High Performance Structures And Materials III

High Performance Structures and Materials III

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Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: [email protected]

WIT Transactions on The Built Environment Editorial Board

E Alarcon Universidad Politecnica de Madrid Spain

C Alessandri Universita di Ferrara Italy

S A Anagnostopoulos University of Patras Greece

E Angelino A.R.P.A. Lombardia Italy

H Antes Technische Universitat Braunschweig Germany

D Aubry Ecole Centrale de Paris France

D E Beskos University of Patras Greece

J J Bommer Imperial College London UK

F Butera Politecnico di Milano Italy

P G Carydis National Technical University of Athens Greece

J Chilton University of Nottingham UK

S Clement Tranport System Centre Australia

M C Constantinou State University of New York at Buffalo USA

G Degrande Katholieke Universiteit Leuven Belgium

A De Naeyer Universiteit Ghent Belgium

W P De Wilde Vrije Universiteit Brussel Belgium

J Dominguez University of Seville Spain

F P Escrig Universidad de Sevilla Spain

M N Fardis University of Patras Greece

C J Gantes National Technical University of Athens Greece

L Gaul Universitat Stuttgart Germany

Y Hayashi Nagoya University Japan

M Iguchi Science University of Tokyo Japan

L Int Panis VITO Expertisecentrum IMS Belgium

W Jager Technical University of Dresden Germany

C M Jefferson University of the West of England, Bristol UK

D L Karabalis University of Patras Greece

E Kausel Massachusetts Institute of Technology USA

K Kawashima Tokyo Institute of Technology Japan

A N Kounadis National Technical University of Athens Greece

W B Kratzig Ruhr Universitat Bochum Germany

A A Liolios Democritus University of Thrace Greece

J W S Longhurst University of the West of England, Bristol UK

J E Luco University of California at San Diego USA

L Lundqvist Unit for Transport and Location Analysis Sweden

M Majowiecki University of Bologna Italy

G D Manolis Aristotle University of Thessaloniki Greece

G Mattrisch DaimlerChrysler AG Germany

F M Mazzolani University of Naples "Federico II" Italy

K Miura Kajima Corporation Japan

G Oliveto Universitá di Catania Italy

E Oñate Universitat Politecnica de Catalunya Spain

A S Papageorgiou Rensselaer Polytechnic Institute USA

G G Penelis Aristotle University of Thessaloniki Greece

A M Reinhorn State University of New York at Buffalo USA

F Robuste Universitat Politecnica de Catalunya - LAMOT Spain

C W Roeder University of Washington USA

J M Roesset Texas A & M University USA

M Saiidi University of Nevada-Reno USA

F J Sanchez-Sesma Instituto Mexicano del Petroleo Mexico

S A Savidis Technische Universitat Berlin Germany

J J Sendra Universidad de Sevilla Spain

Q Shen Massachusetts Institute of Technology USA

A C Singhal Arizona State University USA

P D Spanos Rice University USA

C C Spyrakos National Technical University of Athens Greece

H Takemiya Okayama University Japan

I Takewaki Kyoto University Japan

E Taniguchi Kyoto University Japan

J L Tassoulas University of Texas at Austin USA

M A P Taylor University of South Australia Australia

R Tremblay Ecole Polytechnique Canada

R van der Heijden Radboud University Netherlands

R van Duin Delft University of Technology Netherlands

A Yeh The University of Hong Kong China

M Zador Technical University of Budapest Hungary

R Zarnic University of Ljubljana Slovenia

THIRD INTERNATIONAL CONFERENCE ON HIGH PERFORMANCE STRUCTURES AND MATERIALS

HIGH PERFORMANCE STRUCTURES AND MATERIALS III

CONFERENCE CHAIRMAN C.A. Brebbia Wessex Institute of Technology, UK

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE K.S. Al Jabri B. Alzahabi W.P. De Wilde G. Belingardi M. Ingber T. Katayama S. Kravanja H. Martikka R.A.W. Mines F. Romano R. Schmidt H. Takagi I. Tsukrov Organised by: Wessex Institute of Technology, UK and Free University of Brussels, Belgium Sponsored by: The High Performance Structures and Materials Book Series

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High Performance Structures and Materials III Editor: C.A. Brebbia Wessex Institute of Technology, UK

C.A. Brebbia Wessex Institute of Technology, UK

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 1-84564-162-0 ISSN: 1746-4498 (print) ISSN: 1743-3509 (on-line) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2006 Printed in Great Britain by Cambridge Printing. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Preface This book contains the edited papers presented at the Third International Conference on High Performance Structures and Materials, held in Ostende, Belgium in 2006. The Conference was dedicated to honouring the distinguished career of Professor Patrick De Wilde, from the Free University of Brussels, Belgium, on the occasion of his retirement. The Conference addressed issues involving advanced types of structures, particularly those based on new concepts or new types of materials. This responds to the need to develop a generation of new materials that are suitable for high performance structures which can easily resist a wide range of external stimuli and react in a non-conventional manner. Professor De Wilde, whose achievements are summarised in the following pages, was the originator and driving force behind this Conference series since it started. The Editor Ostende, 2006

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Dedication Professor W. Patrick De Wilde This book is in homage to the long and distinguished career of my friend and colleague Professor Patrick De Wilde from the Vrije Universiteit Brussel (VUB). I was fortunate enough to meet Patrick for the first time many years ago when we were both young researchers. This was on the occasion of the Seminar on Finite Elements that I organised in Southampton in 1970. He was then a Research Assistant at the Department of Applied Continuum Mechanics of VUB and I had just been appointed a lecturer at the Department of Civil Engineering at the University of Southampton. Since then we kept in constant contact and this led me to a better appreciation of his outstanding scientific career and the depth of his knowledge and imagination. Patrick always brought a new and fresh perspective to the most challenging engineering and scientific problems. His academic career has been centred at VUB where he obtained his PhD in Applied Sciences in 1976, rising from lecturer to Head of the Laboratory of Structural Analysis and to full Professor in 1984. He also served with distinction as Chairman of the Research Council and Vice Rector of VUB, and is now about to retire from the position of Head of the Department of Mechanics of Materials and Constructions. Patrick has always been characteristically generous with his knowledge, lecturing part time at many institutions in Belgium and abroad. An appointment that reflects his range of interests was as a Professor at the Higher Institute of Architecture Victor Horta, the link between architecture and engineering being always close to his heart. He also contributed substantially to the enhancement of training and research capabilities in South American and African countries to mention but a few of his international activities.

Always prepared to share his time, he headed up and motivated several generations of young researchers at VUB as well as contributing to the development of many institutions around the world, including our own Wessex Institute of Technology where he has served on the Board of Directors since its beginning. Patrick’s written contributions comprise many papers in learned journals and conference meetings in the field of Computer Aided Engineering, finite element methods; composite materials, mechanics of materials and structures. He has participated in many scientific committees and in the advisory board of different publications and international conferences. He is permanent Co-chairman of the international conference on High Performance Materials and Structures as well as one of the most active members of the Scientific Committee of the Conference on Structural Repairs and Maintenance of Historical Buildings. In spite of his substantial research activities he has not neglected to collaborate with industrial partners and as a result has carried out consulting work for different companies and is currently Chairman of the Technical Board of SECO (Civil Engineering Certification Company). It has been a privilege for me as well as for many other colleagues around the world to be associated with Patrick all the years he has been working at VUB. Soon his teaching days will be over and we hope this will herald the beginning of a new period for him: concentrating on unfinished research and original work which was interrupted by the demands of university duties. Carlos A. Brebbia The New Forest, 2006

Contents Section 1: Conceptual design and structural analysis (Special session organised by W. P. De Wilde) Conceptual design of lightweight structures: the role of morphological indicators and the structural index W. P. De Wilde ......................................................................................................3 Influence of dynamic loads on the optimum design of trusses J. Van Steirteghem, W. Ponsaert, W. P. De Wilde & Ph. Samyn........................13 Modular grid-based design concept for fibre reinforced composite shells E. De Bolster, H. Cuypers, W.P. De Wilde & J. Wastiels...................................21 Influence of stiffness constraints on optimal design of trusses using morphological indicators T. Vandenbergh, W.P. De Wilde, P. Latteur, B. Verbeeck, W. Ponsaert & J. Van Steirteghem......................................................................31 Variations in form and stress behaviour of a V-shaped membrane in a foldable structure M. Mollaert, N. De Temmerman & T. Van Mele ................................................41 Section 2: Composite materials and structures A new composite material based on natural fibres and a thermoset: technology, applications and properties G. Wuzella ...........................................................................................................53 Experimental test of threaded steel rods glued-in hardwood with epoxy D. Otero, J. Estévez, E. Martín & J. A. Vázquez.................................................63 Predicting the mechanical behaviour of large composite rocket motor cases N. Couroneau ......................................................................................................73

Micromechanical modeling of random or imperfect composites M. Šejnoha & J. Zeman.......................................................................................83 Flexural behaviour of ferrocement roof panels A. S. Alnuaimi, A. Hago & K. S. Al Jabri............................................................93 Finite element modeling of actuated fibre composites M. Martinez, A. Artemev, F. Nitzsche & B. Geddes..........................................103 CFRP strengthening of prefabricated timber panel walls M. Premrov & P. Dobrila .................................................................................111 Evaluation of the structural integrity of a sandwich composite train roof structure K. B. Shin, B. J. Ryu, J. Y. Lee & S. J. Lee........................................................121 Measurement of the fiber stress distribution during pull-out test by means of micro-Raman spectroscopy and FEM analysis K. Tanaka, K. Minoshima & H. Yamada ..........................................................131 Section 3: Natural fibre composites (Special session organised by T. Katayama and H. Takagi) Effect of surface treatment to tensile static and creep properties for jute fiber reinforced composite K. Takemura......................................................................................................143 Effects of forming conditions on mechanical properties of resinless bamboo composites H. Takagi & H. Mori.........................................................................................151 Compression moulding of jute fabric reinforced thermoplastic composites based on PLA non-woven fabric T. Katayama, K. Tanaka, T. Murakami & K. Uno ...........................................159 Quality control of fibers end-milled from bamboo pipe using spiral tool path K. Ogawa, E. Aoyama, T. Hirogaki, Y. Tomioka & H. Nakagawa ...................169 Characteristic behaviors of CFRP and GFRP at cryogenic temperature under static and cyclic loadings S. Kubo, K. Okubo & T. Fujii............................................................................179 Mechanical properties of loosing natural fiber reinforced polypropylene K. Mizuta, Y. Ichihara, T. Matsuoka, T. Hirayama & H. Fujita.......................189

Section 4: Material and mechanical characterisation Identification of strain-rate sensitivity parameters of steel sheet by genetic algorithm optimisation G. Belingardi, G. Chiandussi & A. Ibba ...........................................................201 Mechanical characterisation of a viscous-elastic plastic material, sensitive to hydrostatic pressure and temperature V. D. Le, M. Caliez, M. Gratton, A. Frachon & D. Picart................................211 Identification of the material properties of composite beams: inverse method approach E. Euler, H. Sol & E. Dascotte..........................................................................225 Full-field optical measurement for material parameter identification with inverse methods J. Gu, S. Cooreman, A. Smits, S. Bossuyt, H. Sol, D. Lecompte & J. Vantomme..................................................................................................239 Multiaxial characterization of the mechanical behaviour of aluminium foam L. Peroni, M. Avalle & P. Martella...................................................................249 Characterisation of the high strain rate properties of Advanced High Strength Steels J. Van Slycken, P. Verleysen, J. Degrieck & J. Bouquerel ...............................259 Evaluation of bond strength in Roller Compacted Concrete under various normal pressures M. Madhkhan & A. Arasteh ..............................................................................269 High performance fibres and the mechanical attributes of cut resistant structures made therewith S. Rebouillat & B. Steffenino.............................................................................279 The study of surface oxidation of tin(II) fluoride and chloride fluoride materials by Mössbauer spectroscopy: to oxidize or not to oxidize, that is the question G. Dénès, E. Laou, M. C. Madamba & A. Muntasar ........................................301 Study of double ionic disorder (cationic and anionic) and disorder of two kinds of tin(II) (ionic and covalent) within the same material: the incredibly complex Ba1-xSnxCl1+yF1-y solid solution and its study by Mössbauer spectroscopy S. Boufas, G. Dénès, J. Kochuparampil, H. Merazig & A. Muntasar...............311

Study on fatigue and energy-dissipation properties of nanolayered Cu/Nb thin films Y.-C. Wang, T. Hoechbauer, J. G. Swadener, T. Darling, A. Misra, R. Hoagland & M. Nastasi................................................................................323 Advances in computational modeling through the use of higher-level microstructure characterization M. Groeber, M. Uchic, D. Dimiduk, Y. Bhandari & S. Ghosh .........................331 Deformation of aluminum alloys AD-1, AMg-6 and D-16 at dynamic compression and temperatures of 25–250oC V. A. Pushkov, S. A. Novikov, V. A. Sinitsyn, I. N. Govorunov & O. N. Ignatova...............................................................................................343 X-ray residual stress measurements on plasma sprayed molybdenum coatings K. Hirukawa, K. Akita, S. Tobe, T. A. Stolarski & S. Ohya ..............................351 Compressive residual stress generation process by laser peening without pre-coating H. Tanaka, K. Akita, Y. Sano & S. Ohya...........................................................359 The scale effect of roughness in contact problems S. Mezghani, A. Jourani & H. Zahouani...........................................................369 Section 5: High performance concretes Copper slag as fine aggregate for high performance concrete K. S. Al Jabri .....................................................................................................381 Application of FRC in tunnel reinforcement P. Procházka .....................................................................................................391 Innovative procedure to produce high performance pretensioned concrete girders combining high strength concrete and normal or special concrete types C. Vázquez-Herrero & F. Martínez-Abella.......................................................401 Properties of high performance concrete: the effect of cracks E. Mňahončáková, M. Jiřičková & R. Černý ....................................................409 Seismic upgrading of square and rectangular RC columns using FRP wrapping M. A. N. Abdel-Mooty, M. E. Issa, H. M. Farag & M. A. Bitar........................419

Arch bridge made of reactive powder concrete D. Cizmar , D. Mestrovic & J. Radic ................................................................429 Elastic and inelastic seismic response comparison of reinforced concrete buildings with normal resistance concrete and with high resistance concrete J. A. Avila & D. Rivera .....................................................................................439 The strength effects of synthetic zeolites on properties of high performance concrete P. Frontera, S. Marchese, F. Crea, R. Aiello & J. B. Nagy ..............................449 Behaviour model and experimental study for the torsion of reinforced concrete members C. E. Chalioris ..................................................................................................459 A study on use of blended ferrocement: a high performance material for repair/strengthening of brick masonry columns T. Kibriya ..........................................................................................................469 Section 6: Damage and fracture mechanics Crush behaviour of open cellular lattice structures manufactured using selective laser melting M. Santorinaios, W. Brooks, C. J. Sutcliffe & R. A. W. Mines..........................481 Stress intensity factors for cracked cold-drawn steel wires under tensile loading B. Lin & G. Lu...................................................................................................491 Void growth and damage ahead of a crack in pressure-sensitive dilatant polymers H. B. Chew, T. F. Guo & L. Cheng ...................................................................501 An orthotropic damage model for crash simulation of composites W. Wang, F. H. M. Swartjes & M. D. Gan .......................................................511 Ring-shaped crack propagation in a cylinder under nonsteady cooling V. A. Zhornik, Yu. A. Prokopenko, A. A. Rybinskaya & P. A. Savochka...........521 Investigation of the hygrothermal performance of wooden beam ends embedded in inside insulated outside walls H. Stopp & P. Strangfeld ..................................................................................529

Bond repair of cracked beams H. Cruz, J. Custódio & D. Smedley ..................................................................539 Microtexture and nanoindentation study of delamination cracking in Al-Cu-Li-X alloys R. Crooks, M. S. Domack & J. A. Wagner ........................................................549 A contribution to the rehabilitation of reinforced concrete structures by non-destructive electrochemical methods J. L. Rovira Santa Olaya & P. Pardo Tràfach..................................................559 Section 7: Adhesion and adhesives Timber specimens parametrized design for numerical analysis E. Martín Gutiérrez, J. Estévez Cimadevila, D. Otero Chans & S. Muñiz Gómez ............................................................................................571 Theory of the elasticity of the materials of the second order V. Shorkin & V. Gordon....................................................................................581 Section 8: Optimal design Optimal design of fibre reinforced tubular structures H. Martikka & E. Taitokari...............................................................................593 MINLP optimization of steel frames S. Kravanja & U. Klanšek.................................................................................605 Optimization of timber trusses considering joint flexibility S. Šilih, M. Premrov & S. Kravanja..................................................................615 The Hendrickx–Vanwalleghem design strategy W. Debacker, C. Henrotay, W. P. De Wilde & H. Hendrickx...........................625 The optimization of a truss facade B. Verbeeck,W. P. De Wilde & Ph. Samyn .......................................................635 MINLP optimization of the single-storey industrial building steel structure T. Žula, U. Klanšek & S. Kravanja ...................................................................643 Genetically optimised placement of piezoelectric sensor arrays: linear and nonlinear transient analysis J. N. Rao, S. Lentzen & R. Schmidt ...................................................................653

Numerical analysis of the process of trapezoidal thread rolling L. Kukielka & K. Kukielka ................................................................................663 Three-dimensional limit analysis of ancient masonry buildings with rigid block models A. Orduña..........................................................................................................673 Section 9: Reliability of structures Analysis of diffusional stress relaxation in submicron Cu interconnect structures using the model with enhanced vacancy diffusivity in grain boundary region I. Tsukrov, W. M. Grich & T. S. Gross..............................................................685 Application of fuzzy sets to structural reliability of existing structures I. Mura ..............................................................................................................695 Non-linear response of combined system, 3D wall panels and bending steel frame subjected to seismic loading M. Z. Kabir, A. R. Rahai & Y. Nassira..............................................................705

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Section 1 Conceptual design and structural analysis (Special session organised by W. P. De Wilde)

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Conceptual design of lightweight structures: the role of morphological indicators and the structural index W. P. De Wilde Department of Mechanics of Materials and Constructions (MeMC), Vrije Universiteit Brussel, Brussels

Abstract Firstly, but in no way exclusively, this paper addresses architectural engineers facing critical design decisions in the phase of so-called “conceptual design”. The resulting design must yield a structure showing a sound behaviour in both the serviceability limit state (SLS) and in the ultimate limit state (ULS), also meaning that three essential criteria should be satisfied: strength, stiffness and stability (the latter including, if relevant, an acceptable dynamic behaviour). However, the question very often remains open as to which of the three criteria is overruling the other ones. This paper, a synthesis of the work of a research group headed by the author, tries to show that a conceptual design methodology can be developed, hereby using the concept of morphological indicators (originally developed by P. Samyn, subsequently P. Latteur and within the research group) and a structural index (introduced by Shanley). This methodology also gives an answer to the question that very often arises when designing lightweight structures: “design for strength” or “design for stiffness”? Examples in subsequent papers, presented during this Conference and thus included in these Proceedings, illustrate the methodology nowadays used by our students in architectural engineering when designing and analysing their structures. Keywords: conceptual design of architectural structures, strength, stiffness, structural stability, structural vibrations.

1

Morphological indicators

The so-called morphological indicators (MI) were introduced by P. Samyn [1] in 1999. As different papers, both within this conference and in other journals and WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06001

4 High Performance Structures and Materials III books (see e.g. a more recent book of P. Samyn [2]), have already or will underline(d) the potentials of these dimensionless numbers, and thus we will restrict ourselves to a very short introduction. P. Samyn essentially introduced two indicators, one related to the minimum volume of material required for a structural typology achieving a fully stressed design, the other related to the maximum displacement in the same structure. The combination of those two indexes, being the volume indicator W and the displacement indicator ∆ allow the designer to select not only a (sub)optimal typology but also its optimal aspect ratio. The indicators are dimensionless numbers, function of very few parameters, the most important being the so-called slenderness of the structure L/H, in which L is the larger and H the lesser dimension of a window, framing the structure. P. Samyn defined them as: σV a) W= , in which σ is the admissible stress (in practice we consider the FL allowable stress in the serviceability limit state (SLS), V the volume of material, F the resultant of (static) forces, loading the structure, and L its span. Eδ , in which E is the elastic modulus and δ the maximum displacement. b) ∆ = σL W = σV/FL

MULTI-WARREN trusses WARREN trusses

ARCH + TIE + ∞ SW 1,700

ARCH + TIE + ∞ PR

1,414

ARCH + TIE

ARCH + ∞ SW

1,155

ARCH + ∞ PR 0,816

ARCH

0,707 0,577

PR = push rods SW = suspension wires

1,633 2,309 2,828 3,138 3,266

L/H

Figure 1. He also shows that they can be expressed as a function of L/H: W = W(L /H) and ∆ = ∆(L /H). This allows one to draw, either analytically or numerically, diagrams showing the values of the indicators in function of the slenderness, for WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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different typologies of structures. The recent book of P. Samyn [2], although not exhaustive, contains an impressive amount of typologies that can be compared: see fig.1 as an example of a comparison of different typologies through the volume indicator: this diagram is taken from [9]. Similar graphs were developed for the displacement indicator ∆ . It is clear that P. Samyn – see the title of his thesis – hereby provided the architect with “a tool allowing to reach a suboptimal design at the stage of the conceptual design”. The fact that he is still using it today, and undoubtedly producing designs of outstanding quality, proves the robustness of the tools he introduced. However, two major objections could be foreseen and they were very soon subject of controversy. Indeed, the two indicators allow for a preliminary design, achieving the required performances of strength and stiffness with a minimum volume of material (a fully stressed design of statically determinate structures, subject to classical load cases)...but what about (in)stability? It is clear that, in its most simple form, the developed method does not take – at least explicitly - into account possible buckling phenomena. However it would be unjust to consider that it completely overlooks the existence of this phenomenon: in both [1] and (more detailed) in [2], P. Samyn shows that “correction factors” can be computed, thus defining an increase of material consumption, and trying to control the stability of the equilibrium. Nevertheless, gradually we were convinced that conceptual design should take into account the totality of the criteria to be satisfied by the structure: •

•

the strength of the structural parts is controlled through the indicator of σV , as it starts from a fully stressed design (at a stress volume: W = FL level σ). In the conceptual design stage, a minimal value of this indicator is aimed at, thus trying to achieve a minimal consumption of material. But, as we shall experience very quickly, a mere choice of the slenderness L/H, corresponding to a minimum of the curve for the chosen typology (see e.g. fig.1) does not solve the problem! The reason is simple: this simple minimisation does not take into account the stiffness, stability...and dynamic requirements of the design. In this sense we could say that each of the three other criteria introduce kinds of “forbidden zones” in the W=W(L/H) diagrams: a forbidden zone for the excessively flexible structures, one for the unstable structures and a last one – if relevant – for the unacceptable vibrations. the stiffness of the structure is evaluated through the indicator of Eδ . As one can see, it is proportional to the ratio displacement: ∆ = σL δ/L, which is generally limited by design codes (see e.g. Eurocodes). As we will show later this indicator, originally introduced by P. Samyn will be of primordial importance when evaluating the risk for resonance of a structure.

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

6 High Performance Structures and Materials III •

through the buckling indicator Ψ =

µσL

, introduced by Latteur [3], in qEF which µ is the length reduction factor (due to end conditions of the I elements), and q = in which I is the (minimal) inertia moment of Ω2 the section and Ω its section, one can evaluate the sensitivity of the structure to (local) buckling: the higher Ψ , the higher the risk. We are convinced that this instrument should be used in conjunction with the other (W, ∆,Θ), as it gives a better estimate of the penalty in material consumption than the correction factors used by P. Samyn. To be mentioned is that J. Van Steirteghem shows in his thesis work [10] that one can evaluate the risk for global instability, starting from this indicator.

Figure 2. Eventually one also has to consider an indicator describing the dynamic behaviour of the structure. This is achieved through the use of the indicator of L  1 = f ,Ψ  , extensively discussed by J. Van the first natural frequency Θ = H  ∆ Steirteghem in [10]. As one sees, there is a direct link with the indicator of displacement ∆. This indicator Θ is also directly dependent on the buckling indicator, which is not surprising: both resonance and buckling phenomena in elastic systems are related with transformation of energy (compression into bending or torsion for buckling, potential into kinetic for resonance). An important observation can be made here: if one accepts all the simplifications introduced by the concept of morphological indicators and if one has a closer look at the indicator Θ , one will notice that it is independent on the volume indicator and thus on the mass of the system. This could bring one to the conclusion that – at least, at the stage of conceptual design – there is little hope that one can improve the dynamic behaviour of the structure through simple WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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addition of mass. This has been confirmed when designing e.g. slender and elegant footbridges: one quickly comes to the conclusion that the penalty in mass, in order to achieve an acceptable spectrum of resonance frequencies is (very) much higher than the one necessary to control its buckling behaviour (see e.g. [10]). A very well known example, proving this observation, is the 350m long Millennium Bridge, designed by Sir Norman Foster in collaboration with Arup Ltd. See e.g. http://www.arup.com/millenniumbridge/, site from which we quote: “...There are two fundamental ways to limit dynamic excitation: * Stiffen the structure, so the frequency of the bridge and our footsteps no longer match; * Add damping to absorb the energy. It was concluded that stiffening the bridge to change its frequency was not a feasible option. The bridge would need to be at least tenfold stiffer laterally to move its frequency out of the excitation range. The additional structure required to do this would dramatically change the appearance of the bridge. It was decided to adopt a damping solution, either active damping or passive damping. Active damping uses powered devices to apply forces to the structure to counteract vibrations. Passive damping relies on harnessing the movements of the structure to absorb energy.” (end of quote). This is only one of the many examples proving that one has to give necessary attention to the dynamic behaviour of lightweight structures, if one wants to design an acceptable lightweight structure! But this also proves that solving the problem with addition of mass and/or stiffness is very seldom the best option. A new challenge thus for designers!

2

“Design for stiffness” versus “design for strength”

If one wants to consume a minimum of material it is logical that he follows a strategy by which the indicator of volume W is set as the hierarchically most important. Given a span L, loads F and material properties (among which σ controls W and E controls ∆) one can select an acceptable slenderness L/H and thus H for a chosen typology of structure. As P. Samyn very rightly points out in [2], one can take into account the stiffness and even the stability requirements, although one can argue that the latter requirement is addressed in a somewhat empirical way. However, from our experience and research in the Vrije Universiteit Brussel, it appears that: • the strategy, putting forward the indicator of volume W as objective function, although yielding performing and lightweight preliminary designs, practically always produces solutions that have to be corrected in order to satisfy the remaining requirements. As the objective is to minimise volume, one very often achieves designs which are too flexible, unstable...and, most of all, dynamically unacceptable. • it would be good to think about the strategy that consists in an a priori selection of the indicator that should be selected as the hierarchically WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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•

•

more important: W versus ∆, or strength versus stiffness. This alternative is also linked to the two analyses in serviceability limit state (SLS) and ultimate limit state (ULS). In fact, one quickly gets a feeling that there are two extreme situations: on one hand the heavy and very stiff structures (e.g. masonry bridges [11]), on the other hand the lightweight and often rather flexible structures (e.g. the modern footbridge, cf. Millennium Bridge in London). The experience acquired during studio work with architectural engineers, who have to design structures of all types, covering a wide spectrum of functionalities and spans, has underlined that a measure of the danger for (= sensitivity to) instability phenomena (both buckling and resonance) is the best guide for the choice between “design for strength” and “design for stiffness”. other empirical observations are important: o when one foresees risks for instability phenomena it is better to approach the problem through “design for stiffness”. o when designing lightweight structures it is almost always the design constraints in the SLS that prevail over those for the ULS. In other words, these constructions have an enormous reserve in strength, and they thus show a much higher value of W than the optimal one! o the penalty imposed by additional material consumption, in order to avoid unacceptable vibrations, is much more severe than the one controlling (in)stability. one could thus look for a kind of “separation line” or a criterion allowing the selection of the adequate strategy. It seems also that, in order to take into account the “scale effects”, it is better to look for a parameter which takes into account the absolute value of both span L and forces F (hereby included the permanent forces on and self-weight of the structure). Van Steirteghem, in his PhD work [10], suggests that

this parameter could be L / F , which in fact has been introduced by Shanley [8], who called the inverse of its square (F/L2) the “structural index”. Large values point to large span structures with relatively low permanent loads, small values to the contrary. In another paper, here presented, he will show the role played by the structural index in some structural choices. This choice is by no means arbitrary as the indicators for buckling Ψ and for the first resonance frequency Θ can both be linked to it: the former Ψ even explicitly, as it is proportional to the structural index, the latter being an implicit function of Ψ . This brings us to an important conclusion: at the stage of conceptual design one should in the first place check which of the two design strategies, being “design for strength” versus “design for stiffness” should be adopted. A reasonable indication can be obtained by the analysis of the mentioned morphological indicators, but most of all by the structural index. The latter will direct you to a design in SLS rather than ULS, if risks for buckling and/or resonance are to be dealt with.

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9

Alternate strategies

Probably one of the most challenging and promising alternatives to the design strategy described in the previous sections consists in the use of “new algorithms”, allowing a search for improved typologies, especially in the case of complicated load conditions. Instead of looking for the optimal typology of structure, given the span L and the loads F, which yields an optimal slenderness L/H for the catalogued typology (e.g. a Warren truss), one can “open the search space” to yet unknown typologies. This can be achieved using e.g. genetic algorithms, evolutionary strategies, etc. Verbeeck et al. show in [14] that genetic algorithms can be very powerful tools in optimisation processes, but above all have shown in [15] that they can yield new, often unsuspected, typologies, better suited for the given problem. Important is to mention that the objective function these algorithms try to minimize is the volume indicator W, the other morphological indicators reducing the search space by inducing side constraints.

4

Preliminary conclusions

As will be shown by Vandenbergh et al. [12] in a subsequent paper, the conceptual design of lightweight structures introduces the need for an a priori selection of a design strategy based on the need to satisfy in the first place a SLS or an ULS. This can be achieved through an adequate use of morphological indicators, among which the structural index is also important: it is the only one taking into account the so-called scale effects. J. Van Steirteghem has convincingly shown in his Ph.D thesis [10] that it is of paramount importance to include the buckling indicator as a full partner when designing structures subject to dominant static loads. As the buckling indicator is directly related to the structural index, it immediately gives indications about the limit state, either SLS or ULS, in which the structure should be calculated. This choice also is linked to the alternative between “design for stiffness” and “design for strength”. The result of the minimisation of the volume indicator W, in an acceptable design subspace, and in which the two other morphological indicators ∆ and Ψ introduce side constraints, combined with the eagerness of modern architects to show their ability to cross very large spans, very often yields designs of structures showing an unacceptable dynamic behaviour. The penalty induced by corrections on mass or stiffness practically always affects the optimum value of W in a much more severe way than e.g. the need to satisfy buckling or stiffness requirements. Ponsaert et al. [13] will show that the unacceptable vibrations can be controlled through active, hybrid or passive damping, rather than through the tuning of stiffness and/or mass. Although it is perhaps a little early to make final statements, I strongly suspect that the direction taken by my student B. Verbeeck in the field of new numerical algorithms will yield interesting results. It is indeed quite logical to think that, as long as you restrict your search for a minimum volume structure

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10 High Performance Structures and Materials III within a given set of typologies, you could miss better solutions! I am thus convinced that there is still an open field and want to explore it. I am quite thrilled by the enthusiasm of the students who discover the potentials of these indicators and indexes, when working for their studio work. At least it proves that it can effectively be used at the stage of conceptual design. And was that not the goal one tries to achieve as a teacher of structural design?

Acknowledgements Scientific research is very rarely the output of a single brain, and it has not been different with the present study. I thus want to pay due respect to colleagues and students. In the first place, two very important contributors to my evolving insights in structural design and analysis: Philippe Samyn, eminent architect and engineer, and Pierre Latteur, a structural engineer with a real feeling for structural behaviour. They triggered my interest for morphological indicators, allowing me to direct several theses at both Master’s and Doctoral level. All of the others will recognise themselves, but I want to underline the outstanding research work of Jan Van Steirteghem, architectural engineer, who sort of “filled some gaps” left by the excellent work of P. Samyn and P. Latteur, and also initiated work in the field of dynamics of lightweight structures: he allowed me to come back to my real passion for structural vibrations. During the last years I have been also collaborating with one of my former students, Marijke Mollaert, who finished her Ph.D in 1984 on tensile structures and, since then, has become a colleague and always has remained a close friend. I still spend a lot of time discussing new ideas with her and her students, all of them involved in this exciting field of form-active tensile structures. Although the discipline of designing these structures is very much different from what we are doing in my research group, the ideas one can pick up during those informal meetings are refreshing and really exhilarating. Thank you, Marijke, for your friendship. WIT Conferences were also regular meeting points, confronting me with new ideas and insights: first CADCOMP, later evolving into these HPSM Conferences, OPTI, but also Structural Repair and Maintenance of Historic Structures. In particular, I want to mention my friends Santiago Hernandez (U of La Coruña, Spain) and Jerry Connor (MIT, USA), with whom it is an enlightening experience to discuss all kinds of matters, even structures! Research needs funding and scholarships and I thus also want to acknowledge the Institutions which helped this research financially: Fund for Scientific Research in Flanders (FWO), Fund for Applied Research in Flanders (IWT), the VUB and several industrial partners. Last but not least a personal word of gratitude for Carlos A. Brebbia, always showing more confidence in my skills than I deserve, and allowing me to meet such fine persons within the Board of Directors of WIT. It is now about 35 years that we met for the first time and it is always with extreme pleasure that we see each other, either in the Lodge or at his Conferences.

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High Performance Structures and Materials III

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References [1]

[2] [3]

[4] [5] [6]

[7] [8] [9]

[10]

[11] [12] [13] [14]

Samyn, P., Etude comparée du volume et du déplacement de structures bidimensionnelles, sous charges verticales entre deux appuis – vers un outil d’évaluation et de prédimensionnement des structures (4 vol.), Ph. D. thesis, Université de Liège, 1999. Samyn, P., Étude de la morphologie des structures à l’aide des indicateurs de volume et de déplacement, Académie royale de Belgique, Classe des Sciences (2004), ISBN 0365-0952. Latteur, P., Optimisation des treillis, arcs, poutres et câbles sur base d’indicateurs morphologiques – application aux structures soumises en partie ou en totalité au flambement (3 vol.), Ph. D. thesis, Vrije Universiteit Brussel, 2000. Schlaich, J., Bergermann, R., Leicht Weit, Light Structures, Prestel (München, Germany), ISBN 3-7913-2918-9, 2005. Latteur P., Samyn P. et De Wilde P., 2001, Optimization des arcs paraboliques et en chaînette – aide à la conception sur base d’indicateurs morphologiques. Revue française de Génie Civil, Vol. 5, n° 1. Latteur P., Samyn P., De Wilde P., 2000, Comparaison des treillis classiques type Warren, Pratt et Howe : optimization et prédimensionnement sur base d’indicateurs morphologiques. Revue française de Génie civil, Vol. 4, n° 4. Shea K., 1997, Essays of Discrete Structures: purposeful design of grammatical structures by direct stochastic search, PhD Thesis, Carnegie Mellon University, USA. Shanley FR. Weight-Strength Analysis of Aircraft Structures. New York: Dover, 1960. Samyn P., Latteur P., Van Vooren, J., Volume of structures: application to classical and harmonic structures, International IASS symposium on Lightweight structures in Architecture, Engineering and Construction, 1998 –October 5-9, Sydney, Australia. Van Steirteghem J., A Contribution to the Optimisation of Structures Using Morphological Indicators: (In)Stability and Dynamics, Ph.D. thesis, Vrije Universiteit Brussel, Mechanics of Materials and Structures, W.P. De Wilde, supervisor, 2006. Audenaert A., Peremans H., De Wilde W.P., Static determination of the internal forces and displacements in arch bridges, Masonry Society Journal, vol.22, n°1, pp.97-111, September 2004. Vandenbergh Th., Van Steirteghem J., De Wilde W. P., Samyn P., Influence of stiffness constraints on optimal design of trusses using morphological indicators, HPSM 2006, Wit Press, 2006. Ponsaert W., De Wilde W. P., Samyn P., Van Steirteghem J., The use of tuned mass dampers in beam structures, HPSM 2006, WIT Press, 2006. Verbeeck B, De Wilde WP, Samyn Ph, Van Steirteghem J., The Use of Genetic Algorithm and Morphological Indicators in the Optimization of 2D Trusses, HPSM 2004, Wit Press, 2004. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

12 High Performance Structures and Materials III [15]

Verbeeck B., Van Steirteghem J., De Wilde W.P., Samyn Ph., The Need of Numerical Techniques for the Optimization of Structures Using Morphological Indicators, Proceedings OPTI2005, WIT Press, 2005.

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Influence of dynamic loads on the optimum design of trusses J. Van Steirteghem1, W. Ponsaert1, W. P. De Wilde1 & Ph. Samyn1,2 1 2

Vrije Universiteit Brussel, Belgium Samyn and Partners, Belgium

Abstract The Theory of Morphological Indicators allows a preliminary optimisation of structures at the stage of conceptual design. Samyn and Latteur developed the Indicator of Volume to determine the efficiency of structures at early design stages. The main advantage of this approach is that we only need a very limited number of parameters. Samyn establishes efficiency curves, with respect to minimum volume of material, for trusses in which he neglects buckling. Latteur establishes efficiency curves in which he accounts for buckling. The displacements are usually checked afterwards to verify if the normative constraints are not exceeded. In this paper we argue that for trusses, loaded dynamically and with large spans, dynamics become the dimensioning criterion. We use the Indicator of the First Natural Frequency to determine the first natural frequency of trusses. We find that for fully stressed trusses this natural frequency is usually near to the excitation frequencies of man induced and wind induced vibrations. Therefore, we need to include dynamics in the optimisation procedure. We show that for trusses with important spans very large stress reductions are necessary to obtain acceptable natural frequencies. This stress reduction comes at the cost of a very important and unacceptable increase of the volume of material. We determine which typology (Warren, Howe, Pratt) is the most efficient (minimal volume) with respect to dynamics. Moreover we show that when dynamics is the dimensioning criterion, the influence of buckling on the optimum design is negligible since an important stress reduction is necessary. Finally we propose a work scheme that allows considering dynamics in the Theory of Morphological Indicators and we provide an example. Keywords: conceptual design, morphological indicators, trusses, dynamics. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06002

14 High Performance Structures and Materials III

1

Introduction to the theory of morphological indicators

Morphological Indicators are design tools allowing the optimisation of structures for a chosen criterion (volume, stiffness) at the stage of conceptual design using a limited number of parameters [1]. The Indicator of Volume W = σV FL allows the comparison of the volume of material used of different structural systems subjected to the same load case. The Displacement Indicator ∆ = Eδ σL compares the displacement of different structural systems for a given stress level. Samyn [1] and Latteur [2] established the analytical expressions of both W and ∆ for trusses, beams, arches, cables, cable stayed structures, masts and frames subjected to a limited number of (simple) load cases and support conditions. For statically determined structures these Morphological Indicators are only function of the geometrical slenderness L H if instabilities, self weight and second order effect are neglected. Latteur [2] introduced the Indicator of Buckling Ψ = µσL

qEF

to take into account the additional volume of

material due to buckling. Ψ is a ‘global’ measure of the sensitivity of a structure to buckling as a function of the material used. Herein, µ is a reduction factor that determines the effective buckling length of the compression member, q = I Ω 2 is a form factor (with I the moment of inertia and Ω the section area) that determines the disposition of the material with respect to the center of gravity of the compression member (buckling efficient sections have large value of q ). Furthermore Samyn [1] and Latteur [2] introduced the Indicator of Self Weight Φ = ρL σ to take into account the influence of self weight on the total volume of material. Herein, ρ is the specific weight of the material used. In [3, 4] the authors show that dynamics can become the dimensioning criterion for structures with large spans and/or small stiffness/strength ratio. Therefore, an Indicator of the First Natural Frequency Θ = 1

2

∆ is introduced [1, 3, 4].

Indicator of the First Natural Frequency

2.1 Classification of loads For a structure spanning a length L , the total resultant of the loads F consists of: • The external live loads Fl •

The external permanent loads F p

•

The self weight F0

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The external live loads are subdivided in non co-vibrating live loads Fl1 = cFl and co-vibrating live loads Fl 2 = (1 − c )Fl , with c the share of the non covibrating live loads in the total live loads. Furthermore, we define the following combinations: •

FP = F0 + F p , the sum of all permanent loads

•

FE = F p + Fl , the sum of all external loads

We can now express the resultant of all loads as follows: F = F0 + FE = FP + Fl

(1)

When calculating the first natural frequency of a structure, we take into account the co-vibrating loads as vibrating mass. We express the total co-vibrating load FD as follows: FD = F0 + F p + Fl 2 = FP + (1 − c )Fl

Fl1 Non co-vibrating part of live loads

cFl

Fl2 Co-vibrating part of live loads

(1-c)Fl

FD = Fl2 + Fp + F0 = zF all co-vibrating loads

(2)

Fl = Fl1 + Fl2 live loads FE = Fl + Fp external loads

F= FE + F0 total loads

Fp external permanent loads

FP = Fp + F0 =

F0 self weight

These loads refer to the Ultimate Limit State (ULS) calculation of the structure. For a dynamic analysis, we must consider loads in a Serviceability Limit State (SLS). Hence, we can express FD in SLS F F (3) FD* = P + (1 − c ) l 1.35 1.50 And we express the vibrating mass m *D of a structure as: m *D =

FD* g

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(4)

16 High Performance Structures and Materials III Finally, we express the total mass m of a structure as: F m= g

(5)

2.2 The first natural frequency of a SDOF system By approximating the structure by an undamped single degree of freedom (SDOF) structure, with m *D the vibrating mass in SLS of the structure and k the static stiffness of the structure, the first natural frequency can be expressed as: c k (6) f 1 = cor 2π m *D By reducing a continuous system to a SDOF system, some approximations are made. c cor is a correction factor that counters these approximations. For beams, it can be determined analytically; for trusses on the other hand, it must be determined numerically. We can express the vibrating mass as a share of the total mass, with z * the ratio of the co-vibrating load FD* in SLS and the total load F in ULS: m *D = z * m

(7)

By substituting eqns (7) and (5) in eqn (6), the expression of the first natural frequency becomes: c gk (8) f 1 = cor 2π z * F With δ = F k the static displacement, eqn (8) becomes: c cor g 2π z *δ or as a function of the Displacement Indicator: c gE f 1 = cor 2π z * βσ∆L f1 =

(9)

(10)

The parameter β indicates the stress level of the structure. The Indicator of the First Natural Frequency becomes: Θ=

1

=

2πf 1 c cor

z * βσ L gE

(11) ∆ When we calculate the natural frequency we need a larger number of parameters • the span ( L ) • the material ( σ , E ) • the stress level ( β ) •

the ratio of co-vibrating load in SLS and total load in ULS ( z * ).

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2.3 Determination of the correction factor c cor An approximate method to calculate the first natural frequency of a structure, yielding very good results, is Rayleigh’s energy method [5]. By using this method we derive the first natural frequency by equating the maximum potential energy to the maximum kinetic energy. For a structure with concentrated loads the fundamental frequency becomes, with δ j the static displacement under the concentrated load F j :

∑F δ

j

∑F δ

2 j

j

1 f = 2π

g

j

j

j

(12) We checked the results of this method numerically with Robobat. The results obtained using eqn. (12) underestimate the numerical ones marginally (maximum 10% for small slendernesses). 2.4 Indicator of First Natural Frequency We can use the Indicator of the First Natural Frequency to determine the first natural frequency of structures. Θ =1

∆

L = 90 m

1,5

L = 70 m L = 50 m

1

Z=1

Ψ = 20

0,5

Ψ=0 1

L

L[m]∆ 5

f [Hz ]

11

7f [ Hz 12 ]

f [Hz ]

16

17

f [Hz ]

15

14

10

11

9

Figure 1:

13

9

10

8

12

8

11

10

7

9

6

8

7

5

6

5

4

4

3

3

2

1

concrete timber

2

1

10

ccor =1,10

8

7

6

5

4

3

2

1

f1 = S355

7

6

5

4

3

2

H

20

z * = 0,44

β = 1,0

S235 9

15

c cor ,1 2π

g

E z*βσ

1 ∆L

1

Graph allowing the determination of the first natural frequency of Warren truss with 10 panels.

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18 High Performance Structures and Materials III In the first quadrant we plot the Indicator of the First Natural Frequency as a function of the overall slenderness. In the second quadrant we plot 1 function of Θ = 1

∆L as a

∆ . Finally in the third quadrant we plot the frequency

abscissa for different material, stress level and value of z * . Since we include the material, stress level and value of z * in the frequency abscissa the first and second quadrant must never be changed for this structure. In this example we determine the first natural frequency of a steel Warren truss with 10 panels and a span of 50m in which we can neglect buckling (Ψ = 0 ) . For this example we obtain a first natural frequency of 1,7Hz.

3

Trusses dynamically loaded

3.1 Scope We can now study the behavior of trusses that are subjected to dynamic loads. We consider three truss typologies (Warren, Pratt and Howe):

Warren

Pratt

Howe We consider three truss typologies (Warren, Pratt and Howe) with: • Three different spans (L = 20m - 50m - 70m) • Three number of panels (n = 4 - 8 - 12) • Three values of z* (z* = 0,15 - 0,44 - 0,74) • Two different materials (timber - steel). Moreover, we impose that the natural frequency of the structure must be larger than 5Hz because the man and wind induced frequencies are usually between 0Hz and 5Hz. For trusses in which we neglect buckling we can express the constraint on the first natural frequency as a constraint on geometrical slenderness. First, we use eqn. (10) compute an upper bound on ∆ : 2

gE   c = ∆ dyn = ∆ max 0 − 5 Hz ⇒ ∆ <  cor   2π × 5  z * βσL

(13)

In [1] we find the expression of the Indicator of Volume of Warren trusses (eqn. (14)) and Howe/Pratt (eqn. (15)) trusses: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

∆ dyn − ∆2dyn − (n + 2 )

2 L ∆ dyn + ∆ dyn − (n + 2 ) < < H (n + 2 2n )

(n + 2 2n) ∆ dyn − ∆2dyn − (n + 1) L < (n + 1 2n ) H

<

19

(14)

∆ dyn + ∆2dyn − (n + 1)

(15)

(n + 1 2n )

On the other hand, if we consider buckling through Ψ this interval is calculated numerically. 3.2 Results and comparison When we select a target slenderness of L/H = 10, the large majority of combinations (typology, L, n, z*, material) need a large stress reduction to obtain a first natural frequency larger than 5Hz. This stress reduction increases when we use: • Pratt or Howe truss instead of Warren truss • A large span • A large number of panels • A large value of z* Howe-Pratt ( Ψ = 0 ): Necessary stress reduction for target slenderness L/H = 8 for different materials, spans, number of panels and z*.

Table 1:

f > 5Hz

Ψ=0

co-vibrating load Material

z* = 0, 74 timber

Number of panels L = 20m L = 50m L = 70m

β = βσ β = βσ β = βσ

Number of panels L = 20m L = 50m L = 70m

β = βσ β = βσ β = βσ β = βσ β = βσ β = βσ

steel

Timber

0,20 0,08 0,05

0,30 0,12 0,08

0,20 0,08 0,05

0,30 0,12 0,08

steel

timber

0,18 0,07 0,05

0,27 0,11 0,07

steel 4

0,34 0,13 0,09

0,88 0,35 0,25

0,34 0,13 0,09

0,88 0,35 0,25

0,31 0,12 0,09

0,81 0,32 0,23

8

12 0,16 0,06 0,04

z* = 0, 15

4

8 0,18 0,07 0,05

Number of panels L = 20m L = 50m L = 70m

z* = 0, 44

4 0,18 0,07 0,05

L H =8

1,00 1,00 0,29 8

12

1,00 1,00 0,29 12

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0,93 0,37 0,26

20 High Performance Structures and Materials III The stiffness/strength ratio of steel and timber is comparable. Table 1 gives an indication of β , the necessary stress reduction to obtain f 1 > 5Hz for a Howe/Pratt truss.

4

Conclusions and further research

4.1 Conclusions In this paper, we showed that we can include dynamics into the Theory of Morphological Indicators. This comes at the cost of a larger number of parameters to consider. However, we believe that in order to be useful we must include the dynamic behaviour of (lightweight) structures. Moreover, we argue that the number of parameters to consider are still small and are all available at the stage of conceptual design. We developed the Indicator of the First Natural Frequency, Θ , to compute the first natural frequency of structures. The expression of Θ clearly shows that the problem is not scale independent anymore. We found that trusses with large spans, large values of z* and small stiffness/strength ratio are sensitive to dynamic loads and need important stress reduction to obtain an acceptable natural frequency. 4.2 Further research In order to make the Theory of Morphological Indicators more powerful in terms of dynamic behaviour, the following topics can be worked out in the future: • Stress reduction due to fatigue • Higher natural modes and frequencies • External damping devices (passif, actif, hybrid)

References [1] Samyn, Ph., Etude comparée du volume et du déplacement de structures bidimensionnelles, sous charges verticales entre deux appuis – vers un outil d’évaluation et de pré-dimensionnement des structures, PhD thesis, Université de Liège, 1999 [2] Latteur, P., Optimisation des treillis, arcs, poutres et câbles sur base d’indicateurs morphologiques – Application aux structures soumises en partie ou en totalité au flambement, PhD thesis, Vrije Universiteit Brussel, 2000 [3] Van Steirteghem, J., A contribution to the Optimisation of Structures Using Morphological Indicators: (In) Stability and Dynamics, PhD thesis, Vrije Universiteit Brussel, 2006 [4] Ponsaert, W., Het gebruik van trillingsdempers in balkconstructies, Master thesis, Vriie Universiteit Brussel, 2005 [5] Fertis, D.G., Mechanical and Structural Vibrations, John Wiley & Sons, New York, 1995 WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Modular grid-based design concept for fibre reinforced composite shells E. De Bolster, H. Cuypers, W. P. De Wilde & J. Wastiels Department of Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Belgium

Abstract When designing a modular system, two main aspects need to be considered: the aspect of one building stone and the aspect of the overall structure. Both will be studied here as a function of geometry, mechanical behaviour and historical background (architectural design, engineering science, etc.). The state-of-the-art of design principles, used in architectural designs over the latest decades, will provide the basic knowledge for the generation of a new kind of modular construction. Combining both the artistic point of view (aesthetics) and the engineering point of view (calculations, FEM), a new design concept will be generated: a system with modular hyperbolic paraboloid building stones, connected to one another through hinged connections. Keywords: architecture, modular construction, design concept, hyperbolic paraboloid.

1

Introduction

In contemporary architecture, one can notice the tendency towards modular lightweight structures. These designs can easily be assembled and disassembled and they require a minimal amount of materials, which makes this type of design very advantageous. Another tendency that can be observed is the frequent use of freeforms. These designs are usually lightweight as well as aesthetically pleasing. The designs that combine the advantages of modular structures and freeform structures are usually covered with textiles. However, composites (textile reinforced cementmatrix composites, for example) could present a good alternative; especially when used as faces in a sandwich construction. Therefore, it would be interesting to perform a feasibility study of such a modular, freeform WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06003

22 High Performance Structures and Materials III construction made of sandwich panels with faces of textile reinforced composites.

2

Hyperbolic paraboloids

When designing a modular system, two aspects need to be considered: the aspect of one building stone and the aspect of the overall structure, consisting of several building stones. Since a modular system is only as strong as its components (building stones and the connection between them), it is important to select them with care. A hyperbolic paraboloid (also called: hypar) building stone (see figure 1) has many advantages over other types of building stones. Firstly, a hypar surface can be applied over any foundation shape (rectangular, triangular, circular,…). Secondly, a hypar surface is a curved surface and has therefore better bearing capacities than flat surfaces, because the forces can more easily be introduced in an efficient way (i.e. through higher normal forces and lower bending moments). Due to its anticlastic doubly curved shape, instabilities are less likely to occur: the convex curve will stiffen the behaviour of the concave curve and vice versa (see figure 1, on top).

Z Y

X

Figure 1: Hyperbolic paraboloid (parabolas, parametric representation and the two distinct independent families of straight lines). In rectangular coordinates, with the origin in the saddle point of the surface (as indicated in figure 1, in the middle), the parametric equation of a hypar surface can be given as: Z = k XY where: k = a measure for the distortion of the hypar surface. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(1)

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Even the fabrication of a hypar surface is not as difficult as one would expect from a three dimensional shape: a hypar surface is a doubly ruled surface and can therefore be formed by two distinct independent families of straight lines (see figure 1, below), which makes the formwork much easier. When manufacturing hypar-sandwich panels with composite faces, an adaptable and reusable cable net could be used as a reconfigurable mould. The net, which is largely based on the ADAPTENT-concept of Hebbelinck [1], then provides an easy, fast, re-usable and economical methodology in fabricating hypar-sandwich panels that are to be used in modular structures (see figures 2 and 3).

Figure 2:

Figure 3:

ADAPTENT-concept: asymmetrical hypar surface [1].

Laminating a hypar sandwich panel on top of the ADAPTENT.

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24 High Performance Structures and Materials III

3

Design principles

There are many basic concepts that underlie the field of design. All basic tenets can be grouped into design elements and design principles. They are applicable to all of the visual arts, amongst which architecture. The design elements are the components or objects of the design: a line, a point, a surface, colour, etc. The design principles on the other hand govern the relationship between the design elements, giving them meaning. There are many additional concepts that are related to the principles of design. The grid-concept is one of these techniques: it provides a framework for all basic principles. 3.1 Grid Geometric patterns have always been a fundamental aspect of human culture. In nature, geometric order is rarely found. It is rather man’s way to understand his surroundings. Science uses the grid as a technique to cope with complexity. The grid is the basis for most techniques which sample and quantitatively analyse the ‘real world’. In archaeology for example, archaeologists use three dimensional grids as a baseline for complex digs. Excavation takes place within trenches or areas of about 1 metre square, which are located within a site survey ‘grid’ so that every point on the excavation site can be given a horizontal coordinate. The information, revealed through laborious excavation can then be gathered and ordered in a rather simple manner. Ecological sampling methods also make use of the grid-method. If one would want to know what kind of plants and animals are in a particular habitat, and how many there are of each species, it is usually impossible to go and count each and every one present. Samples are taken instead. A method of random sampling is used to map the area under survey and to overlay it with a numbered grid. In short, the fundamental technique for filtering the complexity of the observable is based on the ‘quadrat’ and ‘transect’: small square or cubic areas (quadrats) are examined in detail along a line or across a grid of evenly distributed sample points (transect). For reasons difficult to explain, the significance of the grid in modern and contemporary architecture (and art) is just as important. 3.2 Modern and contemporary architecture The grid is one of the oldest architectural design tools. A grid can help a designer control the positions of built and space elements, making the layout task more systematic. Especially in laying out plans for new towns and cities, the use of grids enables the designers to make decisions at the urban scale, nevertheless providing relative freedom at the block and lot scale for individual developers and house designers. Taking it one step further, the grid can also be applied to a single building. This formal and structural approach to design was seen in the Swiss International Style. The term “International Style” was first used in New York City in WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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conjunction with the 1932 Museum of Modern Art exhibition of contemporary European architecture, entitled “The International Style: Architecture Since 1922”. The term is often used to refer to all Modern architecture. According to Philip Johnson and Henry-Russell Hitchcock (who wrote the catalogue that accompanied the exhibition of the International Style), Modern architecture is characterized by three main principals [2]: architecture as volume, regularity and avoidance of applied decoration. The rigid, simple, geometric and asymmetrical style of the International Style arose from the work of Walter Gropius, Ludwig Mies van der Rohe and of course Le Corbusier and lived on in the work of – for example – Skidmore, Owings & Merrill. Many architects retained the earlier mentioned modernist philosophies, aiming to integrate modern technology and formal elements derived from the basic grid. Mies used minimalistic, linear forms, evoking the language of the idealistic International Style philosophy, called “less is more”. He embraced the glass and steel skyscraper that showed its building materials as a form of ornamentation. His buildings were carefully laid out on a grid, totally embracing the concept of regularity. This careful grid system lined up everything including the tiles in the plaza and even the lighting in the ceiling of the lobby. One of his well known accomplishments is the Barcelona Pavilion in 1929. The grid-system is undeniably present. The ideal modernistic point of view became so stringent that a rupture between architects and this architectural style was inevitable. Postmodernism rebelled against the purity and rigidity of forms with lots of excess and exuberance. Nowadays, in the rise of spirituality and fundamentalism however, architecture has thrown off the ornamentation of postmodernism and returned to some of the values of the International Style. In his book “Supermodernism: Architecture in the Age of Globalization” [3], Hans Ibelings defines the postpostmodernism as a tech-inspired aesthetic movement that reacts against postmodernism and adopts the philosophy of computer product design. Architects are drawn towards the idea of “tabula rasa”; the idea that each new design starts as a pure, empty plane. Structures appear portable and therefore disconnected from their surroundings. Similar to a computer, all the details are inside, while exteriors are neutral and unassuming. High-tech knowledge is used to create spaces that are valued for their visual and spatial sensations only. Since high-tech architecture strongly relates to science, the same techniques of handling complexity are likely to be used. It seems evident that – yet again – the grid-system is a widely-spread designing tool.

4

Overall design

Many have said grid-systems can suffocate creativity, as can modularity. However, this doesn’t have to be the case. Grid-systems can facilitate creativity by providing a framework and some predefined answers to questions such as “What span to use?”, “Which module to use?”, etc. The grid simplifies decisionmaking by limiting the placement of elements to certain places, but if the grid

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26 High Performance Structures and Materials III doesn’t permit a sufficiently rich range of variation, the designer must react and redesign it. 4.1 Design concept In this paper a design concept is formulated in which a grid is used as an integration system (through hinged line connections) of the individual hypar surfaces in the overall “roof”-design. A two-dimensional grid is formed that defines the horizontal coordinates of the four corner points of each hypar surface. The vertical coordinate of the corner points depends on the formal design of the overall structure and is thus insofar independent of the grid. Once the position of four non-coplanar corner points are chosen, a hypar surface can be created. With this systematic method, a variety of shapes can be made modular in five steps: 1. 2. 3. 4. 5.

Choose a mathematical function z=f(x,y), representing the formal aesthetics you - as designer - wish. Choose/note the dimensions of the horizontal rectangular base that should be covered. Divide this base in a number of elements (each element having the same dimensions in the x- and y-direction). Calculate the x- en y-coordinates of each corner point (on the base). Use these coordinates and the mathematical equation of the threedimensional function to calculate the z-coordinate of each corner point.

y z

Figure 4:

x

Generating a modular hypar-system, based on a grid.

The two-dimensional horizontal grid not only defines the overall design, but also facilitates the prefabrication of each hypar surface (see section 2). In figure 4, it was chosen to work with a single regular grid to which each building stone is related. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Since any mathematical function can be chosen as initial aesthetic design, a variety of freeform constructions can be created without the fear of losing oneself in an infinite amount of choices. Due to the grid and the specific shape of the basic building stone, one is better capable of handling the complexity of freeform design: a framework has been set that still leaves room for ones own creativity and originality. 4.2 Modularity science The abovementioned design idea has similarities with the finite element method. In FEM, a complex object is simplified by dividing it in areas (finite elements) and reducing each finite element to his corner points. Calculations of the stresses are done in specific, discrete Gauss points. In the presented modular design system, an overall shape is divided into pieces according to a horizontal grid and approximated by the discrete corner points of each piece. The design is dictated by the hypar surfaces that are drawn through the discrete points. The similarities between the finite element method and designing method are clarified in figure 5: a geodesic dome of Buckminster Fuller is shown. An aluminium sphere with radius of 34.14m is used as roof for the former aviation museum at the airport of Schiphol in the Netherlands. Starting from the formal design (figure 5, top left), calculation methods are used to “mesh” the object under study (figure 5, middle right). Each “mesh” is then extracted from the overall object and studied individually: the module is born (figure 5, bottom left) to which all calculations are related.

MESHING

MODULARISING Figure 5:

From global concept to discrete solutions (National Aviation museum Aviodome-Schiphol, Buckminster Fuller, 1971).

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28 High Performance Structures and Materials III As with FEM, the approximation of the original object is best when the griddimensions (or in case of FEM: the mesh-sizes) reduce to zero. Similar to FEM, it is expected that with a certain grid(mesh)-size a convergence can be reached. This convergence limits the amount of hypar surfaces that needs to be created and therefore limits the time needed for design, fabrication and erection of the modular structure. In the given analogy between finite elements and the modular design system under study has a slight discrepancy at the end. When the division of an arbitrary object in finite elements has occurred, calculation is performed in discrete points. When the division of an arbitrary roof design in modular elements has occurred, calculation is performed on each module. In other words, in finite elements calculation is done in discrete points, while in the presented modular designconcept calculation is said to be done on a surface. In reality, calculation of each module will be done through the use of finite elements. The modular hypar surface building stone becomes the arbitrary object that will be divided in a certain amount of finite elements and calculated in the discrete Gauss points of each finite element (see figure 6). Division

Modules

FEM

DESIGN SYSTEM

Object

Finite element Figure 6:

5

Meshing

Object

Level-wise calculation of an arbitrary design shape.

Conclusion

The grid-system has always been an important design principle: not only in science, art, etc, but also in architecture. Modern and contemporary architecture have turned it into an undeniable part of building concepts. The evolution of scientific knowledge has turned mankind’s urge of controlling his surroundings, and thus the grid, into an unmistakable part of building realisations. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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The presented design concept of modular hypar surface-structures originates from the combination of a two-dimensional, horizontal grid and a chosen, aesthetically pleasing, three-dimensional design-shape. The concept shows a lot of similarities with the finite element method: approximating continuous objects (the overall design shape) by discrete objects (the modular building stones, which are hyperbolic paraboloid surfaces). On a second level, it even makes use of the finite element method: calculations are done in discrete points and no longer on continuous surfaces.

Acknowledgment Financial support from the Institute for the Promotion and Innovation by Science and Technology in Flanders (“IWT-Vlaanderen”) is gratefully acknowledged.

References [1] Hebbelinck, S., A generating system for temporary, adaptable and reusable nets and tensile structures, PhD-thesis Vrije Universiteit Brussel, Belgium, 2002 [2] Leland, M., About the modern Style, http://www.michael.leland.name/ modern/index.html [3] Ibelings, H., Supermodernism: architecture in the Age of Globalization, Nai Publishers, Rotterdam, 1998

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Influence of stiffness constraints on optimal design of trusses using morphological indicators T. Vandenbergh1, W. P. De Wilde1, P. Latteur 2, B. Verbeeck1, W. Ponsaert1 & J. Van Steirteghem1 1

Faculty of Engineering Sciences, Department of Mechanics of Materials and Structures, Vrije Universiteit Brussels, Brussels, Belgium 2 Agronomic Faculty of Gembloux and Setesco NV, Belgium

Abstract Within the framework of sustainable development we strive for structures with a minimum volume of material. When we only consider criteria on resistance and buckling, Samyn and Latteur prove that even at the stage of conceptual design a clear hierarchy among the different truss typologies can be established. Up to now, stiffness constraints - such as the upper limit on static displacements - were not considered. However, an optimum obtained by minimising the volume, only considering the strength criterion, often results in solutions which violate the stiffness constraint(s). To avoid large displacements a stress level reduction can be imposed. However, this comes at the cost of a significant volume increase. With an optimisation process that involves the stiffness constraints at the stage of conceptual design, an optimum can be obtained without the necessity to alter the structure drastically afterwards, which partly annihilates the main objective of minimal use of material. This approach compares the different truss types on a new priority scale, generating new optima. This implicates a non-negligible change in the truss choice at conceptual design stage. The solutions are logically depended on the displacement criterions. This approach forms a first step to a new design philosophy that considers all the stiffness constraints (static displacements, resonance, local and global buckling) at conceptual design stage and is called design for stiffness. Keywords: morphological indicators, stiffness, strength, truss, steel, static displacements, optimisation. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06004

32 High Performance Structures and Materials III

1

Introduction

1.1 Morphological indicators Morphological Indicators (MI) are design tools allowing the optimisation of structures for a chosen criterion (volume, stiffness) at the stage of conceptual design using a limited number of parameters [1]. The indicator of volume W = σ V FL allows the comparison of the volume of material used of different structural systems. It is the volume of an isomorphic structure with unit span L, with at least one section dimensioned on its unit allowable stress σ , subjected to a system of loads with unit resultant F. The displacement indicator ∆ = Eδ σ L compares the displacement of different structural systems. It is the maximum displacement of an isomorphic structure with unit span L in a material with unit Young’s modulus E, with at least one section dimensioned on its unit allowable stress σ , subjected to a system loads with unit resultant F. The analytical expressions of both W and ∆ have been established by Samyn [1] and Latteur [2] for trusses, beams, arches, cables, cable stayed structures, masts and frames subjected to a limited number of (simple) load cases and supports. For statically determined structures those MI are only function of the geometrical slenderness L/H if instabilities, self weight and second order effects are neglected. Efficiency curves depicting the geometrical slenderness with respect to minimum volume material can be established (Figure 1). The indicator of buckling Ψ = µσ L qEF is developed by Latteur [2] to take into account buckling in compression elements. It is the image of the buckling tendency of the compression elements in a structure with span L, composed of bars with a form factor q = I Ω 2 (with I the moment of inertia and Ω the section area) in a material with Young’s modulus E, with at least one section dimensioned on its allowable stress σ with a system of load with total resultant F. µ is the proportion of the buckling length of the compression bars over their geometrical length (which depends on the connection type). 1.2 Stiffness constraints Up to now, stiffness constraints have never been considered in the use of MI. Only the resistance criterion was met with volume as the objective function. This strategy is called design for strength and often results in lightweight structures with a problematic lack of stiffness, which implicates a non-negligible volume increase to meet the (imposed) stiffness criteria. In this paper we only consider one stiffness constraint: the upper limit on the static displacement. We develop an optimisation process that involves this stiffness constraint at the stage of conceptual design and obtain an optimum without the necessity to change the structure drastically afterwards. Taking stiffness into account compares the different truss types on a new priority scale, generating new optima. This implicates a non-negligible change in WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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the truss choice at conceptual design stage. As expected, we show that the solutions are dependent on the displacement criteria.

Figure 1:

2

Indicator of volume as a function of the slenderness for some classical structure typologies, neglecting buckling, Samyn [1].

Upper limit on static displacements

In Eurocode 3 [4] we can find the normative constraint on static displacements. This is usually expressed as an upper limit on δ L . δ is the maximum total displacement of the structure and L the span of the structure. Those are only indicative values; the commissioner of the project can impose sterner limits.

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

34 High Performance Structures and Materials III There are 2 types of constraints: those on the displacements due to the total loads and those on the displacements due to the live loads only. The determinative criterion depends on the ratio of live loads to the total loads (self weight included). We can demonstrate that when this percentage is smaller than 74% we must consider the constraints on the static displacements due to the total loads. Since this is almost always the case, from now on we only consider the displacement constraints due to the total loads. Considering the nature of the displacement constraints, those can very easily be translated to an upper limit on the displacement indicator, ∆. Imposing an upper value on δ L results in imposing an upper value on ∆. However the constraint on ∆ becomes material dependant. The larger the proportion of Young’s modulus over the allowable stress, the larger the allowable value of ∆ and the less severe the constraint. Table 1 shows the basic material properties of the classic structure steels. Table 1:

Material properties of (common types of) structure steel.

Name

Specific weight ρ

S235 S275 S355

(N/m³) 78500 78500 78500

Young’s modulus E (GPa) 210 210 210

Allowable strength σ (MPa) 235 275 355

Eσ 894.6 763.6 591.5

σ ρ

(m)

2993.6 3503.2 4522.3

Finally it is important to mention that, according to the Eurocodes, a calculation of static displacements is a Serviceability Limit State (SLS) calculation as opposed to a strength calculation which is an Ultimate Limit State (ULS) one. The difference between, SLS and ULS resides in the partial safety coefficient imposed on the loads: in ULS we typically use 1.5 for live loads and 1.35 for permanent loads, in SLS, on the other hand 1.0 for both load types. At conceptual design stage, we can accurately transform loads in ULS to SLS by dividing the total loads in ULS by 1.4 to obtain the total loads in SLS. Since the static displacements are linearly proportional to the loads, we must divide the displacement indicator calculated in Samyn [1] and Latteur [2] by 1.4.

3

Indicator of volume of structures subjected to stiffness constraint

3.1 Strategy We violate the displacement constraint for a large interval of the slenderness and for common typologies of trusses. Figure 2 depicts the indicator of displacement ∆ for Warren trusses with minimal W. We observe that for common values of δ L ( 1 200 and 1 500 ) an upper value for ∆ exists which we cannot exceed. This value of ∆ determines a validity interval for the slenderness.

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4.5

∆

4

= region of classic upper limits on ∆, corresponding to relative displacements between 1/500 en 1/200

3.5 3 2.5 2 1.5 1

Ψ = 20 Ψ = 30

0.5

Ψ = 10

L/H

0 0

Figure 2:

5

10

15

20

Displacement indicator of the lightest Warren trusses and common, displacement constraints for S355 steel.

The designer is presented with different solutions: Change the slenderness. This often results in a smaller slenderness. However, in most cases a maximal height is imposed which results in a minimum slenderness • Change the typology (number of meshes and/or truss type). This solution is also subjected to technological and architectural limitations. • Introduce an initial camber: this can compensate the displacement caused by the permanent loads. • Reduce the stress level: by doing so we increase the stiffness of the structure. However this comes at the cost of an increase in volume of material. The scope of these solutions and their respective impact on the increase of volume of material must be considered at conceptual design stage. A possible approach is the calculation of W and ∆ for every structure. Then we must compare the calculated indicator of displacement to its imposed upper limit. When the calculated displacement is allowable we use this structure. If, on the other the hand we violate the constraint we impose a stress reduction to increase the stiffness and meet the criterion on static displacements. We use β to denote the necessary stress reduction and define β as the ratio of the applied stress level to the maximum allowable stress level. Finally we determine the indicator of volume for this reduced stress level. This leads to a relatively accurate prediction of the volume of the lightest structure considering the strength requirement and the constraints on static displacements. The ‘new’ optimum curves depend on these stiffness constraints and therefore on δ L and the material of the structure. The number of parameters increases from 2 ( Ψ and L H ) to 4 ( Ψ , L H , δ L and material). The main disadvantage is that we cannot draw general design curves valid for all structures •

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36 High Performance Structures and Materials III as those set up in Samyn [1] and Latteur [2]. Nevertheless, the number of parameters remains small being the main advantage of the MI.

Figure 3:

Optimum indicator of volume and corresponding slenderness as a function of the indicator of buckling for Warren truss loaded on their lower chord [2].

3.2 Results In [2] Latteur selects the optimal (i.e. the lowest) indicator of volume W and its corresponding slenderness L H opt for every value of the indicator of buckling Ψ . We plot W and L H as a function of Ψ (Figure 3). This allows us to

determine the optimal value of W and L H if Ψ is known. However, the main disadvantage is the absence of sensitivity! Those curves only show the optimal values. For another value of the slenderness, no information is available. Considering that most optima are found for slendernesses between 2 and 12 [2] and that usually trusses have values of the slenderness between 8 and 18 [3], working at the optimum values is not always possible. Therefore we clearly divide the input information in variables and parameters. The material parameters and the displacement constraint are considered as being input parameters, since their values are fixed. For this set of parameters, we plot WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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efficiency curves for the indicator of volume as function of the slenderness for Ψ = 0,10, 20, 30. Ψ = 0 corresponds to the theoretical value of Samyn [1], in which buckling is neglected. Common values of Ψ vary between 10 and 30. Between the curves of Ψ = 10, 20 and 30 linear interpolation is allowed, certainly at conceptual design stage. 10 9

W

without constraint on displacement with constraint on displacement

8 7 Ψ = 30 Ψ = 20

6 5

Ψ = 10

4

Ψ=0

3 2 1 L/H

0 0

2

4

6

8

10

12

14

16

18

20

1

β 0.9

Ψ=0 Ψ = 10

0.8

L/H

Ψ = 20 Ψ = 30

0.7

0.6

0.5

18

0

2

4

6

8

10

12

14

16

18

20

n opt

16

Ψ = 30

14 12

Ψ = 20

10 8 6 4 2

Ψ = 10

0

Figure 4:

L/H

Ψ=0

0 2

4

6

8

10

12

14

16

18

20

Design curves of Warren truss of S355 steel loaded on their lower chord, with (bold) and without the upper limit on static displacement of 1 300 .

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

38 High Performance Structures and Materials III We also provide the optimal stress level reduction as a function of the slenderness for the same values of Ψ , illustrating the impact of the stiffness constraint on the ‘exhaustion level’ of the elements. To limit the amount of curves those calculations are repeated for values of the slenderness between 0 and 20 and n (number of meshes) between 2 to 18. Only the trusses with optimal (minimum W) number of meshes nopt , are selected. Finally, for every slenderness, the corresponding nopt is plotted, representing the optimal truss. This yields 3 design graphs for each set of input parameters (Figure 4, bold curves). The noise is the result of numerical calculations. A corresponding trend line indicates clearly the tendency. 10

W

without constraint on displacement with constraint on displacement

9 8

Ψ = 30

Ψ = 20

7 6

5,46

5 4 3

4,04 3,75 3,16 2,42

2 1 Ψ=0

0 0

Figure 5:

4

2

4

6

8

10

12

14

16

L/H

18

20

Impact of design for stiffness versus design for strength on optimisation of volume.

Practical example

We design a S 355 steel Warren truss with a span of 50m and a maximal height of 4m. We must limit the maximal static displacement to 1 300 . If we only consider the resistance criterion [1] the slenderness and the mesh number are the only variables and we obtain a truss with a slenderness of 12,5 and 2 meshes when we limit the volume of material. In the strategy of Latteur [2], in which buckling is considered explicitly, more input information is needed to determine the value of Ψ : the selected material, the section type of the bars and the total load. S355 steel, tubes with thickness/diameter ratio of 0.04 and a total load of 2500kN , result in an indicator of buckling of Ψ = 22.62. We obtain an optimum truss with 11 panels and a slenderness of 12,5. Finally, the constraint on the static displacement does not influence the slenderness, on the other hand the number of panels decreases to 5. For this truss we obtain W = 3.75. Since we include more phenomena (buckling, static constraints), the indicator of volume obtained by Samyn [1] increases (from 2.42 to 5.46). The solution obtained by WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Latteur [2] also increases due to the constraint on the static displacement (from 3.16 to 4.04). Those results are shown in Figure 5 and clearly illustrate the importance of considering stiffness constraints at conceptual design stage. Here we obtain a gain in volume of 8% in comparison with Latteur [2] and up to 46% with the result of Samyn [1].

5

Conclusions

We found that trusses with a large stiffness are composed of a small number of panels. Even though a larger number of panels reduces the buckling length of compression members, we observe that when we consider the constraint on static displacements we obtain an optimal solution with a smaller number of panels [1]. Moreover, stiff trusses are usually stocky (typically L H between 1.5 and 5), which confirms the results obtained by Samyn [1], guiding us towards small slendernesses. We observe that these stiff trusses are often very light (especially when the truss is not sensitive to buckling). The more buckling sensitive the truss, the smaller the necessary stress level reduction, since the material against buckling also provides stiffness against displacements. This can be noticed on the curves by the larger β’s at larger Ψ ' s for the same slenderness. On the other hand, a stress level reduction against displacements means a less buckling sensitive truss. This explains why at larger slendernesses (the most compliant geometry), the curves of the different Ψ’s join. For very large L/H’s and/or very strict displacements constraints, we can assume that the optimal structures become independent of Ψ. The optimal stress level is not always β = 1 . This clearly shows that fully stressed design doesn’t always give the lightest structure. Finally, Howe and Pratt trusses do not present any advantage in comparison with Warren trusses, since they not only need more material but are less stiff.

6

Further research

The above mentioned strategy, which considers the constraints on static displacements, can be applied to other truss topologies (K, Long, Smith,…) and to different typologies (arches, beams…). Hence, we can consider a new design hierarchy with respect to the minimisation of volume This approach constitutes a first step to a design philosophy that considers all stiffness constraints (static displacements, resonance, local and global buckling) at conceptual design stage, called design for stiffness. Finally, we must consider the influence of the connections (fixed or pinned) on the stiffness and on the volume of material.

Acknowledgement I would like to acknowledge the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT Vlaanderen), which helps this research financially. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

40 High Performance Structures and Materials III

References [1] Samyn P., Etude Comparée du Volume et du déplacement de Structures Isostatiques Bidimensionnelles sous Charges Verticales entre Deux Appuis. Vers un outil d’évaluation de prédimensionnement des structures (Tome 1 à 4), PhD thesis, Université de Liège, Belgium, 1999. [2] Latteur P., Optimisation et Prédimensionnement des Treillis, Arcs, Poutres et Câbles sur Base d’Indicateurs Morphologiques (Tome 1 à 4), PhD thesis, Vrije Universiteit Brussel, Belgium, 2000. [3] Orton A., The way we build now (Chapter 2). Building Structures, E & FN Spon , pp. 42-44, 1994. [4] Eurocode Steel/3, NBN ENV 1993-1-1:1992 pp. 49-52, 1992.

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Variations in form and stress behaviour of a V-shaped membrane in a foldable structure M. Mollaert, N. De Temmerman & T. Van Mele Department of Architecture, Vrije Universiteit Brussel, Belgium

Abstract Adaptable tensile structures are often considered to be either completely opened or completely closed. The current study is part of a research project studying adaptable tensile structures which demonstrate stable behaviour within a wide range of opened positions. In this paper a simple V-shaped membrane is studied during the unfolding process. Starting from an initially flat folded membrane, which is not pre-tensioned, a slight curvature is obtained when it is unfolded due to the fact that along the folding line a curved section is cut out of the fabric. The tension introduced in the transverse direction implies a tension in the longitudinal direction too. Two cases are analysed: one with a high curvature in the diagonal cable (~5% sag) and one with a low curvature (~1.3% sag). Based on computer simulations the form and the tensions are verified for different opening angles. The deformation under loading is checked for the shape with a low curvature of the diagonal cable at an opening angle of 70º. The results indicate that the membrane could be used as a fabric roof. Further refined analysis is needed to be able to implement the presented concept for real applications. Keywords: tensile structure, formfinding, coated textiles, adaptable shelter, foldable structures.

1

Introduction

Tension structures, made of high-strength coated fabrics, are well-suited to create adaptable buildings that can react (open/close) to changing environmental conditions [1]. Several solutions are possible, but only a few techniques keep the membrane tensioned while folding or unfolding. A first possibility is to start from a wave WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06005

42 High Performance Structures and Materials III form. The chosen reference equilibrium form is characterised by an intermediate opening angle (Fig. 5) and an appropriate pre-tension. When the wave form opens (Fig. 6) or closes (Fig. 4), the membrane tension does not change too much as long as the variation in the curvature is not too important and the membrane can be used as a roof in the intermediate positions [2, 3].

Figure 1:

Parallel sliding.

Figure 2:

Central sliding.

Figure 3:

Circular sliding.

Figure 4: Folded Figure 5: Reference Figure 6: Unfolded configuration. configuration. configuration.

Figure 7: Folded rhombus Figure 8: Opened Figure 9: shape. rhombus shape.

Curved diagonal.

In the current study the reference equilibrium form is a folded configuration without pre-tension. To test the basic idea a small model of a rhombus shape with straight edges and covered by fabric was made (Fig. 7, Fig. 8 and Fig. 9). The longitudinal diagonal divides the rhombus in two isosceles triangles with an apex angle of 120º (see Fig. 13 for the definition of the angle). The longitudinal diagonal is the folding axis. At this folding axis a part of the fabric was cut out along an arc that represents the cutting line. As a consequence the longitudinal diagonal is slightly curved. During the unfolding process the membrane is WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

43

tensioned in the transverse direction, and due to the bi-axial behaviour of the material, the longitudinal direction is tensioned too. The opening angles ranging from 60º to 80º are the most appropriate (in terms of covered area: see Fig. 11, Fig. 12) if one wants to use the unfolded tensioned membrane as an architectural roof.

Figure 10: Elevation for Figure 11: opening angles ranging from 50º to 90º.

Covered Figure 12: Covered area area for an for an opening angle of 80º. opening angle of 70º.

The cladding material is a coated fabric and the longitudinal diagonal is reinforced by a belt or a cable. The membrane itself is modelled by a mesh of 25cm x 25cm. The boundaries are considered to be stiff beams and modelled as fixed points on the boundary lines. Starting from a tensionless folded state, called the reference equilibrium form, the stress behaviour is analysed, while unfolded along the longitudinal diagonal.

2

Defining the reference equilibrium form

The reference equilibrium form is characterised by an opening angle of 1º and an initial tension in the membrane of 0.02kN/m, considered as a tensionless state. Calculations are made with the EASY software of Technet, based on the Force Density Method [4]. The formfinding based on the force-density method calculates an equilibrium form that is independent of the stiffness of the elements in the net. Different force values in the diagonal cable can be considered. The force values depend on the ratio of the force-density set for the longitudinal cable to the force-density set for the internal net. Five cases have been calculated to be able to estimate the curvature of the diagonal cable for different values of the force in the diagonal cable. The curvature is characterised by the ratio of the height H divided by the span S (see Fig. 13), the span being 6m. For the numeric modelling a rhombus of 6.0m x 3.464m is considered. The following figure illustrates the geometry and basic parameters: For two cases the unfolding has been analysed: Case 1, having a higher curvature in the diagonal cable, will be considered with a low stiffness of 1kN of the internal net elements, which represents the situation of a ‘stretchable’ net Case 5, having a lower curvature in the diagonal cable, will be considered with a more realistic stiffness of 150kN of the internal net elements, both in the longitudinal and in the transverse direction WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

44 High Performance Structures and Materials III Used symbols β θ H arc AA’

arc BB’

Figure 13:

Table 1:

Case

Geometry and used symbols.

S = AA’

apex angle of triangular module opening angle of triangular module height of the longitudinal arc arc in longitudinal direction (along Xaxis) arc in transversal direction (along Yaxis) span of the longitudinal arc

The height to span ratio of the diagonal for different forces in the diagonal cable. Elevation

Force [kN]

Height [m]

H/S [%]

5

2.1453

0.0775

1.29

4

1.7173

0.0952

1.59

3

1.2895

0.1233

2.06

2

0.8622

0.1750

2.92

1

0.4359

0.3013

5.02

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45

2.50

Force [kN]

2.00 1.50 1.00 0.50 0.00 1.29

1.59

2.06

2.92

5.02

H/S of the diagonal arch [% ]

Figure 14:

Force in the diagonal cable (at 1º opening angle).

It is obvious that the smaller the force in the diagonal cable, the greater the curvature of the diagonal arc will be (for the same pretension in the membrane of 0.02kN/m). But on the other hand, the greater the curvature of the diagonal cable, the higher the stresses in the transverse direction will be when unfolding.

3

Tension in the membrane for different opening angles

3.1 Case 1: Higher curvature in the diagonal cable The influence of the opening angle on the forces in the membrane and the diagonal cable is verified for a low stiffness of the internal net (1kN) and a stiffness of 5000kN of the diagonal cable. The first form given in Table 2 is the reference equilibrium form (with an opening angle of 1º). The other configurations have been obtained by a static analysis, starting from the reference equilibrium form, with the fixed nodes displaced according to the considered opening angle.

Tension [kN/m]

0.50 0.40 0.30 0.20 0.10 0.00 10

20

30

40

50

60

70

80

Opening angle [º]

Figure 15:

Tension in the transverse direction due to variation of the opening angle.

From the calculations with a stiffness of 1kN in the internal net it can be concluded that - in the longitudinal direction the tension in the membrane remains at a value of about 0.02kN/m - in the direction perpendicular to the folding axis the tension increases from 0.02kN/m to 0.44kN/m for larger opening angles. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

6.00

3.40

5.00

3.30 3.20

4.00

H/S [%]

Force [kN]

46 High Performance Structures and Materials III

3.00 2.00

3.10 3.00 2.90 2.80

1.00

2.70

0.00 10

20

30

40

50

60

70

80

10

20

Opening angle [º]

Figure 16:

30

40

50

60

70

80

Opening angle [º]

Force in the diagonal Figure 17: H/S of the diagonal cable cable due to variation of due to variation of the opening angle. the opening angle.

When unfolding, the force in the diagonal cable increases up to a maximum value. Then the force in the diagonal cable decreases for larger opening angles. Since the force in the cable is proportional to the resultant of the membrane tension at both sides of the cable, two different facts interact: - for larger opening angles the resultant decreases if the tension in the membrane remains the same - for larger opening angles the membrane tension in the transverse direction increases

Figure 18: A higher resultant force Figure 19: A lower resultant force with higher membrane with lower membrane tensions at 70º tensions at 60º. 3.2 Case 5: Lower curvature in the diagonal cable The influence of the opening angle on the stresses in the membrane and the diagonal cable is verified for a more realistic stiffness in the internal net of 150kN. The stiffness for the diagonal cable remains 5000kN. The 150kN per 0.25m corresponds to a value of 600kN/m valid for a typical PVC coated polyester membrane. From the calculations where the opening angle varies between 60º and 80º can be concluded that: - in the longitudinal direction the tension in the membrane varies from a value of about 0.6kN/m up to 2.2kN/m for larger opening angles - in the direction perpendicular to the folding axis the tension increases from 0.4kN/m to 20.0kN/m for larger opening angles Similar to the membrane tensions in the longitudinal direction, the forces in the diagonal cable increase up to a maximum value and then decrease due to variation of the opening angle. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 2:

47

Changes in form and stresses by changing opening angle θ [º] for case 1 – low stiffness.

θ

Elevation

Front view

1

40

2.50

Tension [kN/m]

Tension [kN/m]

80

2.00 1.50 1.00 0.50 0.00 60

65

70

75

80

25.00 20.00 15.00 10.00 5.00 0.00 60

Opening angle [º]

65

70

75

80

Opening angle[º]

Figure 20: Tension in the longitudinal Figure 21: Tension in the transverse direction due to variation direction due to variation of the opening angle – case 5. of the opening angle – case 5. Table 3:

θ

Changes in form and stresses by changing opening angle θ [º] for case 5 – higher stiffness. Elevation

1

65 80

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Front view

30.00 25.00 20.00 15.00 10.00 5.00 0.00

5.00 4.00 H/S [%]

Force [kN]

48 High Performance Structures and Materials III

3.00 2.00 1.00 0.00

60

65

70

75

80

60

65

Figure 22:

Force in the diagonal cable due to variation of the opening angle – case 5.

80

Front view

Equilibrium form at an opening angle of 70º. Plan

3D-view with loading

Elevation

Front view

Figure 25:

4

75

Figure 23: H/S of the diagonal cable due to variation of the opening angle – case 5.

Plan and Elevation

Figure 24:

70 Opening angle [º]

Opening angle [º]

Structure under snow load (0.4kN/m2).

Tension in the membrane at an opening angle of 70º under external loading

Still for case 5, with a stiffness of 150kN in the internal net and 5000kN in the diagonal cable, external loading will be applied for an opening angle of 70º. Two representative load cases have been chosen [5]: a uniform snow load of 0.4kN/m2 and an upward wind load of 0.6kN/m2. For the equilibrium form with an opening angle of 70º without external loading the average tension in the membrane is 2kN/m in the longitudinal direction and 4kN/m in the transverse WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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direction. The static analysis under external loading combines the displacement of the fixed nodes according to the opening angle of 70º with the application of load vectors for the selected load case on the equilibrium form without external loading. A uniform snow load of 0.4kN/m2 has been applied. In this case the maximum deflection is 2cm. In the transverse (load bearing) direction the average tension increases from 4 up to 5kN/m. Next an upward wind load of 0.6kN/m2 (suction) has been applied. In this case a maximum deflection of 4cm occurs. In the longitudinal direction the average tension increases from 2 up to 3.1kN/m. Plan

3D-view with loading

Elevation

Front view

Figure 26:

5

Structure under upward wind load (0.6kN/m2).

General remarks

The simulations only give a first estimation of the structural behaviour: the model does not take into account the shear stiffness of the coated fabric, the material behaviour is considered to be linear and independent of the stress level and frequently folding and unfolding will also influence the material behaviour. For more precise results a refined model is required. Moreover, an optimised configuration could be found by choosing a different stiffness or selecting another boundary geometry in plan view.

6

Conclusion

For the same external loading on a membrane roof a more expressive double curvature will imply lower tensions in the membrane. Nevertheless more and more applications use a ‘flat appearing’ curvature to cover box-like spaces. The rhombus-shaped membrane elements will be used as cladding components in a mobile foldable shelter consisting of articulated bars as the primary load bearing structure [6]. Based on the simulation of a rhombus shaped membrane it is shown that the unfolding of flat membrane pieces can, within a certain range of opening angles, create a slightly curved tensioned membrane which can resist, in WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

50 High Performance Structures and Materials III an acceptable way, representative external wind and snow loads. Several parameters should still be optimised (size, geometry, stiffness) for a specific application of the V-shaped foldable membrane.

References [1] [2]

[3]

[4]

[5] [6]

F. Otto, B. Baier and S. Meyer-Miethke, IL 14 Adaptable Architecture, Karl Krämer Verlag, Stuttgart (1975). M. Mollaert, N. De Temmerman, T. Van Mele, Ph. Block Philippe, F. Daerden, Adaptable Tensioned Coverings, Proceedings of the IASSAPCS International Symposium on New Perspectives for Shell and Spatial Structures, Taipei, Taiwan pp.204-205, (full paper on cd-rom) (2003). M. Mollaert, T. Van Mele, N. De Temmerman, Kinetic Structures: Architectural Organisms as a Design Concept, Proceedings IASS Symposium Shell and Spatial Structures from Models to Realization, Montpellier, p.410, (full paper on cd-rom) (2004). E. Moncrieff, Extreme Patterning: Lessons From the Cutting Pattern Generation of the Mina Valley Tent City and the Federal Chancellery Projects, TensiNet Symposium: Designing Tensile Architecture, pp.6681, Brussels (2003). B. Forster, M. Mollaert, European Design Guide for Tensile Surface Structures, TensiNet (2004). De Temmerman, N., Mollaert, M., Decorte, W., Parametrization and development of a foldable mobile shelter system, Vrije Universiteit Brussel, Research Working Paper No. 2, (2005).

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Section 2 Composite materials and structures

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A new composite material based on natural fibres and a thermoset: technology, applications and properties G. Wuzella Kompetenzzentrum Holz GmbH c/o FunderMax Industrie GmbH, Klagenfurter Straße 87-89, A-9300 St. Veit/Glan, Austria

Abstract The aim of this work was the development of a new composite material based on natural fibres as reinforcement and an acrylic thermoset as binder to investigate the material properties for various applications. All applications were made by forming and curing the composite material in a hot forming press. In a first step the technology for formation of such a composite material was tested: Based on the nonwoven-process natural fibre mats were made, only composed of natural fibres without binder and with different area weights. After that, different technologies of application of binder and additives like hydrophobizing agents were tested. In the second step the material properties (flexural and impact strength, water uptake and moisture expansion, testing for climatic extremes and sound-absorbing properties) were investigated for different applications as function of the density of the pressed parts, the mixing ratio of binder and additives. The following properties can be given as very special features of the material: In comparison with composites made of usual thermoset binders the fabricator has not to apply the resin immediately before processing. Secondly the material can be stored easily, because the curing is started beyond 100°C, at lower temperatures the binder is thermoplastic. In comparison with composites made of thermoplastic binders, a lesser fraction of binder is necessary and the pressed parts have a very high flexural stiffness. Finally the critical properties like the water uptake and moisture expansion were improved by the addition of hydrophobizing agents. Keywords: natural fibres, nonwoven, composites, acrylic resin, binding technology.

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54 High Performance Structures and Materials III

1

Introduction

Actual industrial demand for natural fibres has developed over the past few years. Today, the use of natural fibres has become common in some applications; the most important customer is the automotive industry. Technically speaking, the use of natural fibres in automotive applications involves primarily pressmolded composites, produced by the pressing of a nonwoven mat with a binder. Typical uses are in door panels, hat racks, and trunk liners. Two production technologies are commercially employed: • a blend of natural and polypropylene fibres is processed into a nonwoven mat and pressed into the desired shape under heat ("thermoplastic matrix"); • nonwoven mats are coated with thermosets, such as epoxy resin or polyurethane and moulded. The ultimate material is generated by polymerization and hardening of the resin ("duromeric matrix"). The main reasons for the use of natural fibres are: • Weight reduction of 10 to 30% • Good mechanical and manufacturing properties • Possibility to manufacture complex structural elements from one material in a single pass • Good performance in accidents (high stability, no splintering) • Superior environmental balance during material and energetic use • Occupational health advantages compared to glass fibres • No emissions of toxic substances • Overall cost advantage compared to conventional construction The aim of this work is the development of a new composite material, based on natural fibres as reinforcement but instead of a thermoplastic binder only a thermoset is used to realise a new range of applications. Instead of the commonly used thermosets like epoxy resins, polyurethanes or aqueous phenolic resins, another thermoset is used, which can be processed easier and which corresponds better to the proposition of an ecological awareness. Due to the machines of the partner company the production of fibre mats is based on the nonwovens technology: dry-laying staple fibres (the fibres are 1 to 20 cm or longer, but not continuous) by an aerodynamic web formation and mechanically bonded by needle-punching. The term nonwoven is defined as a manufactured sheet, web or bat of directionally or randomly oriented fibres, bonded by friction, and/or cohesion and/or adhesion, excluding paper or products which are woven, knitted, tufted stitch bonded incorporating binding yarns or filaments, or felted by wet milling, whether or not additionally needled. The fibres may be of natural or man-made origin. They may be staple or continuous or be formed in situ [1, 6].

2

Materials

For composites composed of natural fibres, blends of natural fibres (e.g. flax and kenaf or flax and hemp) are particularly interesting. The finer flax fibres impart WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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high stability, but because they impede complete soaking with thermosetting binders, fractures may develop. Only the mixture with coarser fibres like kenaf fibres achieves an optimum balance between stability and complete saturation with the binder. In this project all nonwovens are a mixture of kenaf and flax at the ratio of 50:50. 2.1 Kenaf fibres Kenaf is the term for the plant Hibiscus cannabinus L., a warm season annual closely related to cotton (Gossypium hirsutum L.) and okra (Abelmoschus esculentus L.). The kenaf fibres are gained from the bark of the stem of the plants. The staple fibres have a length of 40 – 60 mm and an average thickness of 25 µm. Because of their mechanical properties they are used as reinforcing fibres in a composite [2]. 2.2 Flax fibres Flax fibres are amongst the oldest fibre crops in the world and the use of flax for the production of linen goes back 5000 years. Flax fibres are soft, lustrous and flexible. They are stronger than cotton fibre but less elastic. Because of their mechanical properties they are used as reinforcing fibres in a composite [3]. 2.3 Matrix The used matrix system in this work is a reactive aqueous acrylic resin, which is free of phenol and formaldehyde. This one-component resin has good storage stability and cross-links upon heating to 180 – 220°C. After the impregnation the fibre mats are dried. The binder begins to cure beyond 100°C, and semi-finished products impregnated with the acrylic resin therefore have excellent storage stability in comparison to other thermosets (e.g. epoxy resins or polyurethane). During the hot press step the impregnated mats are formed and the resin is crosslinked. Composite parts bound with the acrylic resin show high stiffness and strength and meet the ecological requirements of the automotive industry. To achieve the best mechanical properties and moisture resistance, press moulding is carried out with tool temperatures of about 200 to 220°C and cycle times of one minute or less [4, 5]. 2.4 Additives In this project two types of additives are used to improve the properties of the composite. The first is a borate, which contributes to avoid the creation of mildew, because of the moisture content in nonwoven composed of natural fibres. The second additive is a paraffin wax, which improves the water resistance (water uptake and moisture expansion, influence of climatic extremes on material properties) of the moulded part. In particular the type of paraffin wax has to be stable against acid, because of the acidity of the used acrylic resin [5]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

56 High Performance Structures and Materials III

3

Technologies of binding

The first task of this work was to find a technology to apply the binder to the natural fibres. In the following second task the found technology was adjusted to find the optimised settings for the application. The application process of the binder can be done in two ways. Either the naturals fibres are impregnated before they are laid to fibre mats or the application is done subsequent to the nonwovens process. All 3 tested technologies in this project follow the second way of impregnation with the following pre-adjustments: • Used binders are liquid, especially aqueous binders • Additives are mixed together with the binder before impregnating 3.1 Foam impregnation technology The binder is mixed to a foam by a mixing machine and applied to the rotating rolls. The nonwoven, composed of natural fibres, is impregnated by pressure between the roller clearance. After that the impregnated nonwoven is dried in an oven and winded to a roll or cut into formatted pieces.

Figure 1:

Scheme of impregnation by foam.

3.1.1 Evaluation of the foam impregnation technology + Energy-saving concerning the drying process of the impregnated nonwoven, because less water is applied to the nonwoven by the foam compared to other impregnation technologies. - Not all binders can be mixed to foam, sometimes a foaming agent is necessary. - Thick materials (area weight>1000 g/m²) require for a complete impregnation high power of the impregnation plant. Experiments demonstrate that the power consumption for thick nonwovens with a width greater than 0.5 m exceeds the engine power; sometimes a thick nonwoven is not continuously impregnated. - Additionally a complete impregnation of thick nonwovens causes a very high compression of the natural fibres in the mat, which can lead to damaged fibres. 3.2 Impregnation technology of glass prepregs The nonwoven is pulled through a basin, which is filled with the liquid binder. After the impregnation the excess of binder and water is squeezed out between WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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two rolls. After that the impregnated nonwoven is dried in a vertical drying tower and winded to a roll or cut into formatted pieces.

Figure 2:

Scheme of impregnation technology of glass pregpregs.

3.2.1 Evaluation of the impregnation technology of glass prepregs + All liquid binder systems can be used + Continuous impregnation for nonwovens in any thickness and width without damaging of fibres or exceeding of engine power limit. - To guarantee a continuous impregnation of the nonwoven the aqueous binder has to be diluted with further water, which cannot be dried by the drying tower. The vertical drying tower is insufficient to dry the nonwoven quickly enough. - During storage of the winded impregnated nonwoven the residual moisture and binder migrate and cause inhomogeneity of the impregnation. 3.3 Foulard dyeing technology The apparatus comprises a foulard in which the nonwoven is provided with the binder in a treatment bath and squeezed off to the binder and moisture content. The impregnated nonwoven is winded to a roll, which can be transported to the dryer.

Figure 3:

Scheme of a foulard dyeing system.

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58 High Performance Structures and Materials III 3.3.1 Evaluation of the foulard dyeing technology + All liquid binder systems can be used + Continuous impregnation for nonwovens in any thickness and width without damaging of fibres or exceeding of engine power limit. + To guarantee a continuous impregnation of the nonwoven the aqueous binder has to be diluted with further water. In fact the horizontal dryer is limited to avoid the curing of the binder but sufficient enough to dry the impregnated nonwoven to a residual moisture of 10%. - Dimensioning of the dryer and the drying process is more expensive than the foam impregnation technology 3.4 Choice of the impregnation technology Due to the evaluation of the three technologies of binding the decision was made in favour of the foulard dyeing technology. This technology is the easiest way to impregnate the nonwoven continuously without risk to exceed the engine power limit and without risk to damage the fibres because of over-compression. The external horizontal dryer allows drying of the nonwoven to the residual moisture of 10%.

4

Applications and properties

The impregnated mats are moulded to composite parts and the following properties complying with DIN standards are tested Table 1:

Measured properties.

Property Flexural strength [N/mm²] Flexural modulus of elasticity (MOE) [N/mm²] Water uptake [%] Increase of thickness of samples due to moisture expansion [%] Impact strength [mJ/mm²]

Standard DIN EN 310 DIN EN 310 DIN 52351 DIN 52351 DIN EN ISO 179

Many types of applications were made with the impregnated material. Besides door panels, hat racks and trunk liners a roofliner as application should be developed. 4.1 Mechanical Properties Because of the processing of fibre mats (laying and needling) the nonwoven as well as the moulded parts have different mechanical properties subject to the direction a sample is taken. Therefore these properties are proved for two directions, one parallel to the direction of the mat production and one transverse to this direction. Samples in direction of production have higher mechanical strength than samples transverse to this direction. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Flexural strength [N/mm²]

High Performance Structures and Materials III

90 80 70 60 50 40 30 20 10 0

25 % resin 25 % resin; transverse 15 % resin 15 % resin; transverse 15 % resin 0.1

0.3

0.5

0.7

0.9

1.1

15 % resin; transverse

Density [g/cm³]

Flexural MOE [N/mm²]

Figure 4:

Flexural strength of moulded composites with 2 different contents of binder as function of density, tested in 2 orthogonal directions.

9000 8000 7000 6000 5000 4000 3000 2000 1000 0

25 % resin 25 % resin; transverse 15 % resin 15 % resin; transverse 15 % resin 0.1

0.3

0.5

0.7

0.9

Density [g/cm³]

Figure 5:

59

1.1

15 % resin; transverse

Flexural MOE of moulded composites with 2 different contents of binder as function of density, tested in 2 orthogonal directions.

Both flexural properties, the flexural strength and the flexural modulus of elasticity, primarily depend on the density of the moulded composite. The higher the density of the moulded part is the higher are the flexural properties. The binder content has only a small effect on the flexural properties; for samples in production direction and with a higher density the properties are higher, if the binder content is 15% instead of 25%. In comparison to other composite materials based on natural fibres the attention should be paid to the high values of flexural properties for samples with a density of 0.9 g/cm³ and above: samples in production direction have a flexural MOE between 7000 and 8000 N/mm and WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

60 High Performance Structures and Materials III a flexural strength between 70 and 85 N/mm², samples transverse to this direction have a flexural MOE about 4500 N/mm² and a flexural strength of about 50 N/mm². The impact strength of all samples is low (10 mJ/mm²) regardless of binder content and of density. 400 25 % resin; Density = 0.6 - 0.9 g/cm

Water uptake [%]

350 300 250

15 % resin; Density = 0.3 - 0.9 g/cm³

200

15 % resin

150 25 % resin

100 50 0 0.1

Figure 6:

0.3

0.5 0.7 Density [g/cm³]

0.9

1.1

Water uptake of moulded composites with 2 different contents of binder but without hydrophobizing agent as function of density.

4.2 Water uptake and moisture expansion The water uptake primarily depends on the density of the moulded composite. The higher the density of the moulded part is the lower is the water uptake. By higher binder content the ability of water uptake can be decreased. This effect is much higher if the density of the moulded composite is low.

Moisture expansion [%]

40

38

35 30 25

25 % resin; Density = 0.6 - 0.9 g/cm³

25

20

15 % resin; Density = 0.3 - 0.9 g/cm³

14 12

15 10 5 0 10

15

20

25

30

Resin content [%]

Figure 7:

Moisture expansion of moulded composites with 2 different contents of binder but without a hydrophobizing agent.

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The increase of thickness of moulded composites due to moisture depends only on the resin content but not on the density of the moulded part. Higher binder content means that the composite is more rigid against the moisture expansion and the fibres are better protected against water. A composite with low density has besides to the hydrophilic behaviour of natural fibres a high porosity. This explains why the water uptake can be high but at the same time the thickness doesn’t increase to the same degree. 4.3 Improvement of water resistance The measured properties like water uptake and moisture expansion are subject of an improvement, mainly for applications with a low density like a roofliner. Hence a hydrophobizing additive can be mixed into the binder. To improve the water resistance of the moulded composites 1.5% of additive is sufficient for moulded parts with a density of 0.9–1 g/cm³. Table 2: Binder [%] 15 15 25 25

Water resistance of composites with density of 0.9–1 g/cm³.

content

Content additive [%] 0 1.5 0 1.5

of

Water [%] 58 22 31 16

uptake

Moisture expansion [%] 30 14 13 6

roofliner - sound absorption

coefficient of absorption

1.20 1.00 0.80 0.60 0.40 0.20 0.00 400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

6300

frequency [Hz] Default material Sample 02; thickness = 5.5 mm Sample 04; thickness = 3 mm

Figure 8:

Sample 01; thickness = 5.5 mm Sample 03; thickness = 3 mm

Sound absorption of roofliners.

4.4 Acoustic properties Nonwovens with an area weight of 750 g/m² and a binder content of 25% are produced by the foulard dyeing technology. The impregnated mats are moulded to roofliners and proved in a reverberant chamber. The measured absorbance of the samples is compared with a default material. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

62 High Performance Structures and Materials III The acoustic properties are an interesting aspect for applications like roofliners. A roofliner is usually a sandwich construction composed of a foamed core layer between two reinforcing laminates as top layers (=default material). The foamed core acts as sound-absorber. The composite material composed of natural fibres and acrylic resin has a good sound-absorbance. If the frequency is higher than 1250 Hz the composite is better than the compared default material.

5

Summary

A new composite material based on natural fibres as reinforcement and an acrylic thermoset as binder was developed with the following results: • The foulard dyeing technology is the best choice to impregnate both thin and thick nonwovens composed of natural fibres continuously without risk to exceed the engine power limit or to damage the fibres. • Advantages for the fabricator: o The fabricator doesn’t need to impregnate the mats immediately before moulding o The impregnated mats have a good storage stability compared to other thermosets o The acrylic resin has a very good environmental impact (no phenol and formaldehyde) • A reduction of weight of moulded parts can be realised (-20%) without losing of mechanical properties. • A low binder content of 15–20% is sufficient for binding. • To improve the water uptake and moisture expansion of the moulded composites a hydrophobizing agent is necessary (1.5% is sufficient). • In comparison to composites with thermoplastic binder the dimensional stability under heat is excellent.

References [1] [2] [3] [4]

[5] [6]

Albrecht W., Fuchs H., Kittelmann W. NonWoven Fabrics Raw Materials, Manufacture, Applications, Characteristics, Testing Processes. Verlag Wiley-Vch. 2003. Clark T. K., R.L. Cunningham, and Wolff I. A. A search for new fiber crops. 1971. TAPPI 54; (1) p. 63-65. Dambroth. Seehuber. Flachs: Züchtung, Anbau und Verarbeitung. 1988. Reck B., Türk J. Thermally curable aqueous acrylic resins – a new class of duroplastic binders for wood and natural fibers. 2nd International Wood and Natural Fiber Composites Symposium. Kassel, Germany. June 28-29, 1999. Oberbach K., Baur E., Brinkmann S., Schmachtenberg E. Saechtling Kunststoff Taschenbuch. 29. Ausgabe. Verlag Carl Hanser. 2004. Bledzki A. K., Gassan J. “Natural Fiber Reinforced Plastics” in Handbook of Engineering Plastics. N.P. Cheremisinoff (editor). M. Deccer Inc. New York. 1997. p. 787-809.

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Experimental test of threaded steel rods glued-in hardwood with epoxy D. Otero, J. Estévez, E. Martín & J. A. Vázquez Department of Construction Technology, Advanced Technical School of Architecture, University of A Coruña, Spain

Abstract This paper summarizes test results of an experimental study on threaded steel rods epoxied into structural hardwood members. Two test campaigns were carried out. In the first one 36 test specimens of Tali sawn timber (Erythropleum ivorense) with threaded steel rods were tested. Tests arose from the need to evaluate the behaviour of this type of joint because of its use in the construction of hollow bars spatial mesh of laminated timber. Three anchorage lengths and two bar diameters were tested. Moreover, two different thicknesses of the glue line, 1mm and 3 mm, were investigated. In order to complement the first tests a new experimental study with 180 test specimens was carried out. In the second tests, double-sided pull-out tests of Spanish chestnut timber saw specimens were performed. Small diameter threaded steel rods (8, 10 y 12 mm) quality 8.8 (yield stress 640 MPa) were used. In the paper the most significant results of both test campaigns are summarized, as well as the comparison with the existing design formulae for of this type of union. Keywords: joint design, glued-in rods, adhesives for wood, hollow bars, spatial mesh, destructive testing.

1

Introduction

Our team has been researching for a long time on space frames. Timber space frames are a particular aspect of this research. Special attention has been paid to the use of hollow sections in the making of this type of structure. Several patents are the outcome of our pioneering work in this field. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06007

64 High Performance Structures and Materials III A Coruña University commissioned the design of the roof structure for a sports hall on the Zapateira Campus. The project became the first built application of the research. The main aim was to span over this hall with a stackable modular space frame made up with hollow timber sections (Figure 1). Achieving an efficient transmission of loads in the bar node is one of the main problems of this type of structures. The use of hollow sections increases even more this problem. The design solution for the bar end connection with the steel node involves gluing solid timber sections of Elondo or Tali (Erythrophleum ivorense). Threaded steel rods (either 24 or 27 mm diameter) are then embedded into the sawn timber and glued with a two component Hilti HIT-RE 500 epoxy resin (epoxidic bisphenol A/F and poliamid alifatic bases, both with inorganic filling) (Figure 2). Given the influence of the efficiency of this connector on the behaviour of the structure, we decided to supplement the theoretical calculations with experimental analysis. This is the main aim of this article.

Figure 1: Stacked space frame modules.

2

Figure 2:

Cut of the end of the bar to show how the Elondo block is glued into the hollow bars of glulam.

Theoretical research of the joint strength

The only rules references for the calculation of this type of unions appears collected as informative annex in the Eurocódigo 5, according to: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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R ax,k = π ⋅ dequ ⋅ Lg ⋅ f vk

(1)

-0,2 f vk = 0,8 ⋅10-3 ⋅ρ1,5 k ⋅d

(2)

For rods glued in grain direction, with: Rax,k dequ Lg fvk ρk d

Characteristic axial load [N] Lowest of: hole diameter and 1,25 times the diameter of the bar [mm] Glued-in anchorage length [mm] Withdrawal strength [N/mm2] Timber average density [kg/m3] Nominal diameter of the threaded rod [mm]

In the results section we will show that experimental values result significantly lower than estimated by Eurocode formulae values. A number of other design equations were used to compare results. Several references that provide equations of design have been consulted, showing so much dispersion in criteria as in results. For the purposes of comparison the formulae provided by pioneering studies of H. Riberholt [1, 2] was used in this case, as following: R ax,k = f ws ⋅ ρ k ⋅ d ⋅ Lg ; to Lg ≥ 200 mm

(3)

R ax,k = f wl ⋅ ρ k ⋅ d ⋅ Lg ; to Lg < 200 mm

(4)

with: Rax,k fws fwl ρk d Lg

3

Characteristic axial load [N] Strength parameter [N/mm1,5] Epoxi type adhesives take the value of 0,520 Strength parameter [N/mm1,5] Epoxi type adhesives take the value of 0,037 Timber average density [kg/m3] Nominal diameter of the threaded rod [mm] Glued-in anchorage length [mm]

Materials characteristics and test devices.

In the first experimental campaign one-sided-push compression tests were carried out over specimens made of sawn timber of Elondo o Tali (Erythrophelum ivorense) with glued threaded steel rods. Rods were glued only in one end of the timber specimen. This solution was directly related to the endelements development for hollow glulam bars. It conditioned, in some ways, the geometric configuration of the specimens and the diameter of the threaded rods utilized. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

66 High Performance Structures and Materials III Sawn timber blocks, with threaded rods glued-in, were glued into the hollow bars subsequently. Cross-section of the blocks was determined from the characteristics of hollow bars. In accordance with this dimensions, cross-section of test specimens were 75x75 and 90x90 mm. For each one of the geometric configurations three anchorage lengths, 200, 250 and 300 mm, were studied. These lengths were inside the compatible levels with the execution of the hollow bars of the mesh. Timber of test specimens achieved an average density of 911 kg/m3 and a characteristic density of 856.36 kg/m3, which corresponds to the 5th percentile. Galvanized threaded steel bars grade 8.8 (with a yield strength of fy=640 N/mm2 and minimum tensile strength of fu=800 N/mm2) have been used. Rod surface was not treated before bonding, because this is not a usual practice in normal construction. Rods with nominal diameters of 24 and 27 mm were used. It address with rod diameters used in the construction of the space frame hollow sections. Nominal rod diameters were dependent on the constructive conditioning of the steel nodes of the mesh too. For that reason they did not agree with values suggested in Eurocode 5. The adhesive was the same as the one used in the construction of the space frame: Hilti HIT-RE 500. Joints glued with two adhesive thicknesses (1 and 3 mm.) have been studied. The combination of these geometric variables gave as a result 12 types of different test specimens, whose characteristics are summarized in the table 1.

Adhesive thickness

Anchorage length

Specimen dimensions

D

e

L

axaxL

[mm] [mm] [mm]

Lb Lm

L

10 mm

d e D a b a

1 2 3 4 5 6 7 8 9 10 11 12

24 24 24 24 24 24 27 27 27 27 27 27

26 30 26 30 26 30 29 33 29 33 29 33

1 3 1 3 1 3 1 3 1 3 1 3

[mm]

200 200 250 250 300 300 200 200 250 250 300 300

[mm]

Edge distance

Hole diameter

d

Embedded rod slenderness

Rod diameter

Geometric characteristics of the specimens tested in the first experimental campaign.

Series

Table 1:

λ L d

a/2 d

75x75x210

8,33

75x75x260

10,42

75x75x310

12,50

90x90x210

7,41

90x90x260

9,26

90x90x310

11,11

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b

1,56 d

1,67 d

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The test sample was longitudinally drilled with an auger drill to the centre of the section. The drill hole went across all the specimen length. The lower end of the sample was then closed with a 10 mm thick piece of elastic material (elastic stopper). The adhesive was blown to fill the drilled cavity with a mixing gun. Afterwards, the threaded bars were inserted manually by continuous pressing and rotating. The particular design of the test sample has been chosen in order to enable the test to be carried out in an INSTRON mod. 8805, 1000kN one-sided pull-compression universal dynamic press (Figure 3). The elastic stopper at the lower end of the test sample allows the threaded bar to glide down when it reaches its load limit. For test reliability purposes, the rod exposed end was fastened to the load cell by means of a nut. Also, the test sample was placed on the base plate with a 3 mm thick neoprene sheet in between, in order to take the imperfections of the support. A picture of the test device is shown in figure 3.

Figure 3:

Test device used in the first experimental campaign.

Figure 4: Test device used in the second experimental campaign.

Like it is shown in results section, experimental results were lower than theoretical values that were calculated with design formulae. In order to complement first tests a new experimental study with 135 test specimens were carried out. In the second tests double-sided pull-out test of Spanish chestnut (Castanea sativa) timber saw specimens were performed. Test specimens were WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

68 High Performance Structures and Materials III made of Spanish chestnut untreated timber. The cross-section dimensions of the different test specimens were six times the width of the bars diameter. Specimens’ lengths were three times the rod anchorage length. Glulam is the timber that has been utilized generally for this kind of joints, and the greatest number of the existing formulations are based on it. Because chestnut timber was a kind of hardwood with an average density very similar to glulam it was chosen for the second experimental campaign. Timber of test specimens achieved an average density of 567.28 kg/m3 and a characteristic density 474.02 kg/m3 (corresponding 5th percentile). Calculating in design formulae a value of 475 kg/m3 was used. Small diameter threaded steel rods (8, 10 and 12 mm) quality 8.8 (yield stress 640 MPa) were used. Five different anchorage lengths were tested, from 60 mm. until 180 mm. three epoxy systems (Hilti, Sika and Locktite) in glueline thickness of 1mm were investigated. Attending to the geometric parameters, there were fifteen different types of test specimens. Geometric characteristics of the specimens tested in the second experimental campaign are summarized in the table 2. As is shown by figure 4, the specimens were tested on double-side-tensile device in the second experimental campaign. To this end, blind drills were carried out in both ends of the specimen and the threaded steel rods were glued inside them. For that, as in the previous case, the adhesive was blown to fill the drilled cavity with a mixing gun and threaded bars were inserted manually by continuous pressing and rotating. Both of the experimental studies were carried out increasing the load with a constant displacement. The failure value was reached in 5±2 minutes.

4

Results

The INSTRON test machine has an electronic device that controls the displacement of the load cell. It also interprets the results with a computer system that gives the digitalized relationship between load, displacement and test time-length. The graphic representation of results showed correlation between stiffness of the three systems adhesives which were used in the second experimental campaign. More repeated failure modes were shear failure in timber, and timber splitting. For specimens with the lowest diameter the main sample failure was tensile failure in steel threaded rods. Using Eurocode 5 and Riberholt formulae the theoretical values for each series were calculated. Both experimental and theoretical values were plotted to assess correlation between them. One representative example of each experimental campaign is shown in the next figures. Solid lines in the first one (Figure 6) represent theoretical values for Elondo specimens, with a characteristic density of 865 kg/m3, a rod diameter of 24 mm and a glueline thickness of 1mm, with anchorage lengths from 150 to 350 mm. Crosses show experimental results, corresponding series 1, 3 and 5.

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Table 2:

69

Geometric characteristics of the specimens tested in the second experimental campaign. Le

La

La

Li

Le

a

d

da

d

da

a

Lb

B1b1 L

Total rod length

Cross-section side

a

Outer rod length

Lb

Timber specimen length

Le

Inner gap length

L

Anchorage length

Li

Adhesive thickness

La

Hole diameter

e

Rod diameter

da

Series

d

[mm]

[mm]

[mm]

[mm]

[mm]

[mm]

[mm]

[mm]

[mm]

1a 1b 1c 1d 1e 2a 2b 2c 2d 2e 3a 3b 3c 3d 3e

8 8 8 8 8 10 10 10 10 10 12 12 12 12 12

10 10 10 10 10 12 12 12 12 12 14 14 14 14 14

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

60 90 120 150 180 60 90 120 150 180 60 90 120 150 180

60 90 120 150 140 60 90 120 150 140 60 90 120 150 140

180 270 360 450 500 180 270 360 450 500 180 270 360 450 500

120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

180 210 240 270 300 180 210 240 270 300 180 210 240 270 300

42 42 42 42 42 60 60 60 60 60 72 72 72 72 72

In view of that theoretical values were referred on characteristic values, experimental results were significantly lower than theoretical ones. These results could be due to the high density of the Elondo timber compared with the glulam density (traditionally used in this kind of joints), and to the small dimensions of the cross-section of the specimens. This cross-section was dependent on the hollow bars dimensions, which lead to an edge distance lower than literature suggested ones.

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70 High Performance Structures and Materials III

Figure 5:

Main failure modes: shear failure in timber, timber splitting, tensile failure in steel threaded rods and shear failure in the timber close interface timber/adhesive.

LOAD [KN]

400

300 EURO

OLT RIBERH

200

100 150

CODE

200

250

300

350

ANCHORAGE LEGTH [MM]

Figure 6:

Experimental results and theoretical values, according to Eurocode and Riberholt formulae, to Elondo specimens with threaded steel rods of diameter 24 mm and a glueline thickness of 1 mm.

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To chestnut specimen’s theoretical values to a characteristic density of 475 kg/m3, a rod diameter of 12 mm and a glueline thickness of 1mm, with anchorage lengths from 40 to 220 mm, are plotted in Figure 7. Crosses show experimental results, corresponding series 1, 3 and 5. It squares with test series 3a, 3b, 3c, 3d, and 3e, which failure loads are represented with crosses. These results are more akin to theoretical values, taking into account that formulae are referred to characteristic values (5th percentile). 60

LOAD [KN]

50 40 30 20 10

T OL ERH RIB DE OCO EUR

0 40

60

90

120

150

180

200

ANCHORAGE LENGTH [MM]

Figure 7:

Experimental results and theoretical values, according to Eurocode and Riberholt formulae, to chestnut specimens with threaded steel rods of diameter 12 mm and a glueline thickness of 1 mm.

Results show that more works needs to be done in joints made with rods glued in high density hardwood timber. This works will allow one to adapt formulae to this material characteristic and will allow one to study the influence of edge distance. Our research team is working about it.

5

Conclusions

Several series of samples have been tested. The test samples were made with threaded steel rods glued into two different kinds of hardwood timber. The rods were glued using three different systems of two-component epoxy-base adhesive. The test samples were made varying the following parameters: glued anchorage length, rod diameter and glue thickness The obtained results have been compared to existing theoretical formulations. The comparison showed a significant difference between the theoretical predictions and the test results when high density hardwood is used. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

72 High Performance Structures and Materials III

Acknowledgment This research is sponsored by the Ministry of Science and Technology through research project titled “UNIONES METÁLICAS ENCOLADAS CON ADHESIVOS EN BARRAS DE MADERA” (Glued Anchored Timber Joints). The financial support is gratefully acknowledged.

References [1] [2] [3] [4]

[5]

[6]

Riberholt H., Glue Bolts in Glulam, Department of Structural Engineering. Technical University of Denmark. Serie R, No 210, 1986. Riberholt H., Glued bolts in glulam-Proposal for CIB Code. CIB-W18 Meeting. Parksville, Vancouver Island, Canada. Paper 21-7-2, 1988. Broughton J.G., Hutchinson A.R, Pull-out behaviour of steel rods bonded into timber, Materials and Structures. ASCE. AUG. 2001. VOL. 127 No. 8, 2001. Estévez Cimadevila, F.J.; Vázquez Rodríguez, J.A. & Otero Chans, M.D, Diseño y dimensionado del nudo extremo de una barra hueca de madera laminada, CIMAD04. 1º Congreso Ibérico A Madeira na Construçao. Universidade do Minho. Guimaraes. Portugal. Pag 689-698, 2004. Estévez Cimadevila, J. & Vázquez Rodríguez, J.A, Spatial truss of hollow bars made of laminated timber supported by walls of reinforced masonry, Journal of the International Association for Shell and Spatial Structures. Vol. 45 (2004) n.1. April n.144. IASS. ISSN: 0304-3622, 2004. Estévez Cimadevila, J.& Vázquez Rodríguez, J.A, Edificio sportivo a La Coruña, Spagna, Construire in Laterizio, n.102. Año XVII. NoviembreDiciembre 2004.Grupo Editoriales Faenza Edotrice S.p.A. Faenza (Ra).

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Predicting the mechanical behaviour of large composite rocket motor cases N. Couroneau DGA/CAEPE, St Médard en Jalles, France

Abstract A method to develop finite element models of the rocket motor cases of a strategic missile is presented. Based on the use of multivariate analyses, this approach is made necessary given the impossibility to control all input data: difficult characterization of filament wound materials, influence of manufacturing and qualification processes, unknown fabrication parameters, etc. An initial reference model is built up using preliminary data and theoretical winding laws. Having compared the models predictions and available structural test results, a sensitivity analysis is carried out to discuss the individual influence of the input data on the accuracy of the predictions. The multivariate analysis finally enables a global assessment of the parametric analysis results. Keywords: rocket motor case, composite filament winding, finite element model, variable scattering, multivariate analysis, test prediction.

1

Introduction

In addition to providing ground test facilities for the development of the future generation of French strategic missiles, the Centre d’Achèvement et d’Essais des Propulseurs et Engins (CAEPE) is responsible for evaluating the degree of performance, durability and security of the solid propellant rocket motors constituting the missiles. The mechanical expertise work is carried out using finite element models able to predict the stresses and strains of the mechanical parts throughout the fabrication and service life. Many difficulties arise when attempting to describe the mechanical behaviour of the rocket motor cases. The filament-wound composite structures have complex geometry and properties especially in the dome area. As the wound layers are added on a cylindrical mandrel, the curvilinear path leads to a WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06008

74 High Performance Structures and Materials III continuous change in the winding angle and thickness. In addition, the fiber angle varies in the thickness direction because the fiber path depends on the surface on which fibers are wound. Other concerns are due to the difficult characterization of the different materials and the variability of their properties. The method chosen here is to take advantage of various complex tests performed on the full scale specimen in order to identify uncertain material properties or geometrical parameters. This approach, known as the experimental / numerical dialog or inverse identification [1], is presented here for the two largest composite structures of a strategic missile.

2

Development of the initial model

2.1 Architecture The solid propellant rocket motors corresponding to the two first propulsion stages of the French strategic missiles share a common outer diameter (> 2 m). Their structure is based on the same architecture and the same materials: - The carbon-epoxy filament-wound pressure vessel, acting both as the propellant tank and the combustion chamber, - The two metal polar mountings located at the aft and front openings to provide connection with the igniter and the nozzle, - The two cylindrical skirts, made of both carbon-epoxy tissues and circumferential windings, ensuring the connection with the rest of the carrier, - Rubber connections between the vessel and the skirts on the one hand, and between the vessel and the polar mountings on the other hand. 2.2 Winding law The composite envelope is constituted of a succession of circumferential windings on the cylindrical part and satellite windings running between the two openings. The laminate lay-up in the cylindrical area is rather simple to describe with an assumed constant value of winding angle and direction of all plies. Conversely, the complex dome geometry involves a rapid change in angle and thickness along a meridian. Different netting theories [2] based on geometrical or mechanical approaches describe this distribution. The planar theory used here assumes that the fiber patterns lie in a plane which is tangent to the polar opening at one end and tangent to the opposite side of the polar opening at the other end. The winding angle calculated (Fig. 3a) with the planar theory at a given location is applied as a constant value to all plies. The thickness distribution (Fig. 3a) shows good agreement with the measured values (Fig. 3b). A preliminary analysis was performed to investigate the effect of an evolution of the winding angle in the thickness direction, as suggested by Park et al [3]. The thickness of the first ply was calculated from the mandrel shape whereas the subsequent plies were calculated with the updated shape. The difference in angle between plies was particularly noticeable near the polar bosses where the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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thickness is more important. However this induced no significant change in the overall calculated displacements.

Figure 1:

Typical components of a rocket motor case (half-view in section).

Figure 2:

Winding path.

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76 High Performance Structures and Materials III

a) Winding angle Figure 3:

b) Thickness

Planar winding law applied to the first stage.

2.3 Finite element models The finite element analyses considering the geometrical non-linearities are performed using ANSYS code. The initial axi-symmetric models are developed over a small angle in order to use the composite dedicated 8-node finite elements, allowing for a direct input of the materials lay-up and fibers orientation. The data are transmitted to Ansys by means of a Fortran routine incorporating geometric parameters, materials properties and winding laws. A larger density of elements is used near the openings where the winding angle and thickness vary abruptly.

a) First stage Figure 4:

b) Second stage 3D mesh using solid elements.

The model contains general boundary conditions for axisymmetry and full displacement constraints on the frames to take account of the inter-stage skirts which are not included in the models. 2.4 Evaluation of the models Two load cases are selected to evaluate the finite element model performance : internal pressure (Fig. 5a) and axial compression (Fig. 5b). The internal pressure load case is corresponding to the pressure proof test carried out to accept the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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structure prior to its filling up with propellant. It is conducted with an internal pressure 15% higher than the operating pressure and an axial load applied on the aft polar mounting to represent the thrust loads. The axial compression load case is applied on the rear frame with the front frame being blocked. It is representative of an aggression of the missile when stored in the submarine and it also represents the load transmitted to the second stage during the first stage flight. These two tests are carried out and analysed by the contractor in charge of the rocket motor cases design and fabrication.

a) Internal pressure Figure 5:

b) Axial compression

Load cases for evaluating the models.

a) First stage

b) Second stage

c) First stage

d) Second stage

Figure 6:

Model evaluation for the internal pressure load case.

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78 High Performance Structures and Materials III The results obtained for the internal pressure load case are shown in Figure 6. The computed and measured displacements are plotted all along a meridian of the composite vessel. This comparison shows a good agreement except in the regions near the polar openings at the rear and front. The average difference is about 1.5 mm for the 1st stage and 1.1 mm for the 2nd stage, with maximum values of 5 to 6 mm. For the axial compression load case, only the axial displacements of the skirts and the cylindrical part of the structure are used, the other values being too small for comparison. No experimental results are available here so the contractor's calculations are used instead. The calculations are all in good agreement with an average difference of about 0.2 - 0.3 mm for the two models.

a) First stage Figure 7:

3

b) Second stage

Model evaluation for the axial compression load case.

Optimization of the initial model

3.1 Parameters influence analysis An analysis is conducted to identify the influence of each parameter considered individually on the response of the model and the difference between calculations and experiments. Only the internal pressure proof test is considered here as there are no experimental data for this load case and the difference with the contractor calculations is very satisfactory. The main reasons for discussing the accuracy of the input data of the model are listed below: 1) Mechanical properties of composite materials : lack of representativity of the characterization tests performed on uni-directionnal laminate plates with respect to the actual fabrication process of the structures, variability of materials properties and possible damage of the matrix and fibers after the pressure proof test [4], 2) Winding law : Unknown fabrication parameters, theoretical law assumed without possibility of a direct validation, expected slippage of the fiber during the winding process, 3) Initial geometry : Possible evolution of the initial geometry during subsequent fabrication stages [5] or after the pressure proof test. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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The different parameters above are modified one by one in the reference model (Tab. 1). For each of them, a possible variation is assumed and each case results in a new computation being performed. The matrix degradation is modelled by a decrease of the transverse modulus of the hoop layers, the winding laws are transformed by an offset or a multiplying coefficient, and the initial geometry is modified by means of a prior pressurization of the model. Table 1:

Description of the sensitivity analysis.

The results of the sensitivity analysis are summarized in Table 2. The modifications leading to an improvement of the results are indicated in bold characters whereas those leading to a deterioration are printed in italics. The major factors of influence given the chosen variations of the different parameters are the structure moduli, the changes applied to the winding laws and the prior pressure deformation. The calculations related to a change in the materials properties tend to indicate an overvaluation of the reference values or a possible matrix degradation. Concerning the winding laws, the results show a significant latitude for improving the initial theoretical laws. The 10 bar prior pressure deformation of the 2nd stage model also results in an important decrease of the difference between calculations and measurements. 3.2 Multivariate analysis The multivariate analysis performed using Matlab allows for a global assessment of the results of the parameters influence analysis. The data processing consists of a linear system to be solved using the least square method as per eqn (1). The output data is the best combination of modified parameters to minimize the difference between experiments and computations : [A].{x} = {b} where

(1)

[A] is a matrix containing the results of parameters influence analysis, {x} is the requested solution, {b} contains for each of the measurement points the difference between experimental and numerical values of displacements.

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80 High Performance Structures and Materials III Table 2:

Results of the parameters influence analysis.

The results of the multivariate analysis (Tab. 3) show, on the opposite of the analysis performed on individual parameters, that no significant modification has to be applied to the reference materials properties. The 1st stage model can be optimized by a slight modification of the winding law which was initially approached by a theoretical law. For the 2nd stage model, the best solution is essentially obtained through the use of a prior deformation of the model with a 18 bar internal pressure. Table 3:

Results of the multivariate analysis.

The verification carried out with this new set of input data confirms a significant improvement of the predictions for the internal pressure load case (Tab. 4), for the 2nd stage model (Fig. 8b, d). For the 1st stage model (Fig. 8a, c), the axial behaviour of the rear dome is much improved at the cost of a slight degradation of radial displacements prediction. The initial calculations for the compression load case (Fig. 7) are not affected by the modifications.

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High Performance Structures and Materials III

Table 4:

Figure 8:

4

81

Comparative predictions for the internal pressure load case.

a) First stage

b) Second stage

c) First stage

d) Second stage

Evaluation of the models for the internal pressure load case.

Conclusion

This study shows the potential interest of multivariate analyses for the development of mechanical models for structures with uncertainties concerning the input data (difficult characterization of filament wound materials, influence of the fabrication and qualification processes, unknown winding parameters, etc.). The method presented here for two structures and two load cases results in a considerable improvement of the predictions with a coherent modification of the input data. This procedure can be broadened to a multiple load case analysis with non-linear fits for each parameter.

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Micromechanical modeling of random or imperfect composites ˇ M. Sejnoha & J. Zeman Czech Technical University in Prague, Faculty of Civil Engineering, Department of Structural Mechanics, Prague, Czech Republic

Abstract A class of heterogeneous material systems often regarded as random or imperfect composites is addressed in this paper. The literature now offers a number of contributions that open the way to the analysis of large material systems with complex microstructures while taking advantage of popular micromechanics based approaches building on periodicity and first order homogenization techniques. Until recently the attention has been mostly limited to rather classical material systems such as unidirectional fibrous composites and textiles with emphasis on various types of imperfections developed during fabrication process on both microscale (level of fiber bundles) and meso-scale (level of textile geometry). From the basic mechanics point of view, however, it appears logical to exploit the essential principles of the proposed procedures in bridging the gap between mechanical and civil engineering applications. In this regard, historical masonry structures classified as systems composed of more than one material component serve as a typical example of civil engineering applications, which may benefit from standard first order homogenization schemes extended to account for possibly irregular arrangement of individual stone blocks. In this contribution, both groups of material systems will be treated on the same footing demonstrating the applicability of basic homogenization techniques as well as similarities between various heterogeneous material systems when referred to as random or imperfect. Keywords: random and imperfect composites, textiles, masonry, periodic unit cell, two-point probability function.

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84 High Performance Structures and Materials III

1 Introduction Techniques of numerical first-order homogenization have acquired a considerable attention particularly in applications where detailed numerical analysis of complex heterogeneous material systems proved to be prohibitively expensive. Natural assumption of the existence of periodic microstructure, e.g., periodic distribution of fibers in the metal matrix or ceramic matrix composites [1], often serve as the point of departure. The resulting homogenized or effective material parameters are then used in large scale structural analysis. Unfortunately, in the vast majority of real material systems the assumption of periodicity generally fails owing to the presence of various types of geometrical imperfections usually developed during fabrication. Giving up the benefit of periodic fields in such cases seems, therefore, reasonable but definitely not very practical. Instead, the recently introduced concept of statistically equivalent periodic unit cell (SEPUC) [2, 3, 4], appears as a suitable method of attack. The leading idea of this approach is to replace a complex non-periodic microstructure by a certain periodic unit cell (PUC), which still optimally resembles the original microstructure in a proper sense. Here, this objective is formalized as a difference between appropriate statistical descriptors related to the original media and the periodic unit cell, respectively. If the original microstructure and the periodic unit cell is described by an identical set of parameters, this problem coincides with reconstruction of random materials. If the periodic unit cell is described by a substantially smaller number of parameters in order to reduce the problem complexity, it leads to a problem of the best approximation within the selected statistical descriptors. It will be shown in subsequent paragraphs that the proposed technique is applicable not only to more or less classical types of composites represented here by textiles, but also to more conventional class of material systems such as historical masonry structures. The principle idea is evident from Figs. 1-3 showing images of real material or structural systems with corresponding simplified representative volume elements (RVE) presented in terms of certain statistically equivalent periodic unit cells. When referring to textiles, Figs. 1 and 2, the crucial sources of imperfections are attributed to generally random arrangement of fibers within the bundle crosssection (microlevel) and the waviness, misalignment and/or non-uniform crosssectional aspect ratio of individual bundles in the longitudinal direction (mesolevel). Qualitatively similar types of imperfections associated with irregular arrangement of stone blocks, both in terms of block sizes and their location, together with a variable thickness of the mortar phase are often encountered when dealing with historical masonry structures. A typical example is plotted in Fig. 3(a) showing a parapet wall of the Charles Bridge in Prague. The stepping stone in the analysis of all systems is the possibility to replace the original color images, Figs. 1(a)-3(a), by their binary counterparts, Figs. 1(b)3(b). The latter representatives of the true micro or meso-structures are further exploited in the next section when deriving various statistical descriptors. MatchWIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 1: a) Micrograph of random fibrous composite, b) Binary image, c) 5-fiber and 10-fiber SEPUC.

Figure 2: a) Plane view of textile composite, b) Binary image of transverse section, c) Mesoscopic SEPUC.

ing the material statistics of the real microstructure with those corresponding to simplified periodic unit cell in the framework of a certain optimization problem then allows for deriving the desired geometrical parameters needed in the construction of individual SEPUCs. Knowing the periodic unit cell then opens the way for the derivation of effective elastic properties using the well known elements of first order homogenization procedure briefly outlined in Section 3. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

86 High Performance Structures and Materials III

Figure 3: a) Parapet wall, b) Binary image, c) Mesoscopic SEPUC.

2 Definition of a periodic unit cell The crucial step in the definition of optimal periodic unit cell relies on the choice of proper description of heterogeneous materials with random or imperfect structure. In the present work, we restrict our attention to two specific descriptors: one- and two-point probability functions Sr and Srs ; see, e.g., [5] for more details. To that end, consider a binary heterogenous material formed by phases r and s and denote the characteristic function of the domain occupied by the r-th phase χr . (When referring to textiles the symbols r and s may represent the fiber (fiber tow) and matrix phases, while for masonry the two symbols are essentially reserved for brick (stone) and mortar phases.) Then, the one-point probability function gives the probability that a point x will be found in a given phase r and the two-point probability function Srs stands for the probability that the points x and y will be located simultaneously in phases r and s, respectively: Sr (x) = P (χr (x) = 1),Srs (x, y) = P (χr (x)χs (y) = 1).

(1)

For the case of statistically homogeneous and ergodic media, information contained in the one-point probability function reduces to the volume fraction of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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a given phase (Sr = cr ). In addition, the two-point probability function then depends on (x − y) only and can be obtained from the relation
 s , r · χ (2) |Ω|Srs = F −1 χ where |Ω| is the area of the analyzed domain,· and F −1 (·) stand for the direct and inverse Fourier transform and · denotes the complex conjugate. For discretized microstructures, Eq. (2) can be rapidly evaluated by the fast Fourier transform even for high-resolution bitmaps. In the present study, the bitmaps of Figs. 1(b)3(b) were employed. Once the original structure has been characterized by an appropriate statistical descriptor, we can proceed with the definition of the idealized unit cell. A particular parameterization considered in this work appears in Figs. 1(c)-3(c). In case of masonry structures (for textile composites we refer the interested reader to [2, 4], Fig. 3(c), the unit cell is fully determined once the width of the unit cell, heights of each layer of bricks and thicknesses of individual joints are specified. In particular, the geometry of the chosen unit cell is determined by twelve parameters. The statistically optimal values of these parameters then follow from minimization of the least square error E=



2 0 (i, j) − Srs (i, j) , Srs

i

(3)

j

0 where Srs is the two-point probability function related to the original microstructure while Srs stands for the two-point probability function of the idealized unit cell. It can be shown that the objective function E is non-convex, multi-modal and discontinuous due to the effect of limited bitmap resolution. Based on our previous works, a stochastic global optimization algorithm based on combination of real-valued genetic algorithms and the simulated annealing method, see [6], is employed to solve this optimization problem. This approach was successful in delivering the desired periodic unit cells for all material systems considered herein.

3 First order homogenization Consider a heterogenous periodic unit cell Y subjected to a uniform macroscopic (mesoscopic) strain E. In view of the periodicity of the unit cell, the strain and displacement fields in the PUC admit the following decomposition   ε(x) = E + ε∗ u∗ (x) . (4) u(x) = E · x + u∗ (x), The first term on the right hand side of Eq. (4) corresponds to a displacement field in an effective homogeneous medium which has the same overall response as the composite aggregate. The fluctuating Y -periodic displacement u∗ and corresponding strain ε∗ enter Eqs. (4) as a consequence of the presence of heterogeneities; see, e.g., [7] and references therein. Note that the periodicity of u∗ further implies WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 4: (a) Localized equivalent strain, (b) Mesoscopic response for 5-fiber and 10-fiber PUCs. that the average of ε∗ in the unit cell vanishes. The local stress fields σ in the PUC are constrained by constitutive equations and equilibrium conditions σ(x) = L(x) : ε(x),

div σ(x) = 0,

(5)

respectively. Note that symbol L stands for the (position-dependent) fourth-order symmetric stiffness tensor. Combining Eqs. (5) and (4) allows us to determine the distribution of fluctuating displacement u∗ within the unit cell as a function of E and, subsequently, to evaluate the average stress in the PUC as  1 σ(u∗ (E)) dY, in particular, Σ = Lhom : E, (6) Σ =
σ = |Y | Y where Lhom is the homogenized stiffness tensor characterizing the equivalent elastic homogeneous medium. 3.1 Mesoscopic response of polymer matrix based fibrous composites Application of Eq. (6) to the derivation of mesoscopic response of graphite fiber tow impregnated by the polymer matrix is shown in Fig. 4. In this particular example an influence of the number of fibers within the PUC on the mesoscopic response was explored. While the elastic behavior is essentially independent of the number of fibers assumed for the PUC (note that even the model with hexagonal arrangement of fibers gives the same elastic response) the nonlinear behavior suggests possible dependency. This may be attributed to significant non-homogeneous distribution of local fields manifested, e.g., by highly localized zones of equivalent local strain, Fig. 4(a), when loading this system beyond the elastic limit. It should be mentioned that in this particular example the response of the matrix phase was assumed to be well described by the generalized nonlinear viscoelastic Leonov model [8]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 5: (a) Experimental setup, (b) Simulations for various material parameters of interface elements, (c) Experimental vs. numerical response, (d) Mesoscopic response.

3.2 Evaluation of effective fracture energy of masonry The second example demonstrates the use of homogenization theory to the derivation of macroscopic fracture energy as a material property needed in the large scale analysis of historical structures such as bridges, where detailed modeling of individual phases (stones and mortar) is essentially impossible. This property is found as the area under the macroscopic stress-strain curve, Fig. 5(d), multiplied by the PUC area and divided by the total crack length as suggested in [9]. In analogy with the smeared crack model assumed for individual phases, see [10] for more details, it can be shown that for the unit cell, Fig. 3(c), loaded by macroscopically uniform stress or strain in the direction of one of the coordinate axis, say width h, (all cracks are then assumed to be perpendicular to the loading direction with the length approaching the other unit cell dimension h1 + h2 + t1 + t2 ) the macroscopic fracture energy receives the value equal to the area under the macroscopic stress-strain curve multiplied by the crack width h. In such a particular case the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

90 High Performance Structures and Materials III two definitions are not only identical, but yet confirm applicability of the homogenization theory even for quasi-brittle materials. To arrive at reliable macroscopic response, however, may prove to be rather complicated. General assumption of perfect and strong bond between individual phases is usually not acceptable due to weakening effect of air bubbles in the stone phase in the vicinity of stone-mortar interface. In our particular case, this zone was represented by interface elements with the behavior governed by the Mohr-Coulomb failure criterion. The model parameters such as cohesion and angel of internal friction were derived by matching numerical simulations, Figs. 5(b)(c), with the results derived experimentally, Fig. 5(a). Although more sophisticated techniques based on inverse approach are available in the literature, a simple trial and error method was exercised here to fit individual parameters of the assumed constitutive model. In particular, a series of possible solutions displayed in Fig. 5(b) was derived based on randomly generated values of the cohesion and angel of internal friction. Material parameters corresponding to the “best” solution, see Fig. 5(c), were then applied in the unit cell analysis to derive the required macroscopic response to a sufficient degree of accuracy.

4 Conclusion A rather general approach to the analysis of heterogeneous materials with either random or imperfect microstructures was reviewed. The basic scheme assumes formulation of a certain periodic unit cell statistically equivalent (up to two-point probability function) to real material systems. It is expected that the periodic unit cell being statistically similar (from the geometrical point of view) to real systems will also provide similar (at best the same) mechanical response. The robustness of this approach has been demonstrated through applications to rather different material systems varying from plane weave textiles over to natural stone masonry. A special attention was further devoted to the derivation of effective fracture energy of masonry systems to support applicability of homogenization techniques also to quasi-brittle materials.

Acknowledgment ˇ Grant No. 103/04/1321 and CEZ MSM The financial support provided by GACR 6840770003 is gratefully acknowledged.

References [1] Dvorak, G.J., Bahei-El-Din, Y.A. & Wafa, A., Implementation of the transformation field analysis for inelastic composite-materials. Computational Mechanics, 14(3), pp. 201–228, 1994. ˇ [2] Zeman, J. & Sejnoha, M., Numerical evaluation of effective properties of graphite fiber tow impregnated by polymer matrix. Journal of the Mechanics and Physics of Solids, 49(1), pp. 69–90, 2001. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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ˇ [3] Sejnoha, M. & Zeman, J., Overall viscoelastic response of random fibrous composites with statistically quasi uniform distribution of reinforcements. Computer Methods in Applied Mechanics and Engineering, 191(44), pp. 5027–5044, 2002. ˇ [4] Zeman, J. & Sejnoha, M., Homogenization of balanced plain weave composites with imperfect microstructure: Part i – theoretical formulation. International Journal for Solids and Structures, 41(22-23), pp. 6549–6571, 2004. [5] Torquato, S., Random heterogeneous materials: Microstructure and macroscopic properties. Springer-Verlag, 2002. ˇ [6] Matouˇs, K., Lepˇs, M., Zeman, J. & Sejnoha, M., Applying genetic algorithms to selected topics commonly encountered in engineering practice. Computer Methods in Applied Mechanics and Engineering, 190(13–14), pp. 1629– 1650, 2000. [7] Michel, J.C., Moulinec, H. & Suquet, P., Effective properties of composite materials with periodic microstructure: A computational approach. Computer Methods in Applied Mechanics and Engineering, 172, pp. 109–143, 1999. [8] Leonov, A.I., Non-equilibrium thermodynamics and rheology of viscoelastic polymer media. Rheol Acta, 15, pp. 85–98, 1976. [9] RILEM TC50 FMC, Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Materials and Structures, 18, pp. 287–290, 1985. Endorsed May 1993. ˇ ˇ [10] Cervenka, V., Jendele, L. & Cervenka, J., ATENA Program Documentation – ˇ Part I : Theory. Cervenka Consulting Company, 2002.

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Flexural behaviour of ferrocement roof panels A. S. Alnuaimi, A. Hago & K. S. Al-Jabri Department of Civil and Architectural Engineering, Sultan Qaboos University, Sultanate of Oman

Abstract This paper presents the experimental results of nine roof panels made of Ferrocement. Two types of channel sections and one type of box section were tested. All panels were 2m long, 470mm wide and 20mm thick. Channel type A had side edge beams 95mm deep and channel type B had side edge beams 50mm deep. The depth of the box section was 95mm. Thin hexagonal wire mesh was used as reinforcement. The number of wire mesh layers was varied between two to six. The wires were impregnated midway through the thickness of the panels. The panels were tested for bending moment with simple supports. The main variables studied were the number of wire mesh layers, the cross sectional shape of the panel and the depth of edge beam. Tests revealed that all panels showed acceptable strength for roofing systems. The increase in the number of wire mesh layers leads to an increase in the flexural strength. The box section showed strength similar to that of the channel section with 95mm edge beam. The channels with 50mm deep edge beams showed strength much less than the ones with 95mm edge beam and box section. Keywords: Ferrocement, fibre reinforcement, slab panels, bending, box section panels, channel panels.

1

Introduction

Ferrocement is a type of thin wall reinforced concrete commonly constructed of hydraulic cement mortar reinforced with completely infiltrated, closely spaced layers of continuous and relatively small size wire mesh. In its role as a thin reinforced concrete product and as laminated cement–based composite, Ferrocement has found itself in several applications both in new structures and repair and rehabilitation of existing structures. This includes all building members like walls, roofs, columns, beams etc. Compared with the conventional WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06010

94 High Performance Structures and Materials III reinforced concrete, Ferrocement is reinforced in orthogonal directions; therefore, it has homogenous properties in two directions. Ferrocement’s low cost, durability and serviceability were recognized by engineers and builders throughout the world. It is frequently used for the construction of housing and buildings, landscape structures, agricultural facilities, public health facilities and transportation. In developing countries, the most critical components in dwelling construction are appropriate roofing, walls and floors. The use of local materials has not been very successful in producing durable and resistant to fire, insects, and flood or earthquake roofing materials. As a result, many developing countries import expensive galvanised iron sheets or use hazardous asbestos cement sheets as roofing material. Ferrocement appears to be an economical alternative material for roofing [1, 2]. The use of Ferrocement as roofing and slab elements has been a subject of investigation by many researchers. For roofing, Ferrocement has been used for channel type section, folded plates, ribbed slabs, cylindrical shells, circular domes, funicular shells etc [3]. The use of hollow box section as a roofing element has been investigated by Mathews et al. [4]. A total of 21 Ferrocement box sections have been tested under symmetrical line loads applied at one third span points. The test results confirm that the Ferrocement box hollow sections have adequate strength, stiffness and other serviceability requirements for residual applications. Also the theoretical values of cracking load, ultimate load, deflection and crack width at working load showed good agreement with experimental values. Kenai and Brooks [5] carried out extensive testing on direct tensile, four point flexural and drop impact tests on specimens reinforced with steel wire meshes (13 and 25mm thick) with varied amounts. They used a simple model based on plastic analysis which was originally proposed by Mansour and Paramasivam [6]. The model employed a rectangular stress block in the compression zone and the neutral axis depth was calculated by considering the equilibrium of tension and compression forces. The ultimate moment was calculated by multiplying any one of the two forces by the lever arm. Such models cannot be used in cases where the reinforcing mesh is dispersed in the middle of the slab. This is because of the small thickness of the Ferrocement slab panels (about 20mm) which makes it practically difficult to control the uniform dispersion of the wire mesh through the depth. Ahmed et al. [7] studied the shear behaviour of Ferrocement channel beams. Their results indicated that cracking load and ultimate shear strength increase with the increase in the volume of wire mesh and mortar strength and decrease with the increase of shear span/depth ratio. Al-Kubaisy and Jummat [8] investigated the use of Ferrocement in improving the behaviour of reinforced concrete slabs. The tension zone of each slab was covered with Ferrocement layer. They studied the effects of the percentage of wire mesh reinforcement in the Ferrocement cover layer, thickness of Ferrocement layer and the type of connection between the Ferrocement layer and the reinforced concrete slab on the ultimate flexural load, first crack load, crack width and spacing, and load-deflection relationship. They concluded that the use of Ferrocement cover slightly increases the ultimate WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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flexural load and increases the first crack load. Considerable reduction in crack widths and spacing was observed with specimen with Ferrocement layers. The performance of Ferrocement panels under normal, moderate and hostile environments was investigated by Masood et al. [9]. They concluded that the flexural capacity of the panel increases with the addition of fly ash. Considerable deterioration of wire meshes fabric was observed due to sustained exposure in saline casting and curing condition. Recently, Hago et al. [10] conducted 6 experimental tests to study the ultimate and service behaviour of Ferrocement roof slab panels. The parameters studied include: the effect the effect of the percentage of wire mesh reinforcement by volume and the structural shape of the panels on the ultimate flexural strength, first crack load, crack spacing and loaddeformation behaviour. The results demonstrated that the monolithic shallow edge Ferrocement beams with the panels considerably improves the service and ultimate behaviour, irrespective of the steel layers used. Also, slabs with channel sections supported larger ultimate loads and behaved better under service loads than their flat slabs counterparts. Due to large deflections experienced by the thin panels, large deflection theory was adopted in the analysis. Good agreement was obtained between the theoretical and experimental ultimate loads using the proposed mathematical model. In this research, nine simply supported slab panels were tested for flexure. The specimens were arranged in three categories based on the cross-section: channel section type A, channel section type B and box section. The aim was to study the effects of the shape of cross-section and the number of wire mesh layers on the behaviour and ultimate capacity of the tested panels. The panels were constructed manually in a simple manner, so similar panels can be constructed and used as roofing system with almost no equipment needed.

2

Test program

The experimental investigation consisted of fabricating and testing, for flexure, nine Ferrocement roof panels. All panels were 20mm thick and were reinforced with thin steel wire meshes sandwiched midway through the thickness. The panels were divided into three groups according to their shape and number of wire mesh layers (Table 1). The first group, Channel A, consisted of three channel-shaped panels. The dimensions were 470mm outer widths and 2100mm length with two edge beams 95mm deep (Figure 1). The second group, Channel B, consisted of three channel-shaped panels similar to the first group except that the edge beam was 50mm deep (Figure 2). The third group consisted of three box section panels with 470x 95mm outer cross section and 2100mm total length. The hollow core was 430x55mm as shown in Figure 3. In all tested panels, the test span was the middle 600mmm of the span.

3

Material used

Ordinary Portland cement and natural sand were used in making the Ferrocement concrete in the ratio of 1:2 respectively with a water to cement ratio of 0.55. The WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

96 High Performance Structures and Materials III average mortar cube compressive strength was 47.1N/mm2 and the average prism flexural strength was 5.23 N/mm2 (Table 1). For reinforcement, a hexagonal wire mesh with closely spaced wires was used in the tested panels. The wire mesh had a diameter of 0.579mm and a spacing of 12.13mm in both directions. The number of wire mesh layers varied from two layers to six layers. The wire mesh was stretched on a frame of 6mm steel bars having yield strength of 250N/mm2 (Figures 1-3). The cement, sand and water were mixed using a power driven drum mixer for about five minutes. The mortar was designed to give 28day strength of about 40N/mm2. Wooden moulds were used to cast the slabs. A layer of mortar of about 10mm thick was first placed in the mould followed by the reinforcement cage and then a second layer of mortar was placed to make the required thickness. Due to the small thickness of the panel, the wire mesh was placed almost at mid thickness. With each panel, six 100mm cubes and two prisms were cast to determine the mortar compressive strength and modulus of rupture. After one day of casting, the panels and cubes were removed from the moulds and were kept under wet Hessian cloth until the day of testing which was about 28 days from the date of casting. Table 1:

Models tested and their material properties.

Model No.

Dimensions (mm)

Depth of edge beam (mm)

No. of steel layers

% Volume of steel

Ch2-A Ch4-A Ch6-A Ch2-B Ch4-B Ch6-B Box1 Box2 Box3

2100x470x20 2100x470x20 2100x470x20 2100x470x20 2100x470x20 2100x470x20 2100x470x20 2100x470x20 2100x470x20

95 95 95 50 50 50 95 95 95

2 4 6 2 4 6 2 4 6

1.36 1.60 1.76 1.36 1.57 1.77 1.24 1.43 1.62

4

Compressive Flexural strength strength (N/mm2) (N/mm2)

54.9 46.0 47.5 42.6 40.6 42.0 44.1 31.0 54.6

5.0 4.8 4.6 6.6 6.8 6.6 5.3 5.5 7.2

Test procedure

All slabs were tested for flexure. They were simply supported with a clear span of 2000mm and test span of 600mm in mid-span. The load was applied as two symmetrically arranged concentrated loads, using a spreader steel beam and a 5ton hydraulic jack. The load was measured using an electric load cell of 50kN capacity and was applied in increments of 0.5kN. The slabs were painted using white emulsion to assist in detecting the cracks. Deflection under the centre of the slab was measured using Linear Variable Displacement Transducers (LVDT). The load cell and LVDT were connected to a data acquisition system. Surface concrete strains were measured using a digital DEMEC gauges. At each load increment, careful search was made for cracks on all sides of the slab with the aid of a magnifying glass and a powerful electric lamp. The crack spacing, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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the number of cracks, the extent of the cracked zone over the length of the slab and the ultimate load were all noted. The failure load considered in this investigation was the load value after which the panel ceases to resist additional load or the load measured just before sudden collapse. Figure 4 shows typical tested panel.

1R6

20m

1R6

95m

Wire mesh layers 1R6

20m

1R6

470mm

Wire mesh layers

1R6

1R6 1R6

50mm

1R6 20m

Channel type A section, Ch A.

20mm

Figure 1:

470mm

Figure 2:

Channel type B section, Ch B.

470mm

Figure 3:

95m

Wire mesh layers

20m

1R6

55m

20mm

Box B section, Box.

Along with each panel, six 100x100x100mm cubes were tested for compressive strength and two 100x100x500mm prisms were tested for modulus of rupture. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 4:

Test rig and typical panel tested.

3.5

Cracking load (kN)

3.0 2.5 2.0 1.5 Channel A

1.0

Channel B Box

0.5 0.0 2

3

4 No. of Mesh Layers

Figure 5:

5

6

Cracking load.

5 Experimental observation 5.1 Cracking load Figure 5 shows that all sections cracked at loads close to each other when the number of layers was 2. The box section cracking load was less than both channel sections when the number of layers was increased to 4 or 6. In general, as the number of wire mesh increases the cracking load increases in all shapes.

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5.2 Failure load The box section and channel A section had almost similar failure loads which were higher than the failure loads of channel B section for all number of layers as shown Figure 6. The ultimate load was increased with the increase of the wire mesh layers in all shapes. The increment was less pronounced in the case of channel B. 9.0

Ultimate load (kN)

8.0 7.0 6.0 5.0 4.0 3.0

Channel A

2.0

Channel B

1.0

Box

0.0 2

3

4 No. of Mesh Layers

Figure 6:

5

6

Failure load.

Max. deflection (mm)

60.0 50.0 40.0 30.0

Channel A

20.0

Channel B

10.0

Box

0.0 2

3

4

5

6

No. of Mesh Layers

Figure 7:

Maximum deflection.

5.3 Deflection Figure 7 shows maximum vertical deflection at mid-span. It is clear that when the number of wire mesh layers was 2, all panels had close to each other deflection values. With exception to channel B, the maximum deflection was decreased with the increase of the wire mesh layers from 2 to 4 and 6. Figure 8 shows that, in all panels, as the number of wire mesh increases the deflection reduces for the same load. Figure 9 shows that there are no major differences in the behaviour of channel A and box section but channel B behaved in a softer manner, more deflection for same load. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Load(kN)

Channel A (95mm edge beam)

8 7 6 5 4 3 2 1 0

Ch-2A Ch-4A

Ch-6A

0

10

20 30 Deflection(mm)

40

50

(a) Channel B (50mm edge beam)

6

Load(kN)

5 4 3

Ch-2B

Ch-4B

2

Ch-6B

1 0 0

10

20

30 40 Deflection(mm)

50

60

(b)

Load(kN)

9 8

Box-2

7

Box-4

6

Box-6

5 4 3 2 1 0 0

10

20

30

40

50

Deflection(mm)

(c) Figure 8:

(a): Effect of the number of wire mesh layers on channel A deflection, (b): Effect of the number of wire mesh layers on channel B deflection, (c): Effect of the number of wire mesh layers on Box deflection.

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Load(kN)

2 wire mesh layers 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

Ch-2A Ch-2B Box-2

0

10

20 30 Deflection(mm)

40

50

(a) 4 w ire mesh layers

7

Load(kN)

6 5 4

Ch-4A

3

Ch-4B

2

Box-4

1 0 0

10

20

30

40

50

60

Deflection(mm)

Load(kN)

(b) 6 w ire mesh layers

9 8 7 6 5 4 3 2 1 0

Ch-6A Ch-6B Box-6

0

10

20

30

40

50

60

Deflection(mm)

(c) Figure 9:

(a): Effect of shape on deflection with 2 wire mesh layers, (b): Effect of shape on deflection of panels with 4 wire mesh layers, (c): Effect of shape on deflection of panels with 6 wire mesh layers.

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102 High Performance Structures and Materials III

6

Conclusion

Nine roof panels were tested for pure bending. Results from two types of channel sections, channel A and channel B, differing in the depth of the edge beam (95mm and 50mm) and one type of box section were compared. The number of wire mesh layers was varied from 2 to 6. Results show that channel type A behaved in a similar way of the box section with close to each other failure loads and deflection while channel type B was softer regardless of the number of wire mesh layers. All panels showed acceptable cracking and failure load for roofing systems.

References [1] [2] [3] [4] [5]

[6]

[7] [8] [9] [10]

Nedwell P.J and Swamy R.N., Ferrocement, St Edmundsbury Press, Bury St, Edmunds, Suffolk, 1994. Swamy R.N, Concrete Technology and design, Bell and Bain (Glasgow) Ltd, 1984. Al-Sulaimani, G.J., and Ahmad, S.F., Deflection and flexural rigidity of ferrocement I and Box beam. Journal of Ferrocement 18(1):1-12, 1988. Mathews, M.S., Sudhakumar, J. Sheela, S. and Seetharaman, P.R., Analytical and experimental investigations of hollow ferrocement roofing units, Journal of Ferrocement 21(1): 1-14, 1991. Kanai, S, and Brooks, J.J, Tensile, Flexual and Impact Behaviour of Ferrocement with chicken wire mesh reinforcement, Proceedings of the Fifth International Symposium on Ferrocement, UMIST, Manchester, pp.342-355, 1994. Mansour, M.A. and Paramasivan, P, Cracking behaviour and Ultimate strength of Ferrocement in flexure, Proceedings of the Second International Symposium on Ferrocement, Bangkok, Thailand, pp.47-59, 1985. Ahmed S.F., Saroash H. Lodi, and Juneid Qureshi, Shear behaviour of Ferrocement thin webbed sections, Cement and Concrete Research, Vol. 25, No. 5, pp.969-979, 1995. Al-Kubaisy, M.A. and Jummat, M.Z, Flexural behaviour of reinforced concrete slabs with Ferrocement tension zone cover, Journal of Construction and Building Materials, Vol 14, pp 245-252, 2000. Masood A., Arif M, Akhtar S. and Haquie M, Performance of Ferrocement panels in different environments, Cement and Concrete Research, Vol. 33, pp. 555-562, 2003. Hago, A.W., Al-Jabri, K.S., Al-Nuaimi, A.S., Al-Moqbali, H., and AlKubaisy, M.A, Ultimate and service behaviour of ferrocement roof slab panels, Construction and Building Materials 19:31-37, 2005.

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Finite element modeling of actuated fibre composites M. Martinez1, A. Artemev2, F. Nitzsche2 & B. Geddes2 1

National Research Council of Canada (NRC), Institute for Aerospace Research, Ottawa, Ontario, Canada 2 Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada

Abstract A finite element method (FEM) study of the actuation and sensing performance of actuated fibre composites (AFC) is presented. The effect of non-continuous fibres on the AFC performance was analyzed for actuation and sensing applications. The results of the FEM analysis of AFC with non-continuous fibres are compared to experimental results obtained in specimens with fibres damaged by large deformation. A study of the change in the polarization state in the fibre, resulting from the formation of a gap, was performed and its results were incorporated into the FEM. The correlation between the available experimental data and simulation results is discussed. Keywords: PZTs, actuated fibre composites, AFCs, piezoelectric fibres, sensors, smart rotor.

1

Introduction

Modern military and civilian aircraft structures require the use of monitoring systems to identify and predict the health of critical components within the aircraft. Health Prognostic Monitoring (HPM) is in the process of being introduced to modern aircraft structures. This concept will allow the aircraft to determine if the life of a critical component is coming to an end, and have the overall system schedule the time and place for the component to be repaired/replaced. Sensors are required to acquire the current state of strain, temperature or other parameters of concern in specific areas of the aircraft, which is a critical part of HPM. Several sensors show promise for these WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06011

104 High Performance Structures and Materials III applications, such as, strain gauges, fibre optic strain sensors and piezoelectric sensors. The use of piezoelectric sensors and actuators embedded in laminated composites is also being studied as a means of reducing noise and vibration generated by helicopter rotors. A “Smart Blade” structure would be capable of reacting to an external electric stimulus measured by AFC sensors and allow for the deformation of the rotor blade via AFC actuation. This paper concentrates on the use of piezoelectric fibres as strain sensors. The “Smart Blade” tailors the aerodynamic load distribution to attenuate both noise and vibration at flight conditions spanning from vertical take-off, manoeuvring flight, high-speed cruise and vertical landing. The use of piezoelectric materials for both actuation and sensing functions has made them attractive for use in micro electro mechanical systems [1] as well as for large-scale applications. For example, piezoelectric materials are being analyzed for active control of airfoils [2–4]. However, bulk piezoelectric materials have several serious drawbacks impeding their applications. In particular, these ceramic materials have inherently low toughness. A number of brittle materials have been used, in the form of fibres, in fibre-reinforced composites. Such materials allow for the use of high strength fibres together with a high toughness developed in a composite structure. Recently, several attempts have been made to develop composite materials with piezoelectric fibres [5–7]. Such actuated fibre composites (AFC) can be used as actuators and/or sensors in a number of different applications. The use of AFC for practical applications is still restricted by the lack of detailed knowledge about the relationship between the structure, manufacturing process, and performance of the material. Currently, for several AFC systems, research is being conducted on the relationships between structures, properties of the constituent components, and the behavior of the system as a whole [8–10]. One particularly challenging problem is the analysis of long-term behavior and degradation of AFC materials. Studies into the effect of fibre damage on the performance of AFC are an integral part of this research [11–12]. The presence of gaps within the fibre has significant consequences for the polarization field of the fibre. This change in the proximity of the gap size has a major impact on the electric potential distribution and, thus, on the overall response obtained from the AFC. This study has been carried out using a custom developed FEM solver [13].

2

Domain patterns in broken fibres

A phase field model of ferroelectric materials developed in [14] was used to study the effect on the polarization of gaps in fibres. In this model the distribution of the order parameter vector η describes the state of the system. The polarization at any point is determined as P = P0 ⋅ η . The free energy is: β 3 F = ∫  ⋅ ∑ ∇η p  2 p =1

(

)2 + f (η) d 3r + 2πP02 ∑3 

3

∑∫

p =1 q =1

d 3k k p k q ~ η p (k )η~q (k )* 3 2 (2π ) k

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where β is the gradient energy coefficient, δ is Kronecker delta symbol, f (η) is the chemical energy, P0 is the equilibrium polarization, η~ is the Fourier transform of η , and * indicates a complex conjugate. We used: 2

3

    f (η) = −0.136 ⋅ ∑ η i2 − 7.076 ⋅ ∑ η i4 + 4.352 ⋅  ∑ η i2  + 1.86 ⋅  ∑ η i2  i i i  i  and other model parameters characterized by the dimensionless combinations

(

)

λ = P02 / ∆f = 10 and β / ∆f ⋅ l 2 = 0.5 , where ∆f is the difference between the chemical energies of paraelectric and equilibrium ferroelectric phases. The unit length is equal to the mesh cell length. A 512x64x64 equilateral rectangular mesh with a fibre (diameter of 48 and length of 512 cells) was used. Fibres with different gap lengths were simulated. Initially the fibres had homogeneous longitudinal polarization (Fig.1(a)). The equilibrium domain patterns were found using the fast Fourier transform for the solution of the stochastic time-dependent Ginzburg-Landau equation [14] for the minimization of free energy.

a

b

c

Figure 1:

(a) Domain patterns in the broken fibre in the initial state and (b) after relaxation in fibres with a 6 cell long gap and (c) with a 9 cell long gap. The polarization direction is shown by arrows.

Figure 1 illustrates a longitudinal section of the system in the initial state (Fig. 1(a)) and after the completion of the simulation (Fig. 1(b) and (c)). The analysis shows that the presence of the gap results in the appearance of 90o closing domains at the ends of the fibre and 180o domains with reverse polarization extending along the fibre from the fibre end. The formation of this domain structure leads to a partial decrease in the overall polarization near the fibre ends. The simulation shows that increasing the gap length results in an increase in the degree of reduction of the average polarization and the increase in the length of the affected zone. Computational constraints do not allow simulation of fibres with realistic dimensions. However, the simulation results clearly indicate that the formation of a polydomain zone with a reduced poling effect can be expected near the ends of broken fibres. The depth of such a zone for real fibres can be estimated as approximately equal to the fibre diameter. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

106 High Performance Structures and Materials III

3

Finite element models of AFC

Two Finite Element (FE) models were constructed. The first model, for the simulation of sensing applications, consisted of an AFC subjected to a strain of 0.9 micro strains. The second model for the simulation of actuation applications represented an AFC subjected to 900V. In both models the effect of gaps with and without a depolarized field was studied. The models were constructed using the material properties for PZT-5A fibres, an epoxy matrix and two Kapton™ layers. The piezoelectric fibres in the AFC model have the following piezoelectric parameters: d31 of 440⋅10-12 C/N, d33 of 185⋅10-12 C/N, and d24 and d15 of 560⋅10-12 C/N. Mechanical properties were represented by C11=120 GPa, C33=110 GPa C44=21 GPa, C66=23 GPa, C12=75.2 GPa and C13=75.1 GPa [13]. The epoxy had a Young’s modulus of 2.58 GPa and Poisson’s ratio of 0.38. The Kapton™ had a Young’s modulus of 2.5 GPa and Poisson’s ratio of 0.34. 3.1 Sensing AFC Model

Figure 2:

(a) The AFC model used for simulation of the sensing application shown with an epoxy matrix, (b) inner components of the AFC model.

The sensing AFC system was set to 2100µm in length, 1200µm in width and 260µm in thickness. It consisted of four piezoelectric fibres of 254 µm diameter and an epoxy matrix contained between two Kapton™ layers, as shown in Fig. 2(a). Figure 2(b) shows the same model with the Kapton™ removed and a transparent epoxy matrix. An assumption was made that due to the presence of a 30 µm gap, the regions of the fibre next to the gap with the length equal to the fibre diameter, had a zero poling effect. This means that piezoelectric constants in these regions were equal to zero; however, elasticity constants were the same as in the rest of the fibre. A defined strain of 0.9 micro strains was applied in an extension mode. The electric potential difference was measured between the top and bottom of the fibres. 3.2 Actuation AFC model The unconstrained actuation AFC system was set to 4600µm in length, 1200µm in width, and 260µm in thickness. It consisted of four piezoelectric fibres, an WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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epoxy matrix, and copper inter-digitized electrodes (IDE) that were covered by a thin layer of epoxy. Figure 3(a) shows a view of the AFC model including the epoxy matrix. Figure 3(b) shows the same AFC with the epoxy matrix made transparent to show the fibres and the copper electrodes. PZT fibres with 250µm diameters were spaced 300µm center to center. The AFC was cantilevered at one end. The fibres had a varying polarization field along their length. The fibre under the electrodes was modeled to have a zero polarization.

Figure 3:

(a) AFC with epoxy matrix, (b) inner components of AFC, (c) cross-sectional view of AFC at the electrodes showing a 60° contact angle between the electrode and the PZT fibre.

The fibres between the electrodes have a polarization along either positive or negative x-direction, as shown in Fig. 3(b). Switching of the polarization direction permits the AFC to extend or contract, since the direction of the electric field also changes from one electrode pair to another. This type of AFC structure and actuation method were presented and studied in [5–6]. The four copper electrodes are spaced 900µm apart and are 200µm wide and 10µm thick. The electrodes in Fig. 3(b) are labeled E1, E2, E3 and E4. The electric potentials for E2 and E4 are set to zero, while E1 and E3 are set to electric potentials of 900, 1350, and 1800 volts for different simulation runs. The copper electrodes make direct contact with the fibres through an angle of 60°, as shown in Fig. 3(c). In order to study the effect of broken fibres on the actuation performance of the AFC, we created models with different numbers of gaps (0, 1, 2, 5, 10 and 20) introduced into fibres. We used models in which such gaps were introduced without changing the polarization state in the rest of the fibre volume, as well as models in which gaps were accompanied by depolarized regions at the ends of the fibres with a length equal to the fibre diameter.

4

Results of FEM analysis

4.1 Results of sensing application analysis Figure 4 demonstrates the electric potential distribution in the AFC under the applied mechanical strain. The results obtained by the simulation of the sensing application of the AFC indicate that the change in the electric potential between

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108 High Performance Structures and Materials III the upper and lower surface of the system in the proximity of the gap drastically differs from the voltages developed in other regions of the AFC.

Figure 4:

(a) AFC under an applied strain, (b) cross-section of AFC through the gap in the fibre.

As seen in Fig. 4, the electric potential difference through the thickness of the fibre is close to zero in the gap and in the depolarized region near it. Such loss of voltage can be a problem since discontinuities in the AFC tend to appear in the proximity of the electrodes used to apply voltage to fibres [15]. If the same electrodes were used for sensing, then they would detect a voltage corresponding to a strain that is significantly lower than the actual strain. In such a case a dual electrode systems can improve sensing performance. The first electrode group may be used for poling of the piezoelectric fibres, while the second electrode system would be used to monitor voltages produced by the piezoelectric fibres. It should be noted that in the AFC model where gaps in the fibres were not accompanied by depolarized regions, the width of the dead zone in the voltage distribution was much smaller (approximately equal to the gap width) and the loss of voltage less severe. 4.2 Results of actuation application analysis The FEM analysis of the AFC under applied voltage (Fig. 5) showed that gaps in fibres can reduce the actuation performance. If a fibre between two IDE’s is broken, then the electrical field is concentrated in the gap, and two pieces of fibre between electrodes have a very low electrical field applied, producing no contribution to the actuation. However, any significant reduction in the overall WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Percentage of Displacement

actuation performance of the AFC is obtained only when a large number of gaps is introduced effectively creating a short fibre composite. It is unlikely that such a density of damage can be developed before overall failure. The FEM results also indicate that there is very little difference between systems with depolarization regions added to the gaps in fibres and systems with gaps only (without depolarization regions).

100.00 80.00 60.00 40.00 20.00 0.00 0

5

10

Number of Gaps

15

20 Fully polarized Depolarized fibers

(b)

Figure 5:

5

(a) Electric potential in the AFC under voltage applied to IDE, (b) the effect of the number of fibre gaps on the actuation performance of the AFC for systems with and without depolarized regions near gaps.

Discussion and conclusions

An experimental study of the degradation of AFC sensing properties caused by the damage produced by large strains [12] has shown that a significant loss of the sensing capacity can be produced. However, the sensing capability is almost completely restored when the applied load is reduced. Simulation results demonstrate a possible mechanism for such a reversible sensing degradation. Under a large imposed strain, broken fibres can contain wide gaps producing large depolarized regions, which result in a significant loss of the sensing function. When the load is reduced, the gaps in the fibres are closed and a fully poled state can be restored, resulting in a better sensing performance even though fibres remain mechanically broken. This simulation also demonstrates that damage in the fibres can be detrimental to both actuation and sensing performance of the AFC; however, any significant loss of the actuation WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

110 High Performance Structures and Materials III capability can be expected only when a large density of gaps in fibres is produced.

Acknowledgement The authors gratefully acknowledge the financial support of Natural Sciences and Engineering Research Council of Canada (NSERC).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Banks H.T., Smith R.C., Wang Y., Smart Material Structures - Modeling, Estimation and Control. John Wiley & Sons, 1996. Bent A. A., “Active Fibre Composites for Structural Actuation,” Ph.D. Thesis, Massachusetts Institute of Technology, January 1997. Hall S.R., Prechtl E.F. Smart Materials and Struct. 5 (1996) 26. Flinn, E.D. Aerospace America 37 (1999) 40. Odegard G.M., “Modeling of Piezoelectric Polymer Composites” National Institute of Aerospace, Hampton, Virginia. NASA/CR-2003-212681NIA Report No. 2003-10 Wilkie W.K., et al., “NASA Langley Research Center Macro-Figer Composite Actuator (LaRC-MFC): Technical Overview,” Brei D., Cannon B.J., Composites Science and Tech. 64 (2004) 245. Sporn D., Schoenecker A., Mat. Res. Innovat. 2 (1999) 303. Pettermann H.E., Suresh S., Int. J. of Solids and Struct. 37 (2000) 5447. Wilkie W., High J., Bockman J., Reliability Testing of NASA Piezocomposite Actuators, U.S. Army Research Laboratory and NASA Langley Research Center, Hampton, Virginia, USA. Wickramasinghe V.K, Hagood N.W., Smart Mater. Struct. 13 (2004) 1155–1165 PII: S0964-1726 (2004) 83334-2. Melnykowycz M., et al., Performance of Integrated Active Fibre Composites in Fibre Reinforced Epoxy Laminates. to appear in Smart Materials Structures. M. Martinez, et al., Finite Element Analysis of Actuated Fibre Composites. Proceedings of CANSMART 2005, Toronto, Canada (2005) 231. Semenovskaya S., Khachaturyan A.G., J. Appl. Phys., 83 (1998) 5125. Wickramasinghe V.K, Hagood N.W., J. of Aircraft, 41 (2004) 931.

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CFRP strengthening of prefabricated timber panel walls M. Premrov & P. Dobrila University of Maribor, Faculty of Civil Engineering, Slovenia

Abstract This paper provides an experimental analysis of timber-framed walls, coated with carbon fibre-reinforced polymers (CFRP) strengthened fibre-plaster boards, usually used as main bearing capacity elements in the construction of prefabricated timber structures. The tensile strength of the boards is lower than the compressive one and essentially lower than the strength of the timber frame, therefore it is convenient to strengthen boards in their tensile diagonal direction with high-strength materials in order to gain a higher capacity. It has been shown that the inclusion of CFRP diagonal strip reinforcement on the load-carrying capacity can be quite high and that it is maximized when the carbon strips are connected to the timber frame. On the other hand, the ductility itself was not significantly improved. The test samples proved an important distinction in behaviour in timber frame-fibreboard connecting area, dependant on the boundary conditions between inserted CFRP strips and timber frame. Keywords: timber, frame walls, CFRP, fibre-plaster boards, experiments.

1

Introduction

There is an increasing tendency worldwide toward building multi-level prefabricated timber structures with timber-framed walls as the main bearing capacity elements. Their load-carrying capacity becomes critical, especially when taller structures are subjected to heavy horizontal forces, particulary with structures located in seismic and windy areas. In this case it is sometimes necessary to reinforce the walls. The treated wall is a composite element consisting of framed panels made from sheets of board-material fixed by mechanical fasteners to one or both sides of the timber frame (Figure 1). There are many types of panel products available which may have some structural WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06012

112 High Performance Structures and Materials III capacity such as wood-based materials (plywood, oriented strand board, hardboard, particleboard, etc.) or plaster boards and, more recently fibre-plaster boards. In the following analysis we limited our attention to the fibre-plaster boards (FPB), recently the most frequently used in Central Europe. One of the most important reasons for an increased application of these types of gypsum products is their relatively good fire protection. Additionally, gypsum is a healthy natural material and is consequently particularly desired for residential buildings. In structural analysis panel walls for design purposes can be regarded separately as vertical cantilever beams with the horizontal force (FH=FH,tot /n) acting at the top (Figure 1). Considered supports approximate an influence of neighbouring panel walls and assure an elastic-clamped boundary condition for the treated wall (Faherty and Williamson [1] or Eurocode 5 [2]). FH,tot

FH

h

⇒

b n·b

FH = Figure 1:

2

b timber frame (the studs)

y

FH ,tot n

coating boards

Static design and cross section of the treated panel wall.

Design models

2.1 Shear model Many design models have been proposed in order to analyse and predict the behaviour of wood-based shear walls and diaphragms subjected to lateral loads. Källsner [3] and Äkerlund [4] proposed an agreeable approach to determine the load-carrying capacity of the wall unit, based on the following key assumptions: -

behaviour of the joints between the sheet and the frame members is assumed to be linear-elastic until failure,

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the frame members and the sheets are assumed to be rigid and hinged to each other. The influence of shear deformations in the fibreboard can be additionally estimated by introducing the shear angle. Additionally, two models are presented based on the assumption that the load-displacement relation of fasteners is completely plastic. Källsner and Lam [5] presented the walls load-carrying capacity as a function of fasteners spacing along the upper horizontal timber member assuming constant fastener spacing along all timber members. Two simplified computational methods are given in the final draft of Eurocode 5 [2] in order to determine the load-carrying capacity of the wall diaphragm. The first simplified analysis – Method A, is identical to the “Lower bound plastic method”, presented by Källsner and Lam [5]. This method defines the wall’s shear resistance (Fv,d) as a sum of all the fasteners’ shear resistances along the loaded edges in the form of: -

bi ⋅ ci s Ff,Rd …. lateral design capacity per fastener, bi ….. wall panel width, s ….. fastener spacing, Fv ,d = ∑ F f ,Rd ⋅

1 for bi ≥ b0  ci =  bi  b for bi ≤ b0  0

where b0 = h/2

(1)

(2)

This is only an approximated and simplified definition, which can be applicable for wood-based panels where the strength is relatively high and the elements tend to fail because of fastener yielding. The second simplified analysis – Method B is applicable to walls made from sheets of wood-based panel products only, fastened to a timber frame. The fastening of the sheets to the timber frame should either be by nails or screws, and the fasteners should be equally spaced around the perimeter of the sheet. According to Method A the sheathing material factor (kn), the fastener spacing factor (ks), the vertical load factor (ki,q) and the dimension factors for the panel (kd) are included in the design procedure in the form of: bi 9700 ⋅ d ⋅ ci ⋅ k d ⋅ k i ,q ⋅ k s ⋅ k n ; s0 = s0 ρk ρk …. characteristic density of the timber frame

Fv ,d = ∑ F f ,Rd ⋅

d … fastener diameter,

(3)

2.2 Composite model All the above mentioned methods are usually unsuitable for treated walls sheathed with fibre-plaster boards (FPB). The main assumptions do not exactly coincide with the real state of FPB, in which the tensile strength is evidently WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

114 High Performance Structures and Materials III lower than the compressive one. Consequently, cracks in a tensile zone usually appear under heavy horizontal loads before stresses on the fasteners reach their yielding point, and the fibreboards do not behave usually as rigid elements (Dobrila and Premrov [6]). However, by employing FPB as a coating material, a horizontal load shifts a part of the force over the mechanical fasteners to the fibreboard and the wall acts like a deep beam. Distribution of the horizontal force by composite treatment of the element depends on the proportion of stiffness. The effective bending stiffness (EIy)eff of mechanically jointed beams taken from Eurocode 5 [2] can be written in the form of: n

(

)

( EI y )eff = ∑ E i ⋅ I yi + γ yi ⋅ Ai ⋅ a i2 = i =1

=

ntimber .

∑

i =1

(E

i

⋅ I yi + E i ⋅ γ

)

2 yi ⋅ Ai ⋅ a i timber

+

nboard

∑ (E i ⋅ I yi )board

(4)

j =1

where n is the total number of elements in the considered cross-section and ai is a distance between global y-axis of the whole cross-section and local yi-axis of the i-th element with a cross-section Ai (see Figure 2). It is evident that the force distribution in this case strongly depends on the stiffness coefficient of the connecting area (γy), which mostly depends on the fasteners slip modulus (Kser) and fasteners disposition, as well as on the type of the connection. An experimental analysis on the influence of fasteners spacing on behaviour of the treated walls can be found in [6].

3

Strengthening of fibre-plaster boards

As described, the FPB are usually a weaker part of the presented composite system, because their tensile strength is evidently smaller than the wood strength of all members in the timber frame. Thus, especially in multi-level buildings located in seismic or windy areas, cracks in FPB usually appear. In these cases the FPB lose their stiffness and therefore their resistance should not be considered at all. Stresses in the timber frame under a horizontal loads are usually not critical. There are several possibilities to reinforce panel walls in order to avoid cracks in FPB: - by using additional boards. The boards are usually doubled: - symmetrically (on both sides of a timber frame), - non-symmetrically (on one side of a timber frame), - by reinforcing boards with steel diagonals, - by reinforcing boards with carbon or high-strength synthetic fibres. 3.1 Strengthening with additional FPB In [6] we presented the first possibility experimentally using additional FPB, which gave higher elasticity of elements, whilst bearing capacity and especially ductility were not improved in the desired range. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3.2 Strengthening with diagonal steel strips With the intention to improve the resistance and especially the ductility of the walls it is more convenient to insert diagonal steel strips, which have to be fixed to the timber frame. In this case only a part of the horizontal force is shifted from boards over the tensile steel diagonal to the timber frame after the appearance of the first crack in the tensile zone of FPB (see [6]). An enlarged effective crosssection of FPB (A*1b) can approximately be computed considering the compatibility conditions between the actual reinforced and fictitious unreinforced element. The computational procedure is described in details in Premrov and Dobrila [7] and will not be presented here. With regard to the fictive enlarged cross-section of FPB we proposed two approximate analytical models using either fictitious thickness (t*) or fictitious width (b*) of fibreboards: t* =

A1*b 1 E 1 = t + ⋅ s ⋅ sin 2 α ⋅ cos α ⋅ A1s ,0 ⋅ b χ Gb b

(5)

b* =

A1*b 1 E 1 = b + ⋅ s ⋅ sin 2 α ⋅ cos α ⋅ A1s ,0 ⋅ t χ Gb t

(6)

In the above equations α represents the angle of inserted steel diagonals with the net area (A1s,0). A non-dimensional coefficient χ is shear cross-section coefficient defined as a proportion between the shear and actual cross-sectional area of the FPB with the shear modulus (Gb). Alongside the steel diagonals’ influence these models enable simultaneous consideration of the fasteners’ flexibility between the board and the timber frame and any appearing cracks in the tensile area of the FPB. Un-reinforced panels can be computed using actual dimensions of the FPB. Numerical results presented in [7] on diagonally steel reinforced elements show good agreement with measurements performed on the test samples. 3.3 Strengthening with diagonal CFRP strips As the tensile strength of FPB is obviously lower than the compressive strength and corresponding capacity of timber frame, the treated elements tend to fail because the cracks are forming in the tensile area of the FPB, therefore this tensile area could be reinforced with high-strength materials. This strengthening concept is such that the composites would contribute to tensile capacity when the tensile strength of FPB is exceeded. No FRP applications on the treated fibreplaster boards were found in the literature.

4

Test experiment: strengthening with CFRP strips

4.1 Test configuration Three sample groups from total of nine test samples were tested in order to carry out appropriate experimental research on the influence of CFRP strengthened WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

116 High Performance Structures and Materials III walls. All test groups consisted of three walls of actual dimensions h=263.5cm and b=125cm. The cross-section presented in Figure 2 was composed of timber studs (2x9x9cm and 1x4.4x9cm), timber girders (2x8x9cm) and Knauf fibreplaster boards (Knauf [8]) of thickness t=15mm. They were fixed to the timber frame using staples of Φ1.53mm at an average spacing of s=75mm. yi

At, Et

y

yi

Ab , E b

t =1.5 Vz

9.0

4.4

9.0

9.0

ai = 58 b =125 cm

Figure 2:

Cross-section of test samples.

The static model according to Figure 1 was used for all groups of test samples. The samples were actually rotated by 900 according to Figure 1 and they were therefore subjected to vertical force acting at the end of the elements (Figure 3a). The FPB were reinforced in the tensile diagonal area using SikaWrap-230C strips (Sika [9]) made from carbon high-strength fibre reinforced polymers of thickness 1.2 mm. Strips with different widths (300 or 600 mm) and of different boundary conditions were glued to the FPB. The first group (G1) of three test samples was additionally reinforced with two CFRP diagonal strips (one in each FPB) of width 300 mm which were glued on the FPB using Sikadur-330 LVP. The strips were additionally glued to the timber frame (Figure 3a,b) to ensure the transmission of the force from FPB to the timber frame. The second group (G2) of three test samples was additionally reinforced with two CFRP diagonal strips of width 600 mm. The strips were glued on FPB and to the timber frame as in G1 (Figure 3b) to ensure the transmission of the force from FPB to the timber frame. The third group (G3) of three test samples was additionally reinforced with two CFRP diagonal strips of width 300 mm as in G1 but they were not glued to the timber frame (Figure 3c). Material properties for the test samples for all groups were the same (Table 1). Values for timber of quality C22 are taken from EN338 [10], the characteristics of fibre-plaster boards from Knauf [8] and for carbon strips Sika [9] data were used. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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a.)

117

b.)

c.)

Figure 3:

a.) G1; the static system, b.) G2; the CFRP strip is glued on the FPB and additionally to the timber frame, c.) G3; the CFRP strip is not glued to the timber frame. Table 1: E0,m

Timber FPB SikaWrap

Properties of used materials. Gm

fm,k

ft,0,k

fc,0,k

[N/mm2]

[N/mm2]

[N/mm2]

[N/mm2]

[N/mm2]

10000 3000 231000

630 1200 /

22 4.0 /

13 2.5 4100

20 20 /

ρm [kg/m3] 410 1050 1920

4.2 Test results and analysis The force forming the first crack (Fcr) in the FPB, the crushing force (Fu), the maximal cantilever bending deflection (w) under the acting force (F) and the slip (∆) in the tensile area between the FPB and the timber frame were all measured. The measured values for the un-strengthened (UNS) test samples were taken from [6] and included for information and comparison only. Average force forming the first crack (Fcr): G1: Fcr,1 = 24.28 kN G2: Fcr,2 = 32.13 kN

G3: Fcr,3 = 35.90 kN UNS: Fcr,uns = 17.67 kN

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118 High Performance Structures and Materials III Average crushing force (Fu): G1: Fu,1 = 40.33 kN G2: Fu,2 = 46.27 kN

G3: Fu,3 = 36.26 kN UNS: Fu,uns = 26.02 kN

It is evident that the elastic resistance (force forming the first crack) essentially increased for all kinds of CFRP strengthened test samples, but mostly for samples G3, where the CFRP strips were not fixed to the timber frame. The CFRP influence was not so obvious at samples G1, where carbon strips of the same dimensions were additionally glued to the timber frame. On the other hand, when comparing the measured results of the crushing force, a greater improvement can be noticed in the groups where the CFRP diagonals were glued to the timber frame. Compared to the un-strengthened test sample, the crushing force in samples G2 was increased by 78%. In samples G3 the crushing force practically coincided with a force forming the first crack, so cracks hardly appeared at all, which is not a good solution to ensure better ductility, necessary for seismic design. For further analysis it is important to present measured maximal cantilever deflections (w) (Figure 4) and slips (∆) in the connecting area (Figure 5). F[kN]

Fcr,3 Fcr,2

UNS G1 G2 G3

Fcr,1

w[mm]

Figure 4:

Measured average bending deflections (w).

Closer look at the graph in Figure 4 at F>17 kN reveals an obvious difference in the behaviour of the test samples when the CFRP strips were glued to the timber frame (samples G1 and G2) or if they were not (samples G3). Beside the fact that samples G1 and especially G2 demonstrated higher load-carrying capacity than samples G3, it is also important to mention that samples G1 and G2 produced substantially smaller slip than samples G3, which never exceeded 1mm at the first crack forming (Figure 5). Therefore it can be assumed that the yield point of the fasteners was not achieved before cracks appeared at all. Consequently, the walls tend to fail because of the crack forming in FPB. In this case of strengthening the ductility of the whole wall element (see Figure 5 for samples G1 and G2) practically coincides with the “ductility” of FPB, as proposed with d1 and d2 coefficients. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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119

F[kN] Fcr,3 Fcr,2 G1 G2 G3

Fcr,1

ǻ[mm]

Figure 5:

Measured average slips (∆) in the connecting area.

In contrast, in G3 model, where the CFRP strips were unconnected to the timber frame, the slip (∆) between the FPB and the timber frame was evidently higher than in samples G1 and G2, and exceeded 3mm when the first crack in FPB appeared (Figure 5). The load-displacement relation (F-∆) of the fasteners was in this case at the force which produced first cracks almost completely plastic. Since the tensile strength of FPB is essentially improved, the walls tend to fail because of fastener yielding and therefore the “Lower bound plastic method” (EC5 Simplified Method A) can be used to determine the wall`s load carrying capacity, (eqn 1).

5

Conclusions

As shown, there is practically no influence on the element stiffness of any reinforcement before cracks appeared in the un-strengthened FPB. However, after the first cracks in un-strengthened FPB appeared, the test samples demonstrated an important difference in behaviour dependant on the boundary conditions between the inserted CFRP strips and the timber frame. If strips are glued to the timber frame the fasteners produced substantially smaller slip, which never exceeded 1mm when the first cracks appeared. Therefore it can be assumed that the yield point of the fasteners is not achieved before cracks appeared at all and the elements tend to fail because cracks appear in the tensile area of FPB. Therefore, it is not recommended to use Eurocode 5 [2] simplified methods to predict the element resistance. Simple mathematical models with a fictive enlarged cross-section of FPB are proposed in [7]. In the case where the CFRP diagonals are unconnected to the timber frame, the slip between the FPB and the timber frame is evidently higher and the loaddisplacement relation of the fasteners is, after the cracks appeared, almost perfectly plastic. Since the tensile strength of FPB is with CFRP highly improved, the walls tend to fail because of fastener yielding, similar as at woodbased sheathing boards. Therefore the “Lower bound plastic method” can be used to determine the wall’s load carrying capacity. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

120 High Performance Structures and Materials III

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Faherty, K.F. & Williamson, Wood Engineering and Construction Handbook, McGraw-Hill Publishing Company: 1989. CEN/TC 250/SC5 N173, Eurocode 5: Design of Timber Structures, Part 1-1 General rules and rules for buildings, Final draft prEN 1995-1-1, Brussels, 2003. Källsner, B., Panels as wind-bracing elements in timber-framed walls. Swedish Institute for Wood Technology Research, Report 56, Stockholm, 1984. Äkerlund, S., Simple calculation model for sheets on a timber frame, Bygg & Teknik, No.1, 1984. Källsner, B. & Lam, F., Diaphragms and shear walls. Holzbauwerke: Grundlagen, Entwicklungen, Ergänzungen nach Eurocode 5, Step 3, Fachverlag Holz: Düsseldorf, pp. 15/1-17, 1995. Dobrila, P. & Premrov, M., Reinforcing methods for composite timber frame-fiberboard wall panels. Engineering Structures 25(11), pp. 13691376, 2003. Premrov, M. & Dobrila, P., Approximate analytical solutions for diagonal reinforced timber-framed walls with fibre-plaster coating material. Construction and Building Materials, 18 (10), pp. 727-735. Knauf Gipsfaserplatten Vidivall/Vidifloor, 2002. Sika, Sicher bauen mit System. Technische Merkblätter. Ausgabe 5, 2003. European Committee for Standardization, EN 338:2003 E: Structural timber – Strength classes, Brussels, 2003.

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Evaluation of the structural integrity of a sandwich composite train roof structure K. B. Shin1, B. J. Ryu1, J. Y. Lee1 & S. J. Lee2 1

Department of Mechanical Design Engineering, HANBAT National University, South Korea 2 Division of Rolling Stock, Hankuk Fiber Glass Co. Ltd., South Korea

Abstract We have evaluated the structural integrity of a sandwich composite train roof which can find a lightweight, cost saving solution to large structural components for rail vehicles in design stages. The sandwich composite train roof was 11.45 meters long and 1.76 meters wide. The FE analysis was used to calculate the stresses, deflections and natural frequencies of the sandwich composite train roof against the weight of air-condition system. The 3D sandwich FE model was introduced to consider the hollow aluminum frames jointed to both sides of the sandwich train roof. The results shown that the structural performances of a sandwich composite train roof under the loading conditions specified were proven and the use of aluminum reinforced frame was beneficial with regard to weight saving in comparison to steel reinforced frame. Also, we have manufactured the prototype of sandwich composite train roof on the basis of analysis results. Keywords: aluminum honeycomb structure, train roof structure.

1

Introduction

The use of composite materials on land transportation is steadily increasing. Railroad cars, mass transit vehicles and a wide range of ground transportation systems offer expanding opportunities for composite materials. Driving forces for the use of composites in ground transportation applications include low manufacturing investment cost, cost reduction through parts consolidation, weight saving, good mechanical properties, excellent durability and dimensional stability [1]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06013

122 High Performance Structures and Materials III The reduction of structural weight of one large component usually triggers positive synergy effects for other parts of the vehicle. For example, a reduction of the mass of a railway carbody could lead to weight savings in the traction system, suspension, brakes and other subsystems. The manufacturing technologies for steel or aluminum carbodies of railway rolling stocks are currently highly developed. Additionally, cost and weight savings and the increase of availability of the rolling stock are difficult to achieve using the conventional metal materials. Therefore, the use of sandwich composite materials for the railway carriage structure has been proposed and recommended in transportation applications because they offer not only high specific stiffness and strength but also low manufacturing investment cost as compared to other conventional materials [2]. Recently, composites have been applied for the lightweight design of the train roof structure, side structure and carbody shell structure. A prototype composite train roof was developed to find a lightweight, cost saving solution to large structural components for rail vehicles. Additionally, a composite roof structure improves on torsion stiffness of a railway carbody [3]. Other advantages included: (1) reduced production costs, (2) reduced part count, (3) elimination of an additional thermal insulation, (4) improved passenger comfort. ‘Turbostar’ in the UK, ‘PUMA’ Train Express in Germany and ‘RARe-520’ in Switzerland are good examples. These examples applied advanced composite material to train roof structure. The objectives of this paper are as following: (1) Evaluation of the structural integrity of developing sandwich composite train roof structure, (2) Manufacturing the prototype of sandwich composite train roof on the basis of analysis results.

2

Design of the lightweight composite train roof structure

2.1 Structure design The sandwich composite train roof structure was 11.45 meters long and 1.76 meters wide. The composite train roof structure was manufactured using a sandwich composite, an inner stiffener, an edge stiffener and a hollow aluminum extrusion as shown in figure 1. The sandwich structure consists of aluminum skin and aluminum honeycomb core. The inner stiffener was used to increase the bending stiffness of transverse direction and longitudinal direction. The edge stiffener improved the twisting stiffness of the train roof structure. Figure 2 shows the section of the lightweight composite train roof. The air-condition unit will be located on the roof structure. The hollow aluminium extrusion frame will be jointed to the side wall structure of carbody as shown in figure 2. 2.2 Construction materials The sandwich structure was composed of an aluminum skin and an aluminum honeycomb core. The same material should be used for the skin and carbody in order to minimise the stress variation at joints and prevent from galvanic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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123

corrosion with aluminum honeycomb core. Table 1 shows the mechanical properties of material used in the structural analysis. The metal stiffener increased the bending stiffness of the sandwich roof structure. The inner stiffener will be manufactured by aluminum or steel material after evaluation of the weight, stress, and deflection of the sandwich composite train roof using structural analysis.

Figure 1:

Figure 2:

3

3D CAD model of sandwich composite train roof structure.

The drawing of frame section of sandwich train roof.

Evaluation of the structural integrity

3.1 3D Sandwich FE model In order to verify the structural integrity of sandwich train roof, ANSYS V9.0 was used at modal and structural analysis. An 8-node sandwich shell element (shell 91) is recommended to analyze the sandwich composite structure in WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

124 High Performance Structures and Materials III ANSYS. However, the structural behaviours of the hollow aluminum extrusion frame that is bonded at both sides of the sandwich train roof structure are important in this project. Therefore, the sandwich train roof structure should be modelled using the proper 3D FE modelling technique to consider the joint between the sandwich roof structure and hollow aluminium extrusion frames. Table 1: Type

The mechanical properties of materials used in the structural analysis. Materials

Skin

Al. 5052

Core

Aluminum honeycomb core

Dimension t=1.2mm

Cell size = 3/8 inch Cell wall thickness = 70µm Cell depth = 32mm

Properties E=69GPa υ=0.33 ρ=2700(kg/m3) E11=8.27MPa E22=1.31MPa E33=1276MPa G12=0.0001GPa G23=117GPa G13=296GPa υ12=0.75 υ23=0.0001 υ13=0.0001 ρ=100(kg/m3)

E : Young’s modulus, G : Shear modulus, υ : Poisson’s ratio, t : Thickness, ρ : Density

Table 2: Cases 1 2 3 4 5

The analysis cases to select the proper 3D sandwich FE model. Elements Sandwich shell 91(8-node) + sandwich option (Reference) Layered solid 46 (8-node) Layered solid 191 (20-node) Shell 63(skin)/Solid 45(core) Shell 93(skin)/Solid 186(core)

The dimension and material properties of sandwich composite are shown in table 1. Five cases were used to select the proper 3D FE models of sandwich train roof as shown in table 2 and they were conducted using modal analysis which can verify its mass and stiffness. The block Lanczos method was used in modal analysis and it was compared and checked by natural frequency. Table 3 shows the shell element for the skin part and the solid element for the core part should be used to replace the sandwich shell element. 3D layered solid element of case 2 and case 3 could not substitute the sandwich shell element because it do not simulate the bending and shear behaviour of sandwich structure. Accordingly, when the sandwich composite roof structure is jointed with hollow aluminium extrusion frame, 3D FE sandwich modelling is necessary to evaluate the structural integrity of the sandwich train roof. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Table 3:

125

The results of natural frequencies for the selected elements at table 2.

Mode 1 2 3 4 5 6 7 8 9 10

Case 1* 1.87(B) 5.16(B) 8.65(T) 10.14(T) 16.8(B) 17.48(T) 25.13(B) 26.68(T) 35.14(B) 36.42(T)

Case 2 1.64(B) 2.22(T) 3.22(B) 4.52(T) 5.33(B) 6.98(T) 7.96(B) 9.64(T) 11.13(B) 12.58(T)

Case 3 1.25(B) 3.45(B) 5.68(T) 6.78(B) 11.25(B) 11.51(T) 16.87(B) 17.61(T) 23.64(B) 24.11(T)

Case 4 1.86(B) 5.12(B) 8.46(T) 10.05(B) 16.62(B) 17.10(T) 24.82(B) 26.11(T) 34.61(B) 35.66(T)

Case 5 1.86(B) 5.12(B) 8.44(T) 10.04(B) 16.60(B) 17.06(T) 24.77(B) 26.05(T) 34.5(B) 35.0(T)

B = Bending mode, T = Twisting mode, * = Reference

3.2 Analysis cases and FE model Structural integrity of sandwich composite roof structure has been evaluated by modal analysis and structural analysis as shown in table 4. We have considered the aluminium and steel stiffener to examine the structural behaviour of sandwich train roof caused by the difference in stiffener material. The constraints of joint parts between the sandwich train roof and the hollow aluminium extrusion frame have been considered for two cases in analysis: (1) contact condition, (2) coupled condition. The contact condition is nonlinear analysis and can simulate the real jointing condition using bonding option in ANSYS. The coupled condition is linear analysis and can save the calculating time. Figure 3 shows the finite element models for the sandwich composite roof structure. The stiffener and hollow aluminum extrusion were modelled using shell element. The sandwich composite was modelled using the shell element (skin) and solid element (core). Table 4:

The analysis cases of sandwich train roof considered in design stages.

Modal Analysis

Structural Analysis

Cases 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8

Element Shell63 /Solid45

Shell63 /Solid45

Jointing condition Bonded contact Coupled set Bonded contact Coupled set

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Stiffener materials Aluminum Steel Aluminum Steel Aluminum Steel Aluminum Steel

126 High Performance Structures and Materials III

Figure 3:

Finite element model for sandwich composite train roof.

Figure 4:

Table 5: Mode(Hz) 1 2 3 4 5 6 7 8 9 10 Weight(kg)

The boundary condition for modal analysis.

Summary of modal analysis using simply supported BC. Case 1-1 64.76 65.46 66.64 69.52 74.15 79.30 87.66 95.90 106.19 117.67 393.58

Case 1-2 50.62 51.71 52.70 55.25 59.36 63.95 71.88 79.81 89.59 101.41 596.26

Case 1-3 64.33 65.07 66.25 69.12 73.76 78.93 87.29 95.55 105.85 117.34 393.58

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Case 1-4 50.24 51.38 52.38 54.93 59.04 63.65 71.56 79.52 89.30 101.13 596.26

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127

3.3 Modal analysis Modal analysis of the sandwich composite roof structure was conducted to evaluate its bending mode and four different cases were considered as shown in table 4. The cant side of the hollow aluminum extrusion was chosen as a boundary condition as shown in figure 4. Table 5 shows the results of natural frequencies. Bending mode only was occurred for these cases. It shows that natural frequencies of a simply supported sandwich train roof could be changed by the material of the stiffener. The changes in natural frequencies due to the material used for the stiffener can affect the whole carbody structure. If the natural frequencies of the whole carbody structure do not meet the design requirements they can be altered by changing the material of the stiffener. 3.4 Structural analysis The objective of this chapter is to investigate the structural integrity of the sandwich composite train roof structure with the weight of air-conditioning unit (ACU). There are four cases as given in table 4.

Figure 5:

The loading and boundary conditions for structural analysis.

Table 6: Summary of structural analysis using bonded contact condition.

Max. deflection(mm) Max. principal stress(MPa) Total weight(kg)

Case 2-1

Case 2-2

(Bonded contact/Aluminum stiffener)

(Bonded contact/Steel stiffener)

0.42

0.39

8.69

8.93

393.58

596.26

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128 High Performance Structures and Materials III 3.4.1 Contact condition (case 2-1 and 2-2) A contact element was used at the joint between the sandwich composite structure and the hollow aluminum extrusion frame. An aluminum and a steel stiffener was considered in each case. Figure 5 shows the boundary and loading condition for the structural analysis. The results of structural analysis using the contact condition are given in table 6. The deflection and stress have similar results using either aluminum or steel stiffeners. The reason is that the bending stiffness of aluminium stiffener is lower than that of steel, but the weight of the aluminum stiffener is comparatively lighter than that of the steel stiffener. By using the aluminum stiffener, we can obtain a weight saving of 34% in comparison with the steel stiffener. Thus in consideration of weight reduction, the use of the aluminum stiffener can be more effective as both stiffeners are safe with regards to stiffness and strength. 3.4.2 Couple condition (case 2-3 and 2-4) The coupled condition was used at the joint, between the composite structure and the hollow aluminum extrusion frame. An aluminium and a steel stiffener was considered in each case. Table 7 shows the results of the structural analysis for the coupled condition. The results of deflection and stress for the couple condition are rather high in comparison with that of the contact condition. The reason is that the analysis of the coupled condition is linear. As with the contact condition, however, there are few differences in the results of deflection and stress depending on whether an aluminum stiffener or steel stiffener is used. When considering the effect of weight reduction, it is preferable to use aluminum stiffener. Table 7: Summary of structural analysis using coupled condition.

Max. Deflection(mm) Max. principal stress(MPa) Total weight(kg)

Case 2-1

Case 2-2

(Coupled set/Aluminum stiffener)

(Coupled set/Steel stiffener)

0.73

0.67

15.89

15.83

393.58

596.26

3.4.3 Comparison of analysis results of contact and couple conditions The values of the deflection and stress of the sandwich train roof using the contact condition at the joint tend to be lower than that of the sandwich train roof using the coupled condition. We found that the gradient of displacement demonstrated a different tendency to each joint marked at figure 6 and 7. When the contact condition was applied in structural analysis, it could simulate the real joining condition for joints between the sandwich composite train roof and the hollow aluminum extrusion frames. However, the coupled condition only shared the nodal degree of freedom (DOF) at the joints, and it could not simulate the real joining condition. So, the contact condition is recommended to analyse the structural behaviour of the real joining parts although it comparatively takes a long time to calculate the results. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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(a) Contact condition

129

(b) Coupled condition

Figure 6:

Contours of deflections at contact and coupled conditions.

Figure 7:

Comparison of deflections at contact and coupled conditions.

Figure 8:

The prototype sandwich train roof.

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130 High Performance Structures and Materials III 3.4.4 Manufacturing of prototype sandwich composite train roof We found that the sandwich composite train roof with the aluminum skin, aluminum honeycomb core and the aluminum stiffener was best choice on the basis of the structural analysis. Figure 8 shows the prototype sandwich composite train roof, which was manufactured using the autoclave moulding process.

4

Conclusions

In this paper, we have obtained the following conclusions. (1) The sandwich composite train roof structure with an aluminum stiffener is beneficial with regard to weight saving and structural integrity. (2) We observed that the quality of the reinforcing material could be altered to achieve the design parameters for natural frequencies of the whole carbody structure. (3) As a result of structural analysis of the sandwich composite train roof structure, there was no striking difference in deflection and stress by changing the quality of the reinforcing material. We found that the advantage of the light weight of the aluminum and the bending stiffness of steel counterbalance each other. Therefore, if we select weight as the priority order, aluminum reinforcing material would be recommended. (4) As a result of analysis of joints classified by contact and coupled conditions, there is no striking difference in natural frequency. However, structural analysis demonstrated a difference. This was caused by the linear assumption of the contact condition in natural frequency analysis. On the other hand, using structure analysis, conditions were calculated by using a nonlinear term at the contact condition and the linear term at the coupled condition. So, structure analysis demonstrated a difference between deflection and stress. We recommended the use of the contact condition for more detailed structural analysis.

References [1] K B. Shin, C G. Kim, C S. Hong., Correlation of accelerated aging test to natural aging test on graphite-epoxy composite materials, Journal of Reinforced Plastics and Composites 22(2003) 849-866. [2] K B. Shin, S H. Hahn., Evaluation of the structural integrity of hybrid railway carriage structures including the aging effects of composite materials, Composite Structure 68 (2005) 129-137. [3] J. Cantrill, R. Mableson. Development and prototyping of a lightweight composite train roof, 14th COMPOSITE Workshop (2003).

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Measurement of the fiber stress distribution during pull-out test by means of micro-Raman spectroscopy and FEM analysis K. Tanaka1, K. Minoshima2 & H. Yamada3 1

Department of Mechanical Engineering, Doshisha University, Japan Department of Mechanical Engineering, Osaka University, Japan 3 Department of Mechanical Engineering, Kyoto University, Japan 2

Abstract A single-fiber pull-out model composite for an aramid/epoxy system was specially designed to measure the stress distribution of the aramid fiber embedded in the matrix using micro-Raman spectroscopy. The stress transfer length of the fiber obtained was about 400-500µm, which was equal to the result of FEM analysis. Just after the initiation and propagation of the fiber/matrix interfacial debonding, the fiber was broken, and the fiber in the matrix had the axial tensile residual stress. The tensile residual fiber axial stress showed the maximum at around the tip of the interfacial debonding. The stress was reduced and became almost equal to zero after being immersed in deionized water at 80 °C for 44h. This behavior agreed with the result of FEM analysis, in which the friction coefficient was introduced in the fiber/matrix interface. Keywords: interfacial properties, micro-Raman spectroscopy, pull-out test, aramid fiber, friction.

1

Introduction

Investigation of the fracture strength and fracture mechanism of the fiber/matrix interface is extremely important, because the mechanical properties of the fiber reinforced composites depend strongly not only on the properties of the fibers and the matrix but also on the fiber/matrix interfacial ones. We carried out the single fiber pull-out tests to evaluate the influence of water absorption on the interfacial properties of aramid/epoxy composite (Tanaka et al. [1]). In this WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06014

132 High Performance Structures and Materials III study, however, the pull-out load, at which the unstable crack propagated through the total embedded fiber length, was used for the measure of the interfacial strength and the stress distribution along the fiber in the matrix was not clarified. Moreover for the study of interfacial crack propagation under fatigue loading, it is extremely important to understand the stress distribution along the fiber close to the crack tip (Minoshima et al. [2]). Raman spectroscopy is a new technique to directly measure the strain or stress distribution along the fiber embedded in a matrix (e.g. Galiotis et al. [3], Patrikis et al. [4], Cervenka et al. [5]). In this work, a single-fiber pull-out model composite for an aramid/epoxy system was specially designed and Raman spectroscopy was used to clarify the difference of stress distribution along the fiber before and after the interfacial debonding.

2

Experimental procedure and FEM analysis

2.1 Specimen preparation and experimental procedure The technique to measure the stress of the fiber using laser Raman spectroscopy is based on the fact that the Raman frequencies are strain (stress) dependent. Therefore, using the calibration curves of the peak wavenumber of Raman spectrum vs. stress of a fiber, the measured Raman peak wavenumber can be converted to an axial stress. In this investigation, the Raman spectra were obtained by means of a laser Raman micro spectrometer (Japan Spectroscopic Co. Ltd, NRS-2000). The 514.5nm line of an argon-ion laser was used and the laser beam was focused to a 1.5µm spot on the fiber by an optical microscope. The aramid fiber used for single fiber tensile tests was Kevlar 49 manufactured by Du Pont, USA. The fiber had an average diameter of 12 µm. Specimens were prepared by following the recommended testing procedure as described in ASTM D3379/JIS R7601. Polyester thin film (Thickness: 100µm) was used for a tab and a single fiber was glued to it, giving a gauge length of 12 mm, as illustrated in fig.1. Quasistatic tensile tests were carried out using a tensile testing machine with a load cell of 1 N Capacity, which can be installed in the laser Raman micro spectrometer. After chucking a specimen, the tab was cut and the load was applied to the fiber. Fiber

2-6

12

Cut off

12

Figure 1:

Single fiber tensile test specimen. All dimensions are in mm.

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The specimen construction for a single fiber pull-out specimen is shown in fig.2. An epoxy resin, Epikote 828 and curing agent Epicure Z (Japan Epoxy Resins Co. Ltd.) were used in 5:1 weight ratio. The resin was put on the center of the tab with a fiber, as illustrated in fig.2(a). Specimens were placed in an oven for two hours at 80°C and then for two hours at 150°C. The epoxy resin was remaining on the tab after curing and the fiber was embedded in the matrix between A and B (fig.2(b)). Pull-out tests were performed using a tensile testing machine in the laser Raman micro spectrometer. Raman spectrum was measured along the fiber for both the embedded fiber and the free fiber at several stress levels, as shown in fig.3. In this study the stress of the fiber adopted was the nominal stress, which was calculated from the value of the load cell.

(a) Specimen construction.

(b) Shape and dimensions of pull-out specimen. All dimensions are in mm. Figure 2:

Single fiber pull-out specimen.

2.2 FEM analysis FEM analysis was conducted using the commercial code ANSYS to estimate the stress distribution during the pull-out test and after fiber/matrix interfacial debonding. Specimen was simplified to the model shown in fig.4(a) and analysis WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

134 High Performance Structures and Materials III was conducted using 2D model without and with the interfacial debonding of 200 µm in length (fig.4(b)). For FEM analysis, thermal expansion during specimen preparation, friction at the fiber/matrix interface and swelling expansion of resin was taken into account. Mechanical properties used in this analysis are shown in table 1 (Hull and Clyne [6], [7], Kawabata et al. [8]). Friction coefficient (µ) and maximum friction stress (τmax) at the interface were calculated from the result of the previous study [1]. µ =0.6 and τmax =11.9 MPa were obtained and used in the analysis. Swelling of the resin was set at 0.6% [1] and the swelling of the fiber was neglected.

Figure 3:

Schematic drawing of the pull-out test procedure (C-C’ cross section of fig.2).

(a) Model of the specimen. Figure 4:

(b) 2D model with boundary conditions.

FEM model with interfacial debonding.

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Table 1:

135

Mechanical properties of the fiber and resin [6-8]: (a) Epoxy resin (Epikote828 + Epicure Z), (b) Fiber (Kevlar 49). (a) Young’s modulus

2.48 GPa

Poisson’s ratio

0.4 60×10-6 °C-1

Coefficient of thermal expansion

(b) Young’s modulus Poisson’s ratio Coefficient of thermal expansion

EL

129.6 GPa

ET

2.49 GPa

νLT

0.62

νTT

0.31

L

-2×10-6 °C-1

T

59×10-6 °C-1

L: longitudinal direction, T: Radius direction.

Shift

Intensity a.u.

10

3

Raman peak of neon lamp

5

0

Figure 5:

0 GPa 2 GPa

1600

1400 1200 –1 1000 Raman shift cm

Examples of the Raman spectrum of Kevlar 49.

Results and discussion

Raman spectra of a Kevlar 49 single fiber without loading and with loading of 2 GPa tensile stress are shown in fig.5. The increase in tensile stress resulted in a clear shift of the spectrum to a lower wavenumber. In this study, a strong band WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

136 High Performance Structures and Materials III around 1615cm-1, which corresponds mainly to the phenyl ring/C-C stretching (Parthenios et al. [9]), was used to evaluate the stress. The influence of the applied load on the Raman peak position for a Kevlar 49 single fiber is shown in fig.6. Four specimens were tested and each result was shown by the same symbol. A linear decrease in the Raman peak position with stress was found, irrespective of the specimens. Therefore, the liner regression line for the measured results shown in the figure was used to estimate the tensile stress of the embedded fiber. The maximum difference between the liner regression line and the tensile stress of each specimen was 0.1 GPa. Therefore our system achieves the stress measurement accuracy of 0.1 GPa. 1616

Peak wavenumber cm

–1

1614 1612 1610 1608 1606 1604 1602

Figure 6:

0

1 2 Tensile stress GPa

3

Relationship between tensile stress of Kevlar 49 single fiber and peak wavenumber.

The stress distribution of the axial tensile stress of the fiber during the pullout test was shown in fig.7 with the result of FEM analysis. The applied nominal stress to the fiber is also plotted in the figure. The origin point of the distance along the fiber is the meniscus point from which the fiber was embedded in the epoxy matrix. The fiber at the positive value of the distance along fiber was embedded in the resin matrix, whereas the fiber at the negative value corresponded to the unembedded free fiber. The stress distribution was measured when the applied stress was 0, 0.47, 0.94, 1.41, 1.89 and 2.36 GPa. The stress of the embedded fiber decreased gradually along the fiber from the meniscus point, which means that the stress was transferred from the fiber to the matrix. Considering from the fact that the fiber axial stress at the distance of 400 µm and that of 500 µm were almost equal, the stress transfer length was about 400-500 µm for the Aramid/epoxy system adopted in our study. This behavior is agreed with the result of FEM analysis. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Before debonding 2.36GPa 1.89GPa 1.41GPa 0.94GPa 0.47GPa Applied stress 0GPa

Fiber axial stress GPa

3

Mesured with Raman FEM spectroscope

Embedded fiber

2

1

0

–1

Figure 7:

137

–200

0 200 400 Distance along fiber µm

Fiber axial stress along the fiber before debonding with the results of FEM analysis.

When the applied stress was increased to 2.65 GPa, the fiber/matrix interfacial debonding was initiated and propagated, and at the same time the fiber was fractured. After this interfacial debonding, the stress distribution in the resin was again measured. The stress distribution after the fiber/matrix interfacial debonding differed from that before the debonding, which is shown in fig.8. Although external stress was not applied to the fiber because of fiber fracture, the fiber axial stress in the debonded region had the residual tensile stress and had the maximum at the distance of around 200 µm. The position of the interfacial crack tip was 180 µm, which was confirmed after Raman measurement by using optical microscope. This crack tip was located almost at the position of maximum residual stress. This residual stress was considered to be caused by the holding of the previous stress by the fiber/matrix interfacial friction acting on the debonded interface. This speculation was supported by the fact that the friction stress plays an important role in the propagation of the fiber/matrix interfacial debonding and the propagation rate of interfacial debonding slowed down and retarded when the a constant fatigue load was applied to the specimen [2]. After debonding, this specimen was kept in dry air (relative humidity: 25%) for 144 h and after that it was immersed in deionized water at 80 °C for 44 h. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

138 High Performance Structures and Materials III Raman spectrum were measured during and after these conditioning. Figure 8 shows the influence of conditioning in dry air and successive hot water on the fiber axial stress along the fiber after debonding. Analytical results are also plotted in the figure. The conditioning in dry air did not affect so much on the stress distribution, but after immersed in hot water the residual stress decreased to almost 0 GPa. The absolute values obtained by the FEM analysis were not equal to the experimental results, but the behavior that the residual stress showed the maximum stress at around the crack tip of interfacial debonding and it decreased by conditioning in hot water agreed with the result of experiment. Experimental results After debonding (without applied stress) Just after debonding 44 h later 48 h later Dry air (RH: 25 %) 96 h later 144 h later In deionized water at 80℃ for 44 h Analytical results Just after debonding Applied stress In deionized water at 80℃ for 44h

Fiber axial stress GPa

3

2

1

0

–1

Figure 8:

4

Embedded fiber

–200

0 200 400 Distance along fiber µm

Influence of conditioning in dry air and successive hot water on the fiber axial stress along the fiber after debonding.

Conclusions

Stress distribution along the fiber of an Aramid fiber/epoxy model composite during the pull-out test was measured using micro-Raman spectroscopy. The stress transfer length of the fiber was about 400-500µm. Although the external stress was not applied to the fiber, after the fiber/matrix interfacial debonding, the fiber in the debonded region had the residual stress caused by the holding of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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139

the previous stress by the fiber/matrix interfacial friction on the debonded interface. This residual stress showed the maximum at the crack tip of the interfacial debonding and the residual stress decreased to almost 0 GPa after immersed in hot water.

References [1]

[2]

[3] [4]

[5] [6] [7] [8]

[9]

Tanaka, K., Minoshima, K. Grela W. & Komai, K., Characterization of the Aramid/Epoxy Interfacial Properties by means of Pull-out Test and Influence of Water Absorption, Composites Science and Technology, 62(16), pp.2167-2174, 2002. Minoshima, K., Tanaka, K., Araki, Y. & Komai, K., Characterization of the propagation of aramid/epoxy interfacial debonding under fatigue loading, JSME Mechanical Engineering Congress, Japan, No.03-1, pp.363-364, 2003. Galiotis, C. Young, R J. Yeung, P H J. & Batchelder, D N., The Study of Model Polydiacetylene/epoxy Composites Part I The Axial Strain in the Fiber, Journal of Materials Science, 19, pp 3640-3648, 1984. Patrikis, A K. Andrews, M C. & Young, R J., Analysis of the Single-Fiber Pull-out Test by means of Raman Spectroscopy: Part I. Pull-out of Aramid Fibers from an Epoxy Resin, Composites Science and Technology, 52, pp 387-3965, 1995. Cervenka, A J. Bannister, D J. & Young, R J., Moisture absorption and interfacial failure in aramid/epoxy composite, Composites Part A, 29A, pp 1137-1144, 1998. Hull D. & Clyne T.W., An Introduction to Composite Materials, Cambridge University Press, 1990. Technical paper of curing agent for Epikote resin, Japan Epoxy Resins Co. Ltd. Kawabata, S., Sera, M., Kotani, T., Katsuma, K., Niwa, M. & Xiaoxin, C., Anisotropic Mechanical Properties of Advanced High Performance Fibers Obtained by Single Fiber Testing System, Proceedings of the Ninth International Conference on Composite Materials (ICCM/9), Madrid, 6, pp.671-677, 1993. Parthenios, J., Katerelos, D. G., Psarras, G. C. & Galiotis, C., Arramid Fibres; a Multifunctional Sensor for Monitoring Stress/Strain Fields and Damage Development in Composite Materials, Engineering Fracture Mechanics, 69, pp.1067-1087, 2002.

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Section 3 Natural fibre composites (Special session organised by T. Katayama and H. Takagi)

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High Performance Structures and Materials III

143

Effect of surface treatment to tensile static and creep properties for jute fiber reinforced composite K. Takemura Department of Mechanical Engineering, Kanagawa University, Japan

Abstract Nowadays, there is a serious environmental problem with waste disposal etc. In fiber reinforced composite, glass fiber reinforced composite is difficult to reuse and recycle. Instead of glass fibers, natural fibers such as bamboo kenaf, jute, and hemp fibers are focused on for environmental friendliness. The mechanical properties for some natural fiber reinforced composites have been studied. The number of papers for green composite is greatly increased. In this study, jute fibers are used as reinforcement because of the huge production and low cost. Polypropylene is used as a matrix because of recycling properties and cost. The strength and creep properties for Jute Fiber Reinforced Plastics (JFRP) are examined. In Japan, natural fibers have been used as fishing nets for a long time. Astringency of a persimmon is used for hemp fiber as a reinforcement and binder. So, the surface treatment is conducted for the fiber using the astringency of a persimmon. The mechanical and creep properties for jute fiber reinforced composite and the effect of surface treatment are investigated. From the results, the following conclusions are obtained. The treatment using the solution of astringency of persimmon is effective to increase mechanical properties. In the case of using the treatment, some densities are effective and 25% solution is most effective. For the creep properties, there is an effect of surface treatment to creep strain. The effect is large when the load is small and the effect emerges in the initial stage. Keywords: jute fiber, polypropylene, surface treatment, strength, creep.

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144 High Performance Structures and Materials III

1

Introduction

Fiber Reinforced Plastics (FRP) is widely used for airplane, automobile parts, fishing pole and any other engineering products. But, recently there are some environmental problems, especially the disposal of glass fibers could not be burned out. So, natural fibers such as jute, bamboo, hemp and flax are focused as reinforcement [1-5]． In Japan, third international workshop for green composite was held in 2005. Many researchers are interested in this issue. Jute is focused from the viewpoint of cost. And it is easy to take jute from general market. So Jute Fiber Reinforced Plastics (JFRP) is used in this study. The static tensile tests for this composite are conducted in order to examine strength and stiffness etc. Natural fiber has been used for fishing net for a long time especially in Japan. Many chemical treatment methods have been used to the natural fibers. Among them, the astringency of a persimmon has been used for the improvement the face of natural fibers. So, the surface treatment method is conducted for this natural fiber using an astringency of a persimmon. So, mechanical and creep properties for the natural fiber reinforced composite and the effect for astringency of a persimmon are investigated.

2

Specimen and experimental procedure

2.1 Specimen Plain woven jute fiber is used as reinforcement. The directions of jute fibers are parallel and perpendicular to a load direction. Polypropylene resin (Shin Kobe Electric Inc. PP-N-AN) is used as matrix. The geometry of specimens is referenced to JIS (Japanese Industrial Standard) 7054. The length of the specimen is about 200mm. The breadth and thickness are about 15mm and 2.5mm respectively. For surface treatment specimen, the plain woven fiber is laid in the solution of an astringency of a persimmon. 2.2 Static tensile test for jute fiber Shimadzu autograph tensile test machine (AG-IS) is used for static tensile test. The test condition is referenced to JIS7054. The surfaces of the composite are observed using Scanning Electron Microscope (SEM) to confirm the effect of surface treatment. 2.3 Creep test for jute fiber Creep Tester 100LER (Toyo Seiki. Co.) is used for this creep test. The test is continued by a failure of specimen. When a specimen does not fail, the creep test stops at 100hours. Three tests can be conducted simultaneously. Creep extension and strain are measured. This test is referenced to JIS7087. The creep load is used for the consideration of static strength. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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145

Results and discussion

3.1 Static test Static load-strain curve for jute fiber is shown in Fig.1. The 50% solution is used for surface treatment. In this figure, the stiffness of treated specimen is lower than that of untreated one. But the maximum stress increases for the treatment. In the case of that using 100% concentration, the stiffness is not improved. So it is found that 50% diluted solution is effective for the strength. Because the fiber is not tensioned at the initial stage, the initial stiffness seems lower. 250

Untreated

Load(N)

200

Treated(50%)

150 100 50 0 0

2

4

6

Strain(%)

Figure 1:

Static tensile results for jute fiber by surface treatment.

Figure 2 shows the stress-strain curves for jute fiber composite. The solution densities for surface treatment are 0, 25, 50, 75, 100%. For the viewpoint of stiffness, 25% solution is most effective. For untreated specimen, the stiffness is lower than that of 25% specimen. But the strength is not low. For 100% solution, the strength and stiffness is worst. So it is understood that the treatment using an astringency of a persimmon is effective for the composite, but the densities are important. For 100% solution, it seems that the fiber has some damage on its surface. The knee point stress is important for its use as application. The knee point stresses for 25% and 50% are almost same. The knee point stresses for 0% and 75% are the same. The knee point stress for 100% solution is obviously lowest. From this figure, 50% solution may seem to be as effective as 25% solution. The photograph of jute fiber is shown in Fig.3. From the surface of treatment, there is much resin on the surface. It is understood that the adhesive strength on the surface is improved. And, the fiber is not straight. It seems that the fiber has ductile properties. So, the stress-strain curve has knee point. And high stiffness is obtained from the surface. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

146 High Performance Structures and Materials III

Figure 2:

Figure 3:

Static Stress-Strain curves of JFRP composite.

Fracture surface of jute fiber composite.

3.2 Creep test Figure 4 shows the relationships between creep strain and time for untreated composite specimen. For smaller load, there seems to be little transit creep region. On the other hand, for bigger load, there is obviously transit creep region. The slopes of constant creep region seem to be same. So, the difference of the curve is emerged in transit region. The transit region continues by 15 hours.

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147

0.04 200N 300N

Strain

0.03

0.02

0.01

0 0

Figure 4:

20

40 60 Time（ｈ）

80

100

Creep curves for untreated JFRP.

Table 1 shows the creep strains for 25% treated and untreated composite specimen. From this table, it is understood that the effect of surface treatment is emerged at lower load. In the case of 300N, the creep strains seem to be same. From that result, there is obviously an effect of surface treatment. But when the load is bigger, the effect is hidden. In the case of 300N, the shapes of the curves are the same. On the other hand, in the case of 200N, the difference is emerging in the transit region. So, the effect of surface treatment is bigger at the initial stage for lower load. At the initial stage, there is the friction on the surface of jute fibre. So, it is thought that the effect of this region is greater. Table 1:

Comparison of creep strains for untreated and 25% treated jute fiber composite at the end of tests. (Unit %). 200N 300N

Untreated 1.4 2.4

Treated 1.2 2.4

Because the effect of surface treatment is obtained in some conditions, the other treatment using katch is conducted. Katch is the obtained from the skin of tree. The result is shown in Fig.5. From this figure, it is understood that any effects are not obtained for 200N case. For 300N case, the strain is bigger. It is not understood that the katch is effective for these conditions. But in any other conditions, it may have any effects. There are any possible other treatment method, so some other methods have to be tried. As the next stage, the surface treatment using water is conducted. Figure 6 shows the relationships between creep strain and time. For 200N, the creep strain is bigger than that of untreated specimen. So, it is not understood that the surface treatment using water is effective for creep properties. For the reason, an astringency of a persimmon has a role to smooth the surface of jute fiber. For only water, the effect of smoothing is not emerging. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

148 High Performance Structures and Materials III 0.04 200N

Strain

0.03

300N

0.02 0.01 0 0

Figure 5:

20

40 60 Time(h)

80

100

Creep curves of katch treated JFRP.

0.04

Strain

0.03 0.02 200N 300N

0.01 0 0

Figure 6:

20

40 60 Time(h)

80

100

Creep curves of water treated JFRP.

The effect of an astringency of a persimmon seems to be tannin. effect of tannin will have to be examined in detail.

4

So, the

Conclusions

In this study, the static and creep properties of jute fiber reinforced composite are examined and the effects of an astringency of a persimmon are also examined. As a result, following conclusions are obtained. (1) There is an effect of astringency of a persimmon to static tensile properties. (2) The effect is dependent on the solutions densities. When the density is 25%, it is more effective. (3) For creep properties, there is an effect of surface treatment to creep strain. The effect is big when the load is small. The effect emerges in the initial stage. (4) The effect of surface treatment using only water is not obtained. On the other hand, in this case, the creep strain becomes big. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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References [1] [2] [3] [4] [5]

D. Nabi Saheb and J.P. Jog, Natural Fiber Polymer Composite: A Review, Advances in Polymer Technology, Vol.18, No.4, 351 (1999). H. Takagi and Y. Ichihara, JSME International Journal, Series A, 47,551 (2004). K. Goda et al., Proc. of the 2nd Int. Workshop on “Green Composites”, 96(2004). B. Singh et al., Composite Science and Technology, 60,571-589(2000). J. Gassan, at. al., Composite Science and Technology, 60,2857 (2000).

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151

Effects of forming conditions on mechanical properties of resinless bamboo composites H. Takagi1 & H. Mori2 1

Department of Mechanical Engineering, The University of Tokushima, Japan 2 Graduate School of Engineering, The University of Tokushima, Japan

Abstract This paper deals with the development of resinless, unidirectional bamboo fibre-reinforced eco-composites. Resinless bamboo composites were fabricated by the hot-pressing of unidirectional bamboo fibres extracted by a steam explosion method. The effects of forming conditions on their mechanical properties were investigated. The forming temperature was changed every 20°C from room temperature to 180°C, and the forming pressure used was 10 MPa and 50 MPa. Except for the conditions below 60°C, their tensile strength and fracture elongation decreased with increasing forming temperature. Average tensile strength and Young’s modulus of the resinless bamboo composites fabricated at 120°C and 50 MPa were 322 MPa and 37 GPa respectively. Keywords: natural fibre, bamboo, resinless, binderless, unidirectional composites, tensile strength, modulus.

1

Introduction

Biodegradable composites that are also known as “green” composites are mostly made from natural fibers and a biodegradable resin. Therefore the composites are one of the most environment-friendly materials, because they are finally biodegradable, and thus can be resolved into carbon dioxide and water through the ideal decomposition stage. Hence, many researchers have been conducting the research on biodegradable composites, and especially on their mechanical properties [1-7]. However, most of the biodegradable resins are more expensive than conventional plastics, such as PP and PE. In addition, extra energy is needed WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06016

152 High Performance Structures and Materials III when the biodegradable resin is made from natural plants, such as corn and potato. Recently much attention has been paid to the characteristics of engineered wood powder products, which are consolidated under a plastic condition derived from a hot-pressing at high-temperature and high pressure [8-11]. This treatment is called “plasticizing.” The wood-plastics made from waste wood powders have also been developed [11], and they have a good prospect of new ecomaterials. However, their strength is low compared with that of conventional engineering plastics. Thus, they need to be strengthened in order to extend their application field. In recent years, because of a domestic demand of bamboo decreases in Japan, many bamboo forests have been going to ruin. The ruin of bamboo forests causes several environmental problems such as landslides and encroachments against other plants; therefore, much autonomy wants to cut down the bamboo and to utilize them as a raw material of commercial products. To overcome these problems, in this paper, we have tried to fabricate resinless, unidirectional bamboo fiber composites by hot-pressing of steam-exploded bamboo fibers (abbreviated as bamboo fibers hereafter) without using any adhesives and binders. Particular focus is placed on the effect of hot-pressing conditions on their tensile properties.

2

Experimental method

2.1 Bamboo fibers Bamboo fibers were extracted from raw bamboo (phyllostachys pubescens) stem by using a steam explosion method. The condition of the steam explosion was 180°C and 40 min. After the steam explosion, bamboo stem became fragile, and long bamboo fibers were easily taken out from the bamboo stem by hand. Soft cells attached on the surface of extracted bamboo fibers were removed by wiping with a wet cloth. 2.2 Preparation of unidirectional resinless bamboo composites First, the bamboo fibers were cut into about 100 mm in length. Then, the bamboo fibers were set into a metallic mould and finally hot-pressed at various temperatures and pressures for 10 min. The dimensions of specimens are about 10 mm × 100 mm × 1 mm. 2.3 Tensile tests Tensile tests were carried out using an Instron universal test machine (Model 5567). The tensile tests were performed at a crosshead speed of 1.0 mm/min, with a gage length of 25 mm. Aluminium tabs in 2 mm thickness were glued at the both ends of the tensile specimen to prevent damages caused by gripping. The shape and dimensions of the tensile specimen are shown in fig. 1.

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153

100

Figure 1:

3

Sample

5

5

10

Aluminium tab

30

1

5

Shape and dimensions of tensile specimens (unit: mm).

Results and discussion

Under the moulding condition of R.T. and 10 MPa, bamboo fibres did not adhere to each other; therefore, it was impossible for this sample to carry out the tensile test. However resinless bamboo composites can be fabricated by hot-pressing at temperatures over 80°C. The colour of surface changed from dark brown to black with increasing the moulding temperature. This colour change might be attributed to the chemical reaction of lignin comprised in bamboo fibres. Figure 2 shows typical stress-strain curves of resinless bamboo composites moulded at 50 MPa as a function of moulding temperature. The initial slope of the curve (corresponding to Young’s modulus) becomes steeper with increasing the moulding temperature, and simultaneously tensile strength also increases. The stress of composites moulded below 180°C gradually decrease in the latter part of the deformation. This phenomenon should be responsible for the cumulative fibre fracture. However, the stress-strain curve for the composites moulded at 180°C becomes a straight line with the same steep slope as 120°C. At the same time, fracture behaviour is brittle, and fracture elongation becomes considerably small value less than 0.005. In the case of moulding pressure of 10 MPa, almost the similar stress-strain curves are obtained as show in fig. 3. The relationship between tensile strength of bamboo composites and moulding temperature is shown in fig. 4. The tensile strength reaches a maximum value of about 300 MPa at 120°C at both of the moulding pressures. It can be seen from this graph that the bamboo composites become brittle at temperatures over 120°C, and bamboo fibres do not adhere to each other at temperatures below 120°C. The relationship between Young’s modulus and moulding temperature is presented in fig. 5. Young’s modulus initially increases with the moulding temperature, and then levels off at temperatures above 120°C. This rise in Young’s modulus is consistent with the increase in the density of samples. This is because modulus as well as stress also increases with the moulding temperature. Figure 6 shows the relationship between fracture elongation and moulding temperature. The fracture elongation is almost constant below 120°C, and WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

154 High Performance Structures and Materials III decreases with increasing the moulding temperature thereafter. This drop in fracture elongation might be attributed to the increased brittleness of bamboo fibres derived from high temperature exposure [12].

500 R.T. 80C 120C 180C

Stress (MPa)

400 300 200 100 0 0 Figure 2:

0.01

0.02 0.03 Strain

0.04

0.05

Stress-strain curves of resinless bamboo composites moulded at 50 MPa.

500 80C 120C 180C

Stress (MPa)

400 300 200 100 0 0 Figure 3:

0.01

0.02 0.03 Strain

0.04

0.05

Stress-strain curves of resinless bamboo composites moulded at 10 MPa.

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155

Tensile strength (MPa)

600 10MPa 50MPa

500

400 300 200 100 0

Figure 4:

0

150 200 50 100 o Moulding temperature ( C)

Variation of tensile strength of resinless bamboo composites with moulding temperature.

Young's modulus (GPa)

80 60 40 20 0

Figure 5:

10MPa 50MPa

0

50 100 150 200 o Mouding temperature ( C)

Variation of Young’s modulus of resinless bamboo composites with moulding temperature.

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156 High Performance Structures and Materials III

Fracture elongation (%)

2.5

1.5 1.0 0.5 0

Figure 6:

4

10MPa 50MPa

2.0

0

50 100 150 200 o Moulding temperature ( C)

Variation of fracture elongation of resinless bamboo composites with moulding temperature.

Conclusions

The tensile properties of resinless bamboo composites were investigated. The major results obtained are summarized as follows: 1) Resinless bamboo composites can be made by hot-pressing at temperatures over 80°C. The optimum moulding temperature is 120°C among the conditions investigated. 2) Young’s modulus of resinless bamboo composites initially increases with the moulding temperature, and then levels off at temperatures above 120°C. This rise in Young’s modulus is consistent with the increase in the density of samples. 3) The fracture elongation of resinless bamboo composites is almost constant below 120°C, and decreases with increasing the moulding temperature thereafter. This drop in fracture elongation might be attributed to the increased brittleness of bamboo fibres.

Acknowledgement The authors greatly acknowledge the partial financial support provided by the president of The University of Tokushima in 2005.

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References [1] [2]

[3] [4] [5]

[6] [7] [8] [9] [10]

[11] [12]

Wollerdorfer, M. & Bader, H., Influence of natural fibres on the mechanical properties of biodegradable polymers, Industrial Crops and Products, 8(2), pp. 105–112, 1998. Luo, S. & Netravali, A. N., Interfacial and mechanical properties of environment-friendly “green” composites made from pineapple fibers and poly(hydroxybutyrate-co-valerate) resin, Journal of Materials Science, 34, pp. 3709–3719, 1999. Mohanty, A. K., Misra, M. & Hinrichsen, G., Biofibers, biodegradable polymers and biocomposites: An overview, Macromolecular Materials and Engineering, 276/277, pp. 1–24, 2000. Shibata, M., Takachiyo, K., Yosomiya, R. & Takeishi, H., Biodegradable polyester composites reinforced with short abaca fiber, Journal Applied Polymer Science, 85(1), pp. 129–138, 2002. Takagi, H. & Ochi, S., Characterization of high-strength “green” composites using Manila hemp fibers and starch-based resin, Proceedings of the Third Japan-Canada Joint Conference on New Applications of Advanced Composites (JCJC-III), pp. 19–27, 2003. Nishino, T., Hirao, K., Kotera, M., Nakamae K. & H. Inagaki, H., Kenaf reinforced biodegradable composite, Composites Science and Technology, 62, pp. 1281–1286, 2003. Takagi, H. & Ichihara, Y., Effect of fiber length on mechanical properties of “green” composites using a starch-based resin and short bamboo fibers, JSME International Journal, Series A, 47(4), pp. 551–555, 2004. Hillis, W. E. & Rozsa, A. N., High temperature and chemical effects on wood stability, Wood Science and Technology, 19, pp. 57–66, 1985. Anglès, M.N., Reguant, J., Montané, D., Ferrando, F., Farriol, X. & Salvadó, J., Binderless composites from pretreated residual softwood, Journal Applied Polymer Science, 73(12), pp. 2485–2491, 1999. Salvadó1, J., Velásquez1, J. A. & Ferrando, F., Binderless fiberboard from steam exploded Miscanthus Sinensis: optimization of pressing and pretreatment conditions, Wood Science and Technology, 37(3-4), pp. 279–286, 2003. Miki, T., Takakura, N., Iizuka, T., Yamaguchi K. & Kanayama, K., Production of long rods by sequential extrusion of wood powders, Journal of Materials Processing Technology, 140, pp. 407–412, 2003. Ochi, S., Takagi, H. & Niki, R., Mechanical properties of heat-treated natural fibers, Proc. of 1st High Performance Structures and Composites, pp. 117–125, 2002.

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159

Compression moulding of jute fabric reinforced thermoplastic composites based on PLA non-woven fabric T. Katayama1, K. Tanaka1, T. Murakami1 & K. Uno2 1 2

Department of Mechanical Engineering, Doshisha University, Japan Marubeni Intex Co., Ltd., Japan

Abstract To solve the problem of the large environmental burden in the disposal of FRP (Fibre Reinforced Plastics), natural fibre and biodegradable resin have received a lot of attention as easily degradable materials in the natural environment. To enlarge the usage of the composites based on natural fibre and biodegradable resin, cost reduction and enhancing its strength are essential subjects. In this study, in order to develop the environment friendly biodegradable composites that have high performance of strength, rigidity and productivity, the non-woven stacking method was proposed and the influence of moulding conditions on impregnating property and mechanical property of Jute Fabric Reinforced Thermoplastic composites (JFRTP) based on PLA non-woven fabric was discussed. It takes several minutes for PLA to impregnate into fibre bundles after the material reaches preset moulding temperature due to its high melt viscosity and twist of fibre bundles. The properly moulded JFRTP specimen reached about 115 MPa in bending strength, which is comparable to GFRTP with 20 wt % long fiber made by injection moulding. Keywords: green-composite, natural fibre, jute fibre, PLA, non-woven fabric, compression moulding, bending property.

1

Introduction

While growing interest in environmental issues, environmental burden of FRP disposal has been a big problem in recent years. The conventional materials for FRP, such as glass fibre or carbon fibre as reinforcement, and epoxy or polyamide as matrix are hardly degraded in natural environment and hardly WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06017

160 High Performance Structures and Materials III incinerated due to their flame resistance. To settle this problem, the composites based on natural fibre and biodegradable resin, called ‘Green-composite’, have got a lot of attention as easily degradable materials in natural environment. To enlarge its usage, reducing the manufacturing cost and enhancing the strength are essential subjects. For the reinforcement of the green composite, natural fibres, such as bamboo fibre, ramie, flax, kenaf jute, are used. Among these natural fibres, jute is one of superior materials due to its low cost, large amount of production and high specific strength and has got a lot of attention (Acha et al. [1]). Polylactic acid (PLA) is usually used for the matrix of the green composite, because of its high strength, rigidity, melting point and productivity compared with other biodegradable resins. In the case of using PLA as matrix, film type PLA were usually used and they were laminated and moulded under compression with sheet type of reinforcing fibres, which is called ‘Film Stacking’, one of the well-applied impregnating methods of FRTP (Fibre Reinforced Thermoplastics) (Plackett et al. [2], Khondker et al. [3]). However, it is difficult to change the thickness of the film and handling of the film has also some problem, therefore a new impregnating method is expected to be developed. Non-woven fabric thermoplastic, which is used well for clothing, filter, diaper and so on, is applied as matrix resin in this study. Using non-woven fabric as matrix, mass per unit area or thickness can be easily changed. In this study, for the purpose of improving the productivity of not only Greencomposite but also conventional FRTP, we propose a impregnating ‘Non-woven Stacking’ method replaced by Film Stacking, in which reinforcing fibre sheets and matrix resin in the state of non-woven fabrics are laminated and moulded under compressive stress. Jute Fabric Reinforced Thermoplastics (JFRTP) using laminated jute plain woven fabrics and PLA non-woven fabrics were moulded under compressive stress and the influence of moulding time and moulding pressure on density and bending property were discussed.

2

Experimental procedure

2.1 Materials and method Plain woven jute fabric (fig.1) of 430g/m2 is used for reinforcement and PLA non-woven fabric sheet (fig.2) made by melt-brown method is used as matrix in this study. The density, melting point and fibre diameter of PLA are 1.25g/cm3, 140 ºC, 2 - 3µm, respectively.

10mm

Figure 1:

Plain woven Jute fabric.

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High Performance Structures and Materials III

50µm

10mm

(a) Figure 2:

161

(b)

PLA non-woven fabric (a) Macroscopic photography and (b) Microscopic photography.

PLA non-woven fabric sheet is manufactured by melt-blown process. Figure 3 shows the schematic view of the melt-blown process. Melted resin with hot air is blown from the nozzles and non-woven sheet will be formed on the conveyer. Nozzle Melting resin Non-woven fabric Conveyer

Figure 3:

Meltblown process.

Compression moulding is carried out by vacuum press moulding machine. Jute fabrics in 450 mm × 450 mm are stacked between four PLA non-woven fabrics in 450 mm × 450 mm shown in fig.4. All the composites are processed at 48 % fibre volume fraction. After setting the materials on the die, moulding is carried out under vacuum condition of 0.1MPa. All the moulding conditions are shown in table 1. After moulding, the composites are air-cooled at laboratory temperature. To evaluate the impregnation, density of the moulded product is measured and compared with theoretical density that is calculated under the assumption of no void in the composites. 2.2 Mechanical testing Bending tests were conducted under a constant displacement rate of 1mm/min using an Instron 5566 universal testing machine. The length and width of the specimen are 60 mm and 15 mm, respectively and three point bending tests were conducted at span length of 40 mm. The thickness of the specimen depended on the moulding condition and varied from 1.8 mm to 3.1 mm.

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162 High Performance Structures and Materials III Non-woven fabric of matrix resin

Figure 4: Table 1:

Laminated condition. Molding pressures and times.

Pressure (MPa) Time (s)

3

Reinforcing fabric

1.0 20

40

60

2.0

3.0

120 180

4.0 300

600

Results and discussion

3.1 Impregnation The relationship between density and moulding time is shown in fig. 5. Before 60 s, the density increases drastically irrespective of the moulding pressure. After 300 s, however, it become stable and the density under moulding pressure of 3.0 MPa almost equal to that under 4.0 MPa. Figure 6 shows the cross section of the moulding product under the moulding pressure of 1.0MPa, whose results shows more clearly the difference of moulding time on the impregnating state than other moulding pressure. Before 180 s, many visible voids are observed. After 300 s, however, matrix is impregnated into fibre bundles. This observation corresponds with the results of measured density change shown in fig. 4. Considering the fact that the higher moulding pressure than 3.0 MPa and longer moulding time than 300 s don’t WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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163

affect so much on the density and impregnation, moulding pressure of 3.0MPa and moulding time of 300 s are optimum moulding value for the material used in our study. 2.0 1.8

3

Density (g/cm )

1.6 1.4 1.2 1.0 0.8

Theoretical Density 1.0 MPa 2.0 MPa 3.0 MPa 4.0 MPa

0.6 0.4 0.2 0.0

Figure 5:

0

100 200 300 400 500 600 700 Molding Time (s)

Relationship between density and moulding time.

Figure 7 shows the temperature change at the centre of the material during moulding under the moulding pressure of 1.0MPa. After 10 s from the start of moulding, the material reaches the melting point of PLA (140 ºC) and after 60 s it becomes stable. In spite of the higher temperature than the melting point of PLA after 10 s, it takes almost 300 s for PLA to impregnate into the fibre bundles. The high melt viscosity of PLA and the twist of jute fibre bundles are considered to be the reason for this. 3.2 Bending property Figures 8 and 9 show the influence of moulding condition on the bending strength and the bending modulus, respectively. Before 300 s, the bending strength and modulus drastically increase but after that they become stable. This behaviour is similar to the result of density shown in fig.5. From the viewpoint of the mechanical properties, moulding pressure of 3.0MPa and moulding time of 300 s are optimum moulding value. Figure 10 shows the stress–displacement curve of bending test of the specimen moulded under pressure of 3.0 MPa. In the case of specimen moulded for 300 s and 600 s, the bending stress is decreased rapidly after reaching the maximum value and they fractured in brittle manner. This behaviour is one of the evidence that the resin was impregnated into the fibre bundles. Figures 11 and 12 show the comparison of the bending strength and modulus between our JFRTP and other materials. Injection moulded composite of glass fibre reinforced polypropylene with 20wt% long fibre and jute fibre reinforced polypropylene with 51wt% long fibre (Tanaka et al. [4]) are shown in these figures. The PLA/JF product in our study has better bending properties than GFRTP with 20 wt % long fiber. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

164 High Performance Structures and Materials III

500µm

500µm

(a) 20s

(b) 40s

500µm

500µm

(c) 1min

(d) 2min

500µm

500µm

(e) 3min

(f) 5min

500µm

(g) 10min Figure 6:

Cross section photos of the moulded product (Moulding pressure: 1.0MPa).

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Molding Temperature ( ℃ )

High Performance Structures and Materials III 240 220 200 180 160 140 120 100 80 60 40 20 0

Figure 7:

165

100

Temperature 1.0MPa).

profile

200 300 400 Molding Time (s)

during

500

moulding

600

(moulding

pressure:

Bending Strength (MPa)

140 120 100 80 60 1.0 MPa 2.0 MPa 3.0 MPa 4.0 MPa

40 20 0

Figure 8:

4

0

100

200 300 400 500 Molding Time (s)

600

Relationship between bending strength and moulding time.

Conclusions

To develop the environment friendly biodegradable composites that have high strength and high productivity, “non-woven stacking” method was proposed and Jute Fabric Reinforced Thermoplastic composites (JFRTP) based on PLA nonwoven fabric were moulded and the influence of moulding conditions on impregnation and mechanical property was discussed. It takes several minutes for PLA to impregnate into fibre bundles after the material reaches preset moulding temperature, due to its high melt viscosity and twist of fibre bundles. The properly moulded JFRTP specimen reached about 115 MPa in bending strength, which is comparable to GFRTP with 20 wt % long fiber made by injection moulding. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

166 High Performance Structures and Materials III 7.0

Bending Modulus (GPa)

6.0 5.0 4.0

2.0 1.0 0.0

Figure 9:

1.0 MPa 2.0 MPa 3.0 MPa 4.0 MPa

3.0

0

100

200 300 400 500 Molding Time (s)

600

Relationship between bending modulus and moulding time.

Bending Stress (MPa)

140 120 20s 40s 60s 120s 180s 300s 600s

100 80 60 40 20 0

0

Bending Strength (MPa)

Figure 10:

1

2

3 4 5 6 7 8 Displacement (mm)

9 10

Relationship between bending stress and displacement.

140 120 100 80 60 40 20 0 PP/GF 20wt% PP/JF 51wt% PLA/JF 53wt% Long Fibre [4] Long Fibre [4] Plain Weave

Figure 11:

Bending strength of the 3 types of FRTP [4].

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Bending Modulus (GPa)

High Performance Structures and Materials III

167

8 7 6 5 4 3 2 1 0 PP/GF 20wt% PP/JF 51wt% PLA/JF 53wt% Long Fibre [4] Long Fibre [4] Plain Weave

Figure 12:

Bending modulus of the 3 types of FRTP [4].

Acknowledgement This study was partially supported by the Academic Frontier Research Project on “NewFrontier of Biomedical Engineering Research” of Ministry of Education, Culture, Sports, Science and Technology.

References [1] [2] [3] [4]

B.A. Acha, N.E. Marcovich, M.M. Reboredo, Journal of Applied Polymer Science, vol.98, No.2, (2005), pp.639-650. D. Plackett, T.L. Andersen, W.B. Pedersen, L. Nielsen, Composites Science And Technology, vol.63, No.9, (2003), pp.1287-1296. O.A. Khondker, U.S. Ishiaku, A. Nakai, H. Hamada, Journal of Polymers and the Environment, vol.13, No.2, (2005), pp.115-126. T. Tanaka, N. Tashiro, Japan Plastics, vol.56, No.7 (additional volume), (2005), pp.95-104.

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Quality control of fibers end-milled from bamboo pipe using spiral tool path K. Ogawa1, E. Aoyama2, T. Hirogaki2, Y. Tomioka2 & H. Nakagawa1 1

Department of Mechanical Systems Engineering, The University of Shiga Prefecture, Japan 2 Department of Mechanical Engineering, Doshisha University, Japan

Abstract The utilization of unused forest products is beneficial, and therefore the application of natural fibers to FRP products has been proposed. Bamboo fibers have attracted particular attention because bamboo has the fastest growth rate among various types of renewable natural fibers. Moreover, bamboo fibers have high specific strength and stiffness appropriate for structural materials. In the present study, we explain a new method designed to obtain high-quality bamboo fibers by effectively end-milling them with a machining center (MC) using numerical control (NC) and an automatic tool changer (ATC), which enables the manufacture of a wide variety of products. NC programs encoding a spiral tool path were used for the bamboo pipe in order to obtain bamboo fibers effectively. The fiber length and diameter were evaluated at various feed rates, cutting speeds, and depths of cut in the fiber direction. Microscopic observation showed that the length of the fibers obtained can be controlled by selecting the depth of cut along the bamboo fiber direction and the diameter can be controlled by adjusting the feed of the end-mill center, which is determined by spindle speed and feed speed of end-mill under a constant end-mill diameter. Thus the desired fiber shape can be controlled with consistently high accuracy using end-milling by adjusting the cutting conditions. Moreover, using spiral tool path end-milling with a straight cutting edge tool can efficiently acquire high quality straight bamboo fibers with no thermal damage. Keywords: bamboo fiber, end-mill, machining center, spiral tool path.

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170 High Performance Structures and Materials III

1

Introduction

Recently, industrial products made of fiber reinforced plastic (FRP) such as glass fiber-reinforced plastic (GFRP), are being utilized extensively because they have high specific tensile strength (that of GFRP is as high as steel) but are significantly lighter. The use of such materials may have a negative environmental impact because of problems associated with their disposal or destruction. Conversely, the utilization of unused forest products is beneficial to the environment, and therefore a method of acquiring and applying natural fibers to FRP products has been proposed [1-7]. Bamboo fibers, in particular, have attracted attention because bamboo has the fastest growth rate among the various types of renewable natural fibers, and is grown throughout Japan. Moreover, bamboo fibers have high specific strength and stiffness as appropriate for structural materials [8, 9]. Bamboo fibers have been obtained from the wood-like plant through various methods, such as crushing or heat steaming the bamboo stems. Using these methods, however, it is difficult to control the diameter, length and other dimensions, as well as to obtain large quantities of a consistent, uniform shape as necessary for high-quality fiber components in industrial materials [10]. This paper attempts to obtain high quality bamboo fibers by endmilling with a machining center (MC) as an alternative to previous conventional methods. The MC makes it possible to manufacture many kinds of products and the proposed method boasts some advantages over conventional methods. The shape of the bamboo fibers can be controlled because all the machining center processes are controlled by digital program. Moreover, the spiral tool path should produce uniform bamboo fibers and allow accurate control of the shape of fibers by variation of the cutting conditions.

2

Experimental method

2.1 Material Naturally growing Mousouchiku bamboo was used as the base material. The bamboo was cut using a metal saw into 100 mm-long bamboo pipes in order to remove the bamboo joints. The bamboo pipes were then set on the table in an MC and end-milled. The waste bamboo obtained was used as the bamboo fibers used in the experiments. 2.2 Machining equipment and conditions A ROBODRILL (Type: T14iDs, FANUC Co., Ltd.) was used as the MC for milling. The end-mill tool used in the experiments was a square type with two straight cutting edges, and made of high-speed steel without coating films on the tool surface. The end-mill diameter was 6 mm. Table 1 shows the machining conditions. The tool followed a spiral path from the perimeter to the center with a cutting depth of 50 µm per one cycle in the radius direction of bamboo, as shown in Fig. 1. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 1:

171

Machining conditions.

Spindle speed S (rpm) Feed speed F (mm/min) Depth of cut in radius direction Rv (mm) Depth of cut in bamboo fiber direction Ad (mm) End-mill radius r (mm) Number of end-mill cutting edge Z

2500, 5000, 10000, 15000, 20000 250, 500, 750, 1000, 2000 0.05 2.5, 5, 8, 10 3 2

Spiral tool path

End-mill

Bamboo pipe

Figure 1:

Spiral tool path. End-mill

Bamboo pipe

Figure 2:

Convex contour cutting.

2.3 Theoretical cutting configuration Figure 2 shows a diagram of the geometrical configuration. Here, fr is the feed of the cutting edge (mm/tooth), fen is the feed of the end-mill center (mm/tooth), r is the end-mill radius (mm), Rv is the depth of cut in the radius direction per cycle WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

172 High Performance Structures and Materials III (mm), Rb is the bamboo radius after cutting (mm), α is the angle between the fr and fen directions (rad), and αen is the cutting engagement angle (rad). The tool path approaches to a circumference path because the depth of cut in the bamboo pipe radius direction is very small comparing with the end-mill radius and the bamboo pipe radius. The cutting arc length L and the maximum thickness h are obtained respectively based on the geometrical configuration shown in Fig. 2. (1) L = r ⋅ sin α en

h = f r ⋅ sin(α en + α )

(2)

Here, the depth of cut in the radius direction per cycle Rv is very small compared with the end-mill radius r and the bamboo pipe radius Rb. As a result, the cutting engagement angle αen is extremely small, so as to be nearly zero. Therefore, eqn (3) can be obtained from eqn (1) using the approximation sinαen≒αen. (3) L = r ⋅ α en Eqns (4) and (5) can be obtained by the geometrical relation of the triangle that consists of the bamboo pipe center point, the end-mill center point, and the sharp end of the end-mill cutting edge, as shown in Fig. 2. (4) r ⋅ sin α en = (Rb + Rv ) ⋅ sin α

r ⋅ cosα en + (Rb + Rv ) ⋅ cosα = Rb + r

(5)

Eqn (6) is obtained from the relation that the ratio of the feed of the cutting edge fr and the feed of the end-mill center fen is equivalent to the ratio of the distance from the spiral path center to the end-mill center point and to the sharp end of the end-mill cutting edge.

f en R +r = b fr Rb + Rv

(6)

Here, the eqn (7) is conducted by eqns (2), (4), (5), and (6).

h = f en ⋅ sin α en

(7)

Therefore, eqns (8) and (9) can be obtained by replacing the parameters utilized in this study shown in Table 1 into eqns (3) and (7).

r − Rv r

(8)

Rv  R  2 − v  r  r 

(9)

L = r ⋅ arccos

h=

F S ⋅Z

2.4 Evaluation method of bamboo fiber quality The length and diameter of one hundred randomly selected bamboo fibers were measured using an optical microscope. Then aspect ratio, calculated by dividing the fiber length by the diameter, was then calculated to give an indication of the length and narrowness of the obtained fibers. Here, a theoretical fiber diameter Dth seems (L+h)/2 in this method. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

15 Theoretical

200 100

Fiber length (mm)

Fiber diameter ( µ m)

300

0

10 5 0

0

100

200

300

400

0

Cutting speed (m/min)

100 200 300 400 Cutting speed (m/min)

(a) Fiber diameter

(b) Fiber length Coefficient of variation

100 Aspect ratio

80 60 40 20 0 0

0.5

3

Fiber diameter Fiber length Aspect ratio

0.4 0.3 0.2 0.1 0.0

100 200 300 400 Cutting speed (m/min)

(c) Aspect ratio Figure 3:

173

0

100

200

300

400

Cutting speed (m/min)

(d) Coefficient of variation

Change of bamboo fiber shape (fen=0.05 mm/tooth, Ad =10 mm).

with

cutting

speed

Results and discussion

3.1 Influence of cutting speed on acquired bamboo fiber quality Figures 3(a), 3(b) and 3(c) show the changes in fiber shape with cutting speed under a constant feed of the end-mill center of 0.05 mm/tooth and a cut of depth in the fiber direction of 10 mm. Figure 3(d) shows the coefficient of variation of the acquired bamboo fiber shapes at various cutting speeds. The coefficient of variation is calculated by dividing the standard deviation by the mean value. Here, the cutting speed V (m/min) and the feed of end-mill center fen can be calculated by eqns (10) and (11), respectively.

V = 2πr ⋅ S F f en = S ⋅Z WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(10) (11)

174 High Performance Structures and Materials III Therefore, the feed speed F was adjusted for various spindle speed in order to match the constant feed of the end-mill center. Under these conditions, the endmilled sections of the fibers are theoretically uniform. That is, L and h are 548.5 µm and 9.1 µm, respectively, from eqns (8) and (9). As a result, the theoretical diameter is about 276.5 µm.

End-mill

Bamboo pipe

(a) Optical view Figure 4:

37 ˚C

40 ˚C

(b) V=94 m/min (S=5000 rpm)

(c) V=188 m/min (d) V=377 m/min (S=10000 rpm) (S=20000 rpm)

Cutting temperature as monitored (fen=0.05 mm/tooth, Ad =10 mm).

67 ˚C

by

thermography

The results verify that the diameter of the end-milled bamboo fiber decreases with cutting speed until about 200 m/min though obtained fiber diameter has good agreement with theoretical one under low cutting speed. However, above this speed, the fiber diameter remains almost constant. On the other hand, the fiber length is almost constant, at 10 mm, for any cutting speed. As a result, the aspect ratio increases with cutting speed until about 200 m/min and then plateaus at a fiber diameter of about 150 µm at the cutting speeds above 200 m/min. Higher cutting speed with greater feed speed improves the machining throughput. However, the coefficient of variation for the fiber diameter, fiber length and aspect ratio show a similar tendency of increasing with cutting speed. Therefore, a cutting speed of about 200 m/min is suitable in order to obtain high efficiency machining and maintain low variation. It seems that the bamboo fiber diameter is controlled by cutting speed under about 200 m/min. On the other hand, the increased cutting heat generated may damage the bamboo fiber in end-milling; the cutting temperature increases with cutting speed. The cutting temperature in steel rises to 800 ˚C or higher. The melting points of bamboo pipe components of cellulose, hemi cellulose, and lignin are about 240, 180, and 420 ˚C, respectively. The temperature in bamboo cutting was monitored by thermography as a non-contact method. It can be seen that the temperature in bamboo cutting also increases with cutting speed as shown in Fig. 4. However, up to a cutting speed of 200 m/min the temperature grows to no more than about 70 ˚C. It can be concluded that under these conditions bamboo fibers end-milled from bamboo will sustain no thermal damage. 3.2 Influence of cut of axial depth along the fiber direction on acquired bamboo fiber quality Figures 5 shows the microscopic images of end-milled bamboo fibers under various axial cut depths. It can be seen that uniform, straight fibers with no WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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175

thermal damage are acquired. However, fibers end-milled appear as powders with smaller axial cut depth.

5mm

(a) Ad =2.5 mm Figure 5:

(b) Ad =5 mm

(c) Ad =10 mm

Microscopic images of fibers (fen=0.05 mm/tooth, V=188 m/min).

200

bamboo

100

10

0

5 0

0

5 10 15 Axial depth of cut (mm)

0

5 10 Axial depth of cut (mm)

(a) Fiber diameter 80 60 40 20 0 0

5 10 15 Axial depth of cut (mm)

(c) Aspect ratio Figure 6:

15

(b) Fiber length Coefficient of variation

100 Aspect ratio

from

15 Theoretical

Fiber length (mm)

Fiber diameter ( µ m)

300

end-milled

0.5 Fiber diameter Fiber length Aspect ratio

0.4 0.3 0.2 0.1 0.0 0

5

10

15

Axial depth of cut (mm)

(d) Coefficient of variation

Change of bamboo fiber shape with depth of cut in fiber direction. (fen=0.05 mm/tooth, V=188 m/min).

Figures 6 shows the changes in fiber shape with axial depth of cut under a constant feed of the end-mill center of 0.05 mm/tooth and a cutting speed of 188 m/min, the conditions determined to be optimum for acquiring high quality bamboo fibers efficiently as described in section 3.1. Moreover, L and h are WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

176 High Performance Structures and Materials III

15

500 400 300 200

Ad=2.5mm Ad=5mm Ad=10mm Theoretical

100 0

Fiber length (mm)

Fiber diameter ( µ m)

respectively 548.5 µm and 9.1 µm, respectively, under the same condition of constant fiber section for various cutting conditions as given in section 3.1.

Ad=2.5mm Ad=5mm Ad=10mm

10 5 0

0.0

0.1 0.2 0.3 Feed of end-mill center (mm/tooth)

0.0

(a) Fiber diameter

Aspect ratio

100

0.1 0.2 0.3 Feed of end-mill center (mm/tooth)

(b) Fiber length Ad=2.5mm Ad=5mm Ad=10mm

80 60 40 20 0 0.0

0.1 0.2 0.3 Feed of end-mill center (mm/tooth)

(c) Aspect ratio Figure 7:

Change of bamboo fiber shape with feed of the end-mill center and the depth of cut in fiber direction (fen=0.05 mm/tooth, V=188 m/min).

It is clear that the fiber diameter measured shows almost constant value, which is almost half of theoretical one, with the axial depth of cut as shown in Fig. 6(a). On the other hand, the fiber length increases with the axial depth of cut as shown in Fig. 6(b). The result shows that the fiber length agrees very well with the axial depth of cut, as the data points lie almost exactly on a line of slope 1 (Fig.6(b)). As a result, aspect ratio increases with axial depth of cut as shown in Fig. 6(c). Fibers with a large aspect ratio are desirable to confer superior material properties to the FRP. It may be effective to increase the axial depth of cut in order to obtain high quality bamboo fibers. However, the coefficient of variation of fiber diameter, length and aspect ratio seem to have minimum values at an axial cut depth from 5 to 8 mm. Therefore, it seems effective to set the axial cut depth from 5 to 8 mm in order to acquire high quality bamboo fibers. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3.3 Influence of feed of end-mill center on acquired bamboo fiber quality In this section, we investigate the change in shape of the fiber section with changes in the feed of the end-mill center adjusting spindle speed and the feed speed based on eqn (11). In this study, cutting arc length L is constant because L is a function of the end-mill radius and the depth of cut in the radius direction per one cycle as described by eqn (8). However, end-mill radius is a constant 3 mm and the depth of cut in the radius direction per one cycle is a constant 0.05 mm. On the other hand, the maximum thickness of h changes with spindle speed and feed speed under a constant end-mill radius, depth of cut in the radius direction per one cycle, and number of the cutting edge of the end-mill as shown in eqn (9). Therefore, the shape of bamboo fibers end-milled under various values for the feed of the end-mill center, determined by dividing the feed of the end-mill center by the spindle speed, and the axial cut depth in the bamboo fiber direction are investigated in this section. Larger values for the feed of the end-mill center give thinner end-milled bamboo fibers. The maximum thickness of h is 3.0, 9.1, and 36.4 µm for values of the feed of the end-mill center of 0.02, 0.05, and 0.20 mm/tooth, that is, the theoretical fiber diameter is 276, 279, and 292 µm, respectively, in this study. Figure 7 shows the experimental results. It is clear that the smaller diameter bamboo fibers can be obtained with smaller values of the feed of the end-mill center, independent of the axial depth of cut in the fiber direction as shown in Fig. 7(a). In the case of a feed of the end-mill center of 0.20 mm/min, the fiber diameter measured agrees with theoretical one of 292 µm. However, the fiber diameter becomes smaller in case of smaller feed of the end-mill center of 0.05 mm/tooth. On the other hand, the fiber length is controlled very well by the axial depth of cut at a feed of the end-mill center of 0.05 mm/tooth or greater. However, the fiber length becomes smaller in the case of a small feed of the endmill center of 0.02 mm/tooth for each axial depth of cut. As a result, the aspect ratio attains a maximum at a feed of the end-mill center of 0.05 mm/tooth when h is 9.1 µm. Therefore, a feed of the end-mill center of 0.05 mm/tooth seems suitable for obtaining superior bamboo fibers in end-milling of bamboo pipe.

4

Conclusions

End-milling using a machining center that follows a spiral tool path was tried for the production of high quality bamboo fibers. The shape of the fibers end-milled from bamboo pipe was evaluated under various cutting conditions, giving the following results. (1) End-milling with a machining center that follows a spiral tool path is effective for obtaining high quality, straight bamboo fibers with no thermal damage. (2) It is possible to accurately control the bamboo fiber length by adjusting the axial depth of cut. (3) A higher cutting speed can provide bamboo fibers with smaller diameter. However, the decrease in bamboo fibers diameter with higher cutting speed saturates at a certain cutting speed. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

178 High Performance Structures and Materials III (4) Suitable conditions were determined for acquiring bamboo fibers of the desired shape.

Acknowledgement This research is supported by the Research and Development Center for Engineering Application of Bamboo Resources at Doshisha University.

References [1]

[2] [3] [4]

[5]

[6] [7]

[8]

[9]

[10]

Ogawa K., Hirogaki T., Aoyama E. and Katayama T., Data mining of optimum condition to acquire bamboo micro-fiber with mechanical method, High Performance Structures and Materials II, WIT PRESS, pp.441-450, 2004. Jana S. C. and Prieto A., On the development of natural fiber composites of high-temperature thermoplastic polymers, Journal of Applied Polymer Science, Vol. 9, No. 28, pp.2159-2167, 2002. Li H. and Sain M. M., High stiffness natural fiber-reinforced polypropylene composites, Polymer-Plastics Technology and Engineering, Vol.42, No.5, pp.853-862, 2003. Luo S. and Netravani A. N., Interfacial and mechanical properties of environment-friendly ‘green’ composites made from pineapples fibers and poly (hydroxybutyrate-co-valerate) resin, Journal of Materials Science, Vol. 34, No. 15, pp.3709-3719, 1999. Ogawa K., Hirogaki T., Aoyama E., Tomioka Y., and Shiomi T., Shape Control of Fibers End-milled from Bamboo with a Machining Center, Proc. of The 3rd International Conference on Leading Edge Manufacturing in 21st Century, pp.565-570, 2005. Thwe M. M. and Liao K., Environmental effects on bambooglass/polypropylene hybrid composites, Journal of Materials Science, Vol. 38, No. 2, pp.363-376, 2003. Martikka H. and Katajisto J., Study of Natural Fiber Reinforced Biodegradable Composite Materials for Designing Optimally Sustainable Products, Proc. of the 23rd Riso International Symposium on Materials Science, pp. 251-258, 2002. Takagi H. and Ichihara Y., Effect of fiber length on mechanical properties of “green” composites using a starch-based resin and short bamboo fibers, JSME International Journal, Series A: Solid Mechanics and Material Engineering, Vol. 47, No. 4, pp.551-555, 2004. A. Varada RaJulu, S. Allah Baksh, G. Ramachandra Reddy and K. Narasimha Chary, Chemical Resistance and Tensile Properties of Short Bamboo Fiber Reinforced Epoxy Composites, Journal of Reinforced Plastics and Composites, Vol. 17, No. 17, pp. 1507-1511, 1998. Seema Jain, U. C. Jindal and Rakesh Kumar, Development and fracture mechanism of the bamboo/polyester resin composite, Journal of Materials Science Letters, 12, pp. 558-560, 1993. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Characteristic behaviors of CFRP and GFRP at cryogenic temperature under static and cyclic loadings S. Kubo, K. Okubo & T. Fujii Department of Engineering, Doshisha University, Japan

Abstract Characteristic behaviors of plain-woven Carbon Fiber Reinforced Plastics (CFRP) at cryogenic temperature were investigated under static and cyclic loading. The test results were compared with those of Glass Fiber Reinforced Plastics (GFRP). Tensile behaviors of monofilaments were also evaluated. The test results showed that two knee-points appeared in the stress-strain curve of GFRP under static load, while that of CFRP was almost linear. Both strength and failure strain of the CFRP at cryogenic temperature were lower than those of CFRP at room temperature, because the epoxy resin was brittle at cryogenic temperature. At cryogenic temperature, the knee-point was also shown in the SN curve of GFRP of the fatigue test but that was not shown in the S-N curve of CFRP. The elastic modulus of CFRP at cryogenic temperature suddenly decreased in the final stage of fatigue, while such change was not observed at room temperature. The thermal fatigue test where no cyclic loads were applied to the specimen was also conducted in order to investigate the damage progression due to temperature change. After the thermal fatigue test, the residual strength of the CFRP and GFRP were measured at room temperature. The change in residual strength was explained by the difference of the coefficient of thermal expansion. This paper also mentioned that, at cryogenic temperature, the stiffness reduction under cyclic loading was related to the local thermal stress by thermal cyclic fatigue. It should be said that the carbon fiber was failed with accompanying critical crack propagation of the matrix due to the brittle of resin at cryogenic temperature. Keywords: fiber reinforced plastics, cryogenic temperature, coefficient of thermal expansion, residual strength, stiffness reduction.

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180 High Performance Structures and Materials III

1

Introduction

Fiber reinforced plastics (FRP) are widely used for the structural component due to their low coefficient of thermal expansion as well as high specific strength and stiffness [1–3]. The FRPs are also expected in use at harsh environment, for example outer space and liquid fuel tank for disposable rocket. The GFRP (Glass Fiber Reinforced Plastics) has been partly used at low temperature of a ship, and Sakurai [4] discussed the strength of GFRP for them. However their use is limited but aluminium alloy is still used for many applications. To apply the FRP, for example, for the fuel tank instead of the aluminium alloy, behaviors of FRP such as CFRP (Carbon Fiber Reinforced Plastics) and GFRP under the cryogenic temperature should be cleared. The purpose of this study is to investigate the mechanical properties of CFRP at room and cryogenic temperature using liquid nitrogen, in comparison with those of GFRP. The mechanical properties of carbon and glass monofilaments at room and cryogenic temperature were also shown. The coefficients of thermal expansions were measured for FRP as well as the epoxy resin for polymer matrix. The tensile tests of CFRP and GFRP were conducted at room and cryogenic temperature. The tension-tension cyclic loading was applied to the CFRP and GFRP with hydraulic servo testing machine having nitrogen cryostat. Thermal fatigue tests (without loading) were also conducted in which environmental temperature was alternated between room and cryogenic temperature. The residual strengths of the damaged specimens were measured at room temperature after the thermal fatigue test.

Aluminium

R70

Cut off 20

8 80 150

Adhesion t = 2 mm

Figure 1:

2

Monofilament

Grip area 25 Diameter 6.2-8.3µm

Dimensions of specimens of composite and monofilament.

Experimental procedure

2.1 Materials and specimens Plain-woven carbon (PYROFIL TR3110M: Mitsubishi Rayon Co., Ltd.), epoxy resin (bisphenol A type) (E-828: Japan Epoxy Resins Co., Ltd.) and Cyclopolyamide (E-113: Japan Epoxy Resins Co., Ltd.) were used as reinforcement, matrix and hardener for CFRP (Vcf=65.3%). Glass cloth (MS253C: Asahi Fiber-Glass Co., Ltd.), polyester resin (5595APT-S: DH Material Inc.), methyl ethyl ketone per oxide (nacalai tesque) and cobalt WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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naphthenate (6% solution) (Nacalai Tesque Co., Ltd.) were used as reinforcement, matrix, hardene and catalyzer for GFRP (Vgf=35.0%), respectively. The specimen was fabricated by hand lay-up method. The dimensions of the dog-bone shaped specimens of the composites were 150 mm long, 20 mm wide and 2 mm thick as shown in Fig. 1. Aluminium tabs with 35 mm long, 20 mm wide and 2 mm thick were glued on the end of specimens. The dimensions of the specimens of carbon and glass monofilaments are also shown in Fig. 1. 2.2 Experiments 2.2.1 Static tensile tests The carbon and glass monofilaments were extracted from their fabrics. In the tensile tests for monofilaments, at first, the diameter of monofilaments was measured using optical microscope. The tensile testing machine for monofilaments was shown in Fig. 2(a). Tensile load was applied to the monofilament at room and liquid nitrogen temperature. The test conditions of monofilaments specimens were referred to Japanese Industrial Standards (JIS R7606). The strain was calculated from the crosshead displacement. The tensile testing machine for the composites and resin were also shown in Fig. 2(b). Extensometer was attached to the specimen to measure the strain.

(a) Figure 2:

(b)

Tensile tests at cryogenic temperature. (a): for monofilament, (b): for composite.

2.2.2 Thermal mechanical analysis (TMA) The coefficient of thermal expansion αm of the pure epoxy resin, αc of the CFRP were measured from –60oC up to room temperature (RT) by the thermal mechanical analysis (SHIMADZU TMA-60) referred to JIS (K7197) [5]. The heating rate was 5oC/min. 2.2.3 Tension-tension fatigue tests At room and cryogenic temperature, the tension-tension fatigue test was conducted with a Shimadzu Co. EHF-UB50kN at constant amplitude of load with sinusoidal wave at a frequency of 5 Hz and stress ratio R (R=σmin/σmax)=0.1 for CFRP and R=0 for GFRP.

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182 High Performance Structures and Materials III 2.2.4 Thermal fatigue tests The conditions are shown in Table 1. The temperature of the specimens was alternated from cryogenic temperature to room temperature. The specimens were alternatively transferred with an aluminum frame into LN2 from ambient air. At first, specimens were cooled down to –196 oC and kept for 2 min. Then they were held in ambient air for 5 min. To reduce the time required for specimens to return to the ambient temperature, a fan was used during the periods when the specimens were held in ambient temperature [6]. The residual stress was measured after thermal fatigue test. Table 1: Materials

Tmax (oC)

CFRP and GFRP

23

(oC)

Thermal fatigue test condition. Tmin (oC) -196

(oC)

Maximum number of cycles 1, 3, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

3 Results and discussions 3.1 Strength and failure strain of monofilaments The mechanical properties of glass and carbon monofilament at room and cryogenic temperature are shown in Table 2. The strength and the failure strain of glass monofilament were increased about 99% and 82% respectively at cryogenic temperature, compared with those at room temperature. The strength of carbon monofilament was also increased about 11%, while the failure strain of carbon monofilament was decreased about 10% at cryogenic temperature. Fig. 3 and 4 show the fracture surfaces of the glass and carbon monofilament at room and cryogenic temperature, respectively. At room temperature, the typical cleavage fracture surfaces were observed in the surfaces of glass and carbon monofilament. On the other hand, at cryogenic temperature, there was rough area in the center of the fracture surface of glass and carbon monofilament. Table 2:

Mechanical properties of carbon and glass monofilaments.

Materials Diameter [µm]

Temperature [oC] 23+2 (Room Temp.)

CF

240

3503

1.34

269

3168

1.22

68.5

1330

1.70

67.7

2650

3.10

6.0～8.5 -196 (Cryogenic Temp.) 23+2 (Room Temp.)

GF

Elastic modulus Tensile strength Failure str [%] [GPa] [MPa]

7.8～14.1 -196 (Cryogenic Temp.)

The reduction in failure strain and strength of carbon monofilament is considered as the brittle behaviors of the carbon fiber at cryogenic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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temperature [7]. However the glass monofilament had rather different characteristics from those of carbon monofilament, in which the failure strain of the glass monofilament was increased at cryogenic temperature even though it has rough fracture surface.

5µm

x7.0 k

x7.0 k

5µm

Figure 3: Fracture surface of carbon monofilament (left: at CT, right: at RT).

5μ m

×9.0 k

×7.0 k

5μm

Figure 4: Fracture surface of glass monofilament (left: at CT, right: at RT).

3.2 Static mechanical properties of CFRP and GFRP Static strengths were listed in Table. 3. The stress strain (S-S) curves of CFRP and GFRP at room and cryogenic temperature were shown in Fig. 5 and 6, respectively. The S-S curves of CFRP were linear at both of room and cryogenic temperature. The S-S curve of GFRP at room temperature was also linear. However, that of GFRP at cryogenic temperature showed two knee-points. The elastic modulus of the resin is usually increased while its failure strain is decreased at cryogenic temperature. Actually, the elastic modulus of CFRP and GFRP at cryogenic temperature increased in comparison with that at room temperature in this test. At cryogenic temperature, the strength and failure strain of GFRP should be contributed by the increase of the strength and failure strain of glass monofilament at cryogenic temperature. Little changes of the mechanical properties of CFRP were found in the tensile test at room and cryogenic temperature. 1000

500 Stress[MPa]

800 Stress[MPa]

600

RT CT

600 400 200

400 300 200

× RT CT

100

0

0

0.0

0.5

1.0 Strain[%]

1.5

0

1

2 3 Strain[%]

4

Figure 5: S-S curves of CFRP at RT Figure 6: S-S curves of GFRP at RT and CT. and CT. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

184 High Performance Structures and Materials III Table 3:

Static strength of CFRP and GFRP at RT and CT. Static strength : σst [MPa]

Stress level γ=σmax/σst [%]

CFRP RT 639.1

CT 630.0

GFRP RT 294.6

CT 564.5

3.3 Failure strain of resin The stress strain curves of the monolithic epoxy resin are shown in Fig. 7. At room temperature, the failure strain was significantly higher than that at cryogenic temperature. It is generally known that the failure strain decreases at cryogenic temperature in comparison with that at room temperature due to the brittle behavior of polymer resin [8]. The observed failure strain of pure epoxy resin used in this study was about 0.21% at cryogenic temperature. According to the coefficient of thermal expansion αm, the thermal strain of monolithic epoxy resin was calculated about 0.41% at cryogenic temperature. It was half in failure strain at cryogenic temperature, compared with that at room temperature.

Stress[MPa]

60 45 30 RT CT

15 0 0.0

0.5

1.0

1.5

2.0

Strain[%]

Figure 7:

S-S curves of pure epoxy resin at RT and CT.

3.4 Effect of thermal expansion coefficient on residual strength Carbon monofilament has small minus number of the coefficient of thermal expansion because carbon monofilament behaves like ceramic [9]. In this study, the coefficients of thermal expansions of CFRP and its pure resin were measured by TMA to estimate the residual stress at –60oC up to room temperature (Table 4). The coefficient of thermal expansion of the epoxy resin was lower than that of the CFRP. The thermal fatigue stress was applied to the CFRP and GFRP specimens [10–12]. The temperature change of the specimens was repeated from cryogenic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

185

High Performance Structures and Materials III

temperature to room temperature. Figure 8 and 9 show the S-S curves of CFRP and GFRP, respectively after thermal fatigue test, in which the thermal stress was applied to the specimens before the static test. Residual strengths [13] of CFRP and GFRP were almost equal to their each static strength even after 100 cycles of thermal fatigue loading. However the residual elastic modulus of CFRP was decreased by the cyclic thermal stress. There was the large difference in the coefficient of thermal expansion of carbon fiber and that of pure epoxy resin. Therefore the epoxy resin in CFRP should be damaged during the thermal fatigue test [14], because the magnitude of thermal free strain was larger than the failure strain of pure epoxy resin at cryogenic temperature as shown in section 3.3. Table 4:

The coefficient of thermal expansion of CFRP and the pure epoxy resin.

Materials CFRP

Tmin (oC)

23

-60

The coefficient of thermal expansion (-oC) 6.8e-5 4.9e-5

700

300

600

250

500

Stress[MPa]

Stress[MPa]

Epoxy resin

Tmax (oC)

400 300 200

0 cycle 100 cycles

100 0

200 150 100 0 cycle 100 cycles

50 0

0

0.5

1 1.5 Strain[%]

2

Figure 8: S-S curves of CFRP after thermal fatigue test at RT.

0

0.5

1

1.5

Strain[%]

Figure 9: S-S curves of GFRP after thermal fatigue test at RT.

On the other hand, in Fig. 9, the residual elastic modulus of GFRP was equal to the static elastic modulus. This is explained by that the resin in GFRP was not damaged at cryogenic temperature, because the coefficient of thermal expansion of glass fiber was almost equal to that of pure vinyl ester resin. 3.5 S-N curves of CFRP and GFRP The S-N curves of CFRP and GFRP at room and cryogenic temperature are shown in Fig. 10 and 11. CFRP had long durability at cryogenic temperature in high cycle fatigue compared with that at room temperature. The fatigue lives of GFRP at cryogenic temperature were also longer than those at room temperature. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

186 High Performance Structures and Materials III

700 650 600 550 500 450

CT RT

400 100 101 102 103 104 105 106 107 Number of cycles to failure Nf

Figure 10:

S-N diagram of CFRP at RT and CT.

Maximum stress σmax [MPa]

Maximum stress σmax [MPa]

The knee-point appeared in the S-N curve of the GFRP at cryogenic temperature. The maximum stress at the knee point in S-N curve was almost coincident to the stress at knee point under static load at cryogenic temperature. It should be said that the stress at static knee-point is a threshold in cyclic fatigue loading which determines the damage progression mode in fatigue of GFRP. The elastic modulus decay diagrams of CFRP was shown in Fig. 12, where γ=90% at room and cryogenic temperature. The elastic modulus of CFRP at cryogenic temperature suddenly decreased in the final stage of fatigue, while such change was not observed at room temperature. Same result was observed in the paper [15]. At the cyclic number to failure in the condition of room temperature, the sharp decreasing in elastic modulus occurred at cryogenic temperature. 600 500 400 300 200 100

CT RT

0 100 101 102 103 104 105 106 107 Number of cycles to failure Nf

Figure 11:

S-N diagram of GFRP at RT and CT.

Figure 13 shows the elastic modulus decay diagrams (γ=90%) of GFRP at room and cryogenic temperature. At cryogenic temperature, the elastic modulus significantly decreased at first cycle of fatigue, while that was not decreased at room temperature. This should be explained by that, at cryogenic temperature, fatigue damage was accelerated in early stage of fatigue because the specimen was subjected to the load over the stress at knee point, even while the damage was not produced by local thermal stress of thermal fatigue test as shown in section 3.4. The fracture surface of CFRP after fatigue loading test (γ=90%) at room and cryogenic temperature is shown in Fig. 14. At room temperature, fracture of the current CFRP was caused with the meta delamination appeared before the failure of fatigue at room temperature in CFRP. On the fracture surface, at cryogenic temperature, the epoxy resin was remained on the carbon fibers, while debonding was found around carbon fiber on the fracture surface of CFRP at room temperature. It should be said that, at cryogenic temperature, the carbon fiber was failed with accompanying critical crack propagation of the matrix due to the brittle of resin. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

120

Normalized modulus EN/E0 [%]

Normalized modulus EN/E0 [%]

High Performance Structures and Materials III

100 80 60 40

RT CT

20 0 100

101

102

103

104

105

Number of cycles N

Figure 12: Elastic modulus decay of CFRP at RT and CT.

187

120 100 80 60 40

RT CT

20 0 100

101 102 Number of cycles N

103

Figure 13: Elastic modulus decay of GFRP at RT and CT

RT

CT

Figure 14: Fracture surface of CFRP after fatigue test (γ=90%).

4

Conclusion

(1) The fatigue lives of CFRP and GFRP at cryogenic temperature were higher than those at room temperature. (2) At cryogenic temperature, the magnitude of thermal strain of pure epoxy resin was higher than the failure strain of the pure epoxy resin. (3) At cryogenic temperature, the epoxy resin adhered carbon fiber remained on the fracture surface of CFRP applied cyclic loading (γ=90%), while debonding was found around carbon fiber on the fracture surface of CFRP at room temperature. (4) At cryogenic temperature, the elastic modulus of CFRP decreased in the final stage of fatigue, while that of GFRP decreases after the first cycle of fatigue. (5) At cryogenic temperature, the stress at static knee-point of GFRP was the threshold in cyclic loading, which determined the damage progression mode in fatigue. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

188 High Performance Structures and Materials III

Acknowledgement This study was supported by the Ministry of Education, Culture, Sports, Science and Technology for RCAST (Doshisha University).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

H. Fukuda and G. Ben, Introduction to Mechanics of Composite, KOKIN Publishers, Tokyo, 1999 T. Ishikawa, H. Kumazawa, Y. Morino, Y. Hayashi, ACCM-1, 1998, 4371-4 F. Tomioka, H. Wada, JSME-A, 1992, 58-550 A. Sakurai, Reinforced Plastic, Vol. 1, 10-17 B. Fiedler, M. Hojo, S. Ochiai, K. Schulte, M. Ochi, Finite-element modelling of initial matrix failure in CFRP under static transverse tensile load, Composites Science and Technology, 61, 2001, 95-105 Satoshi Kobayashi, Kazuhiro Terada, Nobuo Takeda, Evaluation of longterm durability in high temperature resistant CFRP laminates under thermal fatigue loading, Composites: Part B, 34, 2003, 753-759 K. Yamada, K. Okubo, T. Fujii, 9th FRC2002, 2002, 201-206 C. Henaff-Gardin, M.C. Lafarie-Frenot, Specificity of matrix cracking development in CFRP laminates under mechanical or thermal loadings, International Journal of Fatigue, 24, 2002, 171-177 D. Hull and T. W. Clyne, An Introduction to Composite Materials, Second Edition, 1996 N. K. Naik, V. K. Ganesh, Thermo-mechanical behaviour of plain weave fabric composites: Experimental investigations, Journal of Materials Science, 32, 1997, 267-277 K. F. Rogers, D. M. Kingston-Lee, L. N. Phillips, B. Yates, M. Chandra, S. F. H. Parker, The thermal expansion of carbon-fibre reinforced plastics, Journal of Materials Science, 16, 1981, 2803-2818 J. M. Gaitonde, M. V. Lowson, Low-temperature Thermal Expansion of PEEK, HTA and Some of Their Composites Reinforced with Carbon Fibres, Composites Science and Technology, 40, 1991, 69-85 M. de Freitas, R. de Carvalho, Residual strength of a damaged laminated CFRP under compressive fatigue stresses, Composites Science and Technology, 66, 2006, 373-378 M.C. Lafarie-Frenot, N.Q. Ho, Influence of free edge intralaminar stresses on damage process in CFRP laminates under thermal cycling conditions, Composites Science and Technology, 2005 Surya D. Pandita, Ignaas Verpoest, Tension-tension fatigue behaviour of knitted fabric composites, Composite Structures, 64, 2004, 199-209

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189

Mechanical properties of loosing natural fiber reinforced polypropylene K. Mizuta1, Y. Ichihara1, T. Matsuoka2, T. Hirayama2 & H. Fujita3 1

Graduate Student, Doshisha University, Japan Department of Mechanical Engineering, Doshisha University, Japan 3 Hyogo Prefectural Institute of Technology, Japan 2

Abstract As interest in studying environmental issues has grown, the mechanical properties of eco-friendly materials have been studied. In our research, a card machine that enables raw materials to loose and to mix uniformly was used. Fibrous polypropylene and cotton or ramie fiber were used as raw materials since many textiles and apparel are made of such fibers. From a viewpoint of reuse, the loosing technique of the card machine can be effective. We designed anisotropic composites by using the card machine followed by the hot press process. The tensile, flexural and compression properties were investigated. Ramie fiber reinforced polypropylene (RP) was stronger than cotton fiber reinforced polypropylene (CP); however, breaking elongation of composites of CP is superior to that of RP because of the character of natural fiber. The density of composites applied to the loosing technique also affected the strength of composites. It was clear that density is an important parameter to describe the performance of a green composite. Moreover, it was found that the initial fracture in flexural behavior is affected by the compression strength of composites using the loosing technique. Keywords: cotton fiber, Ramie fiber, loosing technique, tensile property, flexural property, compression property, density.

1

Introduction

The increasing concerns for environmental issues and the growing eco-friendly society direct universities and makers to development of ‘Green composites’ [1-4]. Green composites should be friendly for both environment and human WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06020

190 High Performance Structures and Materials III body. Materials that can cause pollution and environmental load are not real green composites. Properties of various green composites have been studied; however, there are few reports about relationship between density and mechanical properties of composites. In our research, composites were made by using the loosing technique that enables raw materials to loose, to disperse and to mix uniformly. Natural fibers vary in their geometrical and mechanical properties. Therefore, it is important to grasp the interrelation between density and mechanical properties of composites. Tensile, flexural, and compression tests on cotton and ramie fiber reinforced polypropylene were performed to investigate these correlations.

2

Experiment

2.1 Material Polypropylene (PP), a matrix resin, is a fibrous resin with a melting point of 165℃. Cotton and ramie fibers were used as reinforcement. Length of cotton fiber was 30 mm, and ramie fiber was 80 mm. 2.2 Production of composite 2.2.1 Loosing method Loosing of fibrous PP resin and natural fiber was performed by a card machine. In mixing of the fibrous resin and the natural fiber, it is important to distribute the fiber uniformly in resin. Therefore, a loosing fibrous resin sheet and a loosing fiber sheet were separately made by the card machine before mixing fibrous rein and fiber. 2.2.2 Molding method Specimens were manufactured by a vacuum compressed molding method as shown in Fig.1. In this research, molding conditions were selected as shown in Table 1. Pressure Heater

Puressre plate Vacuum bag Spacer Sealant

Composites Heater Pressure

Figure 1:

Vacuum pump

Vacuum compressed molding method.

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High Performance Structures and Materials III

Table 1:

Molding condition.

Weight fraction of fiber [%]

0

30

50

Molding pressure [MPa]

1

10

15

Molding temperature [℃] Molding time [min]

191

210 10

2.3 Experimental method 2.3.1 Static tensile test The test pieces were cut to a rectangle (20×130×2 mm) from a plate, and gage length was 80 mm. The static tensile tests were carried out at the test speed 1.0 mm/min under a room temperature by an INSTRON testing machine (Type 4467). 2.3.2 Flexural test Dimensions of specimen were 15×50×2 mm. A three points flexural test was carried out at the test speed 1.0 mm/min under a room temperature by an INSTRON testing machine (Type 4467). Span length was 30 mm. 2.3.3 Compression test Dimension of specimens were 3×2×55 mm, and gage length was 17 mm. The compression tests were carried out at the test speed 1.0 mm/min under a room temperature by a testing machine (Ez Test-500, SHIMADZU. Corp.). Specimen holding assembly was used to avoid a buckling.

3

Results and discussion

3.1 Tensile test 3.1.1 Effect of loosing technique Fig.2 shows typical S-S curves of cotton fiber reinforced PP (CP) and ramie fiber reinforced PP (RP) in loosing direction. The number following CP or RP means fiber contents. RP is stronger than CP in any fiber contents, while deformation ability of CP is superior to that of RP by more than 50%. It was caused by properties of natural fiber; cotton fiber has many crimps and twists complexly each other, while ramie fiber is relatively high strength and modulus. Fig.3 shows the effect of the loosing technique, where θ represents the angle between loosing direction and tensile direction. The loosing technique could order fiber orientation in both CP and RP, and anisotropic composites could be obtained. 3.1.2 Effect of fiber length on loosing process Strong orientation couldn’t be seen in short ramie (SRP: fiber length=30 mm) in Fig.3. This result indicates the difference of the effect of fiber length on loosing WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

192 High Performance Structures and Materials III degree. It was clear that to order orientation of short ramie fibers was more difficult than long fibers by measuring fiber orientation angle from flatwise direction of specimen. It was one of the reasons that made SRP weak compared to RP.

80 RP50 SR50 RP30 SRP30

70 Stress [MPa]

60 50 40

CP50

CP30

30 PP

20 10 0 0.0

1.0

Figure 2:

2.0 Strain [%]

3.0

4.0

S-S curves of composites.

80 70

SRP(θ = 0° )

Stress [MPa]

60

RP(θ = 0° )

50

SRP(θ = 90°) CP (θ = 0° )

40

RP(θ = 90°) CP(θ = 90°)

30

PP PP

20 10 0 0.0

1.0

2.0

3.0

4.0

Strain [%]

Figure 3:

Anisotropy of loosing natural fiber reinforced polypropylene (Fiber contents: 30 wt.%).

3.2 Flexural test Fig.4 shows flexural behavior of composites. Table 2 shows flexural strength and flexural modulus of composites. Flexural modulus was improved in all materials, while, flexural strength was not increased well on the whole. However, density is an important parameter to estimate the loosing natural fiber reinforced WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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193

polypropylene. The relationship between density and strength properties is discussed in the next section.

Flexural strength [MPa]

80 R50

70

R30 PP

60

C50

50 40 30

C30

20 10 0 0

Figure 4: Table 2:

1

2

3

4

Displace me nt [mm]

5

6

Flexural behavior of NFRP.

Flexural strength and modulus of NFRP.

Flexural strength [MPa] Flexural modulus [GPa]

PP

C30

C50

R30

R50

65 1.9

55 2.2

64 2.7

69 4.4

66 3.8

3.3 Effect of density on mechanical properties 3.3.1 Correlation between density and flexural strength Fig.5 shows the effect of density on flexural strength of CP and RP. Composites using loosing technique have low density because the loosing sheet that consists of fibrous PP and natural fiber contains a lot of voids. However, there is a specific positive correlation between them. Strength of composites, which had densities from 0.75 to 0.85 g/cm3, was almost same to that of PP. That means specific strength increased by around 20% since the density of PP is 0.91 g/cm3. It was also suggested that high density composites yield high strength. Therefore, high density composites were manufactured applying higher pressure. In this paper, we classified into low density materials (LD) and high density materials (HD). As a result, composites with around 1.0 g/cm3 of density were obtained. Flexural strength of high density composites HD C30 and HD R50 could be improved except for HD R30 as shown in Fig.5. 3.3.2 Correlation between density and tensile strength Static tensile tests were also carried out to investigate tensile behavior of high density composites. As a result, tensile strength of high density composites decreased (Fig.6). In this decrease effect, the higher the natural fiber content was, the lower the tensile strength of the composites was. This indicates the effect of molding pressure on natural fiber. In the high pressure molding process, massive WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

194 High Performance Structures and Materials III damage of natural fibers could be the cause of this result. Observation of natural fibers that were exposed to fracture surfaces is shown in the next section. The relationship between some parameters such as strength, crystallization degree and density of PP resin is nearly linear [5]. It is thought that the difference of crystallization degree between low density composites and high density composites is also one of the reasons for these results.

Flexure strength [MPa]

100 90 80 70

LD C30 LD C50 LD R30 LD R50 HD C30 HD C50 HD R30 HD R50

PP

60 50 40 30 0.60

0.70

0.80

0.90

1.00

3

1.10

Density [g/cm ]

Figure 5:

Correlation between density and flexural strength.

Tensile streng th [M Pa ]

70 60 50

LD C30

40

LD R30

LD C50

30

LD R50 HD C30

20

HD C50

10

HD R30

HD R50

0 0.60 0.70 0.80 0.90 1.00 1.10 1.20 3

Density [g/cm ]

Figure 6:

Correlation between density and tensile strength.

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195

3.4 Difference of fracture surface between low & high density material Fig.7 shows SEM photograph of tensile fracture surface of low density and high density CP, respectively. There are many pull out single fibers in both, however these forms are different. A notable point is the difference among their lengths. In the high density composite, fibers were evenly broken shortly. Therefore, it showed relatively flat fracture surface. It was also observed in the fracture surface of CP. The form of the tensile fracture surface is determined by the relative degree and friction force between fiber and resin. If the ratio of tensile strength to interfacial shear strength is relatively small, then the fracture surface is flat [6]. In this research, high pressure was applied to manufacture high density composites. Fatal damage caused by high molding pressure decreased the potential of reinforcement by the natural fiber. Therefore, it is thought the massive damage on the crossing point of fibers made high density composites low strength.

(a) Low density CP Figure 7:

142µm

(b) High density CP

142µm

Fracture surface of CP (Fiber contents: 50wt.%).

3.5 Compression test Strengthening was not found in flexural strength in spite of the improvement of tensile strength. Therefore it is thought that compression property weakens the flexural strength. To investigate the cause of this degradation, compression test was carried out. Fig.8 shows compression strength of composites. As a result, compression strength decreased by 30-50% compared to polypropylene, however, high density materials were superior to low density materials only in compression strength. It was identified compression failure causes in flexural behavior. More detail of relationship between compression strength and flexural behavior is discussed in the next section. 3.6 Effect of compression strength on flexural behavior According to Bazhenov [7], flexural stress-displacement curve can be divided three parts as Fig.9. The transition point from region I to region II is called Knee WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

196 High Performance Structures and Materials III

Compression strength [MPa]

point. At Knee point, composite reaches compression yielding point, and then relationship between flexural stress and displacement starts non liner behavior. Fig.9 shows flexural stress at Knee point in flexural behavior.

50 45 40 35 30 25 20 15 10 5 0

Low de nsity High density

PP

C30

C50

R30

R50

Material

Flexural stress

Figure 8:

Ⅰ

Effect of density on compression strength.

Ⅱ

Ⅲ

Ⅰ. Elastic Ⅱ. Compression yielding Ⅲ. Tensile fracture

Knee point

X-head displacement Figure 9:

Schematic illustrating three stages in flexural deformation.

Compared Fig.8 with Fig.10, we confirmed correlationship between flexural stress at Knee point and compression strength. Compression stress was nearly to the flexural stress at Knee point about both CP and RP. Furthermore, high density materials tend to be superior to low density materials in Fig.8. Reflecting such a tendency, Knee point stress of high density materials is superior to that of low density materials (Fig.10). Therefore, it is necessary to keep compression strength as much as possible to improve flexural strength of loosing natural fiber reinforced polypropylene. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Stress at Knee point [MPa]

50

197

Low density

45

High density

40 35 30 25 20 15 10 5 0 PP

C30

C50

R30

R50

Material

Figure 10:

4

Stress at knee point in flexural test.

Conclusion

Cotton fiber and ramie fiber reinforced polypropylene were manufactured using loosing technique. The experimental data showed that: (1) Loosing technique enables to uniform fiber orientation in both cotton fiber reinforced polypropylene and ramie fiber reinforced polypropylene. (2) Long fibers (fiber length=80 mm) were easier than short fiber (fiber length=30 mm) to orient natural fiber uniformly in ramie fiber reinforced polypropylene. (3) There was a positive correlationship between mechanical properties such as tensile strength and flexural strength and density in low density materials. (4) Damage of natural fiber by high pressure caused weakness of tensile strength of composites. (5) Initial fracture in flexural behavior of loosing natural fiber reinforced polypropylene was affected by compression strength. Therefore, improvement of compression strength is necessary to obtain higher performance of loosing natural fiber reinforced polypropylene.

Acknowledgements Authors would like to express special thanks to CHISSO Ltd. for providing fibrous PP resin in this study. This study was supported by the Academic Frontier Research Project on “New Frontier of Biomedical Engineering Research” of ministry of education, culture, sports, science, and technology.

References [1]

Kenji Ohnishi, Kazunori Umeoka, Hideyuki Ando, and Wenhai Liu. Matsushita Electric Works, Ltd., Technical Report, pp58-63, 2003. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

198 High Performance Structures and Materials III [2] [3] [4] [5] [6] [7]

Teruo Kimura, J. Soc. Mat. Sci, Japan, Vol.50, No10, pp.1158-1163, 2001. Hiroshi Yamamoto, Takashi Matsuoka, Kazuhiko Sakaguchi, and Hiroyuki Fujita, Proceedings of Third International Workshop on Green Composites, pp.95-100, 2005. Anil N Netravali, Proceedings of Third International Workshop on Green Composites, pp.11-15, 2005. Kaneyuki Takagi, Heizo Sasaki, Polypropylene resin, Nikkan Kogyo, Tokyo (1970). D. Hall, An Introduction to Composite Materials, Baifukan Ltd., Tokyo (1984). S.L. Bazhenov, Composites 26, pp.757-765, 1995.

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Section 4 Material and mechanical characterisation

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High Performance Structures and Materials III

201

Identification of strain-rate sensitivity parameters of steel sheet by genetic algorithm optimisation G. Belingardi, G. Chiandussi & A. Ibba Dipartimento di Meccanica, Politecnico di Torino, Italy

Abstract The influence of the loading speed on mechanical response of structural materials can be accounted by means of strain-rate sensitivity parameters. The aim of the present work is to show a numerical technique based on an inverse approach to determine strain-rate sensitivity parameters of steels for car body constructions. This technique is based on the numerical simulation of a simple test according to the ASTM D5420/96 standard by means of a finite element explicit code. The test consists of a falling tup with a spherical head impacting on a thin sheet. Some experimental tests are conducted at different speeds, from quasi-static to impact loading conditions, on a specimen made of XE280P steel. A series of simulations are performed, changing the strain-rate sensitivity parameters in each run according to a genetic algorithm strategy. The strain-rate parameters that lead to the best fit of the experimental load-displacement curve with the numerical result are the assumed material characteristic parameters. The Cowper-Symonds and Johnson-Cook strain rate models have been taken into consideration. Keywords: strain-rate sensitivity, optimisation, genetic algorithm, finite element analysis.

1

Introduction

Since the second half of the past century, research in the automotive industry focused on safety improvement. Increasing customer interest about passive safety and the ever stricter regulations, both in the US and EU, pushed towards more reliable vehicle structure design and a deeper insight into material behaviour. The finite element method applied to crash simulations with explicit WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06021

202 High Performance Structures and Materials III codes and the increasing computing capability showed the way to new possibilities for prediction of passive safety performance of vehicles. But these opportunities have to be accompanied by adequate information on material mechanical behaviour and by proper material modelling: both are fundamental for correct numerical simulation. As a consequence material characterisation and model identification through experimental tests have gained a primary role. One of the most critical uncertainties in crash simulations relates to the influence of the loading speed on the material mechanical characteristics. This influence can be accounted by means of the strain-rate sensitivity parameters that are substantial for most materials. Large strain-rate sensitivity of low-carbon steels, such as the deep-drawing steels of most automotive body, is very well known. Strain rate sensitivity characterisation is quite complex because of the variety of tests that could be performed and the difficulty in data acquisition. Uniaxial tensile tests can be performed only at low strain rate values with traditional hydraulic testing machines, moreover it is not possible to perform tests at strictly constant strain rates and often the strain rate is not uniform on the specimen. Other type of tests and testing machines, as for example the split Hopkinson pressure bar, have been developed to characterise strain rate sensitivity of materials. The instrumented drop dart test has proven to give interesting opportunities in pointing out the material sensitivity to the strain rate and has been successfully used for this type of characterisation. In the present work an optimisation technique, based on a genetic algorithm, is applied in order to determine parameters for strain-rate sensitivity models for a high strength steel to be adopted in car body constructions. It is based on the numerical simulation, by means of a finite element explicit code model, of the simple drop dart test performed at different strain rates and the interactive changing of material parameters until an optimum fit of experimental data is reached. The Johnson-Cook and the Cowper-Symonds strain rate sensitivity models are analysed in the present work. The considered material is the steel XE280P laminated in sheets of 1.5 mm thickness.

2

Experimental tests and numerical models

Quasi-static uniaxial tensile tests and a series of simple bending tests at different strain rates were performed to characterise the XE280P steel sheets and to identify parameters of models. Both types of test are necessary because they involve different mechanisms of deformation and help the stability and completeness of the identification. The uniaxial tensile tests were performed by means of a general purpose hydraulic testing machine (DARTEC HA100). The usual dog bone shaped specimens were manufactured from a laminated sheet of 1.5 mm thickness. The obtained true stress vs. true strain curve is fundamental for the implementation of the material behaviour in FE codes and was used for a first direct characterisation (fig. 1).

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203

600

True stress [MPa]

500 400 300 Elastic modulus: Yield stress: Maximum stress: Failure stress:

200 100 0 0,00

Figure 1:

0,05

0,10

≈180 GPa ≈280 MPa 517 MPa 396 MPa

0,15 0,20 True strain [-]

0,25

0,30

Experimental stress-strain curve from quasi-static uniaxial tensile test. 35 v = 76,5 mm/s

30

Force [kN]

25

v = 6,26 m/s

20 15

v = 0,1 mm/s

10 5 0 0

Figure 2:

5

10 15 Stroke [mm]

20

25

Experimental force-displacement curves of bending tests performed at different loading speeds.

Bending tests based on the ASTM D5420/96 standard were performed in order to characterise the strain rate sensitivity parameters of the steel. The test consists on a falling tup with a spherical head impacting a steel sheet. The specimens are square shaped but are completely constrained on a circumference of diameter 76 mm by means of a blank-holder plate. The tests are conducted at different speeds, from quasi-static loading conditions by means of the hydraulic testing machine, to impact loading conditions, by means of a drop-dart testing machine characterised by settable impact speed up to 6.26 m/s. The falling mass (including the dart) is 20 kg. Each test gives a load-stroke curve. Force values are measured with a load cell placed at the dart head while displacement is obtained by LVDT direct measurement for the quasi-static testing conditions and by double integration of the acceleration in time for the impact testing conditions. Direct speed measurements are used in impact test for confirmation. Figure 2 shows some experimental curves. Modifications induced by the load application speed are evident. Numerical simulations of the bending tests were WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

204 High Performance Structures and Materials III carried out by means of the code RADIOSS. Different meshes and element formulations were tested, and both demonstrated to have little influence on the final results. The default Belytschko formulation with global plasticity algorithm of Radioss was chosen for shell elements. Two models were used: a simplified model with a complete constrain on the clamping circumference and a more complete model with rigid plates holding the square specimen (fig. 3). Both models take advantage of the symmetry of the system.

Figure 3:

On the left: simplified model of the bending test; the external circumference is completely constrained. On the right: exploded view of the complete model with the specimen holding plates.

Beyond the yield stress, the static plastic behaviour of the material is modelled, as usual, by the following Hollomon formulation: σ s = A + Bε pl n

(1)

where εpl is the plastic component of the total strain, while A, B and n are material constants. For the dynamic behaviour two strain rate sensitivity models have been considered. Both models modify the static stress-strain curve by adding a multiplying factor depending on the strain rate and both models are characterised by the presence of two parameters that have to be identified. The Cowper-Symonds model has the following formulation: 

1p

ε σ = σ s (ε ) ⋅ 1 +    

 D  

   

(2)

where σs(ε) is the static stress and D and p are material constants. The JohnsonCook model includes a further factor to account for the temperature dependence that is not considered in this work:   ε σ = σ s (ε ) ⋅ 1 + C ln   ε 0 

  T   ⋅ 1 −     Tm

where C and ε 0 are the material constants. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

  

(3)

High Performance Structures and Materials III

3

205

Identification by genetic algorithm optimisation

The identification of the static and strain rate sensitivity parameters was performed by means of an optimisation technique. A set of simulations were executed, changing the values assigned to the material characterising parameters in each run according to a genetic algorithm strategy. For each run a complete generation of a defined number of models was analysed. The parameters that lead to the best fit of the numerical result with respect to the experimental result have been selected as the optimal material model parameters. In particular the following objective function to be minimised was used: N

(

)

F par1 ,..., parq =

∑ (y

exp,i

− y mod,i

)2

i =1

(4)

N

where parq are the parameter values to be identified, yexp,i and ymod,i are the experimental and numerical responses respectively, and N is the number of curve points. The objective function was evaluated on the stress-strain curve for the uniaxial tensile test and on the force-displacement curve for the bending tests. In both cases curve data up to the maximum of, respectively, the stress or the force were considered. The genetic algorithm is defined by the mechanisms of selection, recombination and mutation. The selection is used to choose the parent models and is performed randomly by means of probability functions derived from the model fitness: the higher is the model fitness the higher is the probability to be chosen as parent of the new generation. The model fitness is defined as the difference between the value of the objective function for the worst model of the generation and the value of the objective function for the model itself. The selection is limited to the models that constitute the 50% of the cumulated fitness, decreasingly ordered. The recombination is the main feature of the genetic algorithm. It defines the son models by combining different parent models:

(

)

x a = (1 − µ α ) ⋅ x α + µ β ⋅ x β ; x b = µ α ⋅ x α + 1 − µ β ⋅ x β

(5)

Two son models (xa, xb) are generated from two parent models (xα, xβ). µα and µβ are random numbers generated by a normal probability function with null

mean value. Finally a mutation, limited to 20% of the admissibility domain, is imposed and the generated models are computed by FE simulations. The best fitness model selected among the previous generation models and the new generated model constitute the present generation. Parameters identification was also performed by a response surface and a gradient method in order to evaluate, by comparison, the performance of the genetic algorithm. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

206 High Performance Structures and Materials III 3.1 Static model parameters identification Parameter identification of the static model of eqn. (1) by means of uniaxial tensile test does not need FE simulations. The good fitting of the model obtained by genetic algorithm is shown in fig. 4, together with the objective function level curves on the A-n parameter plane. These curves highlight a large region of A and n values with a nearly constant value of the objective function that means possible instability in parameter identification. Multiple optimum couples of values of these parameters are possible and for this reason a fixed A parameter defined by a direct identification have been used. 600 0,44

500

B=491,678 MPa

Experimental curve

300

7-9 9-11 11-13 13-15 15-17 17-19 19-21 21-23 23-25 25-27

0,36 0,32

Optimised model curve

0,28

200

n [-]

Stress [MPa]

0,40

400

0,24 0,20

100

0,05

Figure 4:

0,10 True plastic strain [-]

0,15

130 145 160 175 190 205 220 235 250 265 280 295

0,16

0 0,00

0,20

0,12

A [MPa]

Identified model by uniaxial tensile test and level curves of the objective function on the A-n parameters plane.

The bending test performed at low speed can also be used for the static parameters identification. In this case different and not uniform mechanisms of deformation are involved and FE simulations are needed to have the numerical force-displacement curve to be compared with the experimental one. 600

Experimental curve Model gradient method Model genetic algorith

25 20

Stress [MPa]

Force [kN]

30

15 10 5 0 0

Figure 5:

5

10

Stroke [mm]

15

20

500 400 300

Experimental curve Model gradient method Model genetic algorith

200 100 0 0,00

0,05

0,10

0,15

Plastic strain [-]

0,20

Simulated static bending test and stress-strain models obtained by optimisation with bending test and simplified FE model.

The parameters identified by means of the genetic algorithm and of the gradient method based on the simplified structural model of the bending test (fig. 3(a)) bring to the force-displacement and stress-strain curves shown in fig. 5. Both optimisation method work well to fit the experimental curve of the bending test, but the identified parameters are rather different from each other and the corresponding stress-strain characteristics are rather far from the experimental one. This confirms the possible criticality of the identification WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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207

method that could lead to the same well simulated curves with different parameter values. Moreover it shows that the FE model of the bending test could be not sufficiently accurate, both for what concern the mesh refinement and for what concern the through thickness integration. 600

Experimental data Model gradient method Model genetic algorith

25 20

Stress [MPa]

Force [kN]

30

15 10 5 0 0

Figure 6:

5

10

Stroke [mm]

15

500 400 300

Experimental data Model gradient method Model genetic algorith

200 100 0 0,00

20

0,05

0,10

0,15

Plastic strain [-]

0,20

Simulated static bending test and stress-strain models obtained by optimisation with bending test and complete FE model.

The use of the complete structural model (fig. 3(b)) for the identification leads to the results shown in fig. 6. A more accurate solution is obtained: the material model fits better the experimental stress-strain curve of the tensile test, while keeping the same quality of fitness for the force-displacement curve of bending test. All the identified parameters are summarised in table 1. Table 1:

A [MPa] B [MPa] N [-] F [MPa, kN]

Optimal values for the static model.

Uniaxial tensile test Genetic Gradient 241.7 213.5 482.5 491.7 0.3298 0.2857 8.328 8.186

Bending test Simplified model Genetic Gradient 238.7 305.2 499.0 521.1 0.5913 0.9116 0.361 0.179

Bending test Complete model Genetic Gradient 282.4 273.1 532.1 528.4 0.5323 0.4967 0.173 0.185

3.2 Strain rate sensitivity model identification For the identification of the strain rate sensitivity parameters drop tower tests were performed with two different falling heights of the dart: 1 m and 2 m. In order to achieve a more suitable solution the static parameters identified by means of the uniaxial tensile test were used and only the strain rate parameters were searched with the described optimisation procedure. The objective function to be minimised take into account the force-displacement curves of both test conditions: N1

F(p1 , p 2 ) =

1 2

∑ (F

exp1,i

− Fmod1,i

i =1

N1

N2

)2 +

1 2

∑ (F

exp 2,i

− Fmod 2,i

i =1

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N2

)2 (6)

208 High Performance Structures and Materials III

6,0 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0

Objective F [kN]

Objective F [kN]

3.2.1 Cowper-Symonds model The identification of Cowper-Symonds parameters by genetic algorithm was performed by means of 16 models for each generation. For the complete model, the evolution of objective function values for each generation in the best and worst cases are shown in fig. 7(a). In figure 7(b) the values obtained by the gradient method for each iteration are shown for effectiveness comparison. The genetic algorithm optimisation shows a more rapid convergence, even if a greater amount of simulations were used. The identified parameters and the final objective function values are summarised in table 2. The force-displacement curves derived from the simulations with the optimised complete model are shown and compared to the experimental ones in fig. 8. DYNAMIC Genetic algorithm Model: Cowper-Symonds

Minimum of generation Maximum of generation 1

2

3

4

5

6

7

8

9

Generation

10

11

12

13

10 9 8 7 6 5 4 3 2 1 0

14

DYNAMIC Gradient method Model: Cowper-Symonds

1

2

3

4

Iterations

5

6

7

Figure 7:

The objective function behaviour with the genetic algorithm and the gradient method.

Table 2:

Optimal values of the Cowper-Symonds strain rate sensitivity model.

D [s-1] p [-] F [kN]

Complete model Genetic Gradient 4987 3013 1.619 1.503 1.817 1.822

Simplified model Genetic Gradient 2399 3006 1.329 1.416 1.187 1.209

3.2.2 Johnson-Cook model The identified parameters of the Johnson-Cook model by means of the genetic algorithm and the gradient method, with both the simplified and complete models are summarised in table 3. Five models were computed for each generation of the genetic algorithm optimisation. Force-displacement curves obtained by simulations with the complete model optimised by the genetic algorithm are shown and compared to the experimental ones in fig. 9.

4

Conclusions

A genetic algorithm optimisation procedure was applied to the identification of strain rate sensitivity models and proved to be effective. Compared to an optimisation procedure based on the gradient method it reached a better quality result in less iterations, even if the number of simulations to be run could be high. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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35 DYNAMIC

30

Genetic algorithm

Force [kN]

25

Model: Cowper-Symonds

20 15 Experimental curve - h=1 m Experimental curve - h=2 m Simulation curve - h=1 m Simulation curve - h=2 m

10 5 0 0

1

2

3

4

5

Time [ms]

Figure 8:

Comparison between experimental force-time curves of dynamic bending tests and simulations with Cowper-Symonds model.

Table 3:

Optimal values for the Johnson-Cook strain rate sensitivity model.

ε0

Complete model Genetic Gradient

Simplified model Genetic Gradient

1.00

1.00

[s-1] 0,01149 1,375

C [-] F [kN]

0,01223 1,372

0,01028 1,329

0,01024 1,330

35 DYNAMIC

30

Genetic algorithm

Force [kN]

25

Model: Johnson-Cook

20 15 Experimental curve - h=1 m Experimental curve - h=2 m Simulation curve - h=1 m Simulation curve - h=2 m

10 5 0 0

1

2

3

4

5

Time [ms]

Figure 9:

Comparison between experimental force-time curves of dynamic bending tests and simulations with Johnson-Cook model.

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210 High Performance Structures and Materials III Simple drop tower bending tests were performed at different loading speeds on specimen made of XE280P steel. Explicit simulations by means of FE models were run in order to identify the material strain rate characteristic parameters according to the Cowper-Symonds and the Johnson-Cook laws. The identification of static model parameters was also performed with the same algorithm and the same bending test at quasi-static loading conditions, but large identification uncertainty and drawbacks were encountered. A classical uniaxial tensile test was chosen for a more reliable and proper identification of static parameters and for the validation of the bending test modelling.

Acknowledgement The authors kindly acknowledge the European Commission that, by funding the ADVANCE project through contract GRD1-200-25914, made this research activity possible.

References [1] [2] [3] [4] [5] [6]

Jones, N., Structural Impact, Cambridge University Press, 1989. Symonds, P.S., Survey of methods of analysis for plastic deformation of structures under dynamic loading, Report No. BU/NSRDC, Brown University, 1967. Jones, N., Some comments on the modelling of material properties for dynamic structural plasticity, Proc. of Int. Conf. Mechanical Properties of Materials at High Rates of Strain, Oxford, pp. 435-445, 1989. Johnson, G.R. & Cook, W.H., Fracture Characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Eng. Frac. Mech., Vol. 21, No. 1, pp. 31-48, 1985. Avalle, M., Belingardi, G., Vadori, R. & Masciocco, G., Characterization of the strain rate sensitivity in the dynamic bending behavior of mild steel plates, Proc. EUROMAT 2000, Tours, pp. 505-510, 2000. Avalle, M., Belingardi, G. & Gamarino, M., An inverse approach for the identification of strain-rate sensitivity parameters of sheet steels, Proc. SUSI 2004 (Structure under Shock and Impact), Creta (Grecia), pp.13-22, 27-29 March 2004.

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Mechanical characterisation of a viscous-elastic plastic material, sensitive to hydrostatic pressure and temperature V. D. Le1, M. Caliez1, M. Gratton1, A. Frachon1 & D. Picart2 1

Laboratoire de Mécanique et Rhéologie, EA 2640, Ecole d’Ingénieurs du Val de Loire, Université François Rabelais de Tours, Blois cedex, France 2 C.E.A. Le Ripault, Monts, France

Abstract This paper deals with the characterization of the static mechanical behaviour of an energetic material. Due to the constituents (crystals and a polymeric binder), the behaviour is influenced by the pressure, the temperature and the strain rate. The temperature, considered varying slowly, is a parameter and the computational problems are uncoupled. Therefore, a complete experimental protocol and a model have been developed. Inspired from the Visco-Scram model, the behaviour is described using a general Maxwell model in which all the branches are affected by an isotropic damage. The first branch takes into account elastic-plastic behaviour. The yield stress is given by a parabolic criterion, characterized using compressive, tensile and tri-axial tests. The hardening is isotropic and the plastic flow rule is nonassociated. The other branches are viscoelastic. A genetic algorithm is used to optimise the viscoelastic parameters, previously obtained using DMA measurements. Comparisons between the model and experiments are proposed for different temperatures, strain rates and pressures. At last, a user material subroutine has been developed in Abaqus Standard and finite element computations of the Brazilian test are compared to the experimental response. Keywords: energetic material, parabolic plastic criterion, genetic algorithm, DMA, viscoelasticity, Isotropic damage, Brazilian tests.

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212 High Performance Structures and Materials III

1

Introduction

The material is made of organic and energetic crystals mixed with a few percentage of a polymeric binder. After an isostatic compaction forming process, the material has a small porosity of a few percent. Samples can be machined in several geometric shapes, which are more than ten times the length of the material heterogeneity. In order to survey the possible aging of this material, an accurate determination of the mechanical properties has to be done. Unfortunately, this material being available in small amounts, the characterization must be made using a reduced number of standard tests. An unusual experimental procedure is proposed in this paper for this kind of material. When monotonic loading paths were used to determine for example the influence of the strain rate, each sample is submitted to complex loading paths including relaxation, recovery and cyclic conditions. The observation of loading-unloading diagram on figure 1 shows some of the main features of the material at room temperature, and entails specific arrangements for the mechanical tests. 1) Hydrostatic pressure sensitivity: to consider it, an initial hydrostatic loading path (0 MPa, 5 MPa and 10 MPa) is made before the run of a uniaxial compression load. 2) Viscosity: different strain rates (5.10-6 s-1 to 10-3 s-1) have been used to observe such effect. The parameters of the viscoelastic part of the behaviour have been determined using a DMA apparatus (Dynamic Mechanical Analysis). 3) The plastic strains are determined using relaxation and recovery delays. 4) Initial elastic behaviour: standard tests made in various material directions show an initial isotropic behaviour. 5) Damaging: systematic cyclic loading-unloading programs have been performed. 6) Dispersion: to ensure a minimum statistical validity, each loading program is repeated five times. The temperature is considered as a parameter in the model. The material been temperature dependent, compressive and tensile tests are perform at four different temperatures: 5°C, 20°C, 35°C and 50°C. The figure 2 shows the effect of the temperature on monotonic compressive experiments. The main models available in the literature ([1–3]) for this kind of material have been developed for transient dynamic behaviour and are not adapted for a quasi-static study. For example, the influence of the pressure is omitted, even as the difference of behaviour observed in tension and compression loading paths. The Maxwell model we have chosen (fig. 3) is close to the constitutive law proposed in [2]. Several damageable viscoelastic branches and one damageable elastoplastic branch are used. The main difference with the Bennett and coworkers model is the presence of the elastoplastic branch. The determination of the plastic (resp. viscoelastic) behaviour is described in the second (resp. third) part of this paper. In particular, a genetic algorithm has been used to optimize the determination of the viscoelastic parameters. The fourth part deals with the damage rule. It can be noted that the characterization of the plasticity, the damage and the viscosity are uncoupled. The damage rule is assumed to affect also the viscoelastic branches. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Lastly, some comparisons are presented between the model response and the available experimental data.

Figure 1:

Cyclic tests for a given Figure 2: strain rate: pressure effect on the global responses of the material (20°C).

Compressive tests for a given strain rate: temperature effects.

Damageable

Ep E1 Ej En

Figure 3:

η1 ηj ηn

Rheologic diagram of the viscoelastic plastic damageable model.

The test procedure is described in [4]. The first stage (when this one exist) of the loading program is a hydrostatic loading phase. Then, five or six uniaxial loading-relaxation-unloading-recovery cycles are done. The test is driven by one of the two longitudinal gages. Relaxation times have been defined to guarantee an almost complete relaxation of viscous stresses. All the tensile tests and the compressive tests have been done for several temperatures (5°C, 20°C, 35°C and 50°C). Some compressive DMA experiments have been realized using samples of 50 mm long, and a cross section of 4x4 mm2. A small initial preload of 10 µm and a strain amplitude of 5 µm are used to stay in the viscoelastic domain. The range of frequencies going from 0.004 to 40 Hz, the strain rate ranges from 2.10-6 s-1 to 2.10-2 s-1.

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214 High Performance Structures and Materials III

2

Elastoplasticity

The macroscopic stress tensor σ is defined as the sum of the stress of each branch:

σ =σ

ep

+

n

∑σ j =1

j

,

(1)

where the subscript “ep” (resp. “j”) denoted the elastoplastic branch (resp. the jth viscoelastic branch). An additive decomposition is assumed for the strain mechanical part (superscript “m”) and the strain thermal part (superscript “th”):

ε = ε m + ε th . (2) An additive decomposition of the mechanical strain of each branch is also assumed between: an elastic part (superscript “e”) and a plastic (superscript “p”) part (resp. viscous) (superscript “p” or “v”): p elasto-plastic branch, (3) ε m = ε eep + ε ep

ε

m

= ε ej + ε vj visco-elastic branches.

(4)

2.1 Thermo-elasticity

E

The elastic part is supposed linear damageable and given by the following equation: E

σ ep = (1 − d ) (θ ) : ε eep ,

(5)

where (θ ) is the elastic tensor of the virgin material and is defined as a function of the temperature, and d a damage parameter (see below). To identify the elastic mechanism, we need to isolate the elastoplastic behaviour (in particular to be sure that the viscous stresses are relaxed). To this end, the ends of the relaxation and recovery phases are used to determine the elastic modulus. The initial Young’s modulus Eep is a function of the temperature and is defined in the table 1. A value of 0,3 is obtained for the Poisson’s ratio for all the temperatures, using longitudinal and transversal strain measurements. This ratio is used for all the viscoelastic branches. The thermal expansion coefficient is c = 50.10-6 K-1. Table 1:

Temperature dependency of the Young modulus.

Temperatures Initial Young modulus Eep (MPa)

5°C 3400

20°C 2900

35°C 2700

50°C 2450

2.2 Yield criterion A review of the main criteria used to describe isotropic plasticity is presented in [4]. A criterion has been developed at Cambridge University in view of soil modeling and is famous today as the “Cam-clay” model [5]. Numerous WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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adaptations of this model were then developed for various applications [6–8]. As the forming process of the material is an isostatic compaction up to a pressure of 200 MPa (which amplitude will never be reached in quasi-static applications), the criteria is supposed open on the hydrostatic negative axis. Open threshold are usually derived from Mohr-Coulomb, Mises-Schleicher [9], Drucker-Prager [10] and more recently Hoek-Brown formulations [11]. Raghava et al. [12] applied the Mises-Schlecher‘s threshold to polymers. The evolution of this criterion is described by two hardening variables, associated to tensile and compressive response. Lastly, a unified model is proposed by Aubertin and Li [5] in order to reproduce all the kinds of criterion (elliptic, parabolic, hyperbolic). Due to a lack of data about the nature of the hardening mechanisms, an isotropic hardening parameter, denoted k, is introduced in the model. Then, a saturation of the hardening mechanism at the maximum stress is taken into account in the model. A nonlinear plasticity criterion reproduces the evolution of the yield stress (fig. 4). The following relation is used: f ( Q , P , k ) = σ eq − k = 0

where Q = (1 3 ) σ

D

:σ

D

with

σ eq = Q 2 +

is the octahedric stress, σ

D

k2 X (k )

P , (6)

the deviatoric stress

and P the pressure. The set of yield curves is completely defined as soon as the function X(k) and the hardening law are defined. The following guidelines help for the determination of the function X(k). First, it is assumed that the yield curves do not cross themselves in the P-Q plane, each one being embedded in those of higher levels, all of them being embedded in the extreme curve. This is a necessary - but not sufficient condition for the phenomenon to be governed by a unique state variable which is the isotropic strain-hardening parameter. Elementary algebra shows that the following relation satisfies the previous assumption: X (k ) = X 0 + ( X m − X 0 )

k − k0 km − k0

,

(7)

where the parameter X0 = 1,5 MPa, positioning the summits of the initial parabola, is supposed not being temperature depending, and the three parameters Xm, k0 and km are temperature depending. The Xm and km parameters are determined using the ultimate yield stress curve relating the maximum stress states in the P-Q plane. The values are given table 2. Table 2: Temperatures Xm (MPa) k0 (MPa) km(MPa) c1 c2

Temperature dependency of the hardening parameters. 5°C 1,635 0,80 3,88 700 150000

20°C 1,620 0,55 3,5 500 100000

35°C 1,610 0,45 3,26 500 100000

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50°C 1,592 0,35 2,82 500 90000

216 High Performance Structures and Materials III The hardening parameter k has to be related to an effective plastic strain variable, denoted p. In order to obtain a unique curve k(p) for all the available experimental data, p is defined as the cumulated deviatoric plastic strain (fig. 5).

Figure 4:

Figure 5:

Initial and saturated yield criterions for the different temperatures.

Hardening parameter k versus the effective plastic strain p for the different temperatures.

For the hardening law, the following hyperbolic relation is used to interpolate the data:

  1 k ( p ) = k0 + ( km − k0 )  1 − 2   1 + c1 p + c2 p  where c1 and c2 are two parameters temperature depending (table 2).

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(8)

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2.3 Flow rule The flow direction is determined using the ratio between volumic and deviatoric effective plastic strain rates. This ratio, usually called “dilatancy” and denoted β, is given by the following relation:

β=

ε eppV ε eppD

,

p with ε eppV = Tr ( ε ep )

and

pD pD : ε ep ε eppD = 3 ε ep , (9)

pD ε ep being the deviatoric plastic strain rates.

The flow rule expression is then: p ε ep

σ

D

β ep = λ  + 3  3Q

3 I  . ,  (1 + β 2 )

(10)

λ being the plastic multiplier and the plastic flow direction being normalized. The dilatancy β is characterized from the experimental results as a constant (β = 0,15). As a result, a nonassociated plastic law is justified.

3

Viscoelasticity

3.1 Dynamic Mechanical Analysis DMA experiments are used to make a first identification of the linear viscoelastic parameters. The stress response to a unit sinusoidal strain solicitation for this kind of model can be break up in an in-phase part (related to the storage modulus Estor) and an out of phase part (related to the loss modulus Eloss). The analytical response of a generalized Maxwell model is known and depends of the distributions of the Young modulus (Ej) values and of the viscosity parameters (ηj) values. In order to limit the number of parameters (here n = 10 branches) for a more accurate determination, the following relations are proposed from the DMA results [4]:

η j = 10

( A1

j −1 n −1

+ A2

)

((

and E j = A3 + A4 e

A5

j −1 n −1

)

)

−1 .

(11)

Then, the number of unknowns decreases to five (A1 to A5). These relations allow reproducing at the same time the storage and the loss modulus. 3.2 Genetic algorithm The previous set of parameters is used here to determine the bounds of each parameter. We are in the case of a combinatory optimisation problem where a large number of solutions could be suitable. We have chosen to perform an inverse identification of the viscoelasticity parameters directly from the experimental tests. Classical optimisation methods, like conjugated gradient, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

218 High Performance Structures and Materials III have been dismissed because of the possible large number of solutions. Then, a genetic algorithm [13] has been used in this study. Genetic algorithms are based on the Darwinian principle of “survival to the fittest”. An initial population of a given size is created from a random selection of parameters values. Each parameter set represents individual chromosomes. Each individual is assigned of a fitness based on how well each individual chromosome allows it to perform in its environment. The algorithm produces new generations by applying three evolution operators: selection, crossover and mutation. For each generation, the fit individuals survive and the weak die. Evolution operators create new individuals (children) from two selected parents, and these children replace the weak individuals for the next generation. Successive generations are created until very fit individuals are obtained. This algorithm offers the advantage of exploring all the solutions space to find a global optimum of an optimization problem. A sensitivity analysis of the parameters is not required. 3.3 Objective function The objective function is a direct measure of the quality of a solution. The goal is here to minimize the gap between the experimental strain-stress curves and the corresponding simulated curves. Due to different times of recording during the experiments, we propose a fitness function that represents the spatial gap, weighted by the segment length Li between two consecutive data, in the stressstrain space, i.e.: nb exp pts

Obj =

∑ i =1

2

(σ ie ( t ) − σ is ( t ) ) . Li ,

(12)

nb exp pts

∑L i =1

i

where superscripts “s” and “e” respectively denote simulations and experiments. 3.4 Results The simulation uses the data corresponding at the end of load, the end of relaxation, the end of discharge and the end of recovery. From the values of the longitudinal strain and time in these points, the strain rate is rebuilt, constant by piece. The strain increment is then given from the strain rate and the step of time of the program. At 20°C, all the tests (compressive tests with different strain rates, tensile tests and triaxial tests) are used to optimise the five parameters Ai. 60 generations constituted of 500 individuals have been performed and leads to the optimized values of the five Ai parameters at 20°C. The restricted quantity of material does not allow making all those tests for all the tested temperatures. So, for the 5, 35 and 50°C temperature values, only the tensile and compressive tests, for only one strain rate, are done. Therefore, we supposed that the temperature dependency of the Young modulus of the viscoplastic branches is the same that the elastoplastic branch one: E j (θ ) = E ep (θ ). E j (20° C ) / E ep (20° C ) . WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Therefore, for the tests at 5°C, 35°C and 50°C, only the parameters A1 and A2 have to be identified with the genetic algorithm. The results are presented in the table 3. Table 3:

Temperature dependency of Ai parameters defining the ηj viscosity parameters.

Temperature 5°C Tests com- Compressive, puted from: one strain rate A1 2,78 A2 2,63 A3 A4 A5

4

20°C All 3,98 1,64 160,31 16,15 1,21

35°C 50°C Tensile/Compressive, Tensile/Compressive, one strain rate one strain rate 2,15 2,62 2,60 2,32

Isotropic damage

Assuming an isotropic damage, experimental data shows that the phenomenon regularly increases with the highest positive principal strain (fig. 6). This observation indicates that the most probable damaging mechanism is the result of the development of internal micro-defects (cavities, cracks) with tension [14– 18]. A damaging factor d is classically defined as: d=

E0 − E E0

,

(13)

where Eo and E are the initial and current Young’s modulus. A constant Poisson’s ratio is assumed here.

Figure 6:

Damage versus the maximum positive strain.

Experimental values of d immediately result from the measurements of E. An hyperbolic relation, eqn. (14), is used to reproduce an average evolution of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

220 High Performance Structures and Materials III the damaging factor, providing that its value is bounded to 1; d1, d2 and d3 are three parameters. The subscript “+” means “the positive part”.

(

)

  1 , (ε I ) > + + d 2 .1d = d 1 . sup < max N N  1 + d 3 sup < max ε I >+  N = 1 3 I to N   time time

(14)

I =1 to 3

The damage rule is reported in the fig. 6. One can see that the model response is identified using the compression measurements. The hydrostatic data are not taken into account because the pressure stops an eventually growth of the microcavities. The introduction of a second damage mechanism, as proposed in [19], to model the tensile damage is in study. The identified values are the same for all the temperatures: d1 = 0,5, d2 = 0,75 and d3 = 75.

5

Model versus experiments

The constitutive law has been implemented in the finite element code Abaqus/standard. The model is compared to experimental data with unloading cycles to access to the plasticity level and the damage level. Those seem quite well reproduced even if the transversal model response does not present enough damage in compression (fig. 7–8). This observation can be associated to an anisotropic damage (which has been neglected here).

Figure 7: Compressive test (3.10-6 s-1; 20°C); model versus experiment.

Figure 8: Compressive test (1.5 103 -1 s ; 20°C); model versus experiment.

The plasticity branch reproduces very well the pressure effect that causes the difference between tensile and compressive responses (fig. 7, 9, 10). The rate effect is also quite well reproduced on the two compressive tests even if the unloading curves do not present the same nonlinearity (fig. 7–8). In the same manner, the viscous effects in the triaxial test with 10 MPa of confinement pressure (fig. 10) is underestimated. Certainly, these phenomenons is associated to an internal friction in the material or a viscosity pressure dependency. The pressure dependency is certainly the reason of the discontinuity of the Ai

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parameters at 20°C (table 3), temperature for which all the tests are computed and in particular the triaxial ones.

Figure 9:

Figure 11:

Tensile test (3.10-5 s-1); model versus experiment.

Figure 10: Triaxial test (3.10-5 s-1; 20°C); model versus experiment.

Global response of the Brazilian test; model versus experiment.

The temperature dependency of the model parameters gives a quite good adequacy of the model versus the experimental results (fig. 9). The implementation of the constitutive law in the finite element code Abaqus has allowed one to compare simulations to more complex experimental configurations as three-point bend tests and Brazilian experiments. The global response of the compression diametrical test is given in fig. 11. One can see that too much nonlinearity appears in the model response. This point is in study, the three-point bend test giving providing first answers.

6

Conclusion

An experimental procedure has been carried out to characterize a complex material behaviour. A multibranch viscoelastic plastic and damageable model and the corresponding identification procedure have been developed. A genetic algorithm optimisation has one allowed to find accurately some of the viscous parameters. This model has been implemented in the finite elements software WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

222 High Performance Structures and Materials III Abaqus using a user subroutine UMAT. The comparisons between simulations and experiments show a good agreement. Our future works are now devoted to the improvement of the damage rule and of the failure threshold. Lastly, the anisotropy observed during the experiments has to be introduced in the model. The pressure dependency of the viscosity is in study.

Acknowledgement The authors address a special thanks to J.L. Brigolle for its contribution to this study, especially for the realization of the experiments.

References [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

F.L. Addessio, J.N. Johnson, A constitutive model for the dynamic response of brittle materials, J. Appl. Phys., Vol. 67, 1990, pp 3275-3286. J.G. Bennett, K.S. Haberman, J.N. Johnson, B.W. Asay, B.F. Henson, A constitutive model for the non-shock ignition and mechanical response on high explosives, J. Mech. Phys. Solids, Vol. 46-12, 1998, pp. 2303-2322. R.M. Hackett, J.G. Bennett, An implicit finite element material model for energetic particulate composite materials, Int. J. Numer. Meth. Engng, Vol. 49, 2000, pp. 1191-1209. M. Gratton, V.D. Le, A. Frachon, M. Caliez, D. Picart, Mechanical behaviour of a viscoelastic plastic granular material: Experimental procedure and modelling, WSEAS trans. on comp., Is.1, Vol.5, Jan.2006, pp.149-156. M. Aubertin, L. Li, A porosity-dependant inelastic criterion for engineering materials, Int. Jour. of Plast., Vol. 20, 2004, pp. 2179-2208. J.H. Prevost, R. Popescu, Constitutive Relations for Soil Materials, Electronic Journal of Geotechnical Engineering, First Issue, 1996. O. Coussy, Mécanique des Milieux Poreux. Editions Technip, 1991. P.Y. Hicher, J-F. Shao, Elastoplasticité des sols et des roches - Modèles de comportement des sols et des roches - 1, Hermès Science Publications, 2002. F. Schleicher, Z. Angew, Math. Mech., 6:199, 1926. D.C. Drucker, A.M. Asce, R.E. Gibson, & D.J. Henkel, Soil Mechanics and Work-Harding theories of plasticity, Transactions American Society of Civil Engineers, Vol. 122, 1957. X.D. Pan, J.A. Hudson, A simplified three dimensional Hoek-Brown yield criterion, Rock Mechanics and Power Plants, M. Romana (ed.), Rotterdam: Balkema, 1988, pp. 95-103. R. Raghava, R.M. Caddekk, G.S.Y. Yeh, The macroscopic yield behaviour of polymers, J. Material Science, Vol.8, 1973, pp. 225-232. D.E. Goldberg, Genetic algorithms in search, Optimization and Machine Learning, Addison Wesley Publishing Company, 1989. J. Mazars, Mechanical damage and fracture of concrete structure, Proc. I.C.F. 5, Cannes, France, 1981, pp. 1499-1506. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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[15] [16] [17] [18]

[19]

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A. Dragon, D. Halm, Th. Désoyer, Anisotropic damage in quasi-brittle solids: modelling, computational issues and applications, Comput. Methods Appl. Mech. Engrg., 183, 2000, pp. 331-352. D. Halm, A. Dragon, An anisotropic model of damage and frictional sliding for brittle materials, Eur. J. Mech., A/Solids, Vol.17, no.3,1998, pp.439-460. J. Mazars, A description of micro and macroscale damage of concrete structures, Engineering Fracture Mechanics, Vol.25, no.5/6, 1986, pp.729-737. J. Mazars, F. Ragueneau, G. Pijaudier-Cabot, Continuum damage modelling for concrete structures in dynamic situations, Continuum Damage Mechanics of Materials and Structures, O. Allix and F. Hild (Editors), Elsevier Science Ltd., 2002. J. Lee, G.L. Fenves, Plastic-damage model for cyclic loading of concrete structures, Journal of Engineering Mechanics, vol.124, no.8, pp.892-900, 1998.

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Identification of the material properties of composite beams: inverse method approach E. Euler1, H. Sol1 & E. Dascotte2 1

Department of Mechanics of Materials and Construction, Vrije Universiteit Brussel, Brussels, Belgium 2 Dynamic Design Solutions NV, Leuven, Belgium

Abstract In load carrying applications one is mainly interested in the stiffness properties of the structural component. Elastic and shear moduli supplemented with section properties render the stiffness properties of the component. Moduli of isotropic materials are well known and well documented. However, this is not the case for composite materials. The developed procedure belongs to the group of mixed numerical experimental methods. The method makes use of modal data to determine the effective orthotropic material properties of composite beams. Modal reference data is experimentally obtained from the beam at hand. The other modal data set is obtained from a finite element model of the same beam. The orthotropic material properties, also called parameters, in the finite element model are then modified in such a way that both sets of modal data match. If those two sets match, the virtual model has “in a global sense” the same mass and stiffness properties as the real model. A program written in FEMtools is applied to five test cases. Results are discussed. Keywords: inverse methods, FEA, orthotropic material properties, composite materials, modal data.

1

Introduction

Most engineers have considerable experience in the design of simple structural components using isotropic materials. However, today more and more composite materials are used for structural elements. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06023

226 High Performance Structures and Materials III In load carrying applications one is mainly interested in the stiffness properties of the structural component. Elastic and shear moduli supplemented with section properties render the stiffness properties of the component. Moduli of isotropic materials are well known and well documented. This is not the case for composite materials. The effective laminate properties depend on fiber material, matrix material, ply orientation, laminate thickness, stacking sequence, etc. Infinite combinations are possible, resulting in a huge amount of different moduli. It is clear that not all possible scenarios are studied in literature. If all the details of the composite material are known, the engineer can calculate effective laminate properties and use them in theories like for example the first-order shear deformation theory for thin-walled laminated beams. If the ply lay-up results in coupling phenomena, the engineer has to use advanced software to simulate the behaviour of the structure. All this requires substantial effort. In some cases, not all the details of the composite material are known. In this case, elastic properties can be determined by experiment. Drawback, these experiments are destructive in nature. Another possibility is to obtain stiffness properties of the whole structure by conducting an experiment. Drawback here, influence of boundary conditions. All this requires substantial effort. This paper presents a method to determine the global effective orthotropic material properties of composite beams by measuring a certain amount of natural frequencies of the structure.

2 Outline: method and program First, experimental modal analysis (EMA) [1] is used to extract the natural frequencies of the physical structure. This modal data is used as reference response data during the procedure. Next, a mathematical model of the structure is created. In the physical model all mass and stiffness related properties are known except for the anisotropic material properties. All known properties are implemented as such into the mathematical model. The real composite material is modelled as a global homogeneous orthotropic material in the mathematical model. This mathematical model is solved for modal data using finite element analysis. Finally, two sets of non matching modal data are available. One set composed of experimentally obtained reference data. The other set contains calculated data from the mathematical model. The orthotropic material properties are then modified in such a way that both sets of modal data match. If those two sets match, the virtual model has “in a global sense” the same mass and stiffness properties as the real model. This principle is called model updating. The procedure to identify orthotropic material properties by natural frequency measurement is automated in the form of a user-friendly FEMtools program. Program flow is visualized in the next flowchart.

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Input Geometry

227

Check Geometry not ok. New Input Geometry

Input initial material Properties Obtain converged mesh with respect to first ten frequencies {freq}analyt. The frequencies {freq}analyt are based on initial material properties. The ten {freq}analyt and {ψ}analyt are shown. The sensitivity matrix material properties versus frequencies {freq}analyt is shown. Determine how many N response frequencies {freq}exp are taken into account. Determine which material properties are identified.

Fem database contains: First N {freq}analyt First N {ψ}analyt Initial Material Properties

Test database contains: N experimental frequencies {freq}exp First N {ψ}analyt Corresponding with “physical” model

Model Update Modified fem model is shown. Reference “physical” test model is shown. MAC-matrix fem versus test is shown. Correlation tracking curve is shown. Modified Material Properties listed.

No Successful Identification

Successful Identification

3

Model updating: mathematics

From a mathematical point of view, the difficulty with model updating is that the relation between output vector and the parameter vector is nearly always nonlinear. This means that updating the parameter values from an initial value to a final value has to be done iteratively. The value of the output for some new parameter values can be estimated with a Taylor expansion. The Taylor series can be cut off after the linear term or can be cut after some higher order terms. Figure 1 illustrates the mathematics involved in model updating. The matrix [S ] that appears in the linear Taylor term is called the sensitivity matrix. It contains the partial derivatives of the output components for the different parameter values. The success of model updating is highly dependant WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

228 High Performance Structures and Materials III from the numerical condition of the sensitivity matrix because [S ] must be inverted in every iteration step to obtain the parameter correction {∆p} .

Figure 1:

Mathematics of model updating (left) and minimizing of cost function (right).

Figure 2:

Simple cost function (left) and (right) a more elaborated cost function.

Convergence from an initial parameter value

{p}0

to the final value is

obtained by minimization of a cost function in every iteration step. Graphically, this means that the cost function evolutes iteratively from an initial point in the (m+1) dimensional parameter space towards a global minimum. The parameter values in the global minimum are the optimal parameter values. Figure 1 illustrates the principle. One of the most simple cost functions leads to a weighted least squares estimator. In this cost function, a weighting matrix [ W ] is pre- and post multiplied with the difference between the measured and the computed response WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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in a point in the parameter space. The weighting matrix [ W ] allows to express a different confidence in the different measured data points. This is illustrated in Figure 2. A more elaborated cost function also takes the initial parameter values into account. A second term is added to the cost function in which a weighting matrix is pre- and post multiplied with the difference between the initial and the current parameter value. Again, a weighting matrix allows to express a different confidence in the different initial parameter values. The matrix [ W ]p represents the weighting matrix expressing the confidence in the model parameters, while

[W]y

is a weighting matrix expressing the

confidence in the reference response test data.

4

Model updating: important considerations

The success of model updating strongly depends on the following considerations. 4.1 Accuracy numerical model A first aspect is the quality of the mathematical model. All known mass and stiffness properties must be correctly represented in the mathematical model. Secondly, this model is solved using the finite element method. Error due to discretization of the mathematical model is introduced. The discretization error must be kept to a minimum. A coarse mesh density is not as flexible as the mathematical model with infinite degrees of freedom. Hence, the calculated natural frequencies will be overestimated. The model updating will result in physically incorrect material properties. Discretization error is estimated by comparing successive solutions with refined mesh density. More elements will result in a more exact solution of the mathematical model. When the difference between successive solutions is minimal, the mesh is “convergence”. The accuracy of the analysis is related to the mesh density. 4.2 Experimental error Incorrect input cannot result in physical correct material properties. Experimental error can be divided into two categories. Random errors can be treated with statistical procedures. Modal analysis software is capable in minimizing random errors. Systematic errors are a lot more difficult to detect and to solve. A damaged accelerometer will produce systematically an error on his output. Experimental tests on an analytical known problem can indicate a systematic error. 4.3 Controllability A numerical model is controllable if it is possible to tune the model output from an arbitrary point {y}0 in the parameter space to a measured point {y}exp with the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

230 High Performance Structures and Materials III selected parameter set. Non-controllability can be turned into controllability by selecting more or more appropriate parameter. A parameter is appropriate if the sensitivity with respect to the response is sufficiently high. 4.4 Observability A numerical model is observable if measurement of the output contains sufficient information for the identification of the selected parameters. A natural frequency does not contain information concerning the colour of the structure. It is impossible to identify a colour by measuring the natural frequency of the structure. Of course, one wants to identify all four orthotropic material properties by measuring a certain amount of frequencies. Unfortunately, depending on the structure, not every parameter will be identifiable. To investigate observability, FEMtools offers sensitivity sum curves. Such a curve sums all sensitivity values for all responses as a function of parameter number. A low sensitivity sum value shows that none of the responses contains sufficient information for the identification of that parameter. If this occurs, one can add more experimental measurements or conclude that the parameter in question can not be identified by frequency measurement. 4.5 Initial parameter values Model updating requires initial parameter values {p}0 . The quality of the initial parameter values can affect both the speed of convergence and whether or not convergence to the “true” parameters is achieved. The parameter estimation problem is often posed as a problem of constraint minimization and in the case of non-linearity in parameters; a particular minimum is sought on a surface which contains many minima and maxima. Usually either a local minimum is sought, when there is confidence in the initial model, or else the problem is to determine the unique global minimum. To obtain initial values for the longitudinal moduli Ex and Ey, the first bending frequency of a beam specimen is determined. To identify Ex a beam model in the span direction of the structure is used. To identify Ey a beam model in the width direction of the structure is used. Knowledge of the bending frequency and dimensions of the specimen renders the elastic modulus of the material. The formula used to estimate the elasticity modulus [2]: E = 7.89e-2 * (frequency)2 * (length)4 * density * transverse section * (Moment of inertia) –1

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5

231

Model updating: FEMtools

5.1 Sensitivity calculation The matrix [S ] that appears in the linear Taylor term is called the sensitivity matrix. It contains the partial derivatives of the output components for the different parameter values. There are two basic approaches to compute sensitivities: (i) using differential sensitivities and (ii) using a finite difference approximation. Which one to use depends on the parameter type. For nonproportional parameters, such as the Young’s modulus for orthotropic materials, FEMtools uses finite difference sensitivities. In this method derivatives are approximated with a forward finite difference approach. This is done using the results of two finite element analysis for two states of the parameter pj. The element ij of the matrix [S ] becomes

∆y i y i (p j − ∆p j ) − y i (p j ) = ∆p j ∆p j

The sensitivities discussed so far are absolute sensitivities. This means that they use the units of the response and parameter value. The absolute sensitivities can be made independent of the units used for the response and parameter values. They are then referred to as normalized sensitivities. A normalized sensitivity shows the percentage change of the response value for one percent change of the parameter value. The element ij of the matrix [S n ] can be written as

Sij(n) =

∆y i p j × ∆p j y i

5.2 Mode shape pairing During model updating, the algorithm will try to drive the predicted analytical response to the experimental reference data in the test database. This implies that the algorithm knows which analytical response has to match with which experimental response. This can be defined using sequential mode shape pairing. Sequential mode pairing means that analytical mode 1 will be paired with experimental mode 1, analytical mode 2 with experimental mode 2, etc. If during model updating a switch of mode shapes occurs, this method fails. The resulting frequencies will probably be close but the mode shapes are different. There is no possibility to connect a mode shape to a particular experimental reference response. To deal with the above problem, the program will copy the analytical mode shapes, predicted by the initial values for parameters Ex, Ey, Gxy and νxy, to the test database. The user of the program knows exactly the sequence of these modes [the program will previously show them]. It is now up to the user to connect the correct experimental reference frequency with the corresponding mode shape. In other words, the test database must reflect the correct physical WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

232 High Performance Structures and Materials III response [natural frequency with corresponding mode shape]. Automatic mode shape pairing can now be used to drive the analytical response to the experimental reference response in the test database. During model updating, automatic mode shape pairing makes a relation between those frequencies which have the highest Modal Assurance Criterion (MAC). The MAC is a measure of the squared cosine of the angle between two mode shapes. To compute the MAC between an analytical and experimental mode shape, the following equation is used:

({Ψ } {Ψ }) )= ({Ψ } {Ψ })({Ψ } {Ψ }) 2

T

MAC(Ψanalyt , Ψexp

a

e

T

a

T

a

e

e

After model updating, a MAC value can be calculated between the updated fem model and the physical test modal. If no mode switch occurred during model updating this MAC matrix is a diagonal matrix. 5.3 Aspects of convergence CCABSOLUTE: Average value of weighted absolute relative differences between predicted and reference resonance frequencies. CCMEAN: Average value of weighted relative differences between predicted and reference resonance frequencies. In model updating, the above correlation coefficients are interpreted as an objective function that needs to be minimized. With each iteration loop the values of the correlation coefficients will be verified to check if a convergence criterion is satisfied. The following criteria are used: (1) The value of the reference correlation coefficient is less than an imposed margin: CCt < ε 1 (2) Two consecutive values of the reference correlation coefficient are within a given margin.

CC t +1 − CC t < ε 2 The iteration loop in model updating will be stopped as soon as one of these tests is satisfied. Default values for ε1 and ε2 are used namely 0.08.

6

Example: symmetric U-profile

6.1 Section properties Dimensions according to figure 3: Moment of inertia Iy: 439635 mm4 Moment of inertia Iz: 251905 mm4 Torsional stiffness factor J: 1990 mm4

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6.2 Volume properties Volume: 867793 mm3 Mass: 1599 gram Density: 1.8426e-9 Mg/mm3 Length: 1470 mm

65 mm 26.5 mm

3.18 mm 19 mm

3.18 mm

3.18 mm

63.5 mm

Figure 3:

Symmetric U-profile.

6.3 Estimation initial values An initial value for E x is obtained by measuring the first bending mode of a beam specimen in the span direction of the structure. An initial value for E y is obtained by measuring the first bending mode of a beam specimen in the width direction of the structure. Typical engineering properties of a glass-polymer composite are used for G xy and νxy [3].

Ex

Initial Method used value to obtain value (N/mm²) 22000 Measured

Ey

11800

Measured

Gxy

5000

Literature

vxy

0,3

Literature

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234 High Performance Structures and Materials III 6.4 Calculated natural frequencies and mode shapes (based on initial values)

Mode 1: 32

Mode 2: 61

Mode 3: 116

Mode 4: 140

Mode 5: 197

Mode 6: 199

Mode 7: 207

Mode 8: 217

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Mode 9: 221

235

Mode10: 235

6.5 Sensitivity matrix The sensitivity matrix shows the sensitivities of the four material parameters versus the first ten resonant frequencies. Parameter 1, 2, 3 and 4 equals respectively Ex, Ey, Gxy and νxy. Figure 4 shows the sensitivity matrix. The sensitivity matrix indicates that the first torsion mode contains information about the shear modulus Gxy. The second mode - a complex bending mode around the Y-axis – is sensitive to a change in value of Ex and Gxy. The parameter Ey can be identified by using resonant frequency five and six. None of the responses are sensitive to a change in value of νxy. No attempt should be made to identify νxy by measuring natural frequencies.

Figure 4:

Sensitivity matrix.

6.6 Model updating results Natural frequencies and corresponding mode shapes are measured using a laser vibrometer as measuring device and a shaker as excitation device. There was no indication, what so ever, of modes five and six in the experimental measured data. Probably due to the fact that a single shaker is not optimal to excite mode five. Additional, both modes are extremely close to each other. Consequently, further discussion will focus on the determination of Ex and Gxy. The first four modes are used to identify Ex and Gxy. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

236 High Performance Structures and Materials III

7

Conclusions

This paper presents a method to determine the global effective orthotropic material properties by measuring a certain amount of natural frequencies of a composite beam structure. The program - developed in FEMtools - is used to determine the properties of composite beams with closed cross-sections and open cross-sections. Some general trends are clearly observed and stated hereafter. Closed box beams behave relatively straightforward. In general, the first modes are bending modes around the principal axes of the cross-section. The torsion mode is found in a higher region since the torsional stiffness factor of a closed cross-section is rather high. The bending modes can be used to identify Ex and the torsion mode can be used to identify Gxy. The natural frequencies of this kind of beams are not sensitive to a change in value of Ey and v xy . It is not possible to determine these parameters by measuring natural frequencies. Beams with open cross-section are much more complex in behaviour and general conclusions can not be drawn. For this kind of beams, certain complex mode shapes are sensitive to a change in value of multiple parameters. Moreover, frequencies exists which are particularly sensitive to a modification of Ey and can be used to identify this parameter. It is not possible to identify values for v xy .

Initial values [N/mm²]

Ex

Parameter selection

Ey Gxy Vxy Ex Ey Gxy Vxy

22000 11800 5000 0,3 X

X

Mode Fem data based on shape initial values switch versus Exp. data [Hz] occur Fem Exp. Description Y N value value mode 32

32.5

1 torsion 1 complex bending Y 116 137.5 1 bending Z 2 complex 140 146.9 bending Y

X

Final parameter values [N/mm²]

Ex

Ey Gxy Vxy

28944 11800 5014 0,3

61 66.25

The program needs initial values for orthotropic material properties, before identification can start. A deviation of 25 % given on estimated initial values, results in the same final updated values for the parameters. Hence, the final updated results are almost not sensitive to a deviation of initial values. The program can also be used to study the influence of the length of the structure on the possibility to identify certain parameters. Consequently, an optimal length can be determined for which the first (two) frequencies are very sensitive to a change in value of a preferred parameter. Making it possible to identify orthotropic material properties in a more easy and structured manner.

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References [1] Modal Testing: Theory and Practice, DJ Ewins, Research studies press [2] Engineering Vibration, DJ Inman, Prentice Hall [3] Stress analysis of fiber-reinforced composite materials, MW Hyer, McGrawHill

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Full-field optical measurement for material parameter identification with inverse methods J. Gu1, S. Cooreman1, A. Smits1, S. Bossuyt1, H. Sol1, D. Lecompte2 & J. Vantomme2 1

Department of Mechanics of Material and Construction, Vrije Universiteit Brussel, Brussels, Belgium 2 Department of Materials and Construction, Royal Military Academy, Brussels, Belgium

Abstract The application of FE simulation in manufacturing processes and virtual prototyping increases every day. In order to allow accurate simulations, correct constitutive models are needed as input to the FE software. A modern and promising way to identify the material parameters in those constitutive models is “inverse modeling”. Full-field measurement is a suitable way to get the necessary experimental data. The technique has many advantages such as large information contents, non-contacting measurement, and versatile size of observation region, among others. However, there is no standardization yet for this kind of measurements. Therefore, there are many disagreements among researchers about how to design DICT experiments and how to correctly collect the data from DICT experiments. This paper will concentrate on discussing the key points of those problems as well as presenting some work experience with the DICT. Keywords: inverse method, FEA, full field measurement, digital image correlation.

1

Introduction

Identification of cracks and defects, and estimation of distributions of material properties from experimental data are inverse problems, which are not well recognized till the middle of 1980’s. Thanks to the development of information technology by leaps and bounds, increasing efforts have been devoted to advance WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06024

240 High Performance Structures and Materials III in inverse problem research, especially at beginning of 1990’s [1]. The main reason is that the production cycles in industry, such as automobile and aircraft industry, become shorter since 1990’s. To face this challenge, many industries take advantage of computer simulations to provide powerful insight into structural behavior of mechanical systems, manufacturing processes, and many other engineering problems, which can reduce the dependency of the manufacturers on expensive and time consuming hardware prototyping [2]. A common requirement for the success of the numerical models used in computer simulation software is the input of the correct material properties. The properties of many materials however can not be found in literature. A modern alternative to find the material properties is inverse modeling. The principle of inverse modeling is shown in Figure 1. The input quantities are assumed to be perfectly known and are the same for the experiment as for the numerical model. The unknown parameters in the numerical model are tuned in such a way that the computed output matches the experimentally measured output, e.g. displacements or strains, as closely as possible.

Figure 1:

The principle of an inverse method for material identification.

In general, the elasto-plastic material parameters are determined by means of standard tests, such as tensile tests, bending tests and torsion tests. However, the stress and strain fields occurring during these traditional tests do not represent the complex stress and strain fields which are generated during real production processes. With those techniques, the specimen can be tested in such a way that the material circumstances are comparable with the circumstances of the material

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during the service life of the construction or during the manufacturing process, e.g. Single Point Incremental Forming process (SPIF). The main problem of adopting more complicated tests in the past was always hindered by the fact that complex displacement fields simply could not be measured. In recent years however, an increasing number of important developments in the field of full-field displacement measuring have been presented. Moreover, modern measurement equipment, as there are Flow Induced Birefringence, Electronic Speckle Pattern Interferometry, Digital Image Correlation techniques, have become commercially available and more accessible than they used to be in the past. Furthermore, more information about material parameters can be caught from one experiment, so called “full field measurement”. In the current article, an example of full-field measurement by means of digital image correlation technique (DICT) will be presented. This technique exploits a good prospect for inverse modeling since it has some important advantages, for instance non-contacting measurement, large observation region, etc. However, there is no standardization for such kind of measurements. Therefore, there are many disputations about how to design DIC experiment and how to correctly collect the data from DIC experiment. In the following the influence of subset size, step size and strain window size on the final strain computation will be discussed in detail. Some experiences for DIC experiment will be introduced for sharing the knowledge.

2

Full field digital image correlation experiment

In general DICT is just a method to measure displacements. However, some extra calculations on the displacement field allow us to extract strain data from the displacement measurement. In this case, this technique will be applied for the measurement of the heterogeneous deformation fields during a forming process 2.1 Principle of DIC Basically, the displacement measurement goes as follows: a random speckle pattern is applied on the surface of the object of interest. During the forming process, a number of pictures of the object of interest are taken with one or more CCD (Charge Coupled Device) cameras (if two cameras are used, it is possible to measure the displacements in three orthogonal directions). Each picture corresponds to a different state of deformation (usually the first picture is taken at zero loading and is called the reference image). Finally the speckle pattern allows us to correlate the different images to each other and as a result the displacement field (relative to the reference image) in the different images can be measured. As was already mentioned, some extra calculations make it possible to extract strain data from this deformation measurement. Figure 2 summarizes the above described technique.

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Figure 2:

Diagram of digital image correlation experiment.

The CCD cameras use a small, rectangular piece of silicon, which has been segmented into arrays of individual light-sensitive cells, also known as “photo sites”. Each cell constitutes one element of the whole picture and is called a pixel. Every pixel stores a certain grey scale value varying from 0 to 255, in agreement with the intensity of the light, reflected by the surface of the tested specimen. Thus, an image can be looked at as a matrix in which every element represents the grey value of the corresponding pixel. In order to run the correlation algorithm, the image is divided in a number of subsets. A subset represents a part of the whole image. The size of a subset can be varied by the user and the choice of a good subset size depends on the deformation. As was already mentioned, this technique uses a random speckle pattern that is simply sprayed onto the surface of the object or that is offered by the texture of the specimen’s material. The objective is to obtain an image with a varied and distinctive pattern, which enables the correlation algorithm to trace the subsets of the reference image in the deformed images [3, 4]. The concept behind the DICT-software 2D matching algorithm is that the distribution of grey values in a rectangular area (subset) in the picture taken of the specimen in the undeformed state, corresponds to the distribution of grey values of the same area in the picture taken at the deformed specimen. In this way the motion and the deformation of the subsets during the deformation of the object are determined. This will finally lead to a displacement field for the whole area of interest. Some additional calculations allow to extract strain data from this displacement field. Figure 3 shows such a subset in the undeformed (left) and deformed (right) configuration.

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Figure 3:

243

A subset in the undeformed (left) and deformed (right) configuration.

Measurements in three dimensional space require a measuring arrangement with two cameras, which should be placed with an angle of approx. 20° between them. A 3D measuring arrangement has to be calibrated prior to a measurement in order to be able to determine the image-forming qualities of the cameras and the lens distortions. With the aid of the determined image-forming qualities the subsequent calculations can be carried out. 10

140

50

280

ø15

Figure 4:

The geometry of the specimen and the generated speckle pattern zone.

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244 High Performance Structures and Materials III 2.2 Description of DIC experiment set-up A tension test on an aluminium beam with a circular hole in the center is selected to study the identification of metallic plastic parameters, for instance, the anisotropic elasto-plastic law based on Hill48 yield locus with kinematic hardening. The geometry of the specimen is shown in Figure 4. The aluminium specimen was loaded in tension up to 80 kN. A speckle pattern was applied to measure the displacement field in the shaded zone by means of two CCD cameras. The images are regularly taken throughout the tension test. The software offered by Limess Company was used to calculate the strains in the shaded regions (see Figure 4).

3

Post processing of DIC results

During the post processing of the DIC experiment, the area-of-interest (AOI) should be selected. Following that, subset size (pixel) and step size (pixel) are chosen for calculating the displacement field. The strain window size is chosen for the strain calculation by Vic 3D software. One image, taken when force applied on the specimen reached a value of 40kN, is selected to study the influence of Subset, Step and Strain window size on the strain value.

Figure 5:

Upper left picture, the contour of strain y calculated with subset 11 pixels; upper right picture, the contour of strain y calculated with subset 17 pixels; lower picture, the contour of strain y calculated with subset 23 pixels.

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3.1 Different subset The subset size controls the area of the image that is used to track the displacement between images. The subset size has to be large enough to ensure that there is a sufficiently distinctive pattern contained in the area used for correlation. To show the influence of the subset size on the strain computation, a subset size of 11, 17 and 23 pixels was applied. The step size and the strain window size were 5 pixels and 15 respectively. It is found that some region with speckles bigger than subset lost the correlation, for instance, the upper left picture in Figure 5. 3.2 Different step The step size controls the spacing of the points that are analyzed during correlation. If a step size of 1 pixel is chosen, a correlation analysis is performed at every pixel inside the area of interest. To show the influence of the step size on the strain computation, a step size of 1, 5 and 11 pixels was applied. The subset size and the strain window size were 23 pixels and 15 respectively. It is found that too small step sizes result in a lot of noise on the calculated strain field. This causes some difficulties to collect strain data. To big step sizes result in too much averaging. Hence the actual strains are underestimated, especially in regions of high strain gradients, e.g. around the hole, as shown in Figure 6.

Figure 6:

Upper left picture, the contour of strain y calculated with step 1 pixel; upper right picture, the contour of strain y calculated with step 5 pixels; lower picture, the contour of strain y calculated with step 11 pixels.

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246 High Performance Structures and Materials III 3.3 Different strain window size The strain window size can be used to adjust the size of the local neighborhood in which the derivatives of the displacement field are calculated. The value indicates the number of neighboring points instead of pixel. To show the influence of the strain window size on the strain computation, a strain window size of 5, 15 and 25 was applied. The subset size and the step size were 23 pixels and 5 pixels respectively. It is found that too small strain window sizes result in a lot of noise on the calculated strain field (Figure 7), whereas too large strain window sizes result in too much averaging, especially in the region of high strain gradients like around the hole.

Figure 7:

4

Upper left picture, the contour of strain y calculated with strain window size 5; upper right picture, the contour of strain y calculated with strain window size 15; lower picture, the contour of strain y calculated with and strain window size 25

Comparison of DIC results and FE simulation results

A FE simulation of this tensile test was performed by means of ABAQUS software, and the obtained strain contours in x and y direction are compared with those from the DIC experiment, respectively (see Figure 8 and Figure 9). Since the specimen is symmetric, only one eighth of specimen was modeled. The results of the FE simulation approach those of the DIC experiment quite well. The FE strains in x direction vary between 0.00011 and -0.00082; the strains in x direction, computed by the DIC software vary from 0.00033 to -0.00092. The FE strains in y direction vary from 0.0024 to 0.000097; the strains in y direction, computed by means of the DIC software vary between 0.0026 and 0. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 8:

Left picture is the contour of strain in x direction obtained from FE simulation. The scale of strain is from 1.11e-4 to -8.19e-4; right picture is the contour of strain in x direction obtained from DIC experiment. The scale of strain is from 3.31e-4 to -9.19e-4.

Figure 9:

Up picture is the contour of strain in y direction obtained from FE simulation. The scale of strain is from 0.0024 to 9.7e-5; the picture below is the contour of strain in y direction obtained from DIC experiment. The scale of strain is from 0.0026 to 0.

5

Conclusion

In the present paper, the feasibility and reliability of data collection from fullfield optical measurement is studied. It has been shown that the size of subset, step and strain window have a large influence on the accuracy of the measured displacements and calculated strains. Unsuitable subset, step size and strain window size will cause experiment data either underestimated or scattered. In the post processing of DIC experiment, generally the larger the subsets are, the better the results are. However, large is a relative concept. It is important that the subset size is chosen in accordance with the expected deformations. It is clear WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

248 High Performance Structures and Materials III that for steep gradients in the displacement or strain field, a large subset will smooth the real behaviour and thus yield erroneous results.

References [1] Shiro Kubo, Inverse problems. Atlanta Technology Publications, ISBN #1883793-01-07 [2] Sol H., Oomens C.W.J., Material identification using mixed numerical experimental methods. 1997 Kluwer Academic Publishers. pp 1-9 [3] Synnergren P., Sjôdagk M., A stereoscopic digital speckle photography system for 3-D displacement field measurements. Optics and Lasers in Engineering, 31, pp 425-433, 1999 [4] Fricke-Begemann T., Burke J., Speckle interferometry: three-dimensional field measurement with a single interferogram. M., A stereoscopic digital speckle photography system for 3-D displacement field measurements. Applied Optics 40(28), pp 5011-5022, 1 October 2001

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Multiaxial characterization of the mechanical behaviour of aluminium foam L. Peroni, M. Avalle & P. Martella Department of Mechanics, Politecnico di Torino (Technical University of Turin), Italy

Abstract In the past ten years many new processes for making foamed metals, mostly aluminium or aluminium alloys, have been developed. As a matter of fact, closed-cell aluminium foam offers a unique combination of properties such as low density, high stiffness, strength, and energy absorption capability. One of the main differences in the mechanical behaviour of cellular materials with respect to classical homogeneous materials such as solid metals is that foam failure is not independent from a hydrostatic state of stress. Therefore, it is not possible to describe the failure surface from a single, usually uniaxial, test but it is necessary to perform tests with different combination of deviatoric and hydrostatic stress components. Within the European Project APROSYS, whose main objective is increasing the safety of all road-users, the mechanical behaviour of aluminium foam, under multiaxial loading, was investigated by the authors. In this paper the results of the hydrostatic and hydro-compression experimental tests are reported. From the results of these tests, it has been possible to obtain the yield locus of the aluminium foam in the deviatoric-hydrostatic stress components space. Keywords: aluminium foam, hydrostatic, hydrocompression.

1

Introduction

It is well know that the mechanical behaviour of a cellular material, like foam, is not independent from a hydrostatic state of stress as, on the contrary, it happens for classical homogeneous materials such as metals. Therefore, it is not possible to describe the failure surface from a single simple uniaxial test (in tension or compression if there is a different behaviour) but it is necessary to perform tests WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06025

250 High Performance Structures and Materials III with different combinations of deviatoric and hydrostatic stress components [1]. A test that gives pure deviatoric stress is, for example, a shear test. A test that produces pure hydrostatic stress in the material is the hydrostatic or triaxial test. A tensile test gives a combination of deviatoric and hydrostatic stress components. In a material insensitive to the hydrostatic component of stress (as usually are considered metals) the failure limit in a deviatoric-hydrostatic space is a horizontal line. When the deviatoric stress reaches the yield line yielding occurs, therefore for metals it is possible to indefinitely increase the hydrostