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.

MODE I BEHAVIOUR OF CONCRETE

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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

MODE I BEHAVIOUR OF CONCRETE

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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 a1тfr 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.

MODELLING, TESTING AND STRENGTH ANALYSIS

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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