrzynek P. and Ingraffea A.R. “Interactive Finite Element Analysis of Fracture Process : An Integrated Approach”, Theoretical and Applied Fracture Mechanics, V 8, 1987, pp. 137–150. ) Tasdemir M.A., Maji A.K. and Shah S.P. “Crack Initiation and Compression in Concrete Under Compression”, Accepted for Publication ASCE J. of EMD, 1989. ) Melville P.H. “Fracture Mechanics of Brittle Materials in Compression”, Int. J. of Fracture, 13, 1977, pp. 532–534. ) Maji A.K. and Shah S.P. “Process Zone and Acoustic Emission Measurements in Concrete”, Experimental Mechanics, March 1988, pp. 27–31. ) Gopalaratnam V.S. and Shah S.P. “Softening Response of Plain Concrete in Direct Tension”, ACI L, May-June 1985, pp. 310–323. ) Miller R.A., Shah S.P. and Bjelkhagen H.I. “Crack Profiles in Mortar Measured by Holographic Interferometry”, Experimental Mechanics, December 1988. ) Hillerborg A., Modeer M, and Petersson P.E. “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements”, Cement and Concrete Research, V5, No. 6, 1976, pp. 773–784.

62

18 19 20 21 22

23 24 25

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) Bazant Z.P. and Oh B.H. “Crack Band Theory for Fracture of Concrete”, RILEM, Materials and Structures, V 16, No. 93, 1983, pp. 155–177. ) Jenq Y.S. and Shah S.P. “Two Parameter Fracture Model for Concrete”, ASCE J. of EMD, V 111, No. 4, 1985, pp. 1227–1241. ) Bazant Z.P. and Tsubaki T. “Slip-Dilatancy Model for Cracked Reinforced Concrete”, ASCE J. of Structures Division, September 1980, pp. 1947–1966. ) Van Mier J.G.M. “Fracture Study of Concrete Specimens Subjected to Combined Tensile and Shear Loading”, presented at Int. Conf. on Measurement and Testing in Civil Engineering, Lyon-Villerurbanne, Sept. 1988. ) Ingraffea A.R. and Gerstle W.H. “Nonlinear Fracture Models for Discrete Crack Propagation”, in Application of Fracture Mechanics to Cementitious Composites, Edited by S.P. Shah, Proc. of NATO Advanced Research Workshop, Martinus Nijhoff Publishers, 1985, pp. 247–285. ) Fenwick R.C. and Paulay T. “Mechanics of Shear Resistance of Concrete Beams”, ASCE J. of Structures Div., V 94, No. ST10, 1968, pp. 2325–2350. ) Divakar M.P., Fafitis A. and Shah S.P. “A Constitutive Model for Shear Transfer in Cracked Concrete”, ASCE J. of Struc. Div., V 113, No. 5, May 1987. ) Maji A.K. and Shah S.P. and Tasdemir M.A “A Study of Mixed-Mode Crack Propagation in Mortar Using Holographic Interferometry”, Proceedings of SEM Spring Conference, Boston, MA, May 1989, pp. 210–217.

6 THERMAL STRESSES IN CONCRETE AT EARLY AGES M.EMBORG Luleå University of Technology, Luleå, Sweden

1 General In freshly cast concrete structures the temperature rise caused by the hydration process is often considerable and may extend over a long period of time. If unrestrained, the concrete would expand and contract during the heating and the subsequent cooling process without stresses being induced. In practice, however, the concrete is nearly always restrained to some degree either by adjoining structures or by different parts within the concrete itself. Thus, due to these imposed restraint conditions, the temperature change will induce compressive and/or tensile stresses in the element. A question of primary interest is of course whether the induced tensile stresses will lead to thermal cracking or not, Emborg (1989). Cracks may be of two kinds: (1) Surface cracks caused by the restrained volume changes within the element. (2) Through cracks over the entire cross section caused by volume changes restrained by external factors such as adjoining structures of previous casts or by the foundation. In order to avoid cracking caused by thermal stresses, different measures are often taken when casting large concrete structures. For example, the temperature rise due to hydration can be lowered by • • • •

using low-heat cements, reducing the cement content, lowering the placing temperature, and/or by using cooling techniques with embedded pipes.

Also, the restraint of an element in a structure can be reduced by • shortening the length of the section being cast, and/or by • arrangement of construction joints

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ANALYSIS OF CONCRETE STRUCTURES

Measures and restrictions in codes against thermal cracking are mainly based on the magnitude of the temperature rise either within the cast section or between sections cast in different stages. Temperature variations provide, however, in many cases only a crude in formation of the risk of thermal cracking. Sometimes, the temperature field does not even reveal whether a part of the structure is in tension or in compression. 2 Thermal stress analysis The most important parameters when analysing early-age thermal stresses in concrete are the temperature development in the element being cast, the mechanical behaviour of the young concrete and the restraint conditions of the element, see Figure 1. The temperature development in the element is dependent on (a) the environment conditions (air temperature, temperature of foundation etc), (b) the conditions at the concreting (placing temperature, form, insulation, cooling etc), and (c) the thermal behavior of the young concrete (heat of hydration, specific heat etc), see the figure. The following properties regarding the mechanical behaviour of the young concrete are of great importance in the thermal stress analysis: (a) elastic and time-dependent behaviour at normal and high stress levels (b) strength development (c) thermal expansion and thermal contraction and (d) fracture mechanics behaviour. In order to increase our knowledge regarding elastic and time-dependent behaviour of the young concrete creep tests and relaxation tests have been perfomed at Luleå University of Technology (Emborg 1989). Some results from these studies will now be given. Figure 2 shows examples of results from standard creep tests with two concrete mixtures for very early ages at application of loads. A marked age-dependence of the creep responses are indicated in the figures specially for ages at application of loads less than about 0.7 days. The relaxation tests provide a phenomenological study of the thermal stresses and thermal cracking. In the relaxation tests a small concrete specimen is placed in a testing machine directly after casting and is heated by surrounding water in a water tank to give a hydration temperature curve representative for a specific concrete structure. The ends of the specimen are fixed and thus simulating 100% restraint. The compressive and tensile forces that are induced in the specimen are recorded with a load cell. Figure 3a shows examples of temperature curves obtained from measurements in wall sections and Figure 3b shows the corresponding thermal stresses obtained in relaxation tests with early compressive stresses during the heating process and subsequent tensile stresses and tensile failure during the cooling process. The thermal expansion and contraction are measured on specimens at the same test-setup as at the relaxation tests except that the concrete is not constrained by the testing machine. Examples of results from the free thermal expansion and contraction tests are shown in Figure 4. It can be seen that the coefficients of thermal expansion and contraction vary significantly during an early-age heating and cooling cycle. The restraint conditions of the element is affected by the type of the element being cast (wall, slab, tunnel etc), the conditions of the concreting at the site (length of casts etc). The restraint may be of two kinds: (a) The internal restraint arise from the varying temperatures within the highly statically indeterminate structure itself. The resulting stresses from this type of restraint are normally significant in large massive sections, but under certain conditions also in thin sections. An example of internal restraint is the case when high temperatures occur in the section leading to the above mentioned surface cracks.

THERMAL STRESSES IN CONCRETE AT EARLY AGES

65

Fig. 1 Analysis of early-age thermal stresses and risks of thermal cracking—influencing factors.

(b) The external restraint results from the limitations of freedom of movement imposed by adjoining structures. The external restraint can be of different kinds such as continuous edge restraint, and restraint and combinations thereof. The most simple form of external restraint is the uniaxial case with inflexible supports as is the case in the relaxation tests. 3 Model The model used when analysing thermal stresses consists of a fracturing element, a viscoelastic element and a thermal movement element in series, see Figure 5. The viscoelastic element is the wellknown Maxwell Chain Model. The model is based on a standard creep formula which has been extended for the behaviour of the young concrete. The creep formula is the Triple Power Law proposed by Bazant and Chem (1985b). The fracturing model has also been proposed by Bazant and Chem (1985a). The model includes the secant modulus and is based on exponential functions which leads to stable solutions even for long time steps. The relation between the stress σ and fracturing strain ξ of the fracturing element is (1) where c(ξ) is the secant modulus of the stress-strain relation, see Figure 6. In the actual analysis only a part of the strain softening diagram is used, i.e. a tensile failure (thermal crack) is assumed to occur when the peak stress is reached.

66

ANALYSIS OF CONCRETE STRUCTURES

Fig. 2 Results from compression creep tests on young concrete with different ages at application of load. The applied load was 20% of the strength, T=19-20° C. Concrete: Swedish Standard Portland Cement (Slite), c=279 kg/m3 (S279), Wo/c=0.61, cement/sand/gravel=1./3.31/2.79 (by weight) =46 MPa, c=400 kg/m3 (S400), Wo/c=0.41, cement/sand/ gravel= 1./1.94/2.31, =66 MPa.

The thermal movement element relates the expansion and the contraction of the element to the temperature in a linear way by coeficients of thermal expansion and contraction, see Figure 4.

THERMAL STRESSES IN CONCRETE AT EARLY AGES

67

Fig. 3 a Temperature-time graphs obtained from mean temperature recordings in 0.7 m thick wall sections. b Stress-time graphs at relaxation tests at 100% restraint. Concrete: Swedish Standard Portland Cement, S—Slite, A— Degerhamn, c=400 kg/m3, see fig 2, c=331 kg/m3 (S331 and A331) Wo/c=0.55, cement/sand/gravel=1./2.61/2.93, =53.8 MPa (S), 47.0 MPa (A).

The total incremental relation including the elastic deformation and creep, fracturing behaviour and thermal deformation may be written as

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 4 Results from free thermal expansion and contraction tests on newly cast concrete specimen. The specimens are heated directly after pouring by water with different temperature curves. Concrete S279, see Fig 2, c=300 (S300), Wo/ c=0.54, cement/sand/gravel=1./2.93/3.28, =46.2 MPa.

(2) where

∆ε ∆εec ∆ξ ∆ε°

is the total strain is the elastic strain and creep is the inelastic fracturing strain is the thermal strain

The viscoelastic response according to the Maxwell Chain Model and thermal strain may be expressed as (3) where

∆σ ∆εec ∆ε” E”

is the stress increment is the strain increment of the Maxwell Chain elemen is the inelastic strain including relaxation strain and thermal strain (∆ε°) is the relaxation modulus of the element

The strain-softening behaviour of the fracturing element is in the used model expressed as (4)

THERMAL STRESSES IN CONCRETE AT EARLY AGES

69

Fig. 5 Rheologic model describing deformation due to strain softening, elastic deformation, creep strain, shrinkage and thermal deformation (after Bazant and Chem (1985a)).

Fig. 6 Stress-strain relation with strain softening curve (after Bazant and Chem (1985a)).

where

∆σ ∆ξ ∆ξ” D

is the stress increment is the strain increment of the fracturing element is the inelastic strain increment is the strain-softening modulus

The total strain increment of the model may thus be expressed as (5) which may be converted into stiffnes form

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ANALYSIS OF CONCRETE STRUCTURES

(6) This is the constitutive equation for the thermal stress increment in a step-by-step analysis. In the analysis of thermal stresses very low rates of thermal loading are present which give rise to tertiary creep and creep failure. This influence of the rate of loading on the behaviour may be considered by adjusting the peak stress (e.g. the tensile strength). This adjustment has been done according to a study by Wittman et al (1987) in which the influence of rate of loading on the strain softening diagram was studied on three-point bending mode tests, see Figure 7. According to Wittman et al the dependence of the tensile strength on the rate of loading may be expressed as (7) where is the rate of loading d1 and d2 are coefficients dependent on the water-cement ratio In this study, the equation above is used for assessing the tensile strength valid for an arbitrary rate of loading on the basis of the strength at a reference strain rate =0.075/min). In this way, the peak stress of the stress-strain relation (Fig. 5) is reduced for the low strain rates that are present in a thermal loading. Normally this implies a reduction of about 0.7–0.8 of the tensile strength obtained at the reference rate. 4 Theoretical computations Comparisons of theoretically computed thermal stresses according to the equation above with the relaxation tests show in many cases rather good agreement, see Figure 8. The model has also been implemented in a computer program where the structure to be analysed is subdivided Into discrete layers. Each of the layers are attributed to different evolutions of temperature and, hence, early-age development of material properties of its own. The program has been applied to some typical situations of concreting such as a concrete-section pured against inflexible supports, differential temperatures across a wall section, a wall section cast on a slab, a tunnel element and casting of a concrete cover on an existing bridge pier. Figure 9 shows one example of the case of differential temperature across a wall section with high temperatures in the centre and low temperatures at surface. In the figure, a 2 m thick wall section is studied and we get about 70° C in the centre and about 40° C at the surface as maximum temperatures. The computed thermal stresses in the program leads to tensile stress at the surface with a maximum value of 1.4 MPa. This implies a stress level σ(t)/fct(t) of about 0.8. For the case of a wall cast on a slab, the structure has been divided into 12 layers, see Figure 10. The temperature distributions when casting the wall are shown in Figure 11a. The magnitude and distribution of computed thermal stresses vary significantly with the degree of freedom of movement. In the case of no bending of the wall and the slab, compressive stresses develop in the entire wall at the temperature rise, see Figure 11b. Later on, after some cooling, tensile stresses occur throughout the wall height. The maximum computed stress level is In this case 0.48.

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Fig. 7 Tensile strength ft as determined directly from load-deflection diagrams by means of fits with FEM-calculations as a function of rate of loading. Three-point bending mode test, a) w/c=0.40, b) w/c=0.65. Full circles represent data of large beams (span=1150 mm) and hollow circle those of small beams (span=800 mm). From Wittman et al. (1987).

If the wall and slab are allowed to rotate it is interesting to note that both tensile and compressive stresses develop during the temperature rise, see Figure 11c. Again on cooling, high tensile stresses occur in the lower parts of the wall with a maximum stress level of 0.37. In the case of casting a new concrete cover layer on an existing bridge pier, see Figure 12, the old concrete is heated by the hydration process in the fresh concrete. The volume change of the old and much stiffer concrete may then counteract the thermal movement of the young concrete leading to large tensile stresses in the layer despite that the temperature rise is moderate. Figure 13 shows recorded temperatures in the layer and pier for a noncooled section. The stress evaluations show that very high tensile stresses develop in the cover layer, see the figure. The maximum stress level from the computations indicate thermal cracking (σ(t)/ fct(t)=1.0). Stress evaluations using recorded temperatures for a cooled section with embedded cooling pipes yield significantly lower tensile stresses and lower stress levels, see Figure 13b.

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Fig. 8 a Temperature-time graphs for a 0.7 m thick wall obtained at mean temperature computations. Tair=20 C. Concrete: A331, see Fig. 3. b Stress-time graphs obtained from relaxation tests and by theoretical computations with a nonlinear rate-type law.

5 Discussion With the model of thermal stress analysis based on an elastic and a viscoelastic model in series with a fracturing model it seems possible to quantify the stresses and to estimate the risk of thermal cracking. The proposed

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Fig. 9 (a) differential temperatures across a concrete section. (Ts, Tc temperature at surface and centre respectively; σs, σc stresses at surface and centre respectively). Distributions of temperatures (b) and thermal stresses (c) at different times from pouring as computed with the HETT and TEMPSTRE-N program . d=2 m, TP=10°C, Tair=10° C, air Concrete: S400 (see Fig. 2).

nonlinear model works rather well for most of the studied relaxation tests. However, for some concrete mixtures there are some discrepancies and there is, thus, a need for further calibration tests. Also, there is a need for studies of the early-age fracture mechanics behaviour in direct tension tests or in three-point bending tests providing a calibration of the fracturing model separately. In the present study it is concluded that, for control of early-age thermal stresses and cracking in a structure, it is highly inadequate to consider only the early-age distribution of the temperature field within the structure. The magnitude of the thermal stresses depends not only on temperature variations but to a

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Fig. 10 Wall cast on a cold concrete slab. Conditions of thermal stress analysis.

critical extent also on the mechanical behaviour of the hardening concrete as well as on the nature of the restraint. 6 References Emborg M (1989): Thermal stresses in concrete structures at early ages Dissertation Luleå University of Technology, Division of Structural Engineering, 1989:73D, Luleå 1989, 280 pp. Bazant Z P, Chem J C (1985 a): Strain softening with creep and exponential algorithm, J Eng Mech Div (ASCE), Vol 111, No 3, 1985, pp 391–415. Bazant Z P, Chem J C (1985 b): Triple Power Law for concrete creep. J Eng Mat, Vol III, No 4, Jan 1985, pp 63–83. Wittman F H, Roelfstra P E, Mihashi H, Huang Y-Y, Zhang X-H, Nomura N (1987): Influence of age loading, watercement ratio and rate of loading on fracture energy of concrete, Materials and Structures (RILEM, Paris), Vol 20, No 116, 1987, pp 103–110.

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Fig. 11 Wall cast on a cold concrete slab. (a) Temperature distributions and (b)-(c) thermal stresses at different times from pouring as computed by the HETT and TEMPSTRE-N programs; (b) no bending (c) bending.

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Fig. 12 Casting of concrete cover on an existing bridge pier.

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Fig. 13 a Casting of a concrete cover layer onto an existing pier: Recorded temperature distributions and computed thermal stresses (non-cooled section).

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Fig. 13 b Casting of a concrete cover onto an existing concrete pier: Recorded temperature distributions and computed thermal stresses (cooled section).

7 GRAIN-MODEL FOR THE DETERMINATION OF THE STRESS-CRACK-WIDTH-RELATION H.DUDA Technische Hochschule Darmstadt, Institut für Massivbau, Darmstadt, Germany

The post failure behavior of concrete is governed by the stress-crack-width-relation (σ-w-relation). Up to now the shape of the σ-w-relation is determined by experiments only. A simple mechanic model is presented to find the σ-w-relation analyzing the aggregate grading curve of the concrete. This model was developed in connection with a research-work sponsored by DFG in framework of the research-program ‘Constitutive Laws in Civil Engineering’. 1 Introduction The strain softening of tensile loaded concrete can be described with the fictitious crack model which was developed by Hillerborg and coworkers [1,2] The fictitious crack model postulates a stress bearing capacity over micro cracks. The concrete tensile stress σc is a function of the crack-width w. This function is the stress-crack-width-relation (σ-w-relation). Stress transfer over a crack, this seams to be a very strange idea. Often the catch-word ‘aggregateinterlock’ is used to explain the stress transfer but it is not clear what really aggregate-interlock is and how it works in connection with tensile loaded concrete. 2 The Grain Model In figure 1 shows a typical fractured surface of a tensile loaded concrete specimen. In most cases the crack runs around the grains. This gives the first hint how the aggregate interlock works. In figure 2 the crack path through a heterogeneous material like concrete is shown in schematic way. The weakest points is the interfac between the cement past and the grains. Because of this the crack runs in a twisted way though the specimen. This is true for normal concrete. Light-weight and high-strength concrete may exhibit a different behavior as the strength of the cement past may reach the strength of the aggregate.

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Figure 1: Rupture surface of a concrete specimen

3 Crack development between cement past and grains 3.1 The disk model The crack development is a three dimensional problem. For simplification we cut a very thin layer out of the specimen. In this case we get a two dimensional problem. A thin layer cut out of a ball-shaped grain is a disk. Using this disk imbedded in cement past we further examine the the crack development. In figure 3 the small window out of figure 2 is shown at different state of cracking. A crack with the crack width w in direction of the of the load causes also a crack width perpendicular to the load, which is named w . The interface between the cement past and the grain is rough. The basic assumption of this model is that stress transfer between both surfaces is possible as long as the crack width perpendicular to the load is less than the the effective roughness r at the contact surface. In figure 3A the crack width w is equal zero. In this case the bearing capacity of the interface is 100%. In figure 3C the minimum crack width perpendicular to the load (w , min), at the point where the crack path meets the surface of the disk, is just equal to the effective roughness r. The crack width (in direction of the load) at this state is call wmax. For w ≥ wmax no further stress transfer is possible. For w < wmax (figure 2B) stress transfer is possible, the aggregate interlock works. We get two points of a σ-w-relation of the mechanism cement past—disk. First: (load=100%, w= 0) and second: (load=0%, w=wmax). Assuming a linear load decrease between this two points we get the σ-wrelation as shown in figure 4. The crack width wmax depend on the geometry and effective roughness r. The geometry is described by the quantities as listed below a

=

eccentricity of crack path

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Figure 2: Crack path

ds a r w w , min φ z

= = = = = = =

diameter disk angle of crack path effective roughness crack width minimum crack width perpendicular to the load direction auxiliary angle as defined in figure 5 auxiliary quantity as defined in figure 5 (1)

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Figure 3: Crack development at the interface between concrete

Figure 4: σ-w-relation of one disk mechanism

(2)

(3) The equations 1 to 3 link the crack width w ,min with the geometry and the crack width w. To find wmax for a given geometry one can enlarge w until w ,min=r is reached. 3.2 Probability formulation As shown above it is possible to find a σ-w-relation for one mechanism disk—cement past. The rupture surface of a concrete specimen is composed of a large number of those mechanisms. The σ-w-relation of the specimen is the superposition of the σ-w-relations of all disk mechanisms working together. Because of the large number of disk mechanisms the problem can be solved by a probability formulation only.

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Figure 5: Geometry disk mechanisms

3.2.1 Distribution law—Disk Diameter We get the disk size distribution out of the aggregate grading curve (=grain size distribution) with some fundamental mathematical operations (for further information [4]). In table 1 a aggregate grading curve and the belonging disk size distribution is listen and plotted in figure 6. Table 1: Grain and disk size distribution (Example) 1

2

3

4

j

di [mm]

Pgrain, j [%]

Pdisk, j [%]

1 2 3 4 5 6 7

0.25 0.5 1 2 4 8 16

3 7 12 21 36 60 100

4.4 8.8 15.2 26.2 44.4 71.6 100

The probabilities P is defined as the probability that a quantity is greater than the lower boundary value and less or equal to the upper boundary value. In the case of the screen undersize the lower boundary value is always zero. The screen undersize is defined as: The probability Pdisk, j as given in table 1 are define as:

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Figure 6: Grain and disk size distribution (Example)

with dK=diameter grain dS=diameter disk dj=upper boundary values (screen hole size) 3.2.2 Distribution law Angle α The angle α is the angle between the crack path and a line perpendicular to the load direction. As the crack path is normal perpendicular to the load direction the mean value of α is zero. Variations in both directions will occur with the same probability. Because of this the Gaussian-distribution is used as distribution law for α. The distribution is described by the mean value which is zero and by the standard deviation σα. 3.2.3 Distribution law—eccentricity of crack path Any eccentricity a (as defined in figure 5) between zero and half the disk diameter ds/2 will occur with the same probability. Eccentricities less than zero are neglected because the crack path will normal propagate on the shorter way around the grain. As a consequence of this the distribution law is a straight line between zero and half the disk diameter ds/2. 3.2.4 Distribution law—effective roughness The effective roughness r is one of the governing parameter of the presented model but no direct information about it is available. Especially the effective roughness in not equal to the real roughness of the grains. During the crack growth most of the real roughness will be smoothed down. In lack of better information the following function was chosen to determine r: (4)

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Both the parameters K1 and K2 can be determiner by experiment only. For all following calculations the values K1=0.02mm and K2=1/500 will be used. 4 Stress crack width relation The whole set of disk mechanisms can be split up into different subsets having approximate the same parameters (α, ds and α). The distributions laws are divided into nα, nd and nα portions by defining the boundary.values αi, ds, j and ak. A subset is defined by: (5) The probability that a quantity is greater than the lower boundary and less or equal to the upper boundary can be calculated by the distribution law. The probability that one disk mechanism is an element of the subset Mαi, dj, ak asdefined in equation (5) is the product of the probability that parameters of the disk mechanism fit between the chosen boundary values. (6) The crack width wmax as defined in section 3.1 can be calculated for any of the subsets as defined in equation (5) taking the median between the boundary values respectively. The total bearing capacity of a specimen say 100% or fct can be split up into the bearing capacities of the different subsets. The percentage bearing capacity of one subset as defined in equation 5 is equal to the probability as defined in equation (6). By calculating the different wmax we know the σ-w-relation from every subset. The σ-w-relation of the whole specimen is the superposition of the σ-w-relation of all the subsets. As an example the σ-w-relation of concrete with a grain size distribution according to German DIN 1045 A16 was calculated with the procedure as described above. The standard deviation σα=10° was chosen. Twelve subsets were created defining the boundary values as given in table 2A–2C. The medians between the boundary values (αi, z, dsj, z, ak, z) are listed also. With this medians wmax was calculated and listed in table 3 together with the belonging probability as defined in equation (6). The result of this calculation is the σ-w-relation plotted in figure 7. The dashed line is the result of the same calculation with refined subsets. Table 2: Boundary values and probabilities

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Table 3: s-w-relation (A16) 1

2

3

4

5

ai, z [º]

dS, j, z [mm]

ak, z [mm]

P1 [%]

wmax [µm]

–14,1 –14,1 –14,1 –14,1 0 0 0 0 14,1 14,1 14,1 14,1

3 3 12 12 3 3 12 12 3 3 12 12

0,375 1,125 1,5 4.5 0,375 1,125 1.5 4,5 0,375 1,125 1.5 4.5

5,69 5,69 2,26 2,26 24,42 24,42 9,68 9,68 5,69 5,69 2,69 2,26

279,7 39,5 732,0 68,4 89,3 22,5 161,4 39,0 46,6 14,3 81,0 24,6

5 Application and Results From own experiments with normal concrete (see [4]) the parameters of equation (4) and the standard deviation sa were chosen as stated above. In [3] a s-w-relation determined from several experiments is published. Figure 8 shows the experimental results, the ‘best fit curve’ from [3] and the result of the grain model. The grain model was applied to the grain distribution published in the mentioned paper without undertaking any further fitting calculation. As the outcome of the grain model fit very good also with this experiments, it seams that the free parameters of the grain model (K1, K2, sa) are uniform for normal concrete. With this assumption the s-w-relation of concrete with equivalent aggregate grading curve but different maximum grain diameter are calculated. The aggregate grading curve are listed in table 4. The s-w-relations are plotted in figure 9. The grain model exhibits the expected (but up to now experimental not verified) results. The areas under the curve (=fracture energy) is greater for greater grain sizes. Table 4: Grading curve A32, A16, A8, A4 and A2 Grading curve

screen undersize [%]/screen hole size[mm]

A32

0,25 2

0,5 5

1,0 8

2,0 14

4,0 23

8,0 38

16,0 62

32,0 100

GRAIN-MODEL FOR THE DETERMINATION

Grading curve

screen undersize [%]/screen hole size[mm]

A16 A8 A4 A2

3 5 9 14

7 13 22 36

12 21 35 57

21 36 61 100

36 61 100

60 100

87

100

Figure 7: s-w-relation (A16)

Figure 8: Comparison grain model—experiment

6 Conclusions In this paper a model is described to determine the σ-w-relation of concrete from the aggregate grading curve. The model exhibits very good agreement with experimental results. The model may deliver the theoretic background for research work in the influence of grading curve and aggregate size to the σ-w-relation and the fracture energy of concrete.

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Figure 9: s-w-relation from grading curve A32, A16, A8, A4 and A2

References 1 2 3

4

[] A.Hillerborg, M.Modeer, P.-E.Petersson: Analysis of Crack Formation and Crack Growth in Concrete by means of Fracture Mechanics and Finite Elements, Cement and Concrete Research, Vol. 6, 1976, pp 773-782. [] P.-E.Petersson: Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials, Report TVBM-1006, thesis, Div. of Build. Mat., Univ. of Lund, Sweden, 1981. [] H.A.W.Cornelissen, D.A.Hordijk, H.W.Reinhardt: Experiments and Theory for the Application of Fracture Mechanics to Normal and Lightweight Concrete, Contributions International Conference on Fracture Mechanics of Concrete, Lausane, 1-3 October, 1985. [] H.Duda: Ermittlung und Anwendung der Zugspannungs Rissbreiten Beziehung von Beton, Dissertation, Darmstadt (in preparation).

PART TWO STRUCTURAL MODELLING

8 SIZE EFFECT AND EXPERIMENTAL VALIDATION OF FRACTURE MODELS M.ELICES, J.PLANAS Universidad Politecnica de Madrid, Departamento de Ciencia de Materiales, Madrid, Spain

ABSTRACT A methodology for comparison of different cohesive crack models is presented in this contribution. This method is based on the asymptotic behaviour of the maximum load size-effect, which is shown to adopt a simple structure valid for any reasonable model. The asymptotic analysis allows a model-independent definition of the specific fracture energy, although the results based on small size specimens are model-dependent and reflect a model parameter rather than a material property. Four cohesive crack models; quasi-exponential softening, linear softening, Bazant’s model and Shah’s model are compared, using three notched geometries and numerical analysis. Experimental results of notched beams of four sizes are analyzed following standard procedures, as well as the proposed method based on size-effect. The essential result is that, for usual geometries and sizes, all models describe well -within experimental scatter- the size effect, but predictions differ when large sizes are considered. This discrepancy is analyzed, and some of the consequences discussed. 1 INTRODUCTION This contribution has two objectives: to review some Fracture Mechanics concepts applied to concrete and the associated experimental techniques —the fracture energy and its determination in particular— and to provide a methodology to compare different fracture models. The procedure is based on the maximum load size effect. In the field of Fracture Mechanics applied to concrete and rocks there is a need for an unambiguous and accurate terminology, and a need to unify experimental methods of measuring different magnitudes. One realises this on glancing through specialized journals and records of proceedings, and efforts towards clarifying these matters should be encouraged. There is also an increasing number of models able to predict the maximum load a structural concrete component can withstand, which makes it difficult to select the most appropriate approach for a given

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problem, so a unified comparison procedure for such models is desirable. Although most models are able to predict the correct maximum load for usual concrete specimen sizes, as has been shown by the authors (1988b), these no longer give accurate estimates for large sizes and, in general, the predicted specific fracture energy GF is model-dependent. One tends to wonder about the need of GF for practical purposes. As Arne Hillerborg reported in this workshop, GF is a material property (like modulus of elasticity or compressive strength) needed to predict structural strength and ductility in fracture-sensitive structural situations such as shear, punching or minimum reinforcement The difficulties arise when GF is measured; different test methods, based on different fracture models, yield different GF estimates. Certainly, for components of size similar to the test specimens, the modelbased experimental value of GF works well, provided one is self-consistent: this means that the same model is used as a basis for the measurements and predictions are made using these results. One should not, for example, use Bazant’s model with a value of GF measured according to the RILEM procedure, which is based on a cohesive crack model. In such circumstances experimental results do not generally agree with model predictions. However, self-consistency is not enough when extrapolation to larger sizes is considered, because then the model-dependent measurement will be adequate only if the model is the right one (LLorca et al., 1989). The problems arising from the situation just described are analyzed in the following pages. The essential theoretical tools required for the analysis are briefly summarized in the next section. In particular, the asymptotic properties of cohesive crack models, the structural size effect, and the size effect based definition of the fracture energy are dealt with in this second section. The third deals with a unified comparison procedure for fracture models based on size effect, which is defined and applied to four models: cohesive with quasi-exponential softening, cohesive with linear softening, Bazant’s size effect equation, and Shah’s two-parameter model. The fourth section describes the main aspects of an experimental research devised to apply all the model-based procedures to the determination of fracture parameters and to analyze the consistency of the different results obtained: RILEM total energy method, Perturbed Ligament modification, Bazant’s proposal to RILEM, Shah’s proposal to RILEM, and the size effect fitting procedure here described. The contribution ends with some general remarks and suggestions for future work. 2 A SIZE-EFFECT DIAGRAM The comparison procedure for Cohesive models is based on the predicted maximum load for geometrically similar specimens of different sizes. The tool is a general expression for this size effect applicable to all cohesive crack models, and is based on their asymptotic properties as reviewed below. 2.1 Asymptotic properties of cohesive crack models. The application of cohesive crack models to concrete fracture was pioneered by Hillerborg (1976) more than a decade ago. The present contribution is based on cohesive crack models with no bulk dissipation, although occasionally some remarks on the influence of bulk dissipation are made. An extensive review of crack models can be found in a recent RILEM report (Elices and Planas, 1989) so only a few aspects are summarized here.

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The characterization of a cohesive crack model includes the definition of the bulk behaviour, the specification of the condition for crack formation, and the specification of the equations for crack evolution. In practice, such conditions are restricted to: H.1. Bulk behaviour: The solid is homogeneous and the behaviour at any point is isotropic linear elastic —with Young modulus E and Poisson’s ratio v—until the major principal stress reaches a critical value ft. H.2. Loading conditions: Loading is growing monotonically and proceeds symmetrically to produce mode I loading. H.3. Crack initiation: When the maximum principal stress reaches ft, fracture is initiated and strain localization takes place in what is called the fracture process zone (FPZ). The FPZ is modelled by a cohesive crack where the strain localization is idealized as a displacement jump or crack opening, while cohesive stresses simulate the softening behaviour. H.4. Crack propagation: Once the cohesive crack has formed, the stress transferred through the crack faces is assumed to depend upon the relative displacement of the crack faces as: (2.1) where F(w) describes the softening behaviour of the material and is called the softening curve and is considered a material function. In practice, the values of F(w) are assumed to be zero for crack openings exceeding a critical crack opening wC. Coplanar, monotonic crack growth is assumed. The work needed to open a crack of unit surface, monotonically, the specific work supply, is also a material function given by (2.2) The specific work supply needed to fully open a unit surface of crack is a material property called the specific fracture energy GF (or simply, fracture energy), i.e.: (2.3) From the material parameters E, v, ft, and GF, two independent parameters having dimension of length are introduced; the first was called characteristic length by Hillerborg and defined as (2.4) where, from now on, E is understood as the generalized Young’s modulus. The second parameter called characteristic crack opening, is defined as

The different cohesive crack models are closely related, at least for some limiting situations, and the simplest models may be regarded as mathematical approximations of more sophisticated ones. Such relations were proved by the authors for precracked structures (precrack length a0) subject to proportional loading and for positive geometries (Planas and Elices, 1986b, 1987,1990). The essential fact is that when

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Figure 1. Asymptotic properties of cohesive crack models: First order approximation.

the size D of certain precracked bodies is allowed to grow without bound while retaining fixed geometrical proportions, all the cohesive fracture models converge to Linear Elastic Fracture Models (LEFM). This limiting condition will hold if the zone where non-linear processes take place remains bounded in size when D grows to infinite. In this case the modifications in the displacement fields due to the presence of the nonlinear zone vanish as R/r, where R is the size of the non-linear zone and r is the radial distance to the crack tip. Hence, for very large r (which implies that D is also very large), the fields are nearly equal to the unperturbed elastic fields and, consequently, the fracture models are equivalent in the precise sense that their far-fields are (nearly) the same, to order 1/D. Moreover, it was shown that the structure of the far field for a cohesive crack model coincides with that of an effective elastic crack field to order 1/D2 (Planas and Elices 1986b, 1987, 1990). It is worth clarifying the strict sense of this statement as illustrated in Fig. 1. The left part of this figure represents the real structure, in which a cohesive zone of size R has developed monotonically in front of the initial crack while the load increased up to P. Of course, the stress in the material is finite everywhere along the crack path, according to the hypotheses stated above. The right part of the figure represents an imaginary (virtual) specimen, purely elastic, subject to the same load P, but with a crack of length a=a0+∆a∞ where ∆a∞ is an effective crack extension, and subindex ∞ stands for the fact that we assume we are approaching the asymptotic limit of infinite size. The equivalence is stated in the following way: one may choose (calculate) ∆a∞ in such a way that the far fields (shaded zones in Fig. 1) coincide except for terms of the order (R/D)2 or higher. The value of ∆a∞ for each loading step may be computed by solving a universal (size-, shape-, and loadingindependent) integral equation, as shown elsewhere (Planas and Elices, 1986a, 1987, 1989). In the virtual specimen case there is a stress singularity at the crack tip with a stress intensity factor KIvirtual (a) given by an equation of the well known form: (2.5)

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ANALYSIS OF CONCRETE STRUCTURES

where σN is a nominal stress defined as P/BD (P is the load and BD a characteristic section area) and S(a/D) is the geometrical shape factor. Note thatσN is the same for the virtual and real specimens, and that the “virtuality” is contained in the shape factor, since in reality we do not have this geometry For the actual cohesive cracked body the stress intensity factor has no meaning as a measure of the stress singularity, because the stress is finite at the crack tip, but for comparative purposes we can define a nominal stress intensity factor as a loading parameter involving load level, size, and initial shape: (2.6) The actual near-tip fields may be related to the virtual ones by using path-independent integrals. In particular we may use the J integral along a path completely surrounding the cohesive zone, as path Γ0 in Fig. 1, which is equal to the J integral performed along a path Г∞ fully enclosed into the far field region (dashed region): (2. 7) Now, since the actual fields are the same in the dashed region for the actual and virtual cases to order 1/D, one can write: (2.8) from which, after taking into account the order of approximation, one concludes that (2.9) where o(1/D) stands for a function approaching zero faster than its argument. To simplify this equation while retaining the same order of accuracy, one can use a two term expansion of Eq. (2.5) and definition (2. 6) to obtain (2.10) where S(a/D) is the shape factor introduced in (2.5), S'(a/D) is its first derivative with respect to the argument, and subindex 0 refers to initial crack length. On the other hand, the actual J integral may be computed along a limiting path defined by the crack (real plus cohesive) faces themselves. A straightforward calculation yields the result (2.11) where the second equality follows from Eq.(2.2), and CTOD indicates the crack opening displacement at the initial crack tip. Equations (2.10) and (2.11) are the bases for the asymptotic analysis of the size effect in cohesive materials. We must emphasize that here we have exploited a theorem regarding asymptotic far field structure which has been proved elsewhere, together with the method to determine the effective crack extension for infinite size ∆a∞ 2.2 Maximum load size-effect It is well known that the maximum nominal stress decreases as the size increases for a set of geometrically similar precracked bodies. Based on the dimensional analysis and on the limiting LEFM behaviour for large sizes, a rather general expression for this size effect was found by the authors (Planas and Elices, 1986a) for

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cohesive crack models. This expression takes the form of a power series expansion and may be written in terms of maximum nominal stress intensity factor as: (2.12) where lch is the characteristic length (2.4) and Ci are dimensionless coefficients that depend on geometrical ratios and on the shape of the softening curve. A subsequent asymptotic analysis performed by the authors (Planas and Elices, 1986b, 1987, 1990) allowed the determination of the internal structure of the coefficient C1 as the product of a geometrical factor times a material constant. This may be done by considering Eqs. (2.10) and (2.11). According to (2.1), (2.2) and (2. 3), there is an absolute maximum for the specific work supply, WF(w), namely GF, and this maximum is reached for the critical crack opening wC. Using this fact together with Eqs (2.10) and (2.11), and accepting that the maximum takes place for some critical value of the effective crack extension ∆aC∞, the asymptotic expression for the size effect may be written as (2.13) where the ∆ac∞ is now the critical effective crack extension, already mentioned, which may be obtained from the softening curve as described in the quoted references. It must be emphasized that ∆aC∞ is not an ad hoc parameter, but is obtained from a universal equation based only on knowledge of the softening function. It is a derived material parameter, but it is independent of the parameters already introduced ft and GF. It depends heavily on the shape of the softening curve, the longer the “softening tail” of this curve the higher it is. From Eq. (2.13) it may be noticed that two cracked samples of the same material are asymptotically equivalent when the factor (2S0’/S0)D-1 is of the same value for both structures. Consequently, we may define an intrinsic size Di (Planas and Elices, 1988b) as (2.14) At a first glance it may appear that Di depends on the selection of D. But it may be easily shown that this is not so, because, (2.14) is equivalent to the definition

where in the partial derivative with respect to crack length a, the load and all remaining dimensions of the specimen remain constant. It should be borne in mind that this analysis was done only for positive geometries (i.e. when the stress intensity factor increases with crack length, or S0’>0). When the specimens considered are all geometrically similar (homothetical) we have a single degree of freedom and the selection of D does not matter. However, when we consider more degrees of freedom the arbitrariness of the selection of the size makes it difficult to compare the results, and the difficulty increases if we consider different specimens and loadings. Therefore, the introduction of an objective measure, the intrinsic size, seems appropriate. With this definition, Eq. (2.13) takes the simpler form: (2.15) A general size-effect relation is sketched in Fig. 2 for cohesive cracks, together with the asymptotic expression (2.15) which, with the variables used, takes the form of a straight line. Two other extreme cases are also depicted in figure 2; when LEFM applies, KINmax is constant and a horizontal line is found; when limit analysis is appropriate, then σNmax is constant, and a straight line through the origin results. For small sizes, cohesive crack models tend to limit analysis, or strength of materials models. This is shown by an asymptota parallel to the limit analysis straight line. For large sizes cohesive crack models tend towards LEFM and should converge to E GF(KINmax)-2=1, and must do so linearly as shown.

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Figure 2. A size-effect plot for the maximum load and first order asymptotic approximation.

This representation is not the usual size effect plot (σNmax versus D), but it is helpful in performing asymptotic analysis because large sizes are kept in focus near the origin. 2.3 Specific fracture energy concepts Within the frame of cohesive crack models with no bulk dissipation, the concept of specific fracture energy is well defined. According to Eq. (2.3), it is the work of the cohesive stresses inverted in completely breaking a unit area, and may be computed directly when the softening function is known. Moreover, since bulk dissipation is assumed to be zero, it may be measured directly, using for example the RILEM method put forward by Hillerborg (1985), when proper measures are adopted to get rid of spurious sources of energy dissipation (Planas and Elices, 1986a, 1988a). However, a cohesive crack with no bulk dissipation is just a model and is not guaranteed to give a perfect representation of reality. We could, for example, have cohesive stresses following equation (2.1) but with some kind of inelastic material behaviour around the cohesive crack, then giving bulk dissipation to be added to the surface energy dissipation. Incidentally, this has been suggested as a source of the dependence of the results of the RILEM procedure on the specimen size (Planas and Elices, 1986a; Elices and Planas, 1989; Hillerborg, 1988 and in discussions in this workshop). We could also think about the existence of similar approaches where an expression like (2.3) for the specific energy cannot be written, because cohesive forces are not introduced in the formulation, or is not unique because the work of the cohesive forces is path-dependent, as for example if one assumes that cohesive stresses depend on triaxiality (on stresses or strains in the plane parallel to the crack), which does not seem absurd.

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It appears, then, that a definition broader than (2.3), which is strongly model-dependent, is needed, mainly because any fracture model, cohesive or not, must include in one or another way, explicit or implicitly, a parameter with dimensions of specific fracture energy. And this definition may be set in a model-independent formulation based on the asymptotic size effect. Indeed, in the preceding section it has been shown that when the size of a specimen grows larger and larger, the maximum nominal stress intensity . While this is proved for cohesive cracks factor approaches more and more the asymptotic value without bulk dissipation, all the fracture models will reach some definite geometry-independent limit for infinite size, and hence we can use this asymptotic property as a definition, and write (2.16) where it is understood that the limit is for fixed positive geometry. The application of (2.16) is based on the assumption that the maximum loads may be determined for every size and one might well suppose that they are experimentally measured. However, the limit indicated in (2.16) cannot be achieved in practice. It can only be approximated because testing sizes are necessarily finite; to develop a test method, based on this procedure, a minimum specimen size that adequately approaches the limit (2.16) should be conventionally adopted. For metals, the deviation from linear elastic behaviour in the large size range is mainly due to the development of a plastic zone surrounding a very small softening (fracturing) zone, so that the use of elastoplasicity provides reliable estimates of the nonlinear zone size and consequently of the minimum size needed for experimental determination of fracture toughness. For concrete, the situation is very different because the deviation from linear elastic behaviour arises from the development of a large softening zone, and to asses such deviation a description of softening is unavoidable. The existing lack of agreement about the best way to describe the fracture (i.e. about which is the best fracture model for concrete) necessarily leads to a lack of agreement about the minimum size required for application of (2.16). However, it seems that for most models the minimum sizes required are well over those acceptable for routine testing (Planas and Elices, 1988b). What is done in practice is to postulate a model, to adjust the parameters of the model to fit the experimental results on small specimens, and then to use the model to extrapolate, by computation, to infinite size. In this way what one obtains is a parameter of the model, called also specific fracture energy, defined as (2.17) It appears, then, that what we “measure” in any of the proposed experimental methods is a parameter of the underlying model. Only if such a model is a close description of reality will this parameter approach the true fracture energy GF, as sketched in Fig. 3. It was found (Planas and Elices, 1988b, c) that if the parameters of the available models are determined so that the maximum load predictions are fitted to experimental results in the small size range, the large size predictions will be, in general, different for the various models. Reciprocally, if the parameters of the models are adjusted to give the same asymptotic maximum load predictions, up to order 1/D (2.15), the maximum load predictions for the small size range will be different. Still, another source of discrepancy may come from incorrect experimental measurements on small specimens, if these values are used to feed models to obtain parameters that will be extrapolated, by computation, to infinite size. In these situations, although using an accurate model, the final specific

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Figure 3. Specific energy concepts; GF, GF, mod, GF, mod, exp.

fracture energy will be different from GF, as shown in figure 3. In such cases, when experimental inputs are dubious, it may be helpful to itemize the measured energy as (2.18) An analysis of possible sources of energy dissipation was attempted previously by the authors (Planas and Elices, 1986a, 1988a) and will be summarized later.

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In closing this section, it is helpful to summarize the different notations for the specific fracture energy also shown in figure 3. The material property GF is given by (2.3) or (2.16) for a cohesive crack model with no bulk dissipation. For different models, (2.16) stands for a universal definition. When, as always in practice, the limit is computed by assuming a particular model, one should specify that model GFmodel (2.17). At this time there is not an agreed ‘“super-model” for concrete, nor a single experimental procedure generally accepted, and it is advisable to check carefully the accuracy of experimental values, and in any case to specify in some explicit form as GF, model, experimental the model and the experimental procedure used. 3 SIZE-EFFECT VERSUS MODELS The procedure developed in the preceding section is applied to four models; a cohesive crack with quasiexponential softening, a cohesive crack with linear softening, to Bazant’s Size-Effect law and to Shah’s twoparameter model. After summarizing the procedure, the numerical results are presented. This section ends with some comments for large sizes. 3.1 Comparison procedures based on the maximum load The methodology to compare different cohesive crack models was suggested by the authors (Planas and Elices, 19885, c; Llorca et al., 1989) and is based on the effect of the specimen size on the maximum load, taking into account that the specimen sizes are limited in practice and that the experimental scatter is high for concrete specimens. Given a set of models, a set of geometries and the range of practical experimental size, the procedure is as follows: 1. Select one of the models as the reference model. In this research we selected a cohesive crack model with quasi-exponential softening, because its curve is more similar to experimental softening curves for concrete than the other models referred to. 2. Select one of the geometries as the reference geometry. The reference geometry is taken to be a notched beam, a well known specimen, as sketched in figure 4. 3. Compute a reference size effect as the size effect for the reference model and the reference geometry in the practical size range. In this research the practical size range is limited to the interval 0.1 m ≤D≤ 0.4 m, where D is the specimen depth. 4. Adjust the parameters of the remaining models so that their size effect curves fit well the reference size effect. The fitting procedure depends on the model and number of parameters to be optimized. The usual procedure is to minimize some error function defined on the experimental interval. 5. Holding constant the previously determined model parameters, make size effect predictions for the remaining sizes and geometries. Particularly important are extrapolations to large sizes (asymptotic analysis). Now, equation (2.15) becomes: (3.1)

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Figure 4. Selected specimen geometries to explore the influence of stress distribution.

Application of this equation will allow a correlation of the values of fracture energy and of critical effective crack extension for the different models. 6. Decide, by inspection of results, whether or not it is possible to discriminate between the different models when experimental scatter is taken into account.

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Four models were chosen for this comparison; two cohesive crack models characterized by their softening curve —a quasi-exponential one (the reference model) and a linear one— Shah’s two-parameter model (Jenq and Shah, 1985) and Bazant’s model based on his Size-Effect law (Bazant, 1984). Three geometries were selected as shown in Fig. 4. The aim was to explore the influence of stress distribution along the cross-section, ranging from bending (in the notched beam) to uniform tensile stress (in the notched strut). 3.2 Numerical results for the four models Reference model: Cohesive crack with quasi-experimental softening The essential parameters of the model are E,ft and GFE, where subindex E for GF refers to Exponential softening. The assumed softening curve is the modified exponential defined as follows in terms of the stress transferred through the crack faces, σ, and the crack opening w: (3.2) where A=0.0082896 and B=0.96020. Although all results are presented in dimensionless form referred to the parameters of this model, we have to assume some particular value at least for the characteristic length. To be precise we set the parameter values of this reference model as follows:

from which a characteristic length lchE=0.3 m is obtained. The infinite size critical effective crack extension, ∆aC∞E, for this particular softening curve, was computed to be (3.3) The size effect for small specimen sizes was computed by an influence method with a numerical program similar to that used by Hillerborg (Petersson, 1981), but with a modified algorithm to allow non-polygonal softening and to speed up the resolution. 100 equal elements are situated on the crack section, and nodal displacements (crack openings) are related to nodal forces through an influence matrix determined once and for all for a each geometry by the Finite Elements Method, using the ANSYS program. Taking the influence matrix as an input, the cohesive crack analysis follows, using the cohesive crack length as the independent variable (i.e. the number of cracked elements). At each step a Newton-Raphson iterative method is used, in which the softening curve and its derivative are analytically defined. The program automatically determines and stores the peak load. The size-effect results for the notched beam and for a selected set of values of a0/D (0.2 ≤ a0/D ≤ 0.5) were plotted in X-Y diagrams, where (3.4) It was found (Planas and Elices, 1988b) that when plotted in this diagram the resulting size effect curves were independent of a0/D to within ±1 per cent. Hence for this model and for the range of crack depths

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Figure 5. Size-effect results for notched beams.

under study, it appears that the size-effect depends only on the intrinsic size, not on the other geometrical parameters. The average size effect curve is shown in Fig. 5. The size-effect results for the compact (CT) and single-edge-notched (SEN) specimens for 0.3 ≤a0/D≤ 0. 5, were also computed. Again, a small dependence on a0/D was found and results for a0/D=0.5 are plotted in figures 6 and 7 for CT and SEN specimens respectively. Cohesive crack model with linear softening The essential parameters of the model are E,ft and GFL, where subindex L for GF refers to Linear softening. The equation of the softening curve is, in this case: (3.5) It is assumed that both elastic modulus and tensile strength are the same as for the reference model, so that only one degree of freedom is allowed for curve fitting. In particular we have for the characteristic length: (3.6) The infinite size critical effective crack extension for linear softening, ∆aC∞L, was computed to be (3.7) The size-effect curves were computed following exactly the same procedure as for the reference model, and the ratio GFL/GFE was adjusted using graphical methods to get a good fit with the size effect curves for the reference model over the practical experimental range.

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Figure 6. Size-effect results for compact specimens.

The results are presented as size effect curves in dimensionless X-Y plots. Notice that parameter values used to make the variables dimensionless correspond to the reference model. With this representation the Yintercept of the size effect curve equals GFE/GFL and the slope of the curve at X=0 equals ∆aC∞L/lchL, since the asymptotic size-effect equation (3.1) in a Y versus X plot takes the form (3.8) Again, it was found that the size effect curves are nearly independent of the initial notch depth (Llorca, Planas and Elices, 1989) and that the fit is quite good over the practical experimental range. Figures 5, 6 and 7 show the comparison between the size effect curves of this model and those of the reference model for the three geometries for a single notch depth: a0/D=0.5. Bazant’s Size-Effect law Bazant’s law, based on the crack band approach and rather general energetic considerations (Bazant, 1984), may be written as (3.9) where GFB is the fracture energy (subindex B refers to Bazant model), and 10 is a parameter depending both on geometry and material. This equation may be rewritten as (3.10) which is an expression similar to Eq. (3.1) except that ∆aC∞B is allowed to depend on geometry. In X, Y coordinates, it reads:

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Figure 7. Size-effect results for notched struts.

(3.11) The values of the parameters appearing in equation (3.11) that give a good fit of Bazant’s equation with the reference size-effect were obtained by linear regression. For SEN bending, a0/D=0.5. A value GFE/GFB=1. 92 was obtained for fracture energies, while for the critical effective crack extension ∆aC∞B, values in the range 0.03 to 0.04 m were obtained, depending on the precise geometry (remember that in the Bazant model this parameter is allowed to depend on the geometry). The resulting size-effect straight lines for the three geometries and for a0/D=0.5 are plotted in figures 5, 6 and 7. Shah’s Two-Parameter Model In Shah’s Two-Parameter Model (Jenq and Shah, 1985) it is assumed that in a precracked structure a slow crack growth takes place under increasing load up to a certain crack extension ∆aC in which the maximum load is attained. This critical situation is assumed to occur when the stress intensity factor at the extended crack tip takes its critical value KSIC and, simultaneously, the crack opening at the initial crack tip, or CTOD, takes its critical value CTODCS, where index S refers to Shah’s Model. The asymptotic analysis performed by the authors (Planas and Elices, 1988b, c) lead to the following relationships between the fracture energy GFS and the infinite size critical effective crack extension ∆aC∞S and the primary parameters KSIC and CTODCS: (3.12)

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(3.13) The analysis is based on the fact that the CTOD after a crack extension ∆a may be written as (3.14) where L(∆a/D; a0/D) is a dimensionless function depending implicitly on geometry, but satisfying the condition (3.15) A parametric expression for the size effect curves, with parameter ∆ac/D, was obtained as (3.16) (3.17) where X1 and Y1 are size independent purely geometrical functions. Function Y1 is related to the geometrical shape function for the stress intensity factor by the equation (3.18) and is known approximately in closed form for most relevant geometries. Function X1 is related to the crack opening shape function L by the equation (3.19) and is not known in closed form except for very special geometries. Pointwise computations of this function were performed using the linear elastic solutions provided by the influence program (no cohesive forces on crack faces). Two term expansion of the size-effect equations (3.16) and (3.17), with the help of (3.18), (3.19) and (3. 15), leads to the asymptotic expression for the size-effect, which may be written as (3.20) proving that the size-effect equation for Shah’s model has the same structure as that found for the other models. The determination of the parameters of Shah’s model giving a good fit with the reference model is rather cumbersome and was described by Planas and Elices (1988c). For SEN bending specimen, a0/D=0.5 and the practical size range, the computed parameters were GFE/GFS=2.08 and ∆aC∞S=0.022 m. Once more, little dependence on the initial notch depth was observed (except for very small sizes). The size-effect curves for the three geometries and for a0/D=0.5 are plotted in figures 5, 6 and 7. 3.3 Behaviour for large sizes The results in figures 5, 6 and 7 show that the four models give size effect predictions for the three geometries below the 5% level for small sizes (within the assumed experimental size range). Such differences are within the usual experimental scatter band for concrete, making it very difficult to distinguish between different models on the basis of the maximum load criterion.

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Figure 8. Values of GF, reference/GF, model for very large sizes. (Reference model; quasi-exponential softening).

To reveal differences between the models, using this method, one needs to test specimens with very large sizes, roughly one order of magnitude higher than is considered in this research. Despite the good agreement of the models in the small size range, their size-effect curves show clear differences for very large sizes. Shah’s model, Bazant’s model, and cohesive crack model with linear softening, are conservative by 31, 28 and 20 per cent, respectively, when compared with the cohesive crack model with quasiexponential softening. This result is sketched in figure 8. In fact this figure is a zoom near the origin of previous figures. Some other results are summarized in Table 1. Table 1. Summary of Results for the Different Models (All Geometries) Model

Size range (cm)

Maxi. difference** in fracture load (%)

GFM/GFE

Linear Softening Bazant Equation Shah’s Two Parameter

10–40 ∞ 10–40 ∞ 10–40 00

±2.0 –20.0 ±2.5 –28.1 ±4.3 –30.8

0.64* 0.64* 0.52* 0.52* 0.48* 0.48*

(*) Fit for SEN-bending, a0/D=0.5, and practical size range. (**) Relative to Cohesive Crack Model with Exponential Softening.

Here it is interesting to recall that during a round robin test, lead by Hillerborg to measure GF, the authors found that the GFRILEM values determined according to RILEM recommendations (on three sizes: D=0.1, 0.2 and 0.3 m) were not significantly different, giving an overall mean value of 122 N/m (Planas and Elices, 1986a). When maximum loads were used to estimate GFmodel, according to Bazant’s model, a value of 53 N/ m was computed, not far from 0.52 GFRILEM=63 N/m, estimated from Table 1 under the hypothesis that the tested concrete could be modelled as cohesive with a quasi-exponential softening.

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4 FEEDING THE MODELS WITH EXPERIMENTAL RESULTS 4.1 Experimental programme In the preceding sections, it has been emphasized that interpretation of the same set of test results using different models may lead to divergent results for the computed fracture energy. This very essential problem is even worse in practice because the parameters of the different models are extracted from different sets of results, even when a single set of tests is used. This is because the recommended experimental methods are model-based: they rely on specific properties of the assumed model, not on general physical principles. Moreover, as pointed out in a recent RILEM report (Karihaloo and Nallathambi, 1988), in most practical cases the tests are carried out in such a way that only partial information is available, and hence not all the possible models can be determined from the experimentation. The purpose of this section is to present the essential aspects of an experimental program devised in such a way that the data collection is enough to determine (even to over-determine) the parameters of the previously envisaged models: Cohesive crack, Shah’s and Bazant’s. The set of experimental results is used to obtain the parameters of the different models following, first, the procedures already proposed in the literature, based on different direct measurements for the various models (total work in RILEM method, maximum loads for Bazant’s method, and peak loads and compliance for Shah’s procedure). A second set of parameter values is obtained by a unified methodology based on the ideas presented in the previous sections: size effect fitting. A RILEM Standard Concrete with 10 mm maximum aggregate size was used, with 400 kg/m3 of rapid hardening Portland cement in a 1÷1.35÷3.02÷0.55 mix (cement ÷gravel÷sand÷water) giving a slump of 5 cm. Aggregates were natural, rounded, and classified as siliceous. The 28 day standard mechanical characteristics and the direct tensile strength and Modulus at the age of testing are given in Table 2. Table 2. Concrete Properties 28day

28day Strength

Tangent Modulus (GPa)

Compressive (MPa)

Splitting (MPa)

Strength (MPa)

Modulus (GPa)

26.6

33.1

2.8

3.14

25.4

Direct Tensile

The specimens were three point bend SEN specimens of four sizes, as depicted in Fig. 9, of constant thickness B=100 mm. The notch-to-depth ratio of 0.33 does fit in the range for which closed form solutions exist for the essential elastic solutions. However, the span-to-depth ratio of 2.5 (dictated by practical limitations in the frame of a broader experimental research) is not one for which closed form solutions exist, and a good deal of work was necessary to determine the functions required for the analysis of experimental results. The central idea was to obtain a rough solution by a combination of the known solutions for pure bending and three point bending (span-to-depth=4) and to refine it by numerical computation, using FEFAP, ANSYS, and weight functions for the notched infinite strip. The detailed procedures are too lengthy to be detailed here and will be presented elsewherc The experimental arrangement included the measurement of the central force F, the beam deflection ∆ and the CMOD, as depicted in Fig. 10. The deflection was measured relative to the upper points right above

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Figure 9. Specimens used in the experimental program.

Figure 10. Experimental arrangement to measure force, beam deflection and CMOD.

the supports, trying to avoid the inclusion of inelastic displacements due to compaction at the lower supports. The beam self weight was compensated by means of prestressed springs, so that the full F-∆ and FCMOD curves could be obtained (for a detailed discussion on test methods, see Planas and Elices, 1988a). The test was run in CMOD control mode to achieve stability. The rate of CMOD was set proportional to the beam depth, and was such that all the sizes went through the maximum load in 30 to 60 seconds, as specified by the RILEM method (RILEM, 1985). To take into account the possibility of having appreciable energy dissipation, calibration tests were performed as suggested by the authors in previous work (1986a, 1988a). Figure 11 summarizes the method and the results. Figure 11a shows the experimental arrangement A test consists in performing a loadingunloading cycle up to a prescribed maximum load Fmax while recording the full F-∆ curve, as depicted in Fig. 11b. The area of the loop is the dissipated energy. Performing this test for different Fmax one obtains the

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Figure 11. Calibration procedure for energy dissipation at the supports.

calibration curve as a plot of dissipated energy versus maximum load. In practice, the scatter is quite important and a broad band is obtained, as shown in Fig. 11c, where the actual results are displayed.

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Figure 12. Determination of the specific fracture energy according the Perturbed Ligament model: GF, PL.

4.2 The results for fracture parameters following existing procedures

RILEM or average energy method and the perturbed ligament method In the RILEM procedure (Hillerborg, 1985; RILEM, 1985) a cohesive crack model with negligible bulk dissipation is postulated. If the model is correct, the external work supplied to totally break the specimen divided by the ligament area must be constant and equal to GF. In practice, dissipation at the supports takes place which must be subtracted from the total work supply. The calibration curve in Fig. 1 1c is the basis for this correction. In our case, subtractions reached as much as 12% of the raw value. In most cases in the literature, the values of the average specific fracture energy obtained by this procedure are found to depend on the size of the specimen (see, e.g. Hillerborg, 1985). This is also the case for our results, the average specific energy for the largest specimens being about 60% larger than for the smallest specimens. To try to give a quantitative account of this scale-dependence of the average fracture energy, a so called perturbed ligament model (PLM) was proposed by the authors. This model rests upon the assumption that when the test starts, the ligament is already perturbed, either by a segregation or wall effect, or by shrinkage and thermal processes. Under the simplified assumption that the perturbed zone has a depth independent of the size of the specimen—for not too small specimens—one gets a linear relationship between the fracture work per unit thickness and the initial ligament length (Planas and Elices, 1988a). The slope of this line is, by hypothesis, the fracture energy of an eventual unperturbed specimen. This model has been applied to the experimental results. In Fig. 12 the fracture work per unit thickness is plotted versus the ligament length, and a straight line fitted to the results. The slope of this line is the fracture energy, GFPL, predicted by the PLM, which has to be compared with the values obtained by the RILEM method, GFR. Figure 13 shows the individual values for GFR, and the curve representing the PLM relationship. Despite the big scatter, the size dependence of GFR is notorious. This size effect may be eliminated with the use of the PLM by attributing it not to an inconsistency of the cohesive crack model, but to a perturbation related to specimen manufacture and handling.

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Figure 13. Specific fracture energy according RILEM, Bazant’s model and Perturbed Ligament model (GF,RILEM, GF,B, GF,PL).

Bazant’s size effect law The fracture energy for Bazant’s model GFB is determined following the methodology proposed to the RILEM TC 89 by Bazant and Pfeiffer (1988). The only modifications are those due to the change in the geometric shape functions. Using the notation herein, the computational procedure consists in using definition (2.6) in Bazant size effect law (3.9) to get Bazant’s form (4.1) In a σNmax−2 vs. D plot, this equation is a straight line of slope A given by (4.2) This slope is obtained from the experimental results by linear regression techniques, and then GFR is computed from Eq. (4.2). For our tests, the computations gave a fracture energy GFB≈40 N/m, as depicted in in Fig. 13 for purpose of comparison. This is roughly one half of the fracture energy determined with the PLM. Shah’s two parameter model In Shah’s proposal, the computations may be done in three steps. In the first step an effective crack length at peak load, aC, is obtained from the CMOD compliance at the peak. In the proposal it is recommended that the unloading compliance at or near the peak be used in the computation, but it is accepted that the secant compliance may be used instead. In our tests no unloading was performed, hence the secant value was used. In general the compliance equation may be written in the form: (4.3) where C* is a dimensionless shape function. If we call Cti the initial tangent compliance and CsC the secant compliance at the (critical) peak point, we may determine aC from the equation: (4.4)

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Figure 14. Specific fracture energy and critical CTOD according to Shah’s Two-Parameter model.

In the second step, KICS is obtained by direct substitution in Eq. (2.5): (4.5) from which the fracture energy is found from Eq. (3.12) as (4.6) In the third step, CTODCS is determined by substitution in Eq. (3.14) with KI = KICS: (4.7) The results found for our tests are plotted in figures. 14a and 14b versus the initial ligament length. It is clear that both the GF’s and the CTODC’s obtained using this method display a strong size dependency. 4.3 The results for fracture parameters following size effect fitting We may exploit the fact that the size effect curves are quite geometry-insensitive when plotted versus the intrinsic size to extend the theoretical analysis performed in section 3 to practical experimental cases. At a first glance it may appear that fitting of the curves is very complex. This is indeed so if an analytical approach is taken. But there is a graphical approach that allows fast, although somewhat rough, fitting. The method is based on the fact that the dimensional plots (KINmax)−2 versus 1/D are obtained from the dimensionless plots in Fig. 5 by scale changes, or, in geometrical terms, by a simple affinity. The scale changes are directly related to the parameters of the model. In a log-log plot the affinity (change of scale) is transformed into a translation, the components of which are related to the scale changes, and hence, to the material parameters. The simplified procedure is the following: (a) Redraw the size effect curve for the chosen model taken from Fig. 5 in a log-log plot, using transparent paper; (b) Draw the experimental results (KINmax)−2 versus 1/Di in a log-log plot on another sheet of paper, (c) Slide the transparent sheet onto the experimental record until a good fitting is obtained (use of a parallel drawing machine simplifies the procedure); (d) Determine the translation vector graphically. The components of the translation vector are {log(lchE), log (E GFE)} from which lchE and GFE are obtained which in turn are related to the parameters of the chosen model through the values given in table 1.

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Figure 15. Size-effect curves for Bazant, Shah and linear softening models obtained by parameter fitting.

This procedure is valid for models with two degrees of freedom, such as Bazant’s and Shah’s. But in cohesive cracks, where E and ft are independently measured, there is only one degree of freedom, namely GF, and the two components of the translation vector are no longer independent. In fact the translation is decomposed into a fixed known translation of components {log(ft−2E), log E} followed by a translationparallel to the {1,1} direction: {log GFE, log GFE}. This method was applied to the experimental results (for Bazant model standard linear regression was used), and the results of the fittings are shown in Fig. 15 together with the values of the parameters. It is clear that all the models fit the experimental results within 10%, but that the fracture energies are different by a factor of two. 4.4 Summarizing the evaluation of experimental results The dependence on size of the results of the various determinations of the fracture energy is summarized in Fig. 16. All the determinations based on curve fitting, either size effect curves or perturbed ligament method, yield a unique value of the fracture energy. It cannot be proved to be size independent, because it is prescribed to be so. The single specimen based methods, RILEM or Shah, give a strong size dependency. The size dependency of the RILEM results is well described by the perturbed ligament method, and the extrapolated value GFPL is in acceptably good agreement with the GFE obtained by size effect fitting using a cohesive crack model with quasi-exponential softening (13% difference, which fits within statistical scatter band). It must be noted that by increasing the length of the tail of the softening curve, one may fit the actual experimental size effect (practical sizes) while getting any desired value for the fracture energy. The size dependence of the fracture parameters of Shah’s model has not been yet constructively explained, and the values obtained by size effect fitting are in the lower limit, corresponding to very small sizes.

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Figure 16. Summary of the evaluation method and effect of size on the specific fracture energy.

The fitting to experimental results confirms what was theoretically predicted: all the models may fit quite well the experimental size effect, but they predict widely different fracture energies. However, the combination with RILEM method indicates that if the perturbed ligament method turns out to be realistic, the only model consistent with the total energy dissipation is the cohesive model with quasi-exponential softening (or any other similar long-tail-softening). This latter fact may be closely related to a recent upper bound theorem proved by the authors for some models (Planas and Elices, 1988a). It is shown that for a wide class of effective crack models, including Shah’s, the fracture energy as defined by the limit (2.16) is an upper bound of the average specific dissipated energy. In the case of an ideal RILEM test, where energy is only dissipated in fracturing the concrete and inelastically deforming the material close to the fracture process zone, then GFRideal≤ GFmodel. The authors feel strongly that this upper bound theorem must be valid always, for any reasonable model, because the nonlinear zone size increases monotonically with specimen size up to a definite upper limit, and hence the energy dissipation should also increase monotonically. If this turned out to be true, one could write GFRideal≤GFmaterial and then any model M such that GFRideal≥GFmodel M would be unequivocally disproved on scientific grounds. (This is not to be confused with the usefulness of the model for engineering purposes. In particular, such a model would be conservative, which is a desirable feature for design purposes while more accurate models are either lacking or too expensive to be used widely).

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5 CONCLUSIONS In this contribution the essential points regarding the size effect behaviour of maximum load is analyzed with particular emphasis on the asymptotic structure, which is shown to adopt a simple structurc valid for any reasonable model. The asymptotic structure allows a model-independent definition of the fracture energy (Eq. 2.16) although the results based on practical (small size) specimens will be model-dependent and will represent a model parameter rather than a material property, as shown in figure 3. A recently proposed theoretical procedure for unified comparison of fracture models is summarized and applied to different models and geometries. The essential result is that for these usual geometries, and for the usual range of sizes tested in the laboratory, all models may describe size effect within experimental scatter, but that extrapolation to large sizes are divergent, the models of Bazant, of Shah, and cohesive with linear softening being conservative by 28, 31 and 20 % with respect to a cohesive model with quasi exponential softening. The results of a series of stable tests on notched beams of four sizes are analyzed following different procedures and plotted in figure 16. From these results the following conclusions may be drawn: 1. The single specimen based procedures (RILEM, and Jenq and Shah’s) lead to values of the fracture parameters strongly dependent on the size of the specimen. 2. The size dependency of the RILEM fracture energy value is well described by the Perturbed Ligament Model, which gives a single value of the fracture energy which corresponds to a virtual unperturbed specimen. 3. The size effect fitting procedure gives results of the parameters of the models in good agreement with what was theoretically predicted. 4. The size effect fitting value of the fracture energy for Bazant’s model, for Shah’s model and for cohesive crack with linear softening is much less than that given by perturbed ligament extrapolation. 5. The cohesive crack model with quasi exponential softening gives a fracture energy value consistent within 13% with the perturbed ligament extrapolation. Concordance could be enhanced by using a softening curve with a longer tail, in concordance with experimental results for softening curves which display an extremely large critical crack opening. Acknowledgements. The authors gratefully acknowledge the help supplied by G.V. Guinea in processing experimental data. They also acknowledge partial financial support from U.S.-Spain Joint Committee for Science and Technology under Cooperative Research Projects 83/071 and by CICYT, Spain, under grant PB-86–0494. References Bazant, Z.P. (198.4), “Size Effect in Blunt Fracture: Concrete, Rock, Metals”, J. Eng. Mech., ASCE, 110, pp. 518–538. Bazant, Z.P. and Pfeiffer, P.A. (1987), “Size Effect Method for Determining Fracture Energy of Concrete from Maximum Loads of Specimens of Various Sizes”, Proposal for RILEM Recommendations. Elices, M. and Planas, J. (1989), “Material Models”, Chap. 3 in Fracture Mechanics of Concrete Structures: From Theory to Applications, (Elfgren, L. Ed.), Chapman and Hall, pp. 16–66. Hillerborg, A., Modeer, M. and Petersson, P.E. (1976), “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements”, Cement Concr. Res., 6, pp. 773–782.

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Hillerborg, A. (1985), “The Theoretical Basis of a Method to Determine the Fracture Energy GF of Concrete” Materials and Structures, 18, pp. 291–296. Hillerborg, A. (1988), “Existing Method to Determine and Evaluate Fracture Toughness of Aggregative Materials: RILEM Recommendation on Concrete”, in Fracture Toughness and Fracture Energy; Tests Methods for Concrete and Rock (Mihashi, H. Ed.). Jenq, Y.S. and Shah, S.P. (1985), “A Two Parameter Fracture Model for Concrete”, J. Eng. Mech. ASCE, 111, pp. 1227–1241. Karihaloo, B.L. and Nallathambi, P. (1988) “Notched Beam Test: Mode I Fracture Toughness”, report to RILEM TC 89-FMT, April 1988, pp. 1–82. LLorca, J., Planas, J. and Elices M. (1989), “On the Use of Maximum Load to Validate or Disprove Models of Concrete Fracture Behaviour” in Fracture of Concrete and Rock: Recent Developments, (S.P. Shah, S.E. Swartz and B. Baw Eds.) Elsevier Science, pp. 357–368. Petersson, P.E. (1981), “Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials” Report TVBM 1006, University of Lund, Sweden. Planas, J. and Elices, M. (1986a), “Towards a Measure of GF: An Analysis of Experimental Results”, in Fracture Toughness and Fracture Energy of Concrete, (Wittmann, F.H. Ed.), Elsevier Science, pp. 381–390. Planas, J. and Elices, M. (1986b), “Un Nuevo Método de Análisis del Comportamiento Asintótico de 1a Propagación de una Fisura Cohesiva en Modo I”, Anales de Mecánica de 1a Fractura, No. 3, pp. 219–227. Planas, J. and Elices, M. (1987), “Asymptotic Analysis of the Development of a Cohesive Crack Zone in Mode I Loading for Arbitrary Softening Curves”, Proceedings SEM-RILEM International Conference, Houston. Planas, J. and Elices, M. (1988a), “Conceptual and Experimental Problems in the Determination of the Fracture Energy of Concrete” in Fracture Toughness and Fracture Energy; Test Methods for Concrete and Rock, (Mihashi, H. Ed.). Planas, J. and Elices, M. (1988b), “Size Effect in Concrete Structures: Mathematical Approximations and Experimental Validation” Proc. France-USA Workshop on Strain Localization and Size Effect due to Cracking and Damage, Cachan, Paris, France, September 6–9. Planas, J. and Elices, M. (1988c), “Fracture Criteria for Concrete: Mathematical Approximations and Experimental Validation” Proc. Int. Conf. on Fracture and Damage of Concrete and Rock, Vienna, Austria, July 4–6. Also in Engineering Fracture Mechanics, in press. Planas, J. and Elices, M. (1990), “A Nonlinear Analysis of a Cohesive Crack”, To be published in J. Mech. and Phys. of Solids.

9 GENERAL METHOD FOR STABILITY ANALYSIS OF STRUCTURES WITH GROWING INTERACTING CRACKS Z.P.BAŽANT Department of Civil Engineering, Northwestern University, Evanston, Illinois, USA

ABSTRACT To analyze stability of equilibrium of an inelastic structure, determine bifurcation states and find the stable equilibrium path, the tangential stiffness matrices for the structure associated with various combinations of loadings and unloadings needs to be determined. The purpose of this brief paper is to present a general new method for obtaining the tangential stiffness matrices for a structure with cracks that are interacting and growing during deformation. Applications are relegated to a subsequent paper. 1 Introduction The present anniversary volume dedicated to Professor Arne Hillerborg offers a welcome opportunity to the writer to present a new general concept for stability analysis of structures with growing interacting cracks, and thus pay homage to this great pioneer in concrete fracture mechanics and leading innovator. A particular characteristic of concrete structures is that they contain many large cracks which interact, not only under overload but also in service stress states. In some situations, correct prediction of the stability of the growth of the cracks and their interaction is necessary for predicting the structural response and the type of failure. The problem of elastic bodies with many cracks has been studied by many investigators, e.g. Budianski and O’Connell (1976) and Budianski and Honig. Recently, Kachanov (1985, 1987) and others focused attention on bodies whose cracks grow during deformation. However, a general method of calculating the tangential structural stiffness matrix of structure whose cracks grow and interact during deformation seems to be unavailable at present. The tangential stiffness matrix is needed for the analysis of stability and identification of the stable path in the manner recently presented by Bazant (1988a, b).

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2 Calculation of tangential stiffness matrix Consider a structure whose state is characterized by displacements qi (i= 1,…, n). The structure also contains N cracks of length ai (i=1,…, N). The cracks are initially at a state of propagation in mode I or II or III. Some cracks, with coordinates a1,…, am , may be growing (m≤N) and the remaining ones, am+1, … , aN may be unloading. The condition that the initial state is a state of propagation may be written as ∂II/∂ak=-Rk(ak) where II is the potential energy of the structure and Rk(ak) is the given R-curve of crack number k (Rk=Gf=constant for linear elastic fracture mechanics). The condition that the cracks remain propagating are δ (∂II/∂ak)=-δRk(ak) where δ is a total variation. This yields, (1) or (2) in which (3) Assuming that the inverse, Ψij, of matrix Φij exists, Eq. (2) may be solved for δaℓ: (4) with (5) The total force changes at propagating cracks are (6) or (7) where ∂fi /∂qj=Ktij=∂2II/∂qi∂qj=current secant elastic stiffness matrix of the structure damaged by cracks, and ∂II/∂qi=fi=force associated with qi. Sustituting now Eq. (4), we obtain a force-displacement relation that may be written in the form

GENERAL METHOD FOR STABILITY ANALYSIS

119

(8) in which (9) or (10) This is the tangential stiffness matrix at growing cracks. As we see, it is symmetric. The second-order work is (11) From this, stability and the post-bifurcation path may be determined (Bażant 1988a, b). Eq. (10) gives admissible values only for such vector of δqi directions for which: (1) δak≥0 for all k=1, .., m, Eq. (4), and (2) δ(∂II/∂ak)≥0 for all k=m+1,…, N or (12) If these conditions are not satisfied, one must analyze other choices of the cracks assumed to propagate, either with the same total number m (but the cracks renumbered), or with a different number m. To characterize the entire surface δ2W as a function of δqi, one must try successively m=N, N-1,…,1,0, and check also all the possible crack numberings for each number m. The second derivatives of II at ai and qi in Eq. (2), (3) and (9) can be approximated by finite differences after II is calculated by finite elements for ai+∆ai, ai−∆ai, qi+∆qi, qi−∆qi. On the basis of δ2W or Ktij , stability and critical states can be analyzed by the methods described in Bazant (1988a, b). Sometimes it is convenient to calculate the partial derivatives of the potential energy with respect to crack lengths in terms of the stress intensity factor Ki of crack i, using the relation (13) in which E=Young’s elastic modulus, if plain stress situation is assumed Consequently, we must have (14)

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ANALYSIS OF CONCRETE STRUCTURES

(15) Aplications in stability analysis of structures with interacting cracks will appear separately (Bažant— Tabbara, 1989). 3 Concluding remark From the preceding formulation it may be noted that the tangential stiffness matrix can relatively easily be evaluated from the second partial derivatives of the potential energy of the structure with respect to both the displacements and the crack length. These derivatives can be evaluated in general with an elastic finite element code. In some cases, the formulas for the stress intensity factor are available in handbooks, e.g. Tada et al. (1985). ACKNOWLEDGEMENTS.- Partial financial support under AFOSR Contract F49620–87-C-0030DEF with Northwestern University is gratefully acknowledged. Applications in finite element programs are conducted under NSF grant MSM-8700830 to Northwestern University. Thanks are due to M.T. Tabbara, graduate research assistant at Northwestern University, for some useful critical suggestions. References Bazant, Z.P. (1988a), “Stable States and Paths of Structures with Plasticity or Damage”, ASCE Journal of Engineering Mechanics 114(12), 2013–2034. Bazant, Z.P. (1988b), “Stable States and Stable Paths of Propaga tion of Damage Zones and Interactive Fractures”, Preprints, France-U.S. Workshop on “Strain Localization and Size Effect due to Cracking and Damage”, J. Mazars and Z. P. Bazant, editors (held at E.N.S., Cachan, France), paper No. 3A1 ; also “Cracking and Damage”, J. Mazars and Z.P. Bažant, editors, Elsevier, London, 1989, 183–206. Bazant, Z.P. , and Tabbara, M.J. (1989), “Stable States and Paths of Structures with Interacting Cracks”, Report, Center for Concrete and Geomaterials, Northwestern University, Evanston. Budianski, B., and O’Connel, R.H. (1976), “Elastic Moduli of a Cracked Solid”, International Journal of Solids and Structures 12, 81–97. Tada, H., Paris, P.C., and Irwin, G.R. (1985), “The Stress Analysis of Cracks Handbook”, 2nd edition, Paris Productions, St. Louis. Kachanov, M. (1987), “Elastic Solids with Many Cracks: a Simple Method of Analysis”, International Journal of Solids and Structures 23(1), 23–43. Kachanov, M. (1985), “A Simple Technique of Stress Analysis in Elastic Solids with Many Cracks”, International Journal of Fracture 28, R11-R19.

10 USE OF THE BRITTLENESS NUMBER AS A RATIONAL APPROACH TO MINIMUM REINFORCEMENT DESIGN C.BOSCO, A.CARPINTERI, P.G.DEBERNARDI Department of Structural Engineering, Politecnico di Torino, Torino, Italy

Abstract The behaviour of reinforced concrete elements with a low steel percentage is described, based on an experimental investigation, carried out on 60 initially uncracked R.C. beams, by means of three-point bend tests. Fracture mechanics is able to represent the actual behaviour of a cross-section of a reinforced concrete beam in flexure, using a simple model which gives theoretical indications in agreement with the experimental results. The brittleness number, NP, is then defined and, for each concrete grade, this dimensionless parameter makes it possible to determine the minimum steel percentage to be provided. The relationship which is established in this manner, calls for decreasing minimum steel percentages with increasing beam depths. 1 Introduction The behaviour of concrete elements with a low steel percentage may differ to a significant extent from the ideal scheme which is usually adopted, in which the moment-curvature diagram of an element in flexure takes on the shape illustrated in Fig. 1. This difference has already attracted attention in the years from 1950 to 1960, when steel yielding moment values exceeding (by as much as 35%) the theoretically calculated values, were experimentally observed. This is hard to explain, all the more so as the phenomenon is obviously seen to occur when the steel yield limit is reached and the evaluation of the internal lever arm displays a modest degree of indeterminacy, due to the low steel percentage. The phenomenon can be justified by referring to a number of aspects which are usually neglected and yet play a significant role in structures with a low reinforcement percentage. Namely, the behaviour of concrete in tension and the stresses that are trasmitted across the crack. The proposal for the next edition of the CEB Model Code (1990) takes into account both the aspects and suggests the schemes shown in Fig. 2. The opening of the crack wc, as defined in Fig. 2b, is a function of the maximum aggregate diameter and W1 is a function of fracture energy GF. This approach is used not only to provide a numerical explanation to hyperstrength values, but also to describe the intermediate stages of the experimental behaviour, which is characterized by a local maximum moment MA at the crack formation, followed by a decrease and then by a

122

ANALYSIS OF CONCRETE STRUCTURES

Fig. 1. Bending moment versus curvature for a normally reinforced beam.

Fig. 2. (a) Stress versus strain diagram in the tensile zone of concrete; (b) Stress versus crack opening diagram in the fracture zone of concrete.

second peak MY at the steel yielding (Fig. 3). Furthermore, it should be kept in mind that the moment MA is also influenced by the non-linear behaviour of concrete. The diagrams shown in Fig. 3 differ to a considerable extent from that in Fig. 1. If the experimental tests are carried out with a direct applied load, it is possible to obtain different kinds of load-deflection or moment-rotation diagrams, which, with reference to Fig. 3, represent: (a) the behaviour of beams with a very low content of steel, which, when the moment reaches the first cracking value, cannot sustain the external load if it is not reduced. Therefore a suddend failure occurs,

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123

Fig. 3. Load-deflection diagrams obtained by controlling the tensile strain or the crack opening at the lower edge of the beam with low steel percentage.

without warning signs. Experimentally it is possible to observe the softening branch by controlling the deformation; (b) the transitional condition, when the moment of concrete cracking is equal to the moment of steel yielding; (c) the condition for which, after cracking, the beam is able to sustain further increase in applied load. Fracture mechanics is able to represent the actual behaviour of a cross-section of a reinforced concrete beam, using a simple model, which gives theoretical indications in good agreement with the experimental results. As a matter of fact, it is possible to define a dimensionless parameter, the brittleness number, which can represent different behaviours, varying both steel percentage and/or cross-section size. For different values of this parameter, it is possible to obtain qualitative diagrams representing unstable behaviour as indicated in Fig. 3 by the curve (a), or stable behaviour, as indicated by the curve (c), or transitional behaviour, as represented by the curve (b). An experimental investigation was carried out on initially uncracked R.C. beams, by means of threepoint bend tests, by controlling the tensile strain on the lower edge of the beam or, after cracking, the crack mouth opening displacement. The analysis of the experimental behaviour supplies useful indications for the determination of the minimum steel percentage to be provided in reinforced concrete structures in flexure. It is necessary, as is well known, to guarantee that the reinforcement will not reach the yield point up to the formation of the first crack (Carpinteri (1987), Levi, Bosco and Debernardi (1987)). From the foregoing, it can therefore be inferred that, in statically determined structures, where cracking is produced by applied loads, the minimum reinforcement percentage must be calculated with reference to the onset of cracking. If the structure is redundant, on the contrary, a loading redistribution takes place in the section where the crack first appears, the moment decreasing to an extent that varies depending on the overall design of the structure. The same applies also when cracking is caused by imposed deformations. The decrease of the bending moment makes it possible to calculate the minimum steel content on the basis of a reduced value of the cracking moment (Jaccoud and Charif (1986)).

124

ANALYSIS OF CONCRETE STRUCTURES

Fig. 4 Cracked reinforced beam element.

Usually, the calculation of concrete cracking moment and steel yielding moment is carried out on the assumption of a linear-elastic behaviour of the materials and by taking into account the cracked state (no tension material). As mentioned before, the theoretical results may therefore differ to a significant extent from the experimental ones, the latter being also influenced, as will be pointed out later on, by the size of the element (size-scale effect) and by the tensile strength and toughness of concrete. Based on these considerations, two series of tests were performed as described in Section 3. 2 Theoretical model Let the cracked concrete beam element in Fig. 4 be subjected to the bending moment M and to an eccentric axial force F due to the statically undetermined reaction of the reinforcement. It is known (Carpinteri (1981) and (1984)), that bending moment M* and axial force F* induce stress-intensity factors at the crack tip respectively equal to: (1-a)

(1-b) where YM and YF are given, for =a/h≤0.7, by: (1-c) (1-d) On the other hand, from Okamura, Watanabe and Takano (1973), (1975), M* and F* produce local rotations respectively equal to: (2-a) (2-b) where (3-a)

(3-b)

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125

Up to the moment of steel yielding or slippage, the local rotation in the cracked cross-section, due to the bending moment M* and to the closing force F*, is equal to zero: (4) Eq. (4) is the congruence condition giving the unknown force F. Recalling that (Fig. 4): (5-a) (5-b) eqs (2) and (4) provide: (6) where:

(7)

If a perfectly plastic behaviour of the reinforcement is considered (yielding or slippage), from eq. (6) the moment of plastic flow for the reinforcement results: (8) However, it should be observed that, if concrete presents a low crushing strength and steel a high yield strength, crushing of concrete can precede plastic flow of reinforcement. The mechanical behaviour of the cracked reinforced concrete beam section is rigid until the bending moment Mp is exceeded, i.e., φ=0 for M≤MP . On the other hand, for M›Mp the M-φ diagram becomes linear hardening: (9) Now crack growth in concrete will be examined: according to the assumptions, it is consequent to the steel plastic flow, i.e., it occurs for a bending moment MF›MP. After the plastic flow of reinforcement, the stress-intensity factor at the crack tip is given by the superposition principle: (10) Recalling eqs (1) and considering the loadings: (11-a) (11-b) the global stress-intensity factor results: (12) The moment of crack propagation MF is reached when KI=KIC:

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 5. Bending moment of crack propagation against relative crack length.

(13) or in dimensionless form: (14) where (15) fY being the steel yielding strength, KIC the concrete fracture toughness, h the beam depth and As/A the steel percentage, referred to the whole cross-sectional area A=bh. At the same time, the rotation at crack propagation is (16) The crack propagation moment is plotted in Fig. 5 as a function of the crack depth and varying the brittleness number NP. For low NP values, i.e., for low reinforced beams or for small cross-sections, the fracture moment decreases while the crack extends, and a typical phenomenon of unstable fracture occurs, while for Np›8.5 only the stable branch remains. The locus of the minima is represented by a dashed line. In the upper zone the fracture process is stable whereas it is unstable in the lower one. It is possible to summarize the behaviour of the reinforced concrete cross-section.

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127

Fig. 6. Statical scheme of complete disconnection of concrete.

(a) Rigid behaviour is available until the moment of steel yielding is reached (0≤M≤MP). The local rotation is equal to zero. (b) Linear-hardening behaviour follows when MP‹M‹MF. (c) The latter stops when crack propagation occurs (M=MF). If the fracture process is unstable, diagram M-φ presents a discontinuity and drops from MF to FP(h-c) with a negative jump. In fact, in this case, a complete and instantaneous disconnection of concrete occurs. The new moment FP(h-c) can be estimated according to the scheme in Fig. 6. The non-linear descending law: (17) is thus approximated by the perfectly plastic one: M=FP(h-c). On the other hand, if the fracture process is stable, diagram M-φ does not present any discontinuity. The transitional value of Np, NPC, is then determined from eq. (13), where we impose MF=FP (h-c), i.e. absence of discontinuity in the diagram M-φ. In dimensionless form it follows: (18) and then, recalling eq. (15): (19) It is then possible, simply on the basis of the cross-section geometrical characteristics, to separate the locus where the fracture process is unstable (Np‹Npc) from the locus where the fracture process is stable (Np›Npc). The moment-rotation diagrams are reported in Fig. 7 for five different values of the Np number =0.1 and c/ h =0.05). For Np‹Npc≈0.7, it is FP(h-c)‹MF and therefore a discontinuity appears in the M-φ diagram (Figs 7-a to c). On the other hand, for NP‹0.7, the curves in Fig. 5 lie in the unstable zone completely. In conclusion, the above presented theoretical model predicts unstable behaviour for low content of steel and/or for large beam depth. It is worth noting that the initial crack of the theoretical model was not present in the experimental beams, although a crack was formed during the loading process. In addition, the reinforcement plastic flow of the theoretical model, in practice was due to the steel bar slippage. Nevertheless, the theoretical ductile-brittle transition was confirmed by the experiments, as will be shown in the following sections.

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 7. Ductile-brittle transition in the mechanical behaviour of R.C. beams, varying the brittleness number Np. (c/h=1/ 20, a/h=0.1).

3 Specimen preparation and testing procedure Two series of experimental tests were carried out at the Department of Structural Engineering of the Politecnico di Torino. The three point bending tests on R.C. beams were realized by a M.T.S. machine. A first series of 30 reinforced concrete beams were tested in 1988, with the geometrical and mechanical characteristics reported in Table 1. The steel cover is, in each case, equal to 1/10 of the beam depth (c/h=0.1). The maximum aggregate diameter is 13 mm. For each beam size (A, B, C) and for each brittleness class (0, 1, 2, 3, 4) two R.C. beams were realized. All the beams were initially unnotched and uncracked. The results are reported in Table 1. In 1989, a supplementary investigation (2nd series) was carried out on 30 beams with three different mixtures and compressive strength of the order of magnitude of normal concrete. One of the mixtures (beams A, B, C, D in Table 2) presents a very low strength for structural applications. All three concrete mixtures have the same maximum aggregate diameter of 15 mm (allowing small-sized specimens, with concrete cover of limited dimensions). Three brittle

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129

Table 1. Description of the reinforced concrete specimens (series N. 1) and related loads of first cracking, steel yielding and final collapse BRITTLE NESS CLASS

SIZES bxh NOMINA ACTUAL (mm) L STEEL % OF CONTENT STEEL

YIELD ACTUAL LIM. OF VALUE STEEL (N/ OF NP mm2)

CRACK. LOAD (kN)

YIELD. LOAD (kN)

ULTIMAT E LOAD (kN)

11.38 12.16 11.77 11.77 12.07 12.95 13.73 13. 34 15.30 14. 52

0.00 0.00 6.43 7.61 14.71 15.70 26.98 28.90 34.14 34.88

0.00 0.00 6.01 5.55 11.77 10.79 22.48 21.64 48.79 46.83

23.08 22.03 21.40 17.66 19.50 22.19 21.19 23. 54 25.74 27. 62

0.00 0.00 10.42 10.18 23.04 23.16 40.79 42.06 64.01 65.89

0.00 0.00 6.10 5.50 17.69 16.59 57.53 55.91 77.31 75.81

44.12 36.28 36.67 – 39.97 37.49 45.50 40.79 49.03 48.83

0.00 0.00 15.69 – 33.08 31.64 55.69 52.16 87.66 81.18

0.00 0.00 8.40 – 25.50 23.29 66.54 63.46 99.18 96.12

(a) Beam size A 0 150×100

0

0.000

–

0

1

150×100

1ø4

0.085

637

0. 097

2

150×100

2ø5

0.256

569

0.261

3

150×100

2ø8

0.653

441

0.514

4

150×100

2ø10

1.003

456

0.847

(b) Beam size B 0 150×200

0

0.000

–

0

1

150×200

1ø5

0.064

569

0.093

2

150×200

3ø5

0.190

569

0.275

3

150×200

3ø8

0.490

441

0.550

4

150×200

3ø10

0.775

456

0.898

(c) Beam size C 0 150×400

0

0.000

–

0

1

150×400

2ø4

0.043

637

0.098

2

150×400

4ø5

0.128

569

0.261

3

150×400

4ø8

0.327

441

0.521

4

150×400

4ø10

0.517

456

0.847

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ANALYSIS OF CONCRETE STRUCTURES

Table 2. Dimensions, span, steel content and brittleness number of the beams of series N. 2, for each concrete grade BEAM TYPE

GRADE OF CONCR ETE

THICK. (mm)

DEPTH (mm)

(a) Beam size A A1 I 150 100 A2 I 150 100 A3 I 150 100 AA1 II 150 100 AA2 II 150 100 AA3 II 150 100 AAA1 III 200 100 AAA2 III 200 100 (b) Beam size B B1 I 150 200 B2 I 150 200 B3 I 150 200 BB1 II 150 200 BB2 II 150 200 BB3 II 150 200 BBB1 III 200 200 BBB2 III 200 200 (c) Beam size C C1 I 150 400 C2 I 150 400 C3 I 150 400 CC1 II 150 400 CC2 II 150 400 CC3 II 150 400 CCC1 III 150 400 CCC2 III 150 400 (d) Beam size D D1(*) I 200 800 D2 I 200 800 D3 I 200 800 DD1 II 200 800 DD2 II 200 800 DD3 II 200 800 (*) Damaged during casting operations

SPAN (mm)

As

As /A (%)

Np

CRACK. YIELD. LOAD LOAD (kN) (kN)

ULTIM. LOAD (kN)

600 600 600 600 600 600 600 600

1ø5 2ø5 3ø5 1ø5 2ø5 3ø5 1ø5 2ø5

0.131 0.261 0.392 0.131 0.261 0.392 0.098 0.197

0.151 0.301 0 .452 0.148 0.297 0.445 0.110 0.220

5.95 5.87 5 .76 11.26 11.10 11.25 10.62 9.75

8.37 12.76 21 .43 11.49 19.80 27.40 8.75 17.50

8.95 16.29 24.04 11.49 21.21 29.23 7.50 17.50

1200 1200 1200 1200 1200 1200 1200 1200

1ø6 2ø6 3ø6 1ø6 2ø6 3ø6 1ø6 2ø6

0.094 0.188 0.283 0.094 0.188 0.283 0.070 0.141

0 .120 9 .67 0 .237 9 .80 0 .356 9 .30 0.117 11.34 0.234 14.05 0 .351 12 .11 0 .086 16 .75 0 .172 18 .95

9 .73 15 .69 24 .44 10.12 17.69 24 .91 9 .75 20 .12

10.99 20.33 28.89 12.26 23.35 33.00 11.25 24.70

2400 2400 2400 2400 2400 2400 2400 2400

1ø8 2ø8 3ø8 1ø8 2ø8 3ø8 1ø8 2ø8

0.083 0.167 0.250 0.083 0.167 0.250 0.083 0.167

0.145 0.291 0.436 0 .144 0.287 0.431 0.141 0.283

15.88 18.46 19.98 19 .94 21.73 18.75 18.25 19.62

17.69 32.21 47.59 20 .06 31.31 47.50 19.00 28.75

20.23 37.20 52.69 23.65 37.99 55.94 23.10 35.00

4800 4800 4800 4800 4800 4800

1ø10 0.049 2ø10 0.098 3ø10 0.147 1ø10 0.049 2ø10 0.098 3ø10 0.147

0 .116 – 0 .231 30 .05 0 .346 44 .27 0.113 35.24 0 .227 43 .37 0 .340 45 .60

– 44 .05 63 .92 27.78 45 .33 60 .60

– 51.48 75.45 29.10 44.40 97.65

USE OF THE BRITTLENESS NUMBER

131

Fig. 8. View of the four beam size-scales.

ness classes were selected: NP˜0.11–0.15, NP˜0.23–0.30, NP˜ 0.35–0.45. The mechanical characteristics of the three mixtures of concrete are given in Table 3, where fcm is the mean cylindrical compressive strength, Ec the Young’s modulus and KIC the critical value of stress intensity factor, which can be obtained by means of the relationship: (20) Table 3. Mechanical and toughness characteristics of the three concrete grades considered in the second investigation. fcm EC GF KIC

(N/mm2) (N/mm2) (N/mm) (N/mm3/2)

Class I

Class II

Class III

16.63 21000 0.143 54.80

29.38 23150 0.134 55.70

31.96 23500 0.137 56.74

The concrete fracture energy GF is determined on three specimens for each grade of concrete, according to the method specified in the RILEM draft recommendation: “Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams”. The value of the normal elastic modulus is determined according to the method specified in ISO 6784. A general view of the four beam size-scales is given in Fig. 8. 4 Experimental results and discussion The load-deflection diagrams, for the first series of beams, are plotted in the Figs 9-a, b and c, for each beam size and by varying the brittleness class (each curve is related to a single specimen of the two considered). As is possible to verify in Table 1, the peak or first cracking load is decidedly lower than the steel yielding load only in the cases 3 and 4, i.e. for high brittleness numbers NP. In the cases 0 and 1, the opposite result is clearly obtained. On the other hand, case 2 demonstrates to represent a transitional condition between hyperstrength and plastic collapse, the two critical loads being very close. Therefore, the same brittleness transition theoretically predicted in Fig. 7, is reproposed by the experimental diagrams in Figs 9.

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 9. Load versus deflection diagrams for the first series of beans. (a) bean depth h=100 mm; (b) bean depth h=200 mm; (c) bean depth h=400 mm;

Specimen C0 (h=400 mm, no reinforcement) presents an evident snap-back behaviour, the softening branch assuming even a positive slope. It was possible to follow such a branch, since the loading process was controlled by a monotonically increasing function of ti me, i.e. the crack mouth opening displacement. If the controlling parameter had been the central deflection, a sudden drop in the loading capacity and an unstable and fast crack propagation would have occured. The dimensionless bending moment versus rotation diagrams are plotted in the Figs 10–a to e, for each brittleness class and by varying the beam size. The local rotation is non-dimensionalized with respect to the value 90 recorded at the first cracking, and is related to the central beam element of length equal to the beam depth h. The bending moment, on the other hand, is non-dimensionalized with respect to concrete fracture toughness KIC and beam depth h, see eq. 14.

USE OF THE BRITTLENESS NUMBER

Fig. 10. Dinensionless bending nonent versus rotation diagrans for the first series of beans. Brittleness number: NP≈0.00 (a); NP≈0.10 (b); NP≈0.26 (c); NP≈00.53 (d); NP≈0.87 (e). Beam depth: (A) h=100 mm; (B) h=200 mm; (C) h=400 mm.

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Fig. 11. Load versus deflection diagrams for the beams of type II (2nd series).

The diagrams in Fig. 10 are significant only for φ/φ0›1, the strain softening and curvature localization occurring only after the first cracking. The dimensionless peak moment does not appear to be the same, when the brittleness class is the same and the beam depth is varied. This is due to the absence of an initial crack or notch. On the other hand, the post-peak branches are very close to each other and present the same shape for each selected brittleness class. The size-scale similarity seems to govern the post-peak behaviour, specially for low brittleness numbers NP (class 0, 1, 2, 3), and for large beam depths h (sizes B, C). In these cases, in fact, it is very likely that the fracture process zone is negligible with respect to the zone where the stress-singularity is dominant, so that the Linear Elastic Fracture Mechanics model (and the nondimensionalization in Figs 10) is consistent with the experimental phenomena. The experimental results obtained from the second series of 30 beams, are summarised in Table 2. The loaddeflection diagrams for the beams of type II with the lowest brittleness number, are reported in Fig. 11, which, with the help of Table 2, shows how to determine the transitional NPC value, for specimens AA1, BB1, CC1. In particular, the transitional values NPC≈0.12 for concrete I (fcm=16.63 N/mm2) and NPC≈0.14 for concrete II (fcm=29.38 N/mm2) are obtained. The results obtained for concrete III (fcm=31.96 N/mm2) give confirmation for the above range of NPC values. The value NPC≈0.26 was observed in the course of the earlier investigation on high strength concrete beams (fcm=75.70 N/mm2). The dimensionless moment versus rotation diagrams are reported in Fig. 12, for beams of type I and II. It should be noted that, for each concrete grade, the transitional value of the brittleness number, NPC, is markedly constant by varying the beam depth h. The minimum reinforcement percentage can then be determined on the basis of this parameter. Therefore, the latter should be defined as accurately as possible, in relation to concrete grade. To this purpose, we can refer to the diagram shown in Fig. 13, which indicates the variation in the transitional number NPC against the compression strength of concrete. 5 Conclusions When a small percentage of steel is required to reinforce high strength concrete, the crushing failure of the beam edge in compression is usually avoided. On the other hand, one or more cracks originate at the beam

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Fig. 12. Dimensionless bending moment versus rotation diagrams for the beams of the 2nd series. Beam type I: brittleness class NP≈0.11–0.15 (a); NP≈0.23–0.30 (b); NP≈0.35–0.45 (c). Beam type II: brittleness class NP≈0.11–0.15 (d); NP≈0.23–0.30 (e); NP≈0.35–0.45 (f).

edge in tension and the material is so brittle in this case that the size of the crack tip process zone is very small if compared with the size of the zone where the stress singularity field is dominant. For these reasons, it is reasonable to sustain that a Linear Elastic Fracture Mechanics model is able to capture the most

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Fig. 13. Transitional brittleness number NPC against the compression strength of concrete.

Fig. 14. Minimum steel percentage against the beam depth h.

relevant aspects and trends in the mechanical and failure behaviour of low reinforced high strength concrete beams in flexure. Such a conclusion seems to be tendentially valid also when normal strength concrete is tested (beams of the second series). Remarkable size-scale effects are theoretically predicted by using the model described above and experimentally confirmed. The brittleness of the system increases by increasing size-scale and/or decreasing steel area. On the other hand, a physically similar behaviour is revealed in the cases where the nondimensional number NP is the same. From the described investigations and theoretical studies (Bosco, Carpinteri and Debernardi (1988), (1989)1 and (1989)2), (Hillerborg (1988)), the demand transpires to analyze the post-peak and ductile behaviour of low reinforced high strength concrete beams, through the concepts of Fracture Mechanics. As is

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demonstrated in the present paper, the possibility of extrapolating predictions from small to large scales, is entrusted to the non-dimensional (brittleness) number NP where, in addition to the traditional geometrical and mechanical parameters, even the concrete fracture toughness KIC, or the concrete fracture energy GF, appears. Moreover, if we take into account the specific problem of the minimum reinforcement and we consider that, for direct applied loads, the minimum amount can be defined when the yielding moment is equal to the first cracking one, we can identify the transitional values of the brittleness number, NPC. Fig. 14 represents the minimum reinforcement percentage, As/A, as a function of beam depth h for the materials used in the testing program. These percentages are compared to the values required by Eurocode 2 (1988) and those prescribed by the American Concrete Institute (ACI 318–83) (1983). While the values supplied by both the standards (Szalai (1988)) are independent of the beam depth h, the brittleness number NPC makes it possible to determine the minimum steel percentage in relation to h. The relationship which is established in this manner, calls for decreasing minimum steel percentages with increasing beam depths. The authors, at the moment, are carrying out further experimental investigations on beams with concrete grade 40–50 N/mm2, to determine an additional point to be inserted in Fig. 13. References Comitė Euro-International du Bėton (C.E.B.)., Model Code 1990. Carpinteri, A. (1987) Catastrophical softening behaviour and hyperstrength in low reinforced concrete beams. 25th CEB Plenary Session, May 11–13, Treviso (Italy), pp. 97–116. Levi, F., Bosco, C. and Debernardi, P.G. (1987) Two aspects of the behaviour of slightly reinforced structures. 25th CEB Plenary Session, May 11–13, Treviso (Italy), pp. 37–50. Jaccoud, J.P. and Charif, H. (1986) Armature minimal pour le contrôle de 1a fissuration. Rapport final des essais sėrie «C», Publication IBAP n.114, Ecole Polytechnique Federale de Lausanne. Carpinteri, A. (1981) A fracture mechanics model for reinforced concrete collapse. IABSE Colloquium on Advanced Mechanics of Reinforced Concrete, Delft, pp. 17–30. Carpinteri, A. (1984) Stability of fracturing process in RC beams. Journal of Structural Engineering (ASCE), 110, pp. 544–558. Okamura, H., Watanabe K. and Takano T. (1973) Applications of the compliance concepts in fracture mechanics. ASTM STP, 536, pp. 423–438. Okamura H., Watanabe K. and Takano T. (1975) Deformation and strength of cracked members under bending moment and axial force. Engineering Fracture Mechanics, Vol. 7, pp. 531–539. Hillerborg, A. (1988) Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams. Presented at the International Conference on Fracture and Damage of Concrete and Rock, July 4–6, Vienna (Austria). Bosco, C., Carpinteri, A. and Debernardi, P.G. (1988) Fracture of reinforced concrete: scale effect and snap-back instability. Presented at the International Conference on Fracture and Damage of Concrete and Rock , July 4–6, Vienna (Austria). Bosco, C., Carpinteri, A. and Debernardi, P.G. (1989) Effetto scala sul progetto di elementi in calcestruzzo ad alta resistenza (In italian). Giornate A.I.C.P. ’89—Napoli, 4–6 maggio, pp. 81–94. Bosco, C., Carpinteri, A. and Debernardi, P.G. (1989) Size effect on the minimum steel percentage for reinforced concrete beams. International Conference on Recent Developments on the Fracture of Concrete & Rock, September 20–22, Cardiff (Wales), pp. 672–681. Eurocode N. 2 (1988) Design of concrete structures, Part 1, Final Draft, Prepared for the Commission of the European Communities, December. American Concrete Institute (1983) Building Code—Requirements for Reinforced Concrete (ACI 318–83), Detroit, Michigan. Szalai, K. (1988) Principle of dimensioning of slightly-reinforced concrete structures. Lectures presented at the 26th CEB Plenary Session, September 20–23, Dubrovnik, pp. 119–134.

11 FRACTURE MECHANICS ANALYSES USING ABAQUS K.GYLLTOFT National Swedish Testing Institute, Borås, Sweden

ABSTRACT The finite element program ABAQUS is used in fracture mechanics studies. Both a discrete and a smeared approach were used. The examples studied were bending of a plain concrete beam and pull-out of an anchor bolt. Also a road bridge which will be studied in the near future was briefly mentioned. INTRODUCTION In the past, various finite element programs have been used in fracture mechanics studies, see eg Gylltoft 1983 [1]. In the work presented here, the finite element program ABAQUS [2] has been used. Only standard elements and standard material properties offered by the program have been used in the analyses. OPTIONS IN ABAQUS The options in ABAQUS used in these analyses concerning fracture mechanics studies of reinforced concrete are: (1) using nonlinear springs and (2) using a concrete model. (1) In the first method, the crack or the fracture process zone is modelled by using nonlinear springs (a discrete approach). The springs are acting per pendicular to the crack when modelling a modus-I crack and parallel to the crack when modelling a modus-II crack. The nonlinear force-displacement relation for the springs in the modus-I case is determined from the fracture energy according to the model of Hillerborg [3], see Fig. 1. (2) In the second method, cracking is included in the material behaviour in the “concrete” option of ABAQUS (a smeared approach). Cracking in tension is assumed to occur when the stress reaches a failure surface, a “crack detection surface”. This failure surface is taken to be a simple Coulomb line written in terms of the equivalent pressure stress and the Mises equivalent deviatoric stress. Once cracking is defined to occur, the orientation of the cracks is stored, and oriented, damaged elasticity is then used to model the existing cracks. Other cracks in the same point are restricted to be in orthogonal directions [2].

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Fig. 1 Nonlinear spring force-relative displacement relationship

The energy release of Hillerborg [3] can be incorporated into the material model as a strain softening behaviour, see the “tension stiffening” shown in Fig. 2. The slope of the strain softening line should be chosen based on this energy release argument, using a volume associated with the integration point [2]. The “tension stiffening” is formulated as a stressstrain relationship of the material. This means, however, that the material model is not definable until the mesh is chosen (the constitutive relations are meshdependent). According to personal communication with the “ABAQUS people”, this shortcoming will be changed in the next version of ABAQUS. EXAMPLES STUDIED Bending of a plain concrete beam The bending failure in a notched beam of plain concrete has been studied, see Fig. 3. The fracture starts at the notch tip and propagates up through the beam section. In one series of analyses the fracture process zone was modelled by nonlinear spring elements with the force-relative displacement relationship according to Fig. 1. In Fig. 4, the force-deflection relation for the beam is shown. In Fig. 5, the stress distribution of the horizontal stresses at a certain load level is shown just as an example. In another series of tests the same beam was analysed by using the concrete model in ABAQUS discribed above, see Fig. 2. Similar results were obtained. Anchor bolt Also an anchor bolt was analysed. The concrete material option (2) in ABAQUS was used. Symmetry of rotation was used and the studied body is shown in Fig. 6. Unfortunately the size of it was not “standard”!

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Fig. 2 “Tension stiffening” model [2]

Fig. 3 Bending of a plain concrete beam

Futhermore, uptil now, no successful results has been obtained! This could maybe depend on errors in the program. Bridge tests A full scale bridge test of a bridge in Stora Höga outside Göteborg has also been conducted. The test was performed during the spring in collaboration between the Swedish Road Administration, Chalmers University of Technology and the Swedish National Testing Institute. In the autumn an additional bridge test will be conducted.

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Fig. 4 Force-deflection relation

Both a bending test and a shear test were performed. Modal analysis was performed in connection with the bending failure. A successively increasing degree of damage was compared with the change of the modal parameters. The bending failure will also be studied using fracture mechanics analyses. Whether fracture mechanics will be used also concerning the shear failures or not will be decided later. CONCLUSIONS So far, the use of the finite element program ABAQUS seems to be useful solving fracture mechanics problems. However, thework has just been started and the success for future fracture mechanics applications will depend on the continued development of the program in this respect. ACKNOWLEDGEMENT The work presented here is supported by the Swedish Council for Building Research and the Swedish National Testing Institute. REFERENCES 1 2

Gylltoft, K., “Fracture mechanics models for fatigue in concrete structures”, Luleå. University of Technology, Division of Structural Engineering, Doctoral Thesis 1983:25D, Luleå 1983, 210 pp. ABAQUS, HKS Hibbitt, Karlsson & Sorensen Inc., Providence, Rhode Island 02906 USA

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Fig. 5 Horizontal stress distribution at a certain load 3 Hillerborg, A., Modéer, M and Petersson, P-E., “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements”, Cement and Concrete Research, Vol.6, 1976, pp. 773–782

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Fig. 6 Anchor bolt. Studied example

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12 DESIGN AND CONSTRUCTION OF CONCRETE DAMS UNDER CONSIDERATION OF FRACTURE MECHANICS ASPECTS H.N.LINSBAUER Institut für Konstruktiven Wasserbau, Technische Universität Wien, Wien, Austria

1 Introduction A multitude of extensive damage in concrete dams may be attributed to three characteristic types of influence: – Inadequate design and construction (notches, corners, jagged dam-foundation interface, abrupt transition of rock and concrete bond, etc.) – Volumetric change (shrinkage, creep, hydration process effects, restraint, chemical incompatibility of concrete components, etc) – Stresses due to unusual load and reaction (earthquake, excessive change of climate, discontinuos supporting conditions, etc.) Some of these influences (e.g. sequence of concrete casting correlated to climatic conditions, dam foundation interaction behavior) are not easy to predict by analysis and may be left up to the engineers experience and intuition. However, certain sources of cracking initiation in concrete dams may be recognized at first sight by the use of fracture mechanics principles which make it possible to analyze stress situations in regions where the conventional strength of material critera fail. Within this contribution some typical examples of undesirable constructive design of concrete dams (mostly based on actual cases of damage) are presented and discussed from the point of view of fracture mechanics. 2 Characteristic examples of faulty design Typical crack initiation locations may be seen in regions which are subjected to stress singularities in the form of notches or (reentrant) corners respectively:

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Fig.1 Cracking caused by earthquake effect a) Geometry b) Crack pattern c) Fracture mechanics investigation /2/

2.1 Reentrant corner on the downstream side (crown area) A classic example showing consequences of a reentrant corner for cracking is represented in the Koyna dam damage /1/ where, a priori, no cause for concern was given in view of normal load considerations: During a severe earthquake of magnitude 6.4–7.0 (Richter scale) cracking with leakage on the downstream side accompanied by an upstream side crack band within the region of the sharp reentrant headdam-transition was observed (Fig.1a, b). An intensified effect was given by the unusual design of the upper part of the dam acting as an oscillatory mass. Fracture mechanics investigations /2/ showed good agreement with the chronological process of cracking during the action of earthquake (Fig.1c). 2.2 Reentrant corner at the heel of the dam In case of deep cutouts as a result of the topography or weak rock quality the designer may be tempted to use the excavated rock slope as an equivalent for the dam mould, thereby producing a reentrant corner at the end of the dam rock interface (Fig. 2a). From a fracture mechanics point of view this reentrant corner produces a stress singularity with high intensity far beyond the tensile strength and, as a consequence, leads to crack initiation and crack propagation as shown in Fig.2b /3/. The Mode 1 Stress Intensity Factor KI representing the opposite term within the fracture criterion is nearly ten times higher than the corresponding value of the material KIC (fracture toughnes of dam concrete). The progressive growth of the Stress Intensity Factor (Fig. 2c) may end up in a complete separation of a wedge-shaped part of the upstream dam body possibly accompanied by an opening of the dam base or a crack spreading into the rock foundation.

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Fig.2 Reentrant corner at the heel of the dam

2.4 Inadequate treatment of the dam foundation area Extensive geological explorations of the proposed site are an essential requirement for design and reliability of dam structures. But a closer insight into the real foundation circumstances will be gained only during excavation. During this construction stage a refined treatment of the foundation area including smoothening of the jagged excavation profile, rock grouting, concrete sealing of more or less widened rock joints, etc. is performed. Disregarding some of these potential of cracking initiation mechanisms may lead to severe damage of the structure. Crack initiation caused e.g. by foundation joint slipping (Fig.3a) leads to an upstream side oriented crack propagation (Figs. 3b, c) as confirmed in a fracture mechanics investigation /3/.

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Fig.2 Reentrant corner at the heel of the dam

Fig. 3 Slip along a foundtion joint

Similar mechanisms are present in the case of an extremely jagged dam foundation interface (Fig. 4a). This is theoretically comparable to a vertical oriented starter crack. A demonstration of the crack path development was performed by a photoelastic experiment /4/ (see Fig. 4b and Fig. 4c). 3 Consequences of analytically non detectable cracking initiation (what if…) Recent events of cracking have given rise to the urgent question whether or not the current rules and standards for the design of dams are sufficient?

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Fig. 3 Slip along a foundation joint

In one case, in order to avoid any cause for cracking, the design was conceived to allow no tensile stress development at the upstream side of the arch dam. This was achieved by the construction of a base joint for the purpose of eliminating any restraint in the heel region of the dam base during the filling of the reservoir. Additional measures were provided by shifting the grout curtain away from the dam foundation interaction region by means of an apron (Fig.5a) with an obvious positive side effect of an uplift-free dam base.

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Fig. 4 Cracking initiation caused by foundation disconinuity (photoelastic investigation) /4/

In spite of all this an initial crack created by mechanisms which have not yet been understood was driven bv the penetrating water up to the middle of the dam. This was made possible only by the lack of opposite uplift pressure. Fracture mechanics investigations /5/ based on a fictive starter crack showed an adequate crack propagation beha vior (Fig.5b) but in this case were unable to predict any kind of crack initiation.

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Fig. 4 Cracking initiation caused by foundation discontinuity (photoelastic investigation) /4/

Fig. 5 Dam heel with upstream base joint

Such a case shows that, in future, inclusion of a “what if…” analysis seems to be justified.

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4 Conclusions The development of crack-like defects in concrete dams is a well-known phenomenon which is manifested in a number of publications in the literature. Some of the cracking initiation mechanisms may be assigned to inadequate design and construction and thus can be avoided from the beginning by the application of fracture mechanics principles as shown in the preceding chapters. REFERENCES 1 2 3 4 5

. Murthy, Y.K. and Thomas, K.C., “Recent trends in investigation, design and construction of large dams in India”, 10thICOLD, Montreal, 1970, Vol.IV, (1001–1025). . Saouma, V.E., Ayari, M.L. and Boggs, H., “Static and dynamic fracture mechanics of concrete dams”, in Fracture Mechanics of Concrete Structures (RILEM Report), Ed.:L.Elfgren, Chapman and Hall, 1989, (336–354). . Linsbauer, H.N., Ingraffea, A.R., Rossmanith, H.P. and Wawrzynek, P.A., “Simulation of cracking in large arch dam: Part I+II”, J. Struct. Eng., Vol.115, 7, 1989, (1 599– 1630). . Linsbauer, H.N. and Rossmanith, H.P., “Direction of slow stable crack-growth.A photoelastic/FE-analysis”, ÖIAZ, 131, 7/1986, (249–251). . Linsbauer, H.N., “Cracking in mass concrete-fracture mechanics assessment”, Proc.Int.Symp.on Analytical Evaluation of Dam Related Safety Problems, Copenhagen, 1, 1989, (107–117).

PART THREE BENDING

13 SIZE DEPENDENCY OF THE STRESS-STRAIN CURVE IN COMPRESSION A.HILLERBORG Division of Building Materials, Lund Institute of Technology, Sweden

Abstract The stress-strain curve has an ascending branch, a peak, and a descending branch. In the ascending branch the deformation can be expressed as a strain, i.e. a distributed relative deformation, which is independent of the gauge length. In the descending branch, also called the post-peak region, this is not the case, because the deformation more or less localizes to certain zones, whereas the material outside these zones is just unloaded. The measured strain will then depend on the gauge length. The descending branch is thus size dependent, which means that it is not a pure material property. It can be demonstrated that this size dependency is of a great importance for the stress-strain curve to be used in the analysis and design of reinforced concrete beams. The beam size is probably more important than the concrete quality for the choice of the stress-strain curve. The size dependency therefore ought to be taken into account in design rules. This is of a particular importance for high strength concrete. The size dependency of the stress-strain curve is of little importance for the moment capacity of underreinforced beams, of some importance for the balanced reinforcement ratio, and of a great importance for the ductility, expressed e.g. as rotational capacity. Keywords: Concrete, Compression, Stress-strain curve, Balanced reinforcement, High strength concrete. 1 Introduction In discussing the difference in performance between ordinary concrete and high strength concrete, the difference in compression stressstrain curves is often emphasized. Fig. 1 shows a typical example of a diagram, where the stress-strain curves for different concrete qualities are compared. It can however be argued that there does not exist any unambiguous complete stress-strain curve, including the descsending branch. A descending branch always leads to an uneven strain, with a more or less pronounced strain localization. This has to be taken into account if the stress-strain relation should be properly understood and interpreted.

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Fig.1. Typical compressive stress-strain curves, Nilson (1987)

Based on these ideas the stress-strain curve for concrete in the compression zone of a reinforced beam will be discussed. 2 The stress-strain curve 2.1 General Due to the localization, the descending branch is best described as a stress-displacement relation, not as a stress-strain relation. The general description of the deformation properties is then given by two curves, one stress-strain curve and one stress-displacement curve according to Fig. 2, where w is the additional deformation within the localized zone. The mean deformation m on a certain length L, containing one localized zone, is (1) where is taken from the unloading branch of the σ- -curve, see Fig. 2. This description is generally accepted for the tensile failure of concrete. Also in compression failure a localization will occur, even though the behaviour in that case is more complicated, due to the lateral deformations, which cause a three-dimensional state of deformations and stresses. When a complete stress-strain curve is based on Eq. (1), it has to be referred to a certain length L, which means that it is size dependent. When using the stress-strain curve for an analysis, it is necessary to determine what length L it shall be based on.

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Fig.2. The stress-displacement curve for the localized zone as part of the complete stress-strain curve for a gauge length L.

Fig. 3. Application of Fig. 2 to two of the curves in Fig. 1, with L=200 mm.

2.2 Concrete in uniaxial compression The curves in Fig. 1 are based on a gauge length of 200 mm corresponding to L in Eq. (1). The ascending parts of the curves are real stress-strain curves. The descending parts can be transformed to the corresponding stress-displacement curves for the localized zone according to Fig. 2 and Eq. (1) if we assume that the unloading curve from the peak is a straight line parallel to the tangent at the origin. The result regarding two of the curves from Fig. 1, one normal strength concrete and one high strength concrete, is shown in Fig. 3. From this figure it is evident that the stress-displacement curve is steeper for high strength concrete. It can also be seen that the ascending branch is more linear for high strength concrete, and the strain at the peak somewhat higher.

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Fig. 4. Proposed stress-strain curve, where

u

depends on the depth c of the compression zone.

Fig. 5. Strains and stresses in a beam section with balanced reinforcement ratio.

2.3 Concrete in the compression zone of a beam In the compression zone of a beam, strain localization also occurs. In the normal analysis, based on the assumption that plane sections remain plane, the formal concrete strain then must be taken as the mean strain over a certain length L in the stress direction. Some assumption therefore has to be made regarding this length. It seems natural to assume that the length L is approximately proportional to the depth of the localized zone, or to the depth of the compression zone, Fig. 5. Based on the latter assumption Hillerborg (1988) has proposed to use a stress-strain curve according to Fig. 4, assumption I, i.e. u=w1/c, where w1 is a material property, and c is the depth of the compression zone. It was demonstrated that this assumption is supported by tests regarding the rotational capacity of under-reinforced beams.

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Another possibility is to assume , assumption II. This is based on an approximate proportionality between L and the depth of the localized zone, Fig. 5. Both assumptions will be discussed and compared below. 3 Consequences for analysis and design 3.1 Balanced reinforcement ratio The balanced reinforcement ratio is easily calculated for the rectangular section in Fig. 5 on the assumption that the reinforcing steel is elastic with a modulus of elasticity ES up to the yield strength fY. It can be expressed as a mehanical reinforcement ratio (2) where AS is the steel area. From Fig. 5 it is found that (3) (4) where σm is the mean concrete compressive stress. Introducing (4) into (2) gives (5) The practical consequences are best illustrated by means of two examples. =0.002, and fY/ES=0.002, Normal strength concrete: For this case it can be assumed that corresponding to a yield strength of about 400 MPA. The values of w1 and w2 are so far uncertain. Based on some evidence a value of 3 mm is chosen for w1 and 2 mm for w2, which has to be somewhat lower than w1. The shape of the curve is assumed to consist of two parabolas according to Fig. 6. can be assumed to be higher, and is chosen to 0.003. Higher High strength concrete: The value of grade steel is often used with high strength concrete. A value of fY/ES=0.003 is chosen, corresponding to a yield strength of about 600 MPa. The values of w1 and w2 are probably lower than for normal strength concrete. A value of 1 mm is chosen for w1, and 0.5 mm for w2: The ascending branch is more linear for high strength concrete. A simple straight line is chosen according to Fig. 6. For the descending branch a parabola is assumed. Based on these assumptions it is easy to calculate how the value of varies with the beam depth. The results are shown in Fig. 7. is also shown for comparison. The value corresponding to a conventional design with From Fig. 7 it is evident that the balanced reinforcement ratio is size dependent. Assumption I regarding u gives a greater size dependency than assumption II. High strength concrete seems to give a greater size dependency than normal strength concrete for small beams. As the analyses are based on very uncertain assumptions regarding the values of w1 and w2, not too detailed conclusions may be drawn from Fig. 7. The conclusion that the balanced reinforcement ratio is size dependent, decreasing with an increasing depth, is

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Fig.6. Assumed stress-strain curves in the numerical examples, with compression zone depth c in m.

Fig. 7. Calculated relation between beam depth d and the mechanical reinforcement ratio wb for the numerical examples.

however probably true. This is important to remember when the design of a large structure is based on the tests of laboratory sized beams. 3.2 Strength of over-reinforced beams As the balanced reinforcement ratio is size dependent according to the used assumption, the strength of over-reinforced beams is also size dependent. One way of checking the validity of the assumption may therefore be by comparisons with tests on over-reinforced beams.

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3.3 Strength of under-reinforced beams The value of u is of no practical importance for the strength of under-reinforced beams. It is therefore not possible to check the validity of the assumption regarding the value of u by means of comparisons with test results regarding the strength of underreinforced beams. 3.4 Ductility The ductility of an under-reinforced beam, expressed for example as rotational capacity, is sensitive to the value of u. This fact has been used for checking the validity of the assumption u=w1/c. From this comparison it will seem that the assumption is realistic for under-reinforced beams, Hillerborg (1988). It has been shown that the rotational capacity is approximately inversely proportional to the beam depth. The ductility is of importance where design is based on the theory of plasticity, e.g. yield line theory or strip method for slabs. As slabs have normally small depths, the size dependency means that slabs have a high ductility. This is one reason why the theory of plasticity, according to a large number of tests, is well applicable to reinforced concrete slabs. The ductility is also of importance for seismic design and the seismic behaviour of structures. The size dependency of the ductility in this case means that design rules based on laboratory sized specimens my lead to unsafe designs of large size structures, if the size dependency is not taken into account. 3.5 Beams with compression reinforcement and stirrups Compression reinforcement and stirrups may change the behaviour of the compression zone in many ways. Closely spaced stirrups may give confinement to the compression zone, increasing the ultimate concrete strain and the ductility, and also increasing the concrete compressive strength. There is however also a risk that compression reinforcement and stirrups may cause spalling of the concrete cover, leading to a reduction of the ultimate concrete strain, and thus a reduction in ductility before spalling takes place. After spalling the remaining section may be rather ductile, but with a reduced moment capacity. 4 Determination of material properties The material properties w1 or w2 in the proposed stress-strain curve in Fig. 4 is so far not known, nor is it known whether assumption I ( u= w1/c) or assumpton II ( = o+w2/c) gives the most realistic results. Further tests are needed in order to check this, and also to check whether the curve can be regarded as superior to conventional assumptions. It has to be pointed out that uniaxial compression tests are not suitable for determining the material properties to be used for the compression zone of a beam. The behaviour of a specimen in uniaxial compression is too different from the behaviour of concrete in the compression zone of a beam. Possible ways of determining the material properties and to check the validity of the assumptions are either to make strength tests with over-reinforced beams, or to measure ductilities of underreinforced

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beams, and compare with calculated values. In order to compare assumptions I and II, tests with overreinforced beams seem to be most suitable. 5 Conclusions The shape of the stress-strain curve to be used for the compression zone of a beam is probably size dependent, so that the ultimate strain u decreases with an increase of the depth c of the compression zone, Fig. 4. This is of particular importance for high strength concrete, where u is a more crucial parameter. The proposed way of describing the stress-strain curve also explains the difference between normal concrete and high strength concrete. The size dependency of the stress-strain curve is of importance for the balanced reinforcement ratio and for the ductility, which both decrease with an increased beam depth. The validity of the assumptions, on which the conclusions are based, are rather uncertain, although some evidence has been found from comparisons with tests regarding rotational capacities. Much more research is needed to verify the assumptions. The consequences of the size dependency of the stress-strain curve are so important, particularly for high strength concrete, that a substantial research effort is justified. 6 References Hillerborg, A. (1988) Rotational capacity of reinforced concrete beams. Nordic Concrete Research, 7, 121–134. Nilson, A.H. (1987) High strength concrete: an overview of Cornell research. Utilization of high strength concrete, Proceedings, Symposium in Stavanger, Norway, 27–38.

14 INFLUENCE OF THE BEAM DEPTH ON THE ROTATIONAL CAPACITY OF BEAMS K.CEDERWALL, W.SOBKO, M.GRAUERS, M.PLOS Division of Concrete Structures, Chalmers University of Technology, Göteberg, Sweden

Abstract A few tests have been performed to investigate the influence of the beam depth on the rotational capacity of reinforced concrete beams. The test results indicates that a size effect exsist but do not show invers proportionality between the beam depth and the rotational capacity. Test specimen and test setup Four beams with two different beam depth have been tested, The beam depth and the span length have been varied so that they are three times greater for beams of type A than for beams of type B, Fig. 1 and table 1. The longitudinal tensional reinforcement ratio is the same for both beam types. The beams have no compression reinforcement in the rotational zone. All of the beams have vertical stirrups with 200 mm spacing along the whole beam length. The beams have been tested as simply supported beams loaded with a point load at midspan. The tests have been deformation controlled. Results Load-deflection curves for the four beams are shown in Fig. 2. One can see that that the ultimate load bearing capacity for the larger beams are approximately three times greater than for the smaller beams. The rotational capacity was calculated as the plastic deflection at midspan (the total deflection minus the elastic deflection) divided by half the span length. The rotational capacity is about 1.3 times greater for the smaller beams than for the larger beams. When looking at the results one must consider that only a few tests have been performed and the scatter in

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Fig.1. Test setup and beam cross section. Table 1. Test specimen data. Number of beams tested

Type

Longitudinal

Stirrups

2 2

A B

L

Concrete Grade

Reinforcement

4.8 1.6

K30 K30

6ø12 Ks600 2ø12 Ks600

ø8 Ks400s ø8 Ks400s

the result is considerable. More tests are therefore intended to be performed. Discussion The discussions following the presentation mainly concerned the fact that the test results did not correspond to professor Hillerborgs theory, how the beams were reinforced and how the tests were performed. It was said that it would be interesting if overreinforced beams were tested. The effect of stirrups on the the rotational capacity was questioned.

INFLUENCE OF THE BEAM DEPTH

Fig.2. Load-deflection curves and rotational capacity.

163

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Fig. 3. Failure at midspan for beam A1.

INFLUENCE OF THE BEAM DEPTH

Fig. 4. Failure at midspan for beam B1.

165

15 NEW FAILURE CRITERION FOR CONCRETE IN THE COMPRESSION ZONE OF A BEAM L.VANDEWALLE, F.MORTELMANS Departement Bouwkunde, Katholieke Universiteit te Leuven, Heverlee, Belgium

1 Introduction It is usual in the elastic as well as in the limite state method to calculate a concrete section for bending and for shear separately. So the principle of the superposition of the action of forces is implicitely accepted. However, about ten years ago, a large research program has been started to calculate a section simultaneously under the action of a bending moment and a shear force (Fig. 1), taking into account also aggregate interlock (Mortelmans [1]).

Fig. 1 . Internal forces in a reinforced concrete cross-section.

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Fig. 2 General view of the criterion of fracture under biaxial stress.

In this paper, only the compressed zone of the section is discussed, According to the CEB Model-Code ). When concrete failure occurs when the maximum strain in the concrete ( ) reaches 3.5 %o ( besides, the section is subjected to a shear force (V) the previous criterion seems us too optimistic. must decrease when the influence of the shear force Without any calculation it can be presumed that in the concrete (Vc) increases with regard to the influence of the compressive force (N’c). 2 Criterions of fracture under biaxial stress If shear stresses (тc) act in addition to longitudinal compressive stresses (σ’c), the well known uniaxial stress passes into a less known biaxial stress. A general view of the criterion of fracture under biaxial stress is shown in Fig. 2; σI and σII stand for the principal stresses. In Fig. 2, four typical areas can be distinguished : I : tension—tension II : compression—tension III : compression—compression IV : tension—compression From Mohr’s circle it follows that when a compressive stress σ’c occurs simultaneously with a shear stress тc, the principal stresses σI and σII are tensile and compressive stresses respectively (areas II and IV). Different intrinsic curves were proposed for the area “compression-tension” (zone II and IV in Fig. 2). Some of these failure criterions are mentioned below : 2.1 Criterion of Walther [1] The intrinsic curves in the areas II and IV are replaced by a straight line:

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Fig. 3. Criterion of Mohr.

(1)

2.2 Criterion of Hruban—Beltrami [2] These investigators supposed that the sollicitation of a material is measured by the quantity of elastic energy that a material is possible to accumulate in a volume-unit just to the moment that the dangerous limit in that point is achieved: (2)

2.3 Criterion of Mohr [2] This criterion states that in a σ-т-diagram, the intrinsic curve is a tangent line on the circle of compression on the one hand and on the circle of tension on the other hand (Fig. 3). However in the case of pure shear, the maximum shear stress at failure is not equal to тc1, (= intersection of the tangent line with the т-axis) but to тc2 (intersection of the circle, with centre in the origin of the coordinate system and that is tangent to the line тc1-A, with the т-axis).

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2.4 Criterion of Bresler-Pister [2] and [3] Bresler and Pister executed tests on concrete cylinders simultaneously subjected to torsion and axial compression, The fracture criterion, derived from the results of these experiments, is given below: (5)

2.5 Criterion of Kupfer [4] Kupfer performed tests on square concrete plates. These specimens. were loaded without restraint by replacing the solid bearing plates of conventional testing machines with “brush bearing plates”. From the results of these experiments, he deduced the formula, mentioned below: (6)

2.6 Criterion of Bruggeling [5] From results of experiments, executed by Nelissen and Kupfer, Bruggeling calculated critical combinations of тc and σ’c (see Table 1): Table 1. Critical combinations of σ’c and тc 1 0

0.86 0.18

0.75 0.20

0.64 0.20

0.54 0.197

0.43 0.187

0.32 0.17

0.22 0.155

0.11 0.132

0.007 0.10

–0.095 0

2.7 Summarizing The different expressions of the maximum shear stress in the case of pure shear failure are presented in Table 2 and Fig. 4.

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Table 2. Maximum shear stress in the case of pure shear failure

From Fig. 4 it follows that : – the criterion of Bresler-Pister is not so suitable to use in the calculation of the compressed zone of a beam because the shear stresses in the specimen of Bresler—Pister are the result of torsion. – the criterion of Walther is identical to that of Mohr in the case of pure shear failure; it is also one of the most safe ones. During the further calculations in this paper, the criterion of Walther has been applied.

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171

Fig. 4. Maximum shear stress in the case of pure shear failure as a function of “fct/f’c”.

Fig. 5. Reinforced concrete beam with rectangular section.

3 Absorbable shear force in the compressed zone of a beam with rectangular cross-section Consider a beam with rectangular section, subjected to a bending moment (M), a normal force (N) and a shear force (V). A part of the concrete section is compressed (Fig. 5). The stress-strain diagram (σ’c/ ) of the CEB Model-Code is used:

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Fig. 6. Compression zone of a beam.

(7)

(8) To every σ’c-diagram (Fig. 6) corresponds a тc-course which can be calculated with the help of Jourawski’s -values (= concrete strain at the top of the beam), smaller than 2 %o, σ’c well known formula. As, with is monotonously rising, тc must be a continuous function of z. is greater than 2 %o, then σ’c remains constant for “z>2/ c”. The result is that тc is zero from If c” to “z=c”. The exact function тc(z) could theoretically be calculable. The correctness of it, “z=2/ however, depends on the σ’c-function. As this is only an approximation, a correct deduction of тc(z) is questionable. Therefore we prefer to maintain a parabolic course for тc, according to the boundary conditions, indicated in Fig. 6. In the point z=0, the tangent to the тc-curve isvertical, as the longitudinal stress is theoretically zero under the neutral axis. The shear stress thus remains theoretically constant there as far as the lower reinforcement. For z=0, σ’c=0; shear failure occurs when тco=тcu (Table 2). Consequently, one can calculate the resultant of the longitudinal and shear stresses at failure in the compression zone: ≤2 %o - for 0< (9a)

(10a) - for 2 %o<

≤3.5 %o

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(9b)

(10b) Vcu represents the absorbable shear force in the compression zone corresponding to a given value of . In the compressed section there is a strain gradient so that it seems logical to replace the tensile strength fct in the formulae (10a) and (10b) by the bending tensile strength fct, f1 (fct, f1≈fct/0, 6) : (11a)

(11b) with

From formulae (11a) and (11b) it follows that Vcu is a function of the concrete quality and the concrete . With the help of formulae (9) and (11) a diagram can be drawn which represents an interaction strain between the normal and shear force at failure in the compression zone of a beam (see line DCBA Fig. 7). Each point of the line corresponds to a certain value of the concrete strain at the top of the beam. Along AB and BC, ncu and vcu can be considered as ultimate values. ncu is constant (= 0.81) and of course an ultimate value; vcu however has a maximum For ultimate value (=0.381) but can decrease to 0, while ncu remains constant. Consequently the line CD represents an ultimate state for ncu (=0.81) but not for vc (vcu= 0.381). are to be considered as ultimate values ( ) under combined moment and The values of shear force. The relation ncu/vcu can also be represented with good approximation by a continuous function: (12)

4 Experimental program In order to try out the Eq. (9) and (11) in reality, a special test set-up was built. The tests have been executed in the laboratory for reinforced concrete of the Katholieke Universiteit Leuven. In the testbeam resting on two supports, the tensioned concrete is left out in a certain place (aa) to simulate the crack effect (Fig. 8). The lower reinforcement is thus not covered with concrete over that part of the beam. At the ends of the beam strengthened consoles are provided. On the consoles act two adjustable prestressing forces (2 jacks of 750 kN): H1 and H2.

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Fig. 7 . Shear/normal force-interaction diagram.

Fig. 8. Test set-up,

The force P is gradually increased, taking care to adjust the forces H1 and H2 in such a way that in “a” =0 and in “b” corresponds to the value intended for the test. At a certain load P the beam collapses. At that moment Ncu is accurately known, and so is Vcu, belonging to it, acting in the cross-section αα at the moment of failure. The test set-up is very delicate and relatively dangerous as the failure is brought about explosively.

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175

Per test, that is per , only one point can be determined in the ncu/vcu-diagram. The test results are indicated in Table 3 and Fig. 7. It follows that the proposed relation ncu/vcu corresponds pretty well to the experimental results. Table 3. Test results.

1,20 %o 1,50 %o 2,74 %o 2,80 %o

0,485 0,562 0,757 0,762

0,763 0,639 0,709 0,517

5 Proposal for concrete failure Let’s consider the Eq. (9) and (11). Dividing the Ncu-formula by the Vcu-formula we obtain: (13a)

(13b) So NcuK/Vcu is only a function of . As a matter of fact is to consider as the ultimate admissible ) corresponding to a certain combination of normal and shear force. strain of the concrete ( The Eq. (13) represents a new criterion for concrete failure under combined compressive and shear force (see Fig. 9). can be given by With a good approximation the relation (14) This formula holds for When increases (>2), Ncu remains constant as more an ultimate value because

(see Eq. (9b)). In that case vc is no

6 Conclusions – In case of pure shear failure, the fracture criterion of Walther is one of the most safe ones. is in our opinion not a sufficient requirement when the cross– The failure criterion section is subjected to combined bending, tension and shear.

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Fig. 9: Proposed failure criterion under combined compressive and shear force.

– The equation of concrete. – This new failure criterion states that respect to the influence of Nc

means an extension of the generally accepted failure criterion must decrease when the influence of Vc increases with

References Mortelmans, F. (1984) “Interaction diagram bending moment/shear force considering the absorbable forces in the cracked and the noncracked concrete”, (Internal report, Departement Bouwkunde K.U.Leuven, 20-ST-13/18.10. 1984). Godycki—Cwirko, T. (1972) “Le cisaillement dans le béton armé”, 1st edn. (Dunod, Paris, 1972). Neville, A.M. (1977) “Properties of concrete”, 2nd edn (Pitman Publishing, London, 1977). Kupfer, H. (1973) “Das Verhaltens des Betons unter mehrachsiger Kurzzeitbelastung unter besonderer Berücksichtigung der zweiachsigen Beanspruchung”, Heft 229 (Ernst & Sohn, Berlin, 1973). Bruggeling, A.S.G. “Het gedrag van betonconstructies—Deel B”, (Stichting Professor Bakkerfonds, Delft). Mortelmans, F. (1985) “Berekening van constructies—Deel 5 : Gewapend beton III”, 1st edn (Acco, Leuven, 1985). Mortelmans, F. (1985) “De relatie tussen buiging en dwarskracht”, Cement 37 (10) (1985) 758–770. Jans, G., Matthé, P. (1981) “Opneembare dwarskracht in de gedrukte zone van balken in gewapend beton”, thesis K.U.Leuven, 1981.

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177

Mortelmans, F. (1988) “Nieuwe ontwerpopvattingen”, Proceeding of the 8th Belgian concrete day, 08.11.88, pp. 11–10.

Discussion * A.Hillerborg: Why didn’t you get failure at the place of the load “P”? L.Vandewalle: The testbeam is not only loaded with the vertical load “P” but there are also two horizontal forces, acting at the place of the two strengthened consoles. Because of this, the maximum sollicitation doesn’t occur at the place of “P” but at the place of αα. * B.Barr: How did you measure the concrete strains at the place of αα? L.Vandewalle: The concrete strains were regularly measured with the aid of a mechanical strain measuring —device “DEMEC” (graduation: 10.10). The gauge length was 8”. * L.Elfgren: How can you apply this failure criterion during the design of a beam ? L.Vandewalle: To determine the failure load of a beam with certain dimensions and reinforcement, you have to write the equilibrium equations of the heaviest loaded section. Besides these equations there is also the expression of the failure criterion. By iteration, one has to find values for Vcu and Ncu that satisfy all the equations.

PART FOUR SHEAR, BOND AND PUNCHING

16 BOND BETWEEN NEW AND OLD CONCRETE YU-TING ZHU Department of Structural Mechanics, Royal Institute of Technology, Stockholm, Sweden

1 Introduction New concrete is placed onto old concrete in many circumstances. In order to ensure full structural interaction between the old concrete and the new concrete it is necessary to have good bond between them. To test the tensile strength at the interface of new and old concrete, direct tension tests are often used. However according to Silfwerbrand (3) the bond strength assessed by pull-off tests are often obscured by failure elsewhere than along the interface. Although in some indirect tension tests such as the Brazilian test used for testing bonding agents (4) specimens may fail almost along the interface, only the nominal tensile bond strength can be obtained because the stress distribution in the interface at the maximum load is uncertain. The aim of this work was to provide useful information regarding strength and behaviour of repaired concrete beams and to develop and test an alternative approach to evaluate the bond strength between new and old concrete. Three-point bend tests were carried out on notched repaired beams with the interfaces at different angles relative to the notch plane. A complete force-deflection diagram for each specimen was obtained, which would contain more information than a single average failure stress from other types of bond tests. Full information about the descending branch in direct tension tests can be obtained only from elaborate test arrangements. When a concrete member, for example a bridge deck, is repaired the deteriorated concrete has to be removed and replaced by the new concrete. Taking into account that the removal of bad parts of the old conrete may introduce microcracks in the remaining concrete, it is neccesary to test the effect on the bond capability of the “half fracture zone” near the surface. Different methods of concrete removal will result in differences in the extent of induced microcracks. In these test series a well defined fracture surface was obtained by breaking the original beam into two halves. Thus the tests should be repeatable. In this paper only the repaired beams with butt joints (the interface was perpendicular to the beam axis) will be described in detail. With the notches in the direction of the interface planes these beams failed almost along the interfaces. From the tests bond strength in relation to a virgin beam can be obtained. The bond capability was quantified in terms of maximum load (after correction for the self-weight of the beam) and

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Fig. 1. Three-point bend specimens: (a) unrepaired notched beam; and (b) repaired notched beam, where the new concrete half beam was cast directly onto the old concrete half.

the fracture energy, GF, defined by Hillerborg (1). From these two parameters, the bond tensile strength rather than modulus of rupture can be predicted by the combination of fracture mechanics and numerical analysis which will be described in detail in a future paper. 2 Test program and test specimens The tests comprised two test series. In the first test series three-point bend tests were carried out on 18 notched unrepaired concrete beams (cast in two batches but with the same mixture) at the age of 47–55 days. Hereafter we will refer to these unrepaired beams as old beams. Of these 18 beams 4 were used to calibrate the experimental arrangement. During the three-point bend tests these old beams eventually broke into halves. The second test series consisted of 3 new notched unrepaired concrete beams (hereafter referred to as new beams) and of 27 notched repaired concrete beams (24 with inclined joints and 3 with vertical joints). All the old and the new unrepaired concrete beams (Fig. 1a) were tested for comparison. The repaired beams with butt joints between the new and the old concrete half beam as shown in Fig. 1b are described here. The results from the repaired beams with interfaces at different angles with respect to the notch planes (potentially approching crack pathes) will be reported in a future paper. During casting of fresh concrete onto the old concrete the contact surface was the old fractured ligament cross section and the old sawn notch surface formed in the previous three-point bend test on the original beam. The old fracture surface was left “as is” but the old notch surface was sandblasted. No bonding agent was used. The ligament casting was positioned upwards as it was in the unrepaired beam. The notch was sawn perpendicular to the beam axis at midspan. The new notch therefore coincided with the old as shown in Fig. 1b. The overall dimensions for all beams in the two test series were as shown in Fig. 1: length L=840 mm, span 1 =800 mm, width b=100 mm, depth d=200 mm, notch depth a=d/2 and notch width t=3.5 mm. Before the above mentioned 27 repaired and 3 new unrepaired beams were cast (in two batches with the same mixture), the halves of the old beams had been stored in the air for about 4 months. When the three-point bend tests of the repaired beams were performed, the age of the new concrete was 51–57 days while the age of the old concrete was 215–224 days. The mix proportions of the concrete used in these test series were the same throughout, except that some red pigment was added into the new fresh concrete in casting the repaired beams. The water-cement ratio was 0.5 and the maximum size of the aggregate was 16 mm. The contents of materials were: cement: 360 ka/m3, sand 0–8 mm: 944/m3, gravel 8–16 mm: 902 kg/m3 and water: 180 kg/m3.

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181

The strength for each batch of concrete was tested at the age of 56 days. The compressive strength for the concrete in the first test series was measured on 6 companion cylinders (3 for each batch) of 150 mm diameter and 300 mm length. The mean cylinder compressive strength was 36 MPa and the corresponding cube (150 mm) compressive strength derived according to Swedish standard SS 137207:1984 was 48 MPa. The measured cube (150 mm) compressive strength of the new concrete in the second test series was 56 MPa (mean of 6 tests). The tensile strength was determined by pull-off tests. The cores drilled from the testing beams were of 75 mm diameter and 100 mm depth. The mean pull-off tensile strength was 2.6 MPa for the first series and 2.3 MPa for the second series. The static Young’s modulus Ec of each batch of concrete used in the first series was determined on cylinder in compression while that of concrete used in the second series was estimated as Ec=4.5 f GPa, in which f is the compressive cube strength inc MPa. Both methodsc gave the same result: Ec=34 GPa. In order to check the gain in strength of the old concrete parts in the repaired beams, the strength tests for the old concrete (222 days old) were also carried out. The mean compressive strength of the old concrete was 49 MPa (determined from 6 sawn cubes of 100 mm side length) and the pull-off tensile strength was 2. 6 MPa. The tests showed that the gain in the strength of the old concrete was negligible. All the unrepaired and repaired concrete beams were kept under wet sackcloth during the first 24 hours after casting and then stored in lime-saturated water until the time for testing. All the tests were carried out on moist specimens to minimize shrinkage effects. Consequently in this study the effect of shrinkage stresses have not been considered. The three-point bend tests on all the notched unrepaired and repaired beams were performed according to the RILEM recommendation (2). All the tests were run in a displacement controlled servo-hydraulic testing machine INSTRON 1340. The displacement at the middle of the beam and the movement at the supports were measured separately. They finally gave the effective deflection of the beam. During the tests, the outputs of the load cell and displacement transducers were read and stored by a micro-computer HP 9133 with an automatic data acquisition system. The test set-up is illustrated in Fig. 2. 3 Test results and discussion 3.1 Maximum load and fracture energy The immediate result of a three-point bend test is a force-deflection diagram. Some diagrams are shown in Fig. 3. The maximum load, Pmax is calculated from Eq. 1: (1) and the fracture energy, GF, is evaluated from Eq. 2 (2): (2) where Fmax is the maximum applied force, WO is the area under the force-deflection curve, mg is the weight of the beam between the supports, δ0 is the deflection at the final failure of the beam. The results are summarized in Table 1.

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Fig. 2. Photo of test set-up Table 1. Three-point bend test results. Type

Repaired with butt joint New unrepaired

Beam

m (kg) 1/L

δ0 (mm)

17A0 40 0.82 18A0 39 0.58 18B0 40 0.80 H1 39 1.1 H2 40 1.11 H3 40 1.1 Old 17 41 1.03 unrepaired 18 40 1.08 Mean of 14 old unrepaired beams from the first series

Fmax (kN)

Pmax (KN)

Mean

GF (N/m)

Mean

2.68 2.58 3.27 3.71 3.75 3.84 3.58 3.66

2.88 2.77 3.46 3.90 3.95 4.03 3.78 3.86 3.82 ±0.3

3.04 ±0.3

75 64 78 127 125 136 140 159 152 ±14

72 ±6

3.96 ±0.1 3.82

130 ±5 150

These results indicate that the repaired beams are somewhat weaker than both the new and the old unrepaired beams. The ratio between the mean value of the maximum loads for the repaired beams and that for the unrepaired beams is 0.78 and for the fracture energy the corresponding ratio is 0.51.

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183

Fig. 3. Force-deflection curves.

3.2 Failure modes Generally speaking, if no bonding materials are used the possible failure modes in the interface zone of the repaired concrete beam are: 1, failure in the new concrete; 2, failure in the interface; 3, failure in the old

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ANALYSIS OF CONCRETE STRUCTURES

concrete. One would expect that case 1 and 3 would mean fracture near the interface due to various disturbances such as suction, wetting, microcracks in the old “half fracture zone” etc. The event which occurs also is dependent upon the interface strength, the virgin strength of the new and old concrete and their relation, and the extent of damage in the old concrete before casting the new concrete overlay. From the tests it was found by inspection (the new and old concrete were made in different colour) that more than 70% failure occured in the interface and very little in the new concrete. The failure areas in the old concrete were: 20% for Beam 17A0, 23% for Beam 18A0 and 10% for Beam 18B0. Fig. 4 shows the new fracture surfaces of the repaired concrete beams, where the shaded area indicate failure in old concrete. We also found that the loading capacity (Pmax) was conversely proportional to the amount of failure area in old concrete. Fig. 5 shows such a relation. The result from these tests showed that the fracture zone caused by the first three-point bending hade some negative effect. The old concrete halves in Beam 18A0 and Beam 18B0 were taken from the same original beam, i.e., Beam18 used in the first tests. The difference between the proportion of the old concrete failure of Beam 18A0 and Beam 18B0, hence also between the strength of them, were probably due to more fracture zone left in one side of the previous separation crack and less left in another side. However, the number of test beams are far too few to allow a firm conclusion. Acknowledgement I wish to thank my teacher professor Sven Sahlin for his invaluable guidance during the course of the work and in the preparation of the manuscript of this paper. I wish also to thank all the staff members at the Department of Structural Mechanics and Engineering at the Royal Institute of Technology in Stockholm for their help. References 1

2

3 4

5

. Hillerborg, A., “Numerical Methods to Simulate Softening and Fracture of Concrete”, Fracture Mechanics of Concrete: Structural Application and Numerical Calculation, edited by G.C. Sih and A. DiTommaso, Martinus Nijhoff Publishers, Dordrecht, 1985, pp. 141–170. . RILEM-Draft-Recommendation (50-FMC), “Determination of the Fracture Energy of the Mortar and Concrete by Means of Three-Point Bend Test on Notched Beams”, Materials and Structures, vol. 18, No. 106, 1985, pp. 285–290. . Silfwerbrand, J., “Theoretical and Experimental Study of Strength and Behaviour of Concrete Bridge Decks”, Bulletin No. 149, Dept. of Structural Mechanics and Engineering, Royal Institute of Technology, Sweden, 1987. . Wall, J.S., Shrive, N.G. and Gamble, B.R., “Testing of bond between fresh and hardened concrete”, Proceedings of the RILEM Symposium on Adhesion between Polymeers and Concrete, Aix-en-Provence 1986, pp. 335–344. . Zhu, Yu-ting, “Evaluation of bond strength between new and old concrete by means of fracture mechanics method”, in preparation.

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Fig. 4. Fracture surfaces of the repaired beams, where the shaded area indicate the old concrete failure.

185

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 5. Observed relation between overall strength of the repaired beam and old concrete failure proportion in the joint.

17 STRENGTHENING OF EXISTING CONCRETE STRUCTURES WITH GLUED STEEL PLATES B.TÄLJSTEN Division of Structural Engineering, Luleå University of Technology, Luleå, Sweden

1 Introduction Concrete is well known for its high compressive strength and its low tensile strength. A beam manufactured of plain concrete without any kind of reinforcement will crack very easely in bending. Failure happens suddenly and catastropically. In practice, situations can arise where a concrete structure is insufficient with respect to its strength. This can be due to the fact that the original structure was made with bad workmanship or that its dimensions were too small. In extreme cases a structure may have to be repaired because of an accident as e.g. a collision with a vehicle. If situations like these arise one should consider wether it is most economical to strengthen the structure or to replace it. Strengthening of an existing structurc in order to reach a higher load carrying capacity is relatively easy to achive for a structure made of steel. In comparision, strengthening of a concrete structure is more difficult. Attempts to strengthen a concrete structure can easily fail, essentially depending on the fact that it is difficult to reach composite action between the element which should give the extra strength and the existing structure. Strengthening of concrete structures “in situ” with steel plates glued to the concrete using a two component epoxy resin has given promising results. Experiments performed by for example Swamy et al (1987), Bresson (1971) and L’Hermite (1967) shows that the necessary adhesion between steel/glue and glue/concrete can be obtained and that we in this way can achieve composite action between the element and the structure itself. The greatest advantages with the method is that it can be applied in a short time even when the structure is in use. The technique has been used in several countries as England, France, Japan and Belgium in bridges as well as in buildings. It has been used not only in the tension zone but also in the compression and the shearing zones see e.g. Jones et al (1980), Ladner (1981), Jones et al (1982), Yuceoglu (1980) and Macdonald et al (1982). It is also important that the preparation of the concrete surface and the steel surface is carefully done before we apply the glue. The surface of the concrete should be relatively smooth, bigger irregularities than 2 mm

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Figure 1 Different kinds of force transfer in a joint

Figure 2 Comparision between shearing and splitting (peeling) forces in a glued joint

should be avoided, depending on the thickness of the glue layer. Generally it can be said that the irregularities of the surface should not be greater than the thickness of the glue layer. 2 Use of epoxy in load bearing structures Epoxy resins are mostly used for repairing purposes, but also in manufacturing and building of new structures. The development during the past 30 years has led to that epoxy now is used for a large range of applications, for example in the aircraft industries.

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189

The curing of the epoxy glue is an exotermic reaction and this reaction depends on the temperature. Generally it can be said that the time for curing will be half as long when the temperature increases 10 °C and it will be twice as long if the temperature decreases 10 °C. When gluing, temperatures below 10 °C should be avoid. In building technique applications resins are often used directly against concrete and cancrete materials. Hence it is important to compare the mechanical and other properties of the glue with them of concrete and steel. Some typical properties are given i table 2.1 Table 2.1 Physical properties for epoxy resin and concrete

Compressive strength [MPa] Tensile strength [MPa] Modulus of elasticity [GPa] Maximum temperature during loading [°C] Temperature expansion coeffecient [1/°C.10-6]

Epoxy

Steel

Concrete

55–100 9–25 0.5–20 40–80 25–30

200–2000 200–2000 ≈200 <500 10–15

20–70 1.5–3.5 20–35 <300 7–12

3 Methods to strengthen concrete structures with steel plates glued to its tension side It is very important that the preparatory work and the gluing procedure is done in a strict and proper way. The reason for this is that we want to reach maximum adhesion and to achieve composite action between the concrete and the steel in the reinforcing system. To start with, a resin system should be used which has greater strength than the concrete to which the steel shall be bonded. Mostly epoxy resins have a very high tensile strength and they can handle tensile forces very well. The capacity to withstand splitting and peeling is more limited. Therefore it is important that joints are designed so that they, if it is possible, do not get exposed to splitting or peeling, see Figure 1. The stress in the vicinity of the end of the steel plate is much higher when a joint is subjected to peeling compared to the action of shear, see Figure 2. Before gluing all latiance and poor concrete should be removed, for example by sand-blasting. If necessary the concrete could be strengthened by some kind of epoxy concrete. The surface of the steel plates should also be blasted and all dust and grease should be removed from the concrete and the plates. A good recommendation is to apply some kind of rust protector or primer to the steel immediately after the blasting and cleaning of the steel plates. When the gluing process is to be started it is useful to apply the glue both on the concrete and on the steel plates. The concrete and the steel plates are then pressed together. How long time pressure should be applied depends on the temperature and the type of resin that is used. Generally the resin reach about 80 % of its full capacity in 24 hours if it cures in 20° C. The ends of the steel plates should in some way be anchored to the concrete structure, Jones et al (1988). One way is to use bolts, another can be to use some kind of other steel plate, as for example L-shaped plates which are glued to the concrete over the steel plates which shall reinforce the concrete structure, see Figure 3 (a) and 3 (b). These are two different kinds of anchorage which will prevent the same physical event, namely peeling in the joint.

190

ANALYSIS OF CONCRETE STRUCTURES

Figure 3 (a) Mechanical anchor system using bolts

Figure 3 (b) Anchor system with L-shaped iron plates

4 Experiences from tests with concrete beams strengthened with resin bonded steel plates It is common in the experiments that have been performed that full composite action have been achived between steel/glue/concrete. The strengthening can be carried out during the time the structure is in use, Swamy et al (1980). There is for the moment not very much documented about the long time behavior for the method described above. This can be explained by the fact that the method is relatively young. However Ladner (1981), has in Switzerland tested beams strengthened with glued steel plates exposed for climatic conditions and external loads. These beams have been exposed for loads during the past 7 years without any big deterioartious having been observed. Further in England Calder (1988) have done exposure tests on strengthened concrete beams to determine the long time durability of the technique. Small unreinforced concrete beams were strengthened by bonding a mild steel plate to one face using a structural epoxy adhesive. They were exposed at sites representing high rainfall, industrial and marine environments. Specimens were also kept under controlled laboratory conditions. During exposure half the specimens at each site were loaded and half were unloaded. To date, loading tests to failure have been carried out on specimens brought back to the laboratory after 1, 2 and 10 years. Light but sometimes extensive corrosion has occurred on all the plates removed from specimens exposed to natural environments. There was significantly more corrosion after 10 years compared with that after 1 and 2 years. Nevertheless there was a small increase in strength with time, but the beams kept outside, failed at marginally lower loads than the laboratory beams. The degree of corrosion was reduced significantly without loss of strength by coating the steel with an epoxy primer paint prior to bonding the concrete Experiments performed in England of Swamy et al (1987) show that the optimal thickness of the glued joint is between 1.5–2.5 mm when steel plates are being glued to concrete structures (in this case usual concrete beams). From a theoretical point of view as thin joint as possible is preferred for the force transfer from the concrete to the steel through the glue joint. It should be considered that the concrete is rough and that the steel plate can be warped. If this is the case it is possible that the steel plates come in direct contact without any resin between steel and concrete with stress concentrations as a consequence. With a thicker glue joint the forces in the glue layer can easier be controlled and the stress concentrations described above can be avoided. Further it is shown by Swamy et al (1987) that a thickness of the steel plate of 1.5–3.0 will influence a concrete beam in such a way that it will reach a ductile failure.

STRENGTHENING OF EXISTING CONCRETE

191

Figure 4 Dimensions of test beams, part A

5 Experiments performed at Luleå University of Technology In order to evaluate the usefulness of the method described above tests have been performed in the Division of Structural Engineering at Luleå University of Technology. The number of tests has been relatively limited and should at this time be considered as pilot tests. The experiments were divided in to three parts (A), (B) and (C). In the first part, part A, six concrete beams were casted, with dimensions as in figure 4. The difference between the beams were that the beams that should be strengthened had a lower grade of internal reinforcement. All of the beams were designed for the same ultimate load. Before the gluing started both the concrete and the steel plate were blasted and all dust and grease were removed. To achieve a distinct joint with a constant thickness, distance separators were used in the glue joint The separators consisted of steel balls with a diameter of 3 mm, the steel plates and the concrete beams were thereafter pressed together. The beams were tested with both static and dynamic loads. Furthermore the Youngs modulus of the resin was varied, see table 5.1. Table 5.1 Physical properties for the epoxy resin used Resin

NM1

NM2

NM3

Compressive strength [MPa] Tensile strength [MPa] E-modulus [GPa]

103 24 6.7

119 25 7.5

88 21 5.5

192

ANALYSIS OF CONCRETE STRUCTURES

The results show, see table 5.2, that we can reach composite action between steel / glue / concrete and in this case we even got about 15 % higher ultimate load for the strengthened beams. Furthermore the strengthening also prevented cracking in the concrete. Remarkable is that the strengthened beam tested in a dynamic test could withstand about 500 times more cycles than the conventional concrete beam. Any effect of different E-moduli in the glued joint could not be seen. All beams showed about the same load-bearing capacity irrespective of differences in Young’s modulus. To get unambiguous results more investigations have to be done in this area. In the second part of the investigations, part B, we studied the effect of cold climate and the effect of different lengths of the moment arm, A, see Figure 6. We also tried to find a test specimen which was easy to handle and not to expensive to produce. We decided to use test specimens as in figure 5. The test beams have no internal reinforcement. The beams were all tested in static three or four point bending, see Figure 6. Table 5.2 Results from experiments part A Beam

Ultimate Load [kM]

Maximum deformation [mm]

Compressive strength [MPa]

Tensile strength [MPa]

A1 A2 glued NM1 A3 glued Nm2 A4 glued NM3

167 189 169 193 Number of cycles [3 Hz] 650 350 000

21.4 14.7 16.6 14.0

50.2 51.9 32.2 54.9

2.7 2.8 1.9 2.8

6.8 11.5

49.3 51.3

2.49 2.50

A5 A6 glued Nm1

STRENGTHENING OF EXISTING CONCRETE

193

Figure 5 Test beams, part B

Figure 6 Testing procedure

All beams and all steel plates were blasted and all dust and grease were removed before gluing. The steel plates were also applied with a primer before gluing. In the joint we used 2 mm steel balls as separators. The glue used was the NM1 used in part A. The beams which were tested for freeze cycles were placed in a cold climat room to (-24 °C) in 24 hours. Thereafter they were placed in a water basin (20 °C) for another 24 hours and so on. The results from the experiments are reported in table 5.3. The test results show that to reach ductile failure som kind of internal reinforcement are needed. The freeze test indicates no detoriation in the glued joint and the ultimate load does not change significantly. The freeze cycles had more effect on the concrete which tended to get microcracks when it was exposed to water and cold. In the third part, part C, lap splices were tested as well as different steel qualities. The length and the with of the overlap were varied. From the results we can draw the conclusion that there is a minimum value of the length of the overlap plate at which failure changes from brittle to ductile. The effect of using steel plates with a relatively high yielding point did not improve the load-carrying capacity due to bond problems.

194

ANALYSIS OF CONCRETE STRUCTURES

Tabel 5.3 Results from experiment part B

indicates mean value from three tests s: small shear plate, 53×100× 2 mm b: big shear plate, 100×220×2 mm

Figure 7 Shear force when peeling appear

6 Theoretical investigations So far most of the calculations have been made with linear elastic theory. Our aim is to use fracture mechanics in the calculations and to compare it with classical theories. It is difficult to calculate the shear force in the contact area between steel an concrete, see Figure 7. One way could be to measure the fracture energy, Gf, for the strengthened beam, see Figure 8. The fracture energy is defined as the area below the τ—δ curve. In a test as in Figure 9 we can write the following relation for the elastic energy:

For the fracture energy, Gf, we have in this case the following relationship:

STRENGTHENING OF EXISTING CONCRETE

Figure 8 Definition of fracture energy

Figure 9 Test model

If we want to have ductile failure (safe side) the following relation should be fullfilled: Or with P=S, δ, where S is the stiffness of the beam

195

196

ANALYSIS OF CONCRETE STRUCTURES

7 Discussion and conclusion So far the investigations that have been performed at Luleå University of Technology have been quite limited. Nevertheless the results are very interesting. For example we see clearly in table 5.2 that the beams with the glued steel plate to its tension side can withstand a somewhat higher ultimate load than an ordinary concrete beam. We see also in the same table the big differences in the dynamic test. The main explanation to the fact that the strengthened beams can withstand more load is probably that the steel plate “keep” the concrete beam together and in that way delays big bending cracks to appear. For the second investigation, part B, there are difficult to draw any distinct conclusions. But we can see that the freeze cycles had a more negative effect on the concrete than on the glue layer. The most dangerous failure for a strengthened concrete beam is when peeling appear. One way to prevent peeling can be use some kind of anchor in the end of the steel plate. In the third part of our investigations we could not improve the load carrying capacity due to bond problems with higher yielding point in the steel plate. It is also shown that there is a minimum value of the overlap plate at which failure changes from brittle to ductile. It is very hard to calculate the peeling forces with ordinary linear elastic analysis. One way to estimate the peeling forces could be with fracture mechanics and for the moment we are doing some investigations in that area. REFERENCES Bresson J. (1971) Nouvelles recherches et applications concernant I' utilisation des collages dans les structures. Beton plaque. Annales de I’Institut technique du Batiment et des travaux Publics, No 278, February 1971. Calder A.J.J. (1988) Exposure tests on externally reinforced concrete beams-performance after 10 years. Transport and road research laboratory, Research Report 129, Crowthorne, Berkshire 1988, pp 10. L’Hermite R. (1967) L’application des colles et re’sines dans lla construction. La beton a coffrage portant. Annales I’Institut technique du batiment et des travaux publics, No 239, November 1967, pp 1481–1498. Jones R., Swamy R.N., Bloxham J. and Bjounderbalaha A. (1980) Composite behaviour of concrete beams with epoxy bonded external reinforcement. The International Journal of Cement Composites, Volume 2, Number 2 1980, pp 91–107. Jones R., Swamy R.N. and Ang T.H. (1982) Under- and over- reinforced concrete beams with glued steel plates. The International Journal of Cement Composites and light-weight Concrete, Volume 4, Number 11982, pp 19–32. Jones R., Swamy R.N. and Charif A. (1988) Plate separation and anchorage of reinforced concrete beams strengthened by epoxy bonded steel plates. The Structural Engineer, Vol. 66, March 1988, pp. 85–94. Ladner M. (1981) Concrete structures with bonded external reinforcement. EMPA, Report 206, Dubendorf 1981, pp 61. Macdonald M.D. and Calder A.J.J. (1982) Bonded steel plating for strengthening concrete structures. Int. Journal of Adhesion and Adhesives, Vol. 2 April 1982, No 2, pp 119– 127. Swamy R.N. and Jones R. (1980) Technical Notes—Behaviour of plated reinforced concrete beams subjected to cyclic loading during glue hardening. The Int. J. of Cement Composites, Vol 2, No. 4, November 1980, pp 233–234. Swamy R.N., Jones R. and Bloxham J.W. (1987) Structural behaviour of reinforced concrete beams strengthened by epoxy-bonded steel plates. The Structural Engineer, Volume 65 A, Number 2. Yuceoglu U. and Updike D.P. (1980) Stress analysis of bonded plates and joints. Journal of the engineering mechanics division, February 1980, pp 37–56.

18 MODELLING, TESTING AND STRENGTH ANALYSIS OF ADHESIVE BONDS IN PURE SHEAR P.J.GUSTAFSSON, H.WERNERSSON Department of Structural Mechanics, Lund Institute of Technology, Sweden

1 Introduction For theoretical strength analysis of joints, different methods may be employed. These methods include linear elastic fracture mechanics (see e.g. Andersson et al., 1977), elastic and elasto—plastic analysis in conclusion with some maximum stress or strain criterion (see e.g. Adams and Wake, 1984) and limit load analysis by means of the theory of ideal plasticity. The applicability of these different approaches is known to depend on the geometry and the physical size of the joint and on the material properties of the adhesive and the adherends. The various methods predict different strength and entirely different sensitivity to changes in joint geometry and material properties. In this contribution a method that unite linear elastic fracture mechanics and plastic analysis is presented. Only the case of pure shear in the bond line is studied. The actual approach is based on non—linear fracture mechanics, taking into account gradual fracture softening and may in some respects be compared with fracture softening analysis of solid concrete. While test results presented concern wood adhesive joints, method of modelling and analysis is not restricted to any certain adherend material and should therefore be valid also for concrete and steel-concrete adhesive joints. 2 Modelling of a bond line In strength analysis of an adhesive joint, the adhesive layer may be modelled in three different ways: a) As a contact surface without any thickness, its properties being described only by strength and/or critical stress intensity (or fracture energy). b) As a layer in which the components of stress do not vary in the direction perpendicular to the layer. c) As a volume of material, no restriction being imposed to variation of stress and strain within the volume.

198

ANALYSIS OF CONCRETE STRUCTURES

Fig. 1 Constitutive relation of a bond line

Modelling according to a) may be reasonsable in some cases but gives no information about significance of deformation, e.g. gradual fracture softening, in the bond line. Modelling according to c) may be required if the length of the active load carrying part of the bond line is only a few times the thickness of the adhesive layer. In addition, by c) detailed modelling of e.g. spew fillets is possible. In the following discussion, restriction is made to modelling according to b) 3 Constitutive relation of a bond line For a bond line in pure shear, modelled according to b), the mechanical constitution of the bond line, Fig. 1, is defined by a relation between local shear stress, т, and local shear slip, δ, across the adhesive layer. This relation is regarded as a constitutive relation and may depend on type of adhesive, bond line thickness, curing conditions, loading rate, surface preparation, interphase between adhesive and adherend, and so on. It is of significance that the constitutive relation comprises the softening branch of the т-δ curve. Existence of a softening branch has been experimentally verified for various adhesives, see Section 6. As an alternative, the bond line model may be redefined by a constitutive relation for the adhesive, supplemented with information about bond line thickness, t3. As a first sub—alternative, the constitutive relation of the adhesive may be defined in the conventional manner by shear stress, т, versus shear strain, 7, assuming constant shear strain perpendicular to the adhesive layer, i.e. that γ=δ/t3. As a second and non— conventional alternative, in analogy with the fictitious crack model it is also possible to define the constitutive relation of the adhesive by one shear stress—strain relation and one shear stress—slip relation, the latter defining the properties of a shear fracture surface within the adhesive layer, assumed to develop when the shear stress equals shear strength of the adhesive. However, as a base for joint strength analysis it will in most cases be more appropriate to use the т—δ curve of the bond line rather than corresponding properties of the adhesive material. The above mentioned constitutive relations for an adhesive entail certain and different predictions regarding influence of bond line thickness, see (Gustafsson, 1988), and, moreover, bond line properties are most probably influenced not only by type of adhesive and bond line thickness but also by surface preparation and properties of the adherends. 4 Fracture energy Fracture energy, Gf, of a bond line is defined as the energy required to bring one unit area of the bond surface from its unloaded state to its completely fractured state. For the case of pure shear, we get by definition:

MODELLING, TESTING AND STRENGTH ANALYSIS

199

Fig. 2 Different idealized shapes of т—δ curve

(1) i.e. Gf corresponds to the area under the т—δ curve. This fracture energy, Gf, must be distinguished from the critical energy release rate, Gc, used in linear elastic fracture mechanics. While Gf represents the energy required to bring a fixed unit area to complete separation, Gc represents the energy dissipated during movement of a distinct tip of a sharp crack. Gf may be determined from the т−δ curve but may also be experimentally determined from the total work done by the external load during testing of a joint. This latter method demands that energy is dissipated only in the bond line and that a gradual and stable performance can be recorded also when the load is decreasing during increase in displacement. Additionally, peel stresses must be avoided, but it is not required that the shear stress is uniformly distributed. It may also be possible to determine Gf indirectly by fitting theoretical load carrying capacity to test results. 5 Normalized constitutive relation It is suitable to normalize the constitutive relation, т versus δ, by dividing т by тf and δ by Gf/тf, where тf is bond strength, defined in Fig. 1. Normalized т−δ curves that coincide are said to possess identical shape. This shape may be described by a function g: (2) Thus the constitutive relation of the bond line is uniquely defined by the two variables тf and Gf and by the shape function g. The significance of these variables shall be studied later on in this paper. Idealized examples of normalized constitutive relations are shown in Fig. 2. The peak value of these curves is 1.0 and also the area under each curve is 1.0.

200

ANALYSIS OF CONCRETE STRUCTURES

Fig. 3 Test set-up

6 Test of т−δ curve of bond line 6.1 Test method The test set-up is shown in Fig. 3. It consists of two steel parts connected by the test specimen and a steel ball placed in a notch. The width of the steel parts and the specimen is 20 mm. Loading is done in compression to simplify the test set—up and to avoid lashes which can induce instabilities. The aim of the test method is to faciliate determination of the complete т−δ curve, including its softening branch. Therefore the test set-up and the testing machine have to be stiff. Pure shear on the bond line is achieved by anti-symmetric loading on the geometrically symmetric specimen which is attached to the steel parts by epoxy resin. If ideal anti—symmetry can be achieved, the peel stresses are identically zero along the bond line. The slip is measured by two clip—gauges placed according to Fig. 3. The recorded deformation is not only the shear deformation of the bond line but that of the whole joint. However, during evaluation of test results, deformation outside the bond line are reduced for by means of testing solid specimens. The short length of the bond line, only 5 mm, has two purposes, to reduce the possibility of failure within the adherends and to give a uniform shear stress distribution. When the test has been carried out and thus peak mean shear stress and fracture energy are known, the assumption of uniform stress distribution can be verified by stress analysis, e.g. by finite element analysis (Barak, 1990) or by approximate analytical estimations. For further details on test method and test result evaluation, see (Wernersson, 1990).

MODELLING, TESTING AND STRENGTH ANALYSIS

201

Fig. 4 т−δ curves for PVAc, tested at different rate of deformation (R) and after different curing time (C).

Fig. 5 т−δ curves for polyurethane, tested at different rate of deformation (R) and after different curing time (C).

6.2 Test results Using test pieces of wood (Pinus sylvestris), three adhesives were tested after curing times 4 or 16 days and at various rate of loading: 1 mm/min, 0.25 mm/min and 0.0625 mm /min. The adhesives tested were a 1component PVAc (Casco 3304), a 1—component polyurethane (Casco 1804) and a 2—component resorcinol/phenol (Casco 1703). In each test series six nominally equal tests were carried out. The mean т−δ curves obtained are shown in Fig. 4, 5 and 6. In Fig. 7 corresponding normalized shapes of the т—δ curves are shown and in Table 1, тf and Gf are indicated. Ratio т2f/Gf, also indicated in Table 1, is a measure of brittleness of a bond line, see Section 7, a high value indicating a brittle bond line.

202

ANALYSIS OF CONCRETE STRUCTURES

Fig. 6 т−δ curves for resorcinol/phenol, tested at different rate of deformation (R) and after different curing time (C).

Fig. 7 Normalized т-δ curves for PVAc, polyurethane and resorcinol/phenol adhesives. R=0.0625 mm/min, C=4 days Table 1 тf and Gf for different adhesives at different rate of deformation (R) and after different curing time (C). Mean ±standard deviation.

R=.25 mm/min C=4 days R=.0625 mm/min C=4 days R=.25 mm/min C=16 days

Adhesive

Conditions

тf [MPa]

Gf [kNm/m2]

т2f/Gf [GPa/m]

PVAc

8.81±0.92

1.86±0.52

45.5

8.78±0.60

R=1 mm/min C=4 days 1.70±0.29

7.36±0.79

2.11±0.29

26.0

7.12±0.72

1.75±0.21

29.2

Polyurethane

R=1 mm/min C=4 days

4.06±1.46

0.78±0.37

21.4

46.2

MODELLING, TESTING AND STRENGTH ANALYSIS

R=.0625 mm/min C=4 days R=.0625 mm/min C=16 days

R=.0625 mm/min C=4 days R=.0625 mm/min C=16 days

Adhesive

Conditions

тf [MPa]

2.84±1.32

0.60±0.29

13.6

3.56±0.92

0.85±0.25

14.9

R/P

8.78±0.60

8.38±0.67

R=.25 mm/min C=4 days 0.70±0.11 ***

98.7

8.28±1.47

0.48±0.04 **

162.5

Gf [kNm/m2]

т2f/Gf [GPa/m]

1.00±0.07 *

80.1

203

* Only 2 stable tests * Only 3 stable tests *** Only 4 stable tests

Fig. 8 Single and double lap joints in tension

From the test results it appears that the elastic deformation in a bond line is very small compared to the shear slip during plastic hardening and softening. The thickness of the adhesive layer being about 0.1–0.2 mm, the formal shear strain is found to become several hundred percent before the shear stress capacity of the bond line is exhausted. 7 Brittleness ratio and strength function for joint In simple analysis of lap joints, Fig. 8, it is often assumed that the adherends are in a state of pure tension or compression with no bending or shear effects, whereas the adhesive layer is in a state of pure shear. Adopting these assumptions normalized load carrying capacity Pmax/(ℓbтf), can be found to be a function of two dimensionless ratios and the shape function, g, of the т—δ curve (Gustafsson, 1987): (3) where b is the joint width, E1 the modulus of elasticity of one of the adherends, α=t1E1/t2E2 the ratio of the normal stiffness of the adherends and where additional parameters are defined by Fig. 1 and 8 and by eq. (1) and (2). For joints with constant α and g, this strength function may be written

204

ANALYSIS OF CONCRETE STRUCTURES

(4) This relation shows that the normalized mean shear stress at failure is governed by ratio (ℓ2т2f)/(t1E1Gf). This important ratio is in the following referred to as the brittleness ratio of lap joints, high values corresponding to brittle characteristics and low values corresponding to ductile characteristics. The transition from ductile to brittle characteristics is gradual. The brittleness ratio depends on the absolute size of the joint through ℓ, its geometrical shape through ratio ℓ/t1, stiffness of the adherend materials through E, and on properties of the bond line through ratio т2f/ Gf. Experimental values of the bond line characteristic ratio т2f/Gf are given in Table 1. The inverse of ratio (ℓ2т2f)/(t1E1Gf) may be separated into E1Gf/т2f and t1/ℓ2. E1Gf/т2f has the dimension length, it may be regarded as a combined intrinsic length parameter for the joint materials and it is related to the characteristic length parameter ℓch=EGf/f2t, used in analysis of fracture softening solid materials (Hillerborg et al, 1976), (DiTommaso and Bache, 1989). Ratio t1/ℓ2 can be directly compared with the experimentally deduced “joint factor”, /ℓ (Marian, 1954). 8 Strength of lap joints At high values of (ℓ2т2f)/(t1E1Gf) joint failure involves propagation of a self-similar fracture region of minor extension compared to ℓ. This means that linear elastic fracture mechanics may be applied, giving, (Gustafsson, 1987), (5) regardless of the shape, g, of the т—δ curve. Moreover, it can be noticed that Pmax is not influenced by the bond strength, тf. Accordingly, for a brittle joint Gf is the only governing bond line parameter. For ductile joints, (ℓ2т2f)/(t1E1Gf) being small, the shear stress is uniform along the bond line, giving (6) In this case Pmax is proportional to the bond strength, but independent of bond fracture energy. In between the extrems represented by (5) and (6), it may in general be necessary to carry out numerical calculations in order to obtain Pmax. However, for bi—linear shapes of the т—δ curve, e.g. curve I, II and III in Fig. 2, it is possible to obtain Pmax analytically by means of a method for non—linear bond analysis presented by Ottosen and Olsson (1988). The result of such calculations, Fig. 9, shows that, in general, the shape of the т-δ curve is of significance. For the transition region between ductile and brittle joint characteristics it is also found that both тf and Gf have influence on Pmax. For a pile-shaped т-δ curve, curve IV in Fig. 2, an explicit expression for Pmax can be obtained:

(7)

MODELLING, TESTING AND STRENGTH ANALYSIS

205

Fig. 9 Normalized joint strength, Pmax/(тf bℓ), versus brittleness ratio, ℓ2т2f/(tEGf). Curves are valid for joints with α=1. 0, i.e. with t1E1=t2E2=tE. Complete curves correspond to joints in tension, Fig. 8. The broken curve corresponds to joints in tension—compression, Fig. 10.

The pile-shaped т—δ curve is optimal in the sense that this shape produces largest possible joint strength for any constant тf and Gf. Comparing eq. (7) with the extreme value strengths, eq. (5) and (6), it is evident the the pile-shape gives an abrupt transition between ideal plastic analysis and ideal brittle analysis. 9 Test for verification 9.1 Joint specimen for testing pure shear In order to verify bond line tests and theoretical analysis, some joint in which the bond line is in pure shear is needed. In lap joints according to Fig. 8, peel stresses exist, although often neglected in analysis. However, in a geometrically symmetric lap joint, loaded antisymmetrically, the peel stress is identical to zero along the bond line. Such a joint test specimen is shown schematically in Fig. 10 and will be more throughout presented by Wernersson (1990). Although this specimen will bend when loaded, theory and methods of calculation of section 7 and 8 may be applied by making a simple substitution of variables. The equations of equilibrium of the adherends, compatibility across the bond layer and bond line constitutive properties are the same whether or not bending is taken into account. Utilizing that the bond layer is in pure shear, by means of ordinary beam theory it is found that the only difference is the constitutive properties of the adherends: while the normal stiffness of an adherend is btE, the equivalent normal stiffness of the actual adherend is btE/4, see Fig. 11.

206

ANALYSIS OF CONCRETE STRUCTURES

Fig. 10 Joint specimen for testing pure shear

Fig. 11 Section of upper adherend a) =N/(btE), b) =N/(btE/4)

Also observing the difference in boundary conditions, i.e. loading in tension—tension, Fig. 8, and tension —compression, Fig. 10, respectively, for shape I, Fig. 2, of the т—δ curve and α=t1E1/t2E2=1.0, it is found that (8) This theoretical strength of the test joint is illustrated by the broken line in Fig. 9. To illustrate the effect of bending, a comparative calculation of a slender specimen such as in Fig. 10, have been performed. With length 400 mm and additional material and geometry data according to section 9.2 below, specimen response to increasing slip may be calculated and the result is given in Fig. 12. In the two analytical calculations idealized material curve shape I in Fig. 2 is used and as expected for a brittle joint, joint strength is overestimated with a factor 2 if bending effect is not considered. In the FE—model a material model more closely corresponding to the real shape of the т—δ curve of a resorcinol/phenol bond line is used. The bond line is modelled with 400 non-linear 4-node plate elements and the result exhibit good agreement to the analytical solution including bending. Testing joints with adherends made of wood or concrete, it may be found that the joint may break due to tensile or compressive fracture of the adherends. Therefore, in order to get fracture in the bond line, only a strip along the joint may be glued. In the actual tests of wood a strip (b/3)xℓ was glued. In theoretical joint strength analysis, influence of reduced width of bond area can be considered in a simple manner, e.g. by taking b/3 as joint width and increase E by the factor total width divided by net width. 9.2 Test result Three sets of wood (Pinus sylvestris)—resorcinol/phenol joints were tested. Joint lengths were 400, 100 and 25 mm, total width 30 mm, net width 10 mm and t=20 mm. For each joint length two tests were made. Together with corresponding theoretical predictions, tested load carrying capacity is shown in Table 2. The theoretical predictions were obtained for E=13100 MPa (Gustafsson and Enquist, 1988) and тf=8.51 MPa and Gf=710 Nm/m2 (Wernersson and Gustafsson, 1987), see also Table 1. The verification test results seem to be in reasonable agreement with the theoretical predictions, both with respect to relative influence of joint length and absolute values of joint load carrying capacity.

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207

Fig. 12 Calculated response of large shear specimen Table 2 Calculated and recorded failure load of large shear specimen, bonded with resoncinol/phenol adhesive Pmax [kN]

ℓ [mm]

25

100

400

Calculated Recorded

1.93 1.60 1.70

3.66 2.55 3.05

3.74 3.80 4.15

For joint length ℓ=400 mm, Fig. 13 shows experimental and theoretical load-displacement response (Wernersson, 1990). The flat part of the curve for this long joint with a high brittleness ratio is due to movement along the joint of a fracture process region. This region is of approximately constant size as it propagates along the joint. . For the long joint, during propagation of the fracture process region load P is proportional to varies along the joint. Accordingly, recorded variation of P during fracture propagation shows how If E is known the long joint may also be used for determination of Gf. From eq. (8), for high values of joint brittleness ratio, independently of shape of bond line т—δ curve, : (9) where Ppr is the load at propagating fracture and where bb and ba are bond line and adherend widths, respectively.

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Fig. 13 Calculated and experimental response of large shear specimens. At relative displacement 1.5–1.6 mm sudden joint failure occurs.

10 Fracture in adherend material 10.1 Discussion For weak adherend materials such as wood and concrete, using a strong adhesive fracture may not develop in the adhesive layer but instead in the adherend material. For orthotropic materials, e.g. wood and fibre reinforced composites, the adherend fracture may develop and propagate in a layer close to the bond line. The adherend fracture may be assumed to develop if тf is greater than the shear strength, тfr, of the adherend material. Just as the bond line, the adherend layer may be assumed to have a т-δ curve and accordingly a fracture energy. Often, however, the ductility and fracture energy of the adherend is small compared to that of the adhesive layer, giving a simplified effective т—δ curve according to Fig. 14. The reduction in effective energy that results from тf>тfr isof particular significance for the load carrying capacity of brittle joints. It can also be noticed that the limit case тf=тfr corresponds to an abrupt change in effective fracture energy. This means that a small increase in adhesive strength, тf, may entail a sudden and drastic decrease in the load carrying capacity of a brittle joint. In an isotropic material such as concrete, adherend fracture may not propagate along a layer close to the bond line. Still, however, the ductility and fracture energy of the bond line may not be activated if the adhesive is strong, resulting in lower load carrying capacity of the joint.

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209

Fig. 14. Simplified effective fracture energy when тf>тfr

Fig. 15. Schematic т—δ curves for two adhesive bond lines

10.2 A test Often adherend fracture is taken as a positive sign, i.e. that a good and suitable adhesive of high quality has been used. Therefore, in order to attain some verification of the above contraditing theoretical discussion a few tests were carried out. 3 times 3 single—lap joints, loaded according to Fig. 8, with wooden adherends, ℓ=160 mm and t1=t2=20 mm, were tested. Three of the specimens were joined by polyurethane, three by resorcinol/phenol and three specimens were made of solid wood without any bond line. The specimens were sawn in one piece from larger pieces of wood and the first 2 times 3 specimens were cut into two adherends subsequently joined by the adhesive. The т—δ curve of the actual bond lines are shown schematically in Fig. 15, compare Fig. 5 and 6. The two bond lines have approximately the same Gf. However, in the event of adherend fracture the effective fracture energy of the weak polyurethane bond line may become significantly greater than that of the strong resorcinol/phenol bond line. In the present tests, adherend fracture developed in all cases and in Table 3 the results are given. Clearly, the actual test series appears to support the theoretical discussion. The weak polyurethane bond line gave the strongest joint. Resorcinol/phenol gave about the same joint strength as the solid specimen. The theoretical discussion as well as the test results also suggest that a structural member can be made stronger if cut into two pieces and then re-joined by means of a suitable adhesive. Table 3 Strength of joints. Mean values from three tests BOND Solid wood Resorcinol/phenol Polyurethane

1.4 1.3 2.1

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11 Concluding remark Testing and analysis of a bond line exposed to short time pure shear have been discussed. In practical applications duration of load as well as peel stress may be of great significance. In spite of this, conclusions obtained from short term pure shear stress analysis may be valid in a qualitative sense also when peel stress is present, e.g. with regard to the importance of considering both strength and fracture energy of the bond line in joint strength analysis. References Adams, R.D. and Wake, W.C. (1984) Structural adhesive joints in engineering. Elsevier Science Publishers Ltd. Andersson, G.P., Benett, S.J. and DeVries, K.L. (1977) Analysis and testing of adhesive bonds. Academic Press Inc. Barak, S. (1990) FEM-beräkning av+i trä, Student graduation work (in preparation), Div. of Structural Mech., Lund Inst. of Techn., Sweden. DiTommaso, A. and Bache, H. (1989) Size effects and brittleness, Fracture Mechanics of Concrete Structures. RILEMreport ed. by L Elfgren, Chapman and Hall, pp 191–207. Gustafsson, P.J. (1987) Analysis of generalized Volkersen—joint in terms of non—linear fracture mechanics, Mechanical Behaviour of Adhesive Joints. Proc. of Euromech Colloqium 227 ed. by Verchery and Cardon, Edition Pluralis, Paris, pp 323–338 Gustafsson, P.J. (1988) Lim och fiktiva sprickor. Inverkan av fogtjocklek, Rapport TVBM-3038 “Byggnadsmateriallära LTH, 1973–1988", Div. of Build. Mat., Lund Inst. of Techn., Sweden, pp 31–41. Gustafsson, P.J. and Enquist, B. (1988) Träbalks h llfasthet vid rätvinklig urtagning, Rapport TVSM-7042. Div. of Structural Mech., Lund Inst. of Techn., Sweden Hillerborg, A., Modéer, M. and Peterson, P.-E. (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research. 6, pp 773–782. Marian, J.E. (1954) Lim och limning. Strömbergs, Stockholm, Sweden. Ottosen, N.S. and Olsson, K.—G. (1988) Hardening/softening plastic analyses of an adhesive joint, J. of Eng. Mech. 114:1, pp 97–116. Wernersson, H. and Gustafsson, P.J. (1987) The complete stress—slip curve of wood-adhesives in pure shear, Mechanical Behaviour of Adhesive Joints. Proc. of Euromech. Colloqium 227 ed. by Verchery and Cardon, Edition Pluralis, Paris, pp 139–150. Wernersson, H. (1990) Licentiate thesis (in preparation), Report TVSM-3012. Div. of Structural Mech., Lund Inst. of Techn., Sweden.

19 CONCRETE SURFACE LOADED BY A STEEL PUNCH H.W. REINHARDT Stuttgart University (formerly at Institut für Massivbau, Darmstadt University of Technology), Germany

1 Introduction There are several cases in structural engineering where concentrated loads act on a concrete surface. Well— known examples are bridge bearings and anchors for post—tensioning. Other examples are modern fixing devices (undercut anchors), but also the impact of hard ob— jects against concrete structures. It is common to all these loadings that the force acts only on an extremely small bearing area and that passive confinement is activated which causes a triaxial state of compressive stresses. This will increase the loading capa— city of concrete far beyond the uniaxial compressive strength. Although this phenomenon is well known, there was little information available about the quantitative relation between stress and displacement of a small rigid punch on a concrete surface. Fur— theron, the range of applicability of concrete design formulae for the size of the concentrated loading area with respect to the size of the loaded structural element seemed to be very conservative. There-fore it was decided to investigate this aspect more thoroughly. It was preferred to tackle the problem in a global engineering way rather than to perform sophisticated finite element analyses. There were various reasons for this: first, the results should be quickly available, second, the influence of parameters such as water-cement ratio and aggregate size should be investigated. There is no doubt that FE analyses produce reliable results if the correct material law is known. But if the material law is not exactly known the FE result may be erroneous and verification tests are undispensible. In our case, the tests have been performed and the results will be used to define a non—linear stress-displacement relation which can be used in simplified static and/or dynamic analyses of structures. 2 Scope of research Most of the research has been devoted to the experimental study of the penetration of a rigid punch into concrete. For this purpose, a testing rig has been designed which allows the displacement control— led penetration of a cylindrical hardened steel bar into a concrete surface. The force-displacement relations are

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Fig. 1. Loading arrangement

evaluated and discus— sed with respect to the material behaviour. Furtheron, the results are normalized, which makes them generally applicable in static and dynamic analyses of local phenomena. 3 Experiments A test setup was chosen in such a way that a semi—infinite body is loaded at the surface by a rigid punch. Fig. 1 shows the arrangement. The penetration is monitored by four LVDTs and used as displacement control in a closed—loop system. The semi—infinite concrete body is a circumferentially reinforced cylinder. It turned out that this cylinder sometimes failed by radial splitting, i.e. in a failure mode which was not intended. The main variables of the experimental programme were compressive strength of the concrete (29 to 57 MPa), punch diameter (13 to 32 mm), and maximum aggregate size (8 to 32 mm). The various concrete compositions led to different amounts of mortar matrix and to different porosities. It will be shown that both aspects influence the load bearing behaviour under concentrated loads.

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Fig. 2. Contact stress vs. displacement for various punch diameters, fc= compressive strength, dk=max aggregate size, D=punch diameter

All results show some common features. The stress-displacement lines of Fig.2 consist of an initial force increase until a peak value which is reached at about 500 MPa. Then, a steep drop follows whereafter the stress increases again until a second maximum is reached which is higher than the first one. Usually, this maximum coincided with beginning of splitting failure of the concrete cylinder. Other concrete compositions showed about the same behaviour as the shape of the stress-displacement behaviour is concerned. But dependent on the concrete strength, there were differences in the value of the maximum stresses and the inherent displacements. All test results are reported by Lieberum (1987). 4 Discussion of results 4.1 Failure mechanism If a hardened steel punch is placed and loaded on a concrete surface stress concentrations will occur at the edge of the punch. Already at small displacements, the average contact stress is high and reaches the uniaxial compressive strength. At unloading (Fig. 3), the displacement is partly irreversible, i.e. the material has already been stressed beyond an elastic limit stress. Subsequent loading cycles show an increase of total and irreversible displacements, whereas the stiffness is almost constant until the envelope curve is reached. This behaviour leads to the assumption that the material under the punch is compacted but no or very few cracks have occured in the vicinity of the punch. Otherwise a large hysteresis would have occured, as has been found by Sinha, Gerstle and Tulin (1964) and Karsan and Jirsa (1969) in cyclic loading with high compressive stresses. The first maximum stress is accompanied by a circular heave around the punch which starts to fracture and leads to a surface spall. Af— ter spalling the penetration increases with simultaneous load drop. At a deeper level in the concrete, the stress increases again. The concrete is compacted again. The load could cause a second spall, but obviously splitting of the cylinder was easier. When the debris at the surface were

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Fig. 3. Loading and unloading curves

removed, typical zones of damage could be distin— guished (Fig. 4): the spall at the surface, pulverized concrete under the spall, concrete into which the punch has penetrated. The mechanism of cratering can be conceived as follows: at small stresses, the concrete is compressed and loaded in triaxial compres— sion due to passive confinement. Subsequently, internal damage occurs which leads to compaction. This process continues until a certain volume has been compacted and converted to pulverized material. Further displacement forces this material to move and to cause hydrostatic pressure to the confining concrete. During the test, tiny radial cracks, 20 to 30 mm in length, occurred prior to the first stress peak. The circular heave arround the punch had a dia ter of about three times the punch diameter. Finally, the weakest part fails, which causes spalls at the concrete surface.

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Fig. 4. Damaged specimen after failure

4.2 Penetration versus porosity Cement matrix is a porous material. Depending on water-cement ratio and degree of hydration, the porosity of hardened cement paste ranges between 30 and 70 %. If the matrix contains fine sand, the porosity of the matrix decreases. High amount of porosity means low strength and low stiffness. As the penetration of a rigid punch is concerned it is expected that high porosity facilitates penetration. The pore volume VP consists of gel and capillary pores and voids due to entrapped air. The porosity has partly been determined by experiments, partly by computation from cement content, water-cement ratio and degree of hydration. Since the aggregates were dense silicious material the compaction could only take place in the matrix. Therefore the ratio between pore volume VP and matrix volume Vm has been chosen as variable. Fig. 5 shows concrete compressive strength as function of cement matrix porosity. A linear relation can be closely fitted to the range of tested concretes. It appears that concrete strength is about 60 MPa at a porosity of 40 % and about 40 MPa at 60 % porosity of the hardened cement paste. Penetration w divided by punch diameter D is plotted versus matrix porosity in Fig. 6. The parameter is the average contact stress. The punch diameter is indicated in the circles. It can be seen that the penetration does not depend on porosity at low stresses up to 100 MPa. This stress is about 2½times the uniaxial strength, but less than a quarter of the first maximum in Fig. 2. At larger stresses—up to 250 MPa—a linear relation appears between penetration and porosity. Combining Fig. 5 and Fig. 6 leads to a linear relation between penetration and concrete strength. If the stresses grow above 250 MPa there is an overproportional increase of penetration with porosity. It appears from Fig. 6 that the linear relation holds for low values of P/Vm , say up to about 57 %, but beyond that point, a nonlinear relation is valid. The explanation for this transition from linear to nonlinear relationship is to find in the failure of the cement matrix. Up to P/Vm<57 % the matrix does not collapse whereas the matrix collapses if P/Vm › 57 %. The value of 57 % corresponds to a uniaxial compressive strength of about 40 MPa. With higher stresses, the transition point from local damage to collapse of material under the whole punch shifts to lower porosity values. Fig. 7 shows that a nonlinear relation exists between relative penetration and porosity at the stress where the concrete spalls off (first maximum in Fig. 2). At this stress, collapse occurs for all concrete strengths. It should be noted that Fig. 7 has also been normalized with respect to matrix volume Vm in concrete volume Vc in order to compare all results, i.e. with different cement content and aggregate size.

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 5. Compressive strength as function of matrix porosity, =porosity of hardened cement paste

Fig. 6. Relative penetration vs. matrix porosity u=displacement

It is difficult to quantify the relation between penetration and porosity from materials composition. An estimation may be the fol— lowing. From experimental evidence a compacted volume exists under and around the penetrated punch. The radius of the fictitious com— pacted cylinder is D. With u=measured penetration, hv the compacted cylinder depth, and pc=concrete porosity, then conservation of mass leads to (1) (2)

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217

Fig. 7. Relative penetration vs. matrix porosity at spalling stress Vm/Vc= volume of hardened cement paste in volume of concrete

With p=0.16 the depth hv≈1.5 u. This would mean that the compacted area stretches under the punch by 0.5 times u. The spall could then be generated in a zone which starts deeper than the actually measured penetration. 5 Normalization of results From the aforementioned it will be clear that the behaviour of concrete under a very concentrated load is a complex matter. It contains multiaxial state of stress, large displacements, and local crushing. A realistic modelling is very difficult and cumbersome and is not appropriate for practical engineering. In order to make the results useful for practise they should be simplified. The first step for that is a normalization in such a way that various punch diameters and concrete qualities can be described by a single rela— tion. The result of normalization is shown by Fig. 8. The vertical axis gives the stress normalized with respect to concrete compressive strength and the horizontal axis shows the displacement normalized with respect to punch diameter and strength. It should be noted that the constant 40 has the dimension of stress in MPa. It can be seen that the stress at spalling increases with the square root of con— crete compressive strength and that the inherent displacement de— creases linearly with compressive strength. Furthermore, the dis— placement increases linearly with punch diameter. This last relation follows already from linear analysis of a rigid punch on an elastic material by Yettram and Robbings (1969). It should be noted that the stress at spalling does not depend on the size of the punch. Although there was a slight tendancy that the maximum spalling stress decreased with punch diameter, this could not be confirmed systematically. The characterizing points of the curve in Fig. 8 are no. 1, which is the limit of linear elastic response, no. 2, which is the first maximum (spalling stress), no. 3, which is the minimum stress after decay, no. 4, which is the second maximum which is also the limit of applicability of the graph. The respective normalizedstresses are 6, 12.5, 3.5, and 25, while the normalized displacements are 0.02, 0.2, 0.5, and 2.

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ANALYSIS OF CONCRETE STRUCTURES

Fig. 8. Normalization of stress penetration relation

The energy consumption during penetration is given by the area under the curve. With the short notations (3) the integral reads (4) After numerical integration the value of the integral is (5) This expression can be used for static or quasi-static problems. Together with eq. (1) the penetration can be calculated for hard impact, for instance Reinhardt (1988). 6 Concluding remarks Tests have been performed with rigid punches on a flat concrete sur— face. There were several features which emerge from the test results:

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219

– the contact stress amounts to 12.5 times the uniaxial compressive strength when the first visible damage occurs – the first stress peak causes spalling of a concrete cone around the punch – further penetration leads to increased stresses – the porosity of the cement matrix is decisive for the displacement and the maximum contact stress – all results could be normalized in such a way that various punch diameters and concrete qualities are represented by a single relation – the stress at spalling does not depend on the size of the punch 7 Acknowledgements Most of the presented research has been performed during the preparation of the doctoral thesis of Dr. Lieberum. His contributions are gratefully acknowledged. The author would like to thank Prof. A. Hillerborg for all the inspiring discussions on concrete failure and for his unceasing effort to clarify complex things by conspicuous models. References Lieberum, K.H. (1987) Das Tragverhalten von Beton bei extremer Teilflächenbelastung. Dissertation Darmstadt, 122 pp Sinha, B.P., Gerstle, K.H. and Tulin, L.G. (1964) Stress-strain relations for concrete under cyclic loadings. J. Am. Concrete Inst., 61, no. 2, 195–211 Karsan, I.K. and Jirsa, J.O. (1969) Behaviour of concrete under compressive loadings. J. Struct. Div. ASCE, 95, no. 12, 2543–2563 Yettram, A.L., Robbings, K. (1969) Anchorage zone stresses in axially post-tensioned members of uniform rectangular section. Mag. Concrete Res. 21, no. 67, 103–112. Reinhardt, H.W. (1988) Assessment of impact penetration by static testing. Darmstadt Concrete 3, 129–139

PART FIVE ANCHORAGE

20 FRACTURE MECHANICS BASED ANALYSES OF PULLOUT TESTS AND ANCHOR BOLTS R.BALLARINI Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio, USA S.P.SHAH NSF Center for Advanced Cement-Based Materials, The Technological Institute, Northwestern University, Evanston, Illinois, USA

Introduction The pull-out failure of rigid anchors embedded in brittle (tension weak) materials is a critical consideration for many design situations. Anchor bolts are often used as connections in concrete structures, roof bolts in rock tunnels and tie backs in rocks. Because failure may occur as a result of the bolts pulling out of the matrix, one of the considerations important for design is the pull-out capacity of the bolts. A second application of short anchor bolts is their use in the pull-out test, a nondestructive technique which appears to offer considerable promise for determining in situ concrete properties. This test consists of embedding an anchor bolt in a concrete structure during casting, pulling it out at desired ages of curing, and inferring the compressive strength of the concrete from the pull-out load. In the design of an anchor bolt what is usually known a priori is the compressive strength (or tensile strength) of the matrix material, and the objective is to predict the tensile capacity of the bolt. In the pull-out test, on the other hand, the pull-out load is measured, and the objective is to deduce a material property which can be used to assess the strength of the material. The first problem can, therefore, be considered the inverse of the second problem. This paper presents a review of analytical and experimental investigations which have been conducted by the authors and others to gain a better understanding of the mechanisms which govern the failure of anchors embedded in brittle matrices. The paper will, for the most part, address the case in which concrete is the matrix material and the loading is monotonically increasing tension. However, it is expected that the conclusions and suggested fracture mechanics modeling could be extended to the study of other brittle materials such as rocks, ceramics, etc. , and to generalized loading conditions including initial prestress and fatigue loading. The Pull-out Test The most critical period in the life of a concrete structure is during construction. During this early age, the concrete may not have gained sufficient strength, and as a result, the structure may collapse if the temporary supports such as form works or shoring are prematurely removed (Lew et al 1981). In the past the strength

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1. Configuration of the Lok-Test

development of concrete has normally been determined by testing field cured concrete cylinders in uniaxial compression. However, the concrete in a test cylinder may differ significantly from that in the structure because of possible different transport, casting, compaction and curing conditions. A considerable interest has recently been directed towards determining in situ concrete properties. Among the many nondestructive techniques being investigated, the pull-out test appears to offer considerable promise (Malhotra and Carette 1980 and Bickley 1981). Pull-out testing of concrete to determine its in-place strength was first suggested by a Russian scientist (Skramtajev 1938). This idea was later extensively explored by Kierkegaard-Hansen at the Technical University of Denmark (Kierkegaard-Hansen 1975 and KierkegaardHansen and Bickley 1978). The increasing importance of the pull-out test in the United States has manifested itself as a tentative standard (ASTM-C900–78T). Kierkegaard-Hansen’s research has led to the Lok-Test, whose configuration is shown in Figure 1. A test bolt, consisting of a stem and a circular steel disc, is mounted inside the form during construction (Figure 1a). After curing, the form is stripped and the stem is unscrewed. A rod having a slightly smaller diameter than the stem is screwed into the disc at the time of testing, and a cylindrical counter pressure is applied (Figure 1b). The rod is then loaded by a pullout force until failure occurs. Because of its shape, the steel insert pulls out a cone of concrete. The precise geometry of the failure surface depends on factors such as the diameter of the disc, the diameter of the counter pressure, the depth of the embedment, and the constitutive properties of the concrete. The test is considered non-destructive because the damage induced by the relatively small pull-out cone does not impair the integrity of the structural member and can be easily repaired. The maximum pull-out load is empirically related to the compressive strength. Such calibration curves have been derived from a large number of laboratory and field studies (Malhotra and Carette 1980, Bickley 1981, Kierkegaard-Hansen 1975, and Mailhot et al 1979). By correlating standard cylinder strength with the

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223

pull-out force at 18 construction sites, Bickley concluded that the empirical relation between the pull-out force and compressive strength should be determined for each site and for each type of concrete and aggregate size. One of the reasons for such recommendations, which require extensive empirical calibration, is that the material properties measured by the pull-out test are not sufficiently understood to allow analysis and prediction of the failure mechanisms. The state of stress in the pull-out test is complex and difficult to analyze. Considerable controversy has arisen over what strength property of concrete is actually being measured and what is the precise mechanism of failure. An attempt to show that the Lok-Test is a measure of compressive strength was made by Jensen and Braestrup (1976) using plasticity theory. They assumed that the failure surface is the frustrum of a cone and that failure occurs by sliding along this surface. By using a Mohr-Coulomb failure criterion, an upper bound solution was found for the pull-out force. With the geometrical dimensions of the Danish pull-out system, they showed that when the angle between the direction of deformation and the failure surface are equal to the angle of friction for the concrete, then the pull-out force is directly proportional to the concrete compressive strength. Their results (which should represent an upper bound), when compared with tests conducted at the Technical University of Denmark, underestimated the experimentally observed pull-out loads. Ottosen (1981) analyzed the Lok-Test by means of axisymmetric nonlinear finite elements. The constitutive model he used was based on non-linear elasticity coupled with either a modified Coulomb criterion or a multiaxial failure criterion. His analysis followed the progression of circumferential and radial cracking by means of an iterative smeared cracking procedure, and showed that circumferential cracks begin at the disc edge at approximately 15 percent of the ultimate load, grow towards the reaction ring, and reach the ring at approximately 65 percent of the ultimate load (Figure 2). Beyond this load, Ottosen observed that large compressive stresses run from the disc edge in a rather narrow band towards the support. He postulated that the remaining load carrying capacity can be attributed to the crushing of this compressive strut of material, and on this basis he concluded that the test measures compressive strength. Realizing the importance of the pull-out test for the safety of concrete structures during construction, and recognizing that insufficient information exists for the exact failure mode in pull-out tests, Stone and Carino (1983) from the National Bureau of Standards (N.B.S.) conducted pull-out tests of enlarged, extensively instru mented specimens. To circumvent the problems of instrumenting the commercially available pull-out units, which are dimensionally small, the N.B.S. tested specimens which were twelve times as large as those in a standard pull-out test. Strains were measured in three mutually perpendicular directions by specially designed micro-embedment strain gauges. The slip between the bottom of the disc and the surrounding concrete was measured with a slip gauge. The N.B.S. study is quite important because it represents the first time that detailed measurements were made of internal strains and slip at various stages of the pull-out test. Some of the results are summarized as follows: (1) By inspecting the strain histories of each embedded gauge a list of all discontinuities and the load stages at which they occur was compiled (Figure 3). Three distinct phases of pull-out behavior were observed. Circumferential cracking near the upper edge of the disc initiates at approximately 30–40 percent of the ultimate load and ends the elastic response. The circumferential crack continues to grow towards the reaction ring with an increase in load, and any resistance to additional load is provided by aggregate interlock. (2) Observed compressive strains adjacent to the failure surface were insufficient to initiate compressive failure.

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2. Crack Development with Increasing Loading (Loading Expressed in Relation to Predicted Failure Load): (a) Loading=15%; (b) Loading=25%; (c) Loading=64%; (d) Loading=98% [From Ref. Ottosen (1981)]

(3) Large tensile strains perpendicular to the failure surface exist near the outer edge of the disc and decrease rapidly toward the reaction ring just before the load at which circumferential cracks initiate. As circumferential cracking progresses, all radial gauges pick up large tensile strains. In contradiction to the theoretical studies of Jensen and Braestrup and Ottosen, the results of the N.B.S. study clearly indicate that stress concentration, crack initiation and crack growth play an important role in determining the pull-out response of concrete. Stone and Carino (1984) also performed a linear elastic, axisymmetric finite element analysis of two pull-out configurations corresponding to the specimens they tested. While they did not extend the analyses beyond the first cracking load, they found good agreement between the experimental and analytical strains up to the load at which cracking first occurred in the laboratory specimens. Because the stresses in the uncracked state were compressive parallel to a line from the pull-out disk to the reaction ring, and tensile in the circumferential and radial directions, they suggested that the failure surface is formed as a result of tensile stresses. They proposed that the good correlation between the pull-out load and the compressive strength of the concrete arises from the strong dependence of the compressive strength on the tensile strength. Another experimental investigation to study the fracture processes in the pull-out test was conducted by Krenchel and Shah (1985). They performed pull-out tests which were in scale with standard equipment. To register microcracking in concrete, acoustic emission (a.e.) activity was measured during the test. Some tests were performed with only partial loading, followed by unloading, making it possible to examine the development of microcracking at various load levels by cutting sections of the unloaded concrete specimens

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3. Discontinuity Histograms for Large Scale Pull-Out Specimens [From Ref. Stone (1983)]

(Figure 4). The following conclusions (which supported the findings of the N.B.S. investigators) were drawn by the authors. (1) Microcracking was detected using acoustic emission techniques at about 30 percent of the ultimate load. (2) For loads up to about 65 percent of the peak load, cracking seems primarily concentrated near the upper corners of the pull-out disk. The angle these cracks make with the horizontal is approximately 15–20°. This cracking system, which appears to be stable, was termed primary cracking. (3) For loads near the peak load, secondary cracks form, running from the upper edge of the disc to the inside edge of the support ring. These cracks propagate at angles approximately 25–45°. This secondary cracking pattern did not fully develop all the way around before the peak load was reached.

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ANALYSIS OF CONCRETE STRUCTURES

Because the experiments conducted by the N.B.S. and Krenchel and Shah showed that failure in the pullout test is dominated by the propagation of discrete cracks, fracture mechanics modeling of the pull-out test has recently gained popularity. Ballarini et al (1985, 1986, 1987) conducted a combined analytical and experimental study of a twodimensional pull-out test using mortar as the matrix material. Since it is not possible to optically observe crack propagation in an opaque material like concrete before the ultimate load is reached during an axisymmetric test, they designed the planar system shown in Figure 5. Using this system crack propagation was observed throughout the experiment (on the surface of the specimens). To determine the influence of geometry the embedment depth and the support reaction distance were varied. To determine the influence of compressive strength the tests were performed at three ages of curing after casting: 1, 2, and 3 days. It should be noted that this setup can model the pull-out test as well as the anchor bolt design problem by the proper adjustment of the parameters d and h. The analysis performed in the study relied on the solution obtained by the author to the idealized elastostatics problem shown in Figure 6. In this two-dimensional analysis, the anchor is modeled as a vertically loaded, partly bonded rigid plate in an elastic half-space. Failure is assumed to arise from cracking which emanates from the corners of the plate. The details of the mathematical analysis can be recovered from Ballarini et al (1987). To characterize crack growth during the pull-out process, stress intensity factors were calculated for several combinations of the parameters involving crack length, The angle of extension, location of the concentrated forces support reactions at the top surface, and embedment depth of the plate. Figures 7–12 present the results of these calculations. Figures 7 and 8 are plots of the stress intensity factors as functions of crack extension angle for two configurations. These results show that for short cracks the maximum values of the opening mode stress intensity factor (K) occur at points where the shear-mode factor KII) is nearly zero. This suggests that both crack initiation and the direction that the extended crack will choose to grow are governed by the opening mode. The anchor pull-out crack initiation direction was therefore assumed to depend upon the direction of maximum KI Figure 9 is a plot showing the maximum value of the opening mode stress intensity factor versus crack length for several test configurations. The effect that the support reactions have on the stability of crack propagation can be clearly seen. For relatively short spacing of the support forces and deep embedments, cracks will grow in a stable manner until they reach 1/c values approximately equal to 0.75; after this point they will continue to grow, but in an unstable manner. On the other hand, for wide spacing and shallow embedments, crack growth is unstable for all crack lengths. The configurations of the Lok-Test and of Krenchel and Shah’s test set-up correspond to the case where d/c=h/c=2.0. These results may explain the stable crack propagation that was observed in their experiments. The predictions of the mathematical model agree very well with the results obtained from the experiments. Figure 10 shows the experimental results for the case d/c=h/c=2.0. The maximum load for this specimen was 1,540 1b. Crack initiation was observed through the microscope and in the load-slip curve at 1,000 1b. The cracks grew in a stable manner until they reached a critical crack length. These results are consistent with those predicted by the model (Figure 9). Figure 11 shows the results for the case h/c=1.5, d/c=3.0. The maximum load, which in this case was equal to the crack initiation load, was 450 1b. The cracks grew in an unstable manner, and this result is also consistent with the model (Figure 9). The stress intensity factors were used to construct the crack paths shown in Figures 10 and 11. It can be seen that cracks will initiate and grow almost horizontally for short lengths. As they become longer, they tend to turn and grow towards the supports. The microscopically observed crack tip locations are also shown in Figures 10 and 11. Again, it can be seen that the theoretical model is consistent with experimental observations.

FRACTURE MECHANICS BASED ANALYSES

227

4. Crack Analysis After Sectioning at: (a) 53% of average peak load; (b) 65%; (c) 93% [From Ref. Krenchel (1985)]

Figures 12 and 13 show the critical stress intensity factors as functions of compressive strength for crack initiation (Figure 12) and for ultimate loads (Figure 13). These figures were constructed by extracting the factor Kc /P from Figure 9 at the point of instability for ultimate load and the factor Kc /P at 1/c=0 for crack initiation. By multiplying these factors by the experimentally observed ultimate loads and crack initiation loads, the factors Kc1 1/2 at ultimate load at the Initiation load are obtained and plotted as functions of

228

ANALYSIS OF CONCRETE STRUCTURES

5. Configuration of Two-Dimensional Pull-Out Specimen [From Ref. Ballarini (1985)]

compressive strength. It can be seen from these figures that for a particular compressive strength, the fracture toughness lies in a relatively narrow band. The figures imply that the tensile capacity is governed by the fracture toughness. The same conclusion was reached by Ryhming et al (1980) in their analysis of a rock breaking machine tested in PMMA, where the failure mechanisms are similar to those in the pull-out test. Because these figures are independent of geometry they can be used to predict either ultimate (or crack

FRACTURE MECHANICS BASED ANALYSES

229

6. Analytical Model of Two-Dimensional Pull-Out Specimen [From Ref. Ballarini (1985)]

initiation) load given compressive strength, or compressive strength given ultimate (or crack initiation) load. The latter procedure may be used to interpret pull-out tests, while the first may be used for anchor bolt design, as will be discussed subsequently. It should be noted that the cracking pattern observed in these experiments correspond to the primary cracking system seen by Krenchel and Shah. A test set-up very similar to the one designed by Ballarini et al was recently suggested by Hillerborg at Lund Institute of Technology and used by Ohlsson and Elfgren at Luleå University of Technology to study pull-out tests on concrete specimens (Ohlsson and Elfgren 1989). Except for the difference in dimensions, the only difference between Ballarini’s and Hillerborg’s specimens is that in the latter (shown in Figure 14) springs were used (perhaps to prevent or measure rotation effects). Ohlsson and Elfgren’s tests were conducted for the cases: 2c=100 mm, t=25 mm, e=15 mm, b=50 mm, a/d=300/200, 150/100, 100/200. Their experimental results are summarized in Table 1 and Figures 15–16. Figure 15 shows that for the

230

ANALYSIS OF CONCRETE STRUCTURES

7. Stress Intensity Factors as Functions of Extension Angle, Concentrated Forces Present [From Ref. Ballarini (1985)]

configuration of specimens 3 and 4 there is not much of an increase in load beyond first cracking, while the remaining configurations lead to a significant amount of stable crack growth. The crack paths observed in these experiments are shown in Figure 16 along with those observed in Ballarini’s tests on mortar and those predicted by his linear elastic fracture mechanics model. The differences between the paths in concrete and mortar can be attributed to cracks propagating around aggregates. In their communication Ohlsson and Elfgren invited scientists in the worldwide engineering community to analyze the tests with their respective methods and to communicate their results back to them. A rough analysis of tests 3 and 4 can be made using Figure 9 since the configuration of these tests corresponds to the case h/c=2.0, d/c=3.0. For this configuration the linear elastic fracture mechanics analysis which led to Figure 9 predicts that crack propagation is stable and that the ultimate load should be approximately 1.1 times the initiation load (.587. 52). The proportional limit for test 4 is approximately 14.6 kN and the ultimate load is 15.4 kN. The ratio between ultimate load and initiation load is approximately 1.06, which is quite close to the ratio predicted by the linear elastic fracture model. From Figure 9 Kc1/2 at initiation is approximately 0.58. The intensity of loading is given by P=14.6 kN/b=14.6 kN/0.05 m, and c=0.05 m. This leads to a value for KIc=846 kN/m3/2 which is a reasonable value for concrete.

FRACTURE MECHANICS BASED ANALYSES

231

8. Stress Intensity Factors as Functions of Extension Angle, Concentrated Forces not Present [From Ref. Ballarini (1985) Table 1. Results from Anchor Bolt Tests—[Ohlsson and Elfgren (1989)] Test No.

a (mm)

d (mm)

F (kN)

1 2 3 4 5

300 300 150 150 200

200 200 100 100 200

16.0 20.1 15.2 15.4 99.5

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ANALYSIS OF CONCRETE STRUCTURES

9. Maximum Mode-I Stress Intensity Factors as Functions of Crack Length [From Ref. Ballarini (1985)]

It can be seen from Figures 12 and 13 that the fracture toughness for a given compressive strength is higher for ultimate loads than for crack initiation loads. This is a result of the process zone which evolves as the crack is growing in a stable manner. This zone leads to an increase in the critical stress Intensity factor as the crack propagates (R-curve behavior).

FRACTURE MECHANICS BASED ANALYSES

233

10. Experimentally Observed and Theoretically Predicted Crack Profiles for Close Spacing of Supports. Load Levels in Experiments Designated at Various Points in Pounds (Kilonewtons) [From Ref. Ballarini (1985)]

To account for the effects of the process zone Hellier et al (1987) analyzed the axisymmetric pull-out test using the nonlinear fracture mechanics finite element program developed by Ingraffea and coworkers at Cornell. In this model the process zone is replaced by a distribution of normal and shear stresses which represent the transfer of load across the process zone. Their analysis was able to predict the stable, primary cracking system previously discussed. This system extends from the outer edge of the insert and is arrested at a point beneath the reaction at approximately 45% of the ultimate load observed in the experiments conducted by the NBS (Figure 17). The stress redistribution which results at this load from the primary cracking system leads to the development of a secondary system which initiates below the top free surface at the inner edge of the reaction ring and propagates towards the outer edge of the insert. Figure 18 shows the tensile stress distribution at the stage when the secondary cracking system has fully developed. As shown in the load displacement plot in Figure 19, their simulation was halted at approximately 80% of the ultimate experimental load when the secondary cracking system is arrested by compressive stresses which lead to a localization whose physics was not included in the finite element model. It was postulated that the formation of the failure surface is completed by a direct shear failure of the uncracked ligament (Figure 20). The research described in this section clearly indicates that the implementation of fracture mechanics principles into available numerical techniques has led to a much better understanding of the failure mechanisms in the pull-out test. The next section presents a brief review of recent investigations aimed at incorporating fracture mechanics principles into design procedures for short anchors and studs embedded in brittle materials.

234

ANALYSIS OF CONCRETE STRUCTURES

11. Experimentally Observed and Theoretically Predicted Crack Profiles for No Supports. Load Levels in Experiments Designated at Various Points in Pounds (Kilonewtons) [From Ref. Ballarini 1985)]

Anchor Bolt Design The use of short anchor bolts and welded studs is wide-spread. Figure 21 (taken from the Prestressed Concrete Institute Design Handbook) shows typical details of connections which rely on anchors to transmit loads. Whether they are used as column to foundation, column to column, beam to column, deck unit to

FRACTURE MECHANICS BASED ANALYSES

235

12. Fracture Toughness at Crack Initiation as a Function of Compressive Strength [From Ref. Ballarini (1985)]

beam, or other connections, predicting the capacity (tensile, shear, moment) of the anchor bolts is a critical consideration in the design process. Because the state of stress is complex and difficult to analyze, it is very difficult to predict the tensile capacity of short anchor bolts, and design procedures Inevitably involve simplifying assumptions. When an anchor bolt fails by pull-out of the concrete, the resistance is calculated by using either of the following criteria (and modifications to them if other factors such as free edge effects come into play): (1) The failure resistance is the resultant of tensile stresses equal to the maximum concrete tensile strength, directed perpendicular to the surface area of the truncated cone shown in Figure 22. (2) The failure resistance is the resultant of tensile stresses equal to the maximum concrete tensile strength, directed parallel to the direction of the applied load. Klinger and Mendonca (1982) performed a literature review on the tensile capacity of short anchor bolts, where the results of available tests were compared with predictions of six design procedures currently available for computing the nominal capacity of tension-loaded short anchor bolts and welded studs.

236

ANALYSIS OF CONCRETE STRUCTURES

13. Fracture Toughness at Ultimate Load as a Function of Compressive Strength [From Ref. Ballarini (1985)]

Klinger and Mendonca showed that these methods are not conservative, and more importantly, reveal considerable scatter (Figure 23). Their investigation suggests that more accurate procedures are necessary to predict safely the ultimate capacity of anchor bolts. As for the pull-out test, fracture mechanics methodology has gained popularity for predicting the load carrying capacity of short anchor bolts embedded in brittle materials. An analytical-experimental program aimed at providing background results for a design guide for anchor bolts was initiated by Elfgren et al (1980). The report which presents the preliminary results of their investigation also provides an excellent list of references for test results, codes and standards, and design rules. As far as the authors know, the analysis carried out by Elfgren was the first to include cracking in the analysis of short anchor bolts. The finite element model of the anchor bolt configuration they analyzed is shown in Figure 24. Figure 25 shows the principal stress contours for various crack lengths. It can be seen from Figures 25(a), 25(d) and 25(g) that the maximum tensile stress decreases for the given load as the

FRACTURE MECHANICS BASED ANALYSES

237

14. Two-Dimensional Pull-Out Specimen Proposed by Hillerborg [From Ref. Ohlsson (1989)1

crack length increases. Although they used a coarse mesh and did not investigate crack stability using a stress intensity factor approach, their results indicate there is a tendency that the applied load must be increased in order to increase the crack length. Elfgren’s results suggested that useful information about the load-carrying capacity of short anchors could be obtained by incorporating discrete cracking into available analysis techniques. This fact led Ballarini et al to investigate the effects of embedment depth and support reaction distance on the tensile capacity and stability of crack propagation using the planar system previously described. Figures 9, 12 and 13 can be considered design curves for the mortar which was tested. Knowing the compressive strength of the mortar, the curve in Figure 12 can be entered and the resulting factor Kc ‘can be obtained. For the desired configuration (d/c, h/c), the factor Kc1/2/P can be obtained from Figure 9, and thus the load P which leads to crack initiation can be computed. The same procedure can be employed for ultimate loads using Figure 13 instead of Figure 12. It should be noted that the analysis conducted by Ballarini et al is twodimensional. However, using the finite element method similar curves could be developed performing similar analyses on axisymmetric models. Moreover, important factors such as generalized loadings, initial prestress, fatigue loadings, etc. could be incorporated. Analytical and experimental work along these lines is being conducted by Eligehausen et al at Universitat Stuttgart (1987, 1988, 1989). In a recent investigation Eligehausen (1989) conducted pull-out tests with headed studs in large concrete blocks (Figure 26). In these experiments strains in the concrete were measured along the anticipated failure surface. Figure 27 shows the distributions of measured strain perpendicular to the expected crack direction at loads equal to 30% and 90% of ultimate for an embedment depth h=520 mm.. These distributions were used to compute the distributions (Figure 28) of normal stress by assuming the strains are the sum of elastic and inelastic (due to crack opening) components. The effect of stress redistribution due to stable crack growth can be clearly seen from this Figure by observing the shift in location of maximum stress with increasing load. Stable crack growth was confirmed through acoustic emission analysis. Eligehausen developed a predictive model using an energy approach. The length of the crack and its direction were

238

ANALYSIS OF CONCRETE STRUCTURES

15. Load-Deflection Diagram of Tests Using the Specimen Proposed by Hillerborg [From Ref. Ohlsson (1989)]

calculated as functions of applied load, using the finite element method, from the variation of the total free energy with crack length and crack angle respectively (Figure 29). The following relation was derived between the force F and crack length a (1)

FRACTURE MECHANICS BASED ANALYSES

239

16. Photos of Specimens After Failure: (a) Specimen 2; (b) Specimen 3 [From Ref. Ohlsson (1989)]

where G is the fracture energy, E is Young’s modulus of the concrete, and lb is the length of the failure cone surface. The function f depends on the crack length and is shown in Figure 30 for a point-load and a circumferential crack inclined at 37.5° (this failure cone was observed in the experiments). It can be seen from this figure that for this configuration crack growth is stable up to a/lb= 0.45, and thus the maximum load is

240

ANALYSIS OF CONCRETE STRUCTURES

17. Deformed Mesh Showing the Primary Cracking System in the Pull-Out Test [From Ref. Helier et al (1987)]

(2) Figure 31 is a comparison between the failure loads predicted using equation (2) and those predicted using the empirical relation derived in Eligehausen et al (1988). The agreement is quite good. Conclusion All of the research described In this brief review clearly indicates that fracture mechanics is an extremely useful tool for characterizing the failure mechanisms of short anchors embedded in concrete and for predicting their load carrying capacity. Moreover, the results strongly suggests that design procedures should be modified to include information obtained from fracture mechanics modeling and testing. Curves

FRACTURE MECHANICS BASED ANALYSES

241

18. Tensile Stress Distribution at the Stage When the Secondary Cracking System has Fully Developed [From Ref. Helier et al (1987)]

such as those presented in Figures 9, 12 and 13 can be used to develop design equations of the type given by Eq. (2), which is based on rational methods of analysis. REFERENCES Ballarini, R. (1985) “An Analytical and Experimental Investigation of a Two-Dimensional Anchor Pull-Out Test,” Ph. D. Dissertation, Northwestern University. Ballarini, R., Shah, S.P. and Keer, L.M. (1986) “Failure Characteristics of Short Anchor Bolts Embedded in a Brittle Material,” Proceedings of the Royal Society of London, A404, pp. 35–54.

242

ANALYSIS OF CONCRETE STRUCTURES

19. Comparison of Analysis with Experimental Load-Displacement Curves for Concrete and Mortar Specimens [From Ref. Helier et al (1987)] Ballarini, R., Keer, L.M. and Shah, S.P. (Feb. 1987) “An Analytical Model for the Pull-Out of Rigid Anchors,” International Journal of Fracture, Vol. 33, pp. 75–94. Bickley, J.A. (1981) in In Situ Non-Destructive Testing of Concrete. edited by V.M. Malhotra, pp. 195–198, American Concrete Institute, Special Volume SP-82. Elfgren, L., Broms, C.E., Johansson, H.E. and Rehnstrom, A. (1980) “Anchor Bolts in Reinforced Concrete Foundations: Short Time Tests,” University of Luleå Research Report 36. Eligehausen, R., Fuchs, W. and Mayer, B. (1987) “Tragverhalten von Dube l befestigungen bei Zugbeanspruchung (Behavior of Fastenings Under Tension Loading),” Betonwerk+Fertigteil-Technik, No. 12, pp. 826–832. Eligehausen, R., Fuchs, W. and Mayer, B. (1988) “Tragverhalten von Dube l befestigungen bei Zugbeanspruchung: Teil 2 (Behavior of Fastenings Under Tension Loading: Part 2),” Betonwerk + Fertigteil- Technik, No. 1, pp. 29–35. Eligehausen, R. (1988) “Bemessung von Befestigungen Stahldubeln-Zukunftiges Konzept (Design of Fastenings with Steel Anchors-Future Concepts),” Betonwerk+Fertigteil-Technik, No. 5, pp. 88–100. Eligehausen, R. and Fuchs, W. (1988) “Tragverhalten von Dube 1befestigungen bei Querzug-Schragzng-und Beigebeanspruchung (Load-bearing Behavior of Anchor Fastentings Under Shear, Combined Tension and Shear or Flexural Loading),” Betonwerk + Fertigteil-Technik, No. 2, pp. 48–56. Eligehausen, R. and Sawade, G. (1989) “A Fracture Mechanics Based Description of the Pull-Out Behavior of Headed Studs Embedded in Concrete,” private communication. Hellier, A.K., Sansalone, M. , Ingraffea, A.R., Carino, N.J. and Stone, W.C. (Summer 1987) “Finite Element Analysis of the Pull-Out Test Using a Non-Linear Discrete Cracking Approach,” ASTM Cement Concrete and Aggregates, pp. 20–29. Jensen, B.C. and Braestrup, H.W. (1976) “Lok-Tests Determine the Compressive Strength of Concrete,” Nordisk Betong, Stockholm, Sweden, Vol. 20, No. 2, pp. 9–11.

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243

20. Deformed Mesh Showing the Primary and Secondary Cracking Systems at the Onset of Failure [From Ref. Helier et al (1987)] Kierkegaard-Hansen, P. , “Lok-Strength,” Nordisk Betong, Vol. 3, pp. 19–28. Kierkegaard-Hansen, P. and Bickley, J.A. (Oct. 29—Nov. 3, 1978) “In Situ Strength Evaluation of Concrete by the LokTest System,” presented at the A.C.I. Fall Convention, Houston, Texas, U.S.A. Klinger, R.E. and Mendonca, J.A. (July-Aug. 1982) “Tensile Capacity of Short Anchor Bolts and Welded Studs: A Literature Review,” Journal of the American Concrete Institute, Vol. 79, No. 4, pp. 270–279. Krenchel, H. and Shah, S.P. (1985) “Fracture Analysis of Pull-Out Testing,” Mater. Struct., Vol. 18, No. 108, pp. 439–446. Lew, H.S., Carino, N.J., Fattal, S.G. and Batts, M.E. (Sept. 1981) “Investigations of Construction Failure of Harbour Cay Condominium in Cocoa Beach, Florida,” National Bureau of Standards. Mailhot, G., Bisaillon, A, Carette, G.G. and Malhotra, V.M. (Dec. 1979) “In-Place Concrete Strength: New Pull-Out Methods,” Proceedings of the American Concrete Institute, Vol. 76, No. 12.

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ANALYSIS OF CONCRETE STRUCTURES

21. Details Which Use Anchor Bolts as Connections Malhotra, V.M. and Carette, G.G. (1980) “Comparison of Pull-Out Strength of Concrete with Compressive Strength of Cylinders and Cores, Pulse Velocity and Rebound Numbers,” Journal of the American Concrete Institute, Vol. 77, pp. 161–170. Ohlsson, U. and Elfgren, L., “Anchor Bolts in Concrete Structures: Two-Dimensional Modeling,” private communication. Ottosen, N.S. (April 1981) “Non-Linear Finite Element Analysis of a Pull-out Test,” ASCE Journal of the Structural Division, Vol. 107, No. 4, pp. 591–603.

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245

22. Failure Surface Used in Currently Available Design Procedures [From Ref. Hellier (1987)]

23. Ratio of Actual to Predicted Tensile Capacity of Short Anchor Bolts [From Ref. Hellier (1987)] Ryhming, I., Cooper, G.A. and Berlie, J. (1980) “A Novel Concept for a Rock-Breaking Machine I: Theoretical Consideration and Model Experiments,” Proceedings of the Royal Society of London, A373, pp. 331–351. Skramtajev, B.G. (1938) “Determining Concrete Strength for Control of Concrete Structures,” Proceedings, American Concrete Institute, Vol. 34. Stone, W.C. and Carino, N.J. (Nov.-Dec. 1983) “Deformation and Failure in Large Scale Pull-Out Tests,” Journal of the American Concrete Institute, Vol. 80, No. 6, pp. 501–513. Stone, W.C. and Carino, N.J. (Jan.-Feb. 1984) “Comparison of Analytical with Experimental Internal Strain Distribution for the Pullout Test,” Journal of the American Concrete Institute, Vol. 81, pp. 3–12.

ACKNOWLEDGEMENT The second author is grateful for partial support from the National Science Foundation under Grant No. NSF-DMR-8808432 (Dr. Lance Haworth, Program Manager).

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ANALYSIS OF CONCRETE STRUCTURES

24. Finite Element Model of a Cracked Anchorage Zone: (a) Whole Mesh; (b) Central Part of Mesh When the Crack has a Length of 42.4 mm. [From Ref. Elfgren (1980)]

FRACTURE MECHANICS BASED ANALYSES

25. Contours of Principal Stresses for Model Shown in Figure 24 [From Ref. Elfgren (1980)]

247

248

ANALYSIS OF CONCRETE STRUCTURES

26. Dimensions of Test Specimen and Headed Studs [From Ref. Eligehausen (1989)]

FRACTURE MECHANICS BASED ANALYSES

27. Distribution of Strains Perpendicular to the Failure Cone Surfaces [From Ref. Eligehausen (1989)]

249

250

ANALYSIS OF CONCRETE STRUCTURES

28. Distribution of Tensile Stress Perpendicular to the Failure Cone Surfaces [From Ref. Eligehausen (1989)]

FRACTURE MECHANICS BASED ANALYSES

29. Application of the Theoretical Model on Headed Studs Embedded in Concrete [From Ref. Eligehausen (1989)]

251

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ANALYSIS OF CONCRETE STRUCTURES

30. Calculated Load as Function of Crack Length [From Ref. Eligehausen (1989)]

FRACTURE MECHANICS BASED ANALYSES

31. Failure Load as a Function of Embedded Depth [From Ref. Eligehausen (1989)]

253

21 ANCHOR BOLTS IN CONCRETE STRUCTURES. TWO DIMENSIONAL MODELLING U. OHLSSON, L. ELFGREN Luleå University of Technology, Luleå, Sweden

Abstract An experimental and numerical investigation of anchor bolts embedded in concrete structures is presented. The investigation was designed in order to check fracture mechanics procedures for analysis of anchor bolts. To be able to study the crack propagation during the tests and to simplify the analysis, the tests were carried out under plane stress conditions on anchor bolts embedded in concrete plates. 1 Introduction Anchor or headed studs can be embedded in, drilled into, or grouted in concrete or rock. Much work is going on in order to develop safe design procedures, Rehm et al (1988), Elfgren et al (1987) and Klinger and Mendonca (1982). Fracture mechanics gives us the possibility to analyse the interaction between an anchor bolt and the surrounding concrete in an accurate way. Contributions to this development have among others been given by Ottosen (1981), Elfgren et al. (1982), Peier (1983), de Borst (1986), Ballarini et al. (1986), Hellier et al. (1987), Rots (1988), Eligehausen and Sawade (1989). 2 Laboratory Tests 2.1 Materials and test specimen Test specimens according to figure 2.1 were cast with normal strength and high strength concrete, see Ordqvist and Soutukorva (1990). The test configuration was proposed by Arne Hillerborg. A similar test set up was earlier used by Shah and Ballarini, see Ballarini et al. (1986). The specimens were cast in steel moulds. The anchor bolts, except on the upper sides of the anchor heads, were treated with a thin layer of form oil to prevent bond between steel and concrete.

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255

Figure 2.1 Test specimen. In the tests presented here 2c=100 mm, t=25 mm, e=15 mm, and b=50 mm. The ratio a/d was 300/200=1.5, 150/100=1.5 and 200/100=2.0. The stiffness k was obtained by a tie in form of two 12 mm reinforcement bars which gives k=EA/L=50 to 100 MN/m depending on the length L.

The concrete was made with Portland cement and natural aggregates, see table 2.1. Cubes (150mm) and notched beams (840×100×100mm) were also cast in order to determine the splitting strength, fcspl, the compressive strength, fcc, and the fracture energy, GFE. The fracture energy was determined with the RILEM three-point bending tests of beams, see Hillerborg (1985). The modulus of elasticity, E, was determined on drilled concrete cores, diameter 70mm, length 150mm, from the tested notched beams, see table 2.2 Table 2.1 Concrete mixes. Concrete

C45 C80

Coarse aggregate 8–12mm

Fine aggregate 0–8mm

Cement

Water

Micro silica

Super-plastisizer

(kg/m3)

(kg/m3)

(kg/m3)

(kg/m3)

(kg/m3)

(kg/m3)

859 1149

933 770

374 450

173 156

– 70

– 10

Table 2.2 Material properties. Concrete type

fcc (MPa)

fcspl (MPa)

Ec (GPa)

GF (N/m)

C45 C80

55±2 (3) 85±7 (3)

3.4±0.2 (3) 5.9±0.3 (6)

38.2±0.9 (3) 40.4±0.9 (4)

91±7 (3) 121±1 (3)

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ANALYSIS OF CONCRETE STRUCTURES

Figure 2.2 Test set-up

The table shows mean values, standard deviations and number of tests performed. 2.2 Test set-up The anchor bolts were tested “upside down” in a servohydraulic testing machine, see figure 2.2. The load, F, the restraining force, T, and the anchor head displacement, d, (the displacement of the anchor head relative to the supports), were registered during the test. The tests on the C45 concrete specimen were loaded by simply pulling the anchor steel rod with a constant velocity. The high strength C80 concrete specimen were tested in displacement control with a constant anchor head displacement velocity. 3 Results from the laboratory tests 3.1 Crack patterns Figure 3.1 shows a tested specimen. The final failure generally consists of three main cracks. Crack A and C form the well known pull-out cone. The tests also show a third bending crack, B. The two 12 mm reinforcement bars, giving the restraining force T, were intended to stop this bending crack but the stiffness

ANCHOR BOLTS IN CONCRETE STRUCTURES

257

Figure 3.1 Test specimen, number 6.

obtained were apparently not enough. The bending crack can be avoided by making the specimen very high compared to the embedment depth, but from a practical point of view the test specimen would then be very difficult to handle. Generally the cracking starts with the forming of a crack, A. After crack A is induced crack B starts to propagate and shortly after that, crack C starts to develope. Some of the high strength, C80, specimen with embedment depth 100 mm showed another crack pattern, figure 3.2. The crack here starts below the anchor bolt and propagates toward the edge without any bending crack. 3.2 Load-displacement curves. Figure 3.3 to 3.6 show examples of load-anchor head displacement curves. For test 1 to 4 two curves are given from each test. They represent measured values from the two sides of the specimens. A load displacement curve may be divided into three different parts: The first part is linear elastic. The crack growth has not yet started. The second part is characterized by a weakening of the load-displacement curve and a peak followed by an unloading part. This part of the curve corresponds to the development of tensile cracks forming the pullout cone. At the end of this second part the crack system is visible. In the third part of the curve a stiffening effect can be seen. This stiffening is caused by shear stresses in the existing tensile cracks. Some of the tested specimen reach the maximum load in this third part, other specimen reach Fmax already in the first peak.

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ANALYSIS OF CONCRETE STRUCTURES

Figure 3.2 Test specimen, number 10.

Figure 3.3 Load-anchor head displacement curves from test number 1 and 2, concrete type C45, embedment depth d=200 mm.

3.3 Peak loads Table 3.1 shows the peak loads obtained from the pullout tests.

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259

Figure 3.4 Load-anchor head displacement curves from test number 3 and 4, concrete type C45, embedment depth d=100 mm. Table 3.1 Peak loads Test#

Concrete type

d (mm)

a/d

Fmax (kN)

1 2 3 4 5 6 7 8 9 10 11

C45 C45 C45 C45 C80 C80 C80 C80 C80 C80 C80

200 200 100 100 200 200 200 100 100 100 100

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.0 2.0

16.0 20.1 15.2 15.4 27.5 31.8 25.8 38.1 23.6 24.8 21.3

4 Finite Element Calculations 4.1 Finite element model The pullout tests of the anchor bolts were modelled with the finite element program “ABAQUS” using a discrete crack approach. The purposes of the finite element calculations were to check the applicability of

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ANALYSIS OF CONCRETE STRUCTURES

Figure 3.5 Load displacement curves from tests number 5 to 7, concrete type C80, embedment depth d=200 mm.

the crack model to this problem and to study the influence of different embedment depths and material properties. Figure 4.1 shows the finite element meshes. Two embedment depths, 100 and 200 mm, were studied. Due to symmetry, only one half of the specimen was modelled. The crack path is marked with solid lines in the figure. In order to simplify the calculations, the crack path was modelled as a straigt line from the upper edge of the anchor head to the support on the top surface. Compared with the laboratory tests, this is a good approximation. The specimen was modelled with four-noded plane stress elements, type CPS4 with four integration points, see ABAQUS User’s Manual (1989). The crack in the concrete was modelled with nonlinear spring elements, type SPRING2. A load-displacement relation in the 1-direction, (crack opening direction), was choosen to obtain the σ-w relation in figure 4.2. In the 2-direction, (crack sliding direction), the stiffness was zero. Perfect bond between steel and concrete was assumed on the upper edge on the anchor head where large compressive stresses occur. No bond between steel and concrete was assumed anywhere else. 4.2 Input data for the calculations In order to study the influence of different material properties, input data was varied according to table 4.1.

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261

Figure 3.6 Load displacement curves from tests number 8 to 11, concrete type C80 embedment depth d=100 mm. Table 4.1 Input data for the finite element calculations. GF (N/m)

fct (MPa)

Ec (GPa)

d (mm)

50, 100, 200, 400 100 100 50, 100, 200, 400 100

3.0 0.75, 1.5, 3.0, 6.0 3.0 3.0 0.75, 1.5, 3.0, 6.0

30 30 15, 30, 60, 120 30 30

200 200 200 100 100

5 Results from the finite element analysis 5.1 Load-displacement cuves 5.1.1 Influence of fracture energy Calculations were made with four different fracture energies GF=50, 100, 200 and 400 N/m. Loaddisplacement curves are shown in figure 5.1. A higher fracture energy gives a higher peak load, a larger displacement at the peak load and a more ductile behaveour after peak load.

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Figure 4.1 Finite element meshes.

5.1.2 Influence of tensile strength Figure 5.2 shows load-displacement curves from calculations with four different tensile strengths, ft=0.75, 1. 5, 3.0, and 6.0 MPa.

ANCHOR BOLTS IN CONCRETE STRUCTURES

263

Figure 4.2 Stress-Displacement relation for the nonlinear spring element.

A higher tensile strength gives a higher peak load and a larger displacement at the peak load. The loaddisplacement curve after the peak load becomes steeper for high tensile strengths. A low tensile strength gives an extremely ductile behavour. 5.1.3 Influence of modulus of elasticity The concrete modulus of elasticity, Ec, was varied between 15 and 120 GPa. The steel modulus of elasticity, Es, was held constant at 210 GPa. Figure 5.3 shows the load-displacement curves. A high modulus of elasticity gives a higher peakload and a stiffer behaviour before peak load. The post peak behaveour depends only of the “crack parameters” GF and ft and is therefore similar for the different curves in the figure. The elastic deformations in a pull-out test does not only depend on the stiffness of the concrete, but also on the stiffness of the steel. Figure 5.4 shows some calculations where also the steel modulus of elasticity is varied. 5.1.4 Influence of embedment depth Figure 5.5 shows results from calculations with two embedment depths, d=100 and d=200 mm. The peak load is higher for the larger embedment depth but the difference is rather small. The ratio between the peak loads is 1.34. The influence of embedment depth is a size effect problem. We can compare the results obtained with analytical solutions based on strength theory and linear elastic fracture mechanics (LEFM). For a true size effect problem a solution based on strength theory gives a ratio between the peakloads of 2. A LEFM solution gives the ratio 1.41. A nonlinear fracture mechanics solution, like this FEM-calculation, should

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ANALYSIS OF CONCRETE STRUCTURES

Figure 5.1 Load-deformation diagrams for varying values of the fracture energy GF

give a result somewhere between these two solutions, but it does not. The reason for this is that the studied pull-out test with two different embedment depths is not a true size effect problem. The size of the anchor bolt is the same for the two embedment depths. This means that the smaller specimen will be stiffer compared to the larger specimen. This stiffness increase will also increase the peak load, just like a higher modulus of elasticity does as shown in section 5.1.3, see figure 5.3 and 5.4. The stiffness increase for small embedment depths can be seen in the elastic, prepeak part of the loaddisplacement curves in figure 5.5. 5.2 Brittleness number for the pullout specimen The FEM-analysis of the anchor-bolt has given theoretical pull-out loads for different combinations of material properties and geometries. It would be convenient to present all results in just one diagram. To be able to do so, we need a new parameter which includes the material properties and the size of the specimen. A useful parameter is the brittleness number B=(If2t) / (EGF) ,where 1 is the size of the structure. The brittleness number is a dimensionless parameter which describes the ratio between elastic energy and fracture energy in a structure. The elastic energy is a measure of the energy that builds up in a structure during loading before fracture occurs. The fracture energy is a measure of the amount of energy that the structure can absorb during the fracture process. When the elastic energy in a structure is large compared to the fracture energy, the structure will have a brittle behaveour, compare Bache (1989) and Di Tommaso (1989). We now try to estimate the brittleness number for our test specimen, an anchor bolt in a concrete plate. The embedment depth is d and the thickness of the plate is b.

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265

Figure 5.2 Load-deformation diagrams for varying values of the tensile strength ft

The elastic energy here consists of two parts. One part is from the concrete and the other part is from the anchor bolt. We start with neglecting the elastic energy in the anchor bolt. Assume that the average concrete stress prior to cracking is kft. According to figure 5.6, the average concrete strain energy can then be written (1) The elastic energy is proportional to the strain energy times the volume of the structure, V. V is proportional to bd2. (2) The fracture energy is proportional to GF times the fractured area, A. A is proportional to bd. (3) The brittleness number B=elastic energy / fracture energy can now be written: (4) Using the above formula, we see that if we double the concrete modulus of elasticity, the brittleness number is reduced to the half, thus the elastic energy is also reduced to the half. This is of course not true since the stiffness and thus the energy stored in the anchor bolt will be the same. One way of modifying the brittleness number is to introduce an effective modulus of elasticity, Eeff, which includes the stiffness of both concrete and steel. By looking at the elastic stiffness we can calculate the effective modulus of elasticity, Eeff, for different combinations of Ec anc Es and different embedment depths, d.

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ANALYSIS OF CONCRETE STRUCTURES

Figure 5.3 Load-deformation diagrams for varying values of Ec.

We start to calculate the initial elastic stiffness, kref, for a specimen with EC=ES=30 GPa. The effective modulus of elasticity, Eeff, for this specimen is 30 GPa. Then we calculate the stiffnesses, ki, for the other combinations of Ec and Es. We then calculate the effective modulus of elasticity with the formula: (5) The brittleness number B can now be written: (6)

5.3 Dimensionless diagram The results from the finite element calculations are summarized in table 5.1. Assume that a formula for calculating the pullout load in the analysis, Fmax, may have the form: (7) Investigating the influence of the brittleness number we plot a dimensionless diagram with the brittleness number, B, versus Fmax/ (dbft), figure 5.7, compare Gustafsson and Hillerborg (1988). Table 5.1 Results from the finite element analysis. Series

GF (N/m)

ft (MPa)

Ec (GPa)

ES (GPa)

Eeff (GPa)

d (mm)

Fmax (kN)

B

Fmax/bdft

A

50

3.0

30

210

43.98

200

11.86

0.8186

0.3954

ANCHOR BOLTS IN CONCRETE STRUCTURES

Series

GF (N/m)

ft (MPa)

Ec (GPa)

ES (GPa)

Eeff (GPa)

d (mm)

Fmax (kN)

B

100 200 400 B 100 100 100 C 100 100 100 D 100 200 400 E 100 100 100

3.0 3.0 3.0 100 1.5 3.0 6.0 100 3.0 3.0 3.0 50 3.0 3.0 3.0 100 1.5 3.0 6.0

30 30 30 0.75 30 30 30 3.0 30 60 120 3.0 30 30 30 0.75 30 30 30

210 210 210 30 210 210 210 15 210 210 210 30 210 210 210 30 210 210 210

43.98 43.98 43.98 210 43.98 43.98 43.98 210 43.98 77.70 134.34 210 57.16 57.16 57.16 210 57.16 57.16 57.16

200 200 200 43.98 200 200 200 24.20 200 200 200 57.16 100 100 100 57.16 100 100 100

14.87 18.29 22.55 200 11.32 14.87 18.51 200 14.87 17.19 19.99 100 11.10 13.86 17.32 100 8.65 11.10 14.22

0.4093 0.2096 0.1023 8.22 0.1023 0.4093 1.6371 12.50 0.4093 0.2317 0.1340 8.80 0.1575 0.0787 0.0394 6.60 0.0394 0.1575 0.6298

0.4956 0.6098 0.7517 0.0256 0.7547 0.4956 0.3085 0.7438 0.4956 0.5729 0.6664 0.3149 0.7399 0.9242 1.1545 0.0098 1.1535 0.7399 0.4740

Fmax/bdft

1.0957

0.4166

0.5867

1.7608

Figure 5.4 Load-deformation diagrams for varying values of the modulus of elasticity for concrete, E, and steel, E.

267

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ANALYSIS OF CONCRETE STRUCTURES

Figure 5.5 Load-deformation diagrams for varying embedment depths, d.

Figure 5.6 Strain energy.

Figure 5.7 shows that there is a correlation between the brittleness number and the peak load obtained from the finite element analysis. The function f(B) can be seen as a kind of reduction factor for Fmax. Brittle structures with high brittleness numbers gives a low f(B), thus reducing the maximum load, Fmax.

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269

Figure 5.7 Results from the finite element calculations. Series A to E are according to table 5.1.

Figure 5.8 Comparison between laboratory tests and finite element calculations.

5.4 Comparison with the laboratory tests. In figure 4.8 the results from the finite element calculations are compared with the laboratory tests. The laboratoy tests also show a correlation between the brittleness number and the peak load, but the scatter is very large. The finite element calculations seem to predict a lower limit for the bearing capacity.

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ANALYSIS OF CONCRETE STRUCTURES

6 Discussion Load-displacement curves and peak loads have been obtained from the laboratory tests. A correlation between the peakloads and the brittleness numbers can be seen. The load-displacement curves can be divided into a linear elastic part, a tensile fracturing part and a friction part. Some specimen reach their maximum load in the tensile fracturing part, other specimen in the friction part Finite element calculations have been performed with different material properties and embedment depths. The peakloads obtained show a good correlation with the brittleness numbers and seem to predict a lower limit for the experimental peak loads. The finite element calculations do not consider the friction in the cracks. The friction part of the load-displacement curve can therefore not be modelled. 7 Acknowledgement The project was supported by the Swedish Council for Building Research (BFR 830900–3). 8. References ABAQUS User’s Manual, Version 4–7 (1989) Hibbit, Karlsson and Sorensen, Inc. 100 Medway Street, Providence, Rhode Island 02906, USA. Bache, H.H. (1989) Fracture mechanics in integrated design of new, ultra-strong materials and structures. Chapter 18 in Fracture Mechanics of Concrete Structures. From theory to applications, a RILEM report ed. by L. Elfgren, Chapman & Hall, London, pp 382–398. Ballarini, R., Shah, S.P. and Keer, L.M. (1986) Failure characteristics of short anchor bolts embedded in a brittle material. Proceedings of the Royal Society of London, Vol. A404, pp 35–54. de Borst, R. (1986) Non-linear analysis of frictional materials. Proefschrift ter verkrijging van de graad van doctor in te technische wetenschappen aan de Technische Hogeschool Delft... Delft, 140pp. Di Tommaso, A. (1989) Size effect and brittleness. Chapter 7 in Fracture Mechanics of Concrete Structures. From theory to applications, a RILEM report ed. by L. Elfgren, Chapman & Hall, London, pp 191–207. Elfgren, L. ed. (1989) Fracture mechanics of concrete structures. From theory to applications. A RILEM State of Art Report.. Chapman & Hall, London, 400 pp. Elfgren, L., Broms C.E., Cedervall, K. and Gylltoft K. (1982) Fatigue of anchor bolts in reinforced concrete foundations. IABSE Colloquium “Fatigue of steel and concrete structures”, Lausanne March 1982, Proceedings, IABSE report, Vol. 37, Zürich, pp 105–117. Elfgren, L., Ohlsson, U. and Gylltoft, K. (1987) Anchor bolts analysed with fracture mechanics. In Fracture of Concrete and Rock, ed. S.P. Shah and S.E. Swartz. SEM-RILEM International Conference, Houston, Texas, June 17–19 1987. Springer, Berlin, 1989, pp 269–275. Eligehausen, R. and Sawade, G. (1989) A fracture mechanics based description of the pull-out behaviour of headed studs embedded in concrete. Chapter 13.2 in Fracture mechanics of concrete structures. From theory to applications, a RILEM report ed. by L. Elfgren, Chapman & Hall, London, pp 281–299. Hellier, A.K., Sansalone, M., Ingraffea, A.R., Carino, N.J. and Stone, W.C. (1987). “Finite element analysis of the pullout test using a nonlinear discrete cracking approach”. ASTM, Cement, Concrete and Aggregates, Vol. 9, No 1, Summer 1987, pp 20–29. Hillerborg, A. (1985) The theoretical basis of a method to determine the fracture energy, GF, of concrete. RILEM. Materials and Structures, Vol 18, No. 106, pp 291–296.

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Gustafsson, P.J. and Hillerborg, A. (1988) Sensitivity in shear strength of lognitudinally concrete beams to fracture energy of concrete. ACI Structural Journal, Vol. 85, May-June 1988, pp 286–294. Klinger, R.E. and Mendonca, J.A. (1982) Tensile capacity of short anchor bolts and welded studs: a litterature review. Technical paper No. 79–27, ACI Journal, Vol. 79, July August 1982, pp 270–279. Ordqvist, C. and Soutukorva, M. (1990) Anchor bolts in high strength concrete. (In Swedish with a summary in English) Division of Structural Engineering, Luleå University of Technology. Diploma Work to be published. Ottosen, N.S. (1981) Nonlinear finite element analysis of the pullout test. American Society of Civil Engineering, ASCE, Journal of the Structural Division, Vol. 107, No. ST4, April 1981, Paper No. 16197, pp 591–603. Peier, W.H. (1983) Model of pull-out strength of anchors in concrete. ASCE, Journal of Structural Engineering, Vol. 109, No 5, May 1983, Paper No. 17949, pp 1155–1173. Rehm, G., Eligehausen, R. and Mallee, R. (1988) Befestigungstechnik (Fastening technology). In German. In Beton Kalender 1988, Ernst & Sohn, Berlin 1988, pp 569–663. Rots, J. (1988) Computational modelling of concrete fracture, Dissertation, Delft University of Technology, Department of Civil Engineering, Delft, 132 pp.

INDEX

This index has been compiled using the keywords assigned to the papers by the Concrete Information Service of the British Cement Association, with some editing and additions to facilitate its use. The assistance of the Association is gratefully acknowledged. The numbers are the page numbers of the first page of the paper referred to. Adhesives 208, 220 Age 199 Aggregate 88 Anchor bolts 152, 245, 281 Anchors 208 Axial loads 32 Beams 133, 152, 171, 179, 183

Cracking 69, 99, 152, 179, 245, 281 Crack initiation 55, 160, 245 Crack length 19 Crack propagation 44, 55, 88, 128, 133, 245, 281 Cracks 1, 19, 32, 44, 88 Crack width 88, 281 Creep 69

Beams, notched 1, 19, 32, 99, 199 Bearing capacity 179 Bending 1, 152, 199 Bending moment 133 Biaxial loads 32, 183 Bond 55, 199, 220 Bond strength 199 Bridges 152 Brittleness 133, 220, 281

Damaged concrete 160 Dams 160 Deflection 69, 133, 152, 179, 199 Deformation properties 1, 208, 220, 281 Displacement 44, 234, 281 Ductility 171 Early age properties 69 Elasticity 128 Elastic properties 220 Embedded bolts 245, 281 Epoxide resins 208

Cement paste 19, 88 Cohesiveness 99 Composite parts 208 Compression zone 171, 183 Compressive strength 32, 69, 234, 245 Computer program 152 Confined strength 32, 55 Construction works 128, 208 Cooling 69 Crack distribution 19

Failure 152, 179, 183, 199, 208, 234, 245 Finite element method 55, 152, 281 Forces 152, 183, 208 Foundations 160 Fracture energy 1, 99, 133, 199, 208, 220, 281 Fracture mechanism 44 272

INDEX

Fracturing 69 Freezing 208 Geometry 44 Green concrete 69 Hardening 220 High strength concrete 171, 281 Hillerborg strip method 1 History 1 Holography 55 Impact loads 234 Interaction properties 183 Interface 55, 199 Interferometry 55 Joints 160, 199, 220 Lapped joints 220 Lightweight aggregate concrete 32 Load bearing capacity 208 Loads 55, 69, 88, 99, 133, 179, 199, 245, 281 Mathematical models 99 Microcracking 245 Models 1, 44, 69, 88, 99, 133, 152, 281 Modulus of elasticity 281 Moment/curvature relations 133 Mortar 19, 32 Movement 69 Non-linear methods 245 Notched beams 1, 19, 32, 99, 199 Penetration properties 234 Photoelasticity 32, 160 Plasticity 1 Plastic properties 220 Plates 32 Porosity 234 Pull-off testing 199 Pull-out testing 245, 281 Punching 234 Reinforced concrete 1, 133, 171, 179, 183 Reinforcement 133, 171, 179 Repair 199

273

Restraint 69 Reviews 1 Rock 245 Rotation 19, 133, 179 Shear 32, 183, 208, 220 Shearing stress 44 Size effects 99 Slabs 1 Softening 19, 69, 88, 99, 220 Spalling 234 Span/depth ratio 133, 179 Splitting 1 , 208 Stability 128 Steel 220, 234 Steel content 133 Steel plates 208 Stiffening 152 Stiffness 19, 32, 128 Stirrups 171, 179 Strain 69, 88, 281 Strength 220 Strengthening 208 Stress 1, 19, 44, 55, 69, 88, 99, 160, 183, 234, 281 Stress distribution 99, 152, 245 Stress strain curves 133, 171 Structural analysis 1, 19, 55, 128, 133, 171, 245, 281 Structural design 160 Structural members 133 Surface zone 234 Temperature effects 69 Tensile strength 1, 19, 32, 69, 88, 199, 208 Tensile stress 133, 152 Testing equipment 32, 44 Test specimens 19, 32, 44, 99, 133, 183, 199, 208, 220, 234 Thermal analysis 69 Thermal properties 69 Toughness 133 Transfer properties 208 Ultimate load 133, 179, 245 Uniaxial loads 19, 171 Viscoelasticity 69 Wood 220

274

INDEX

Yield value 133 Young’s modulus 128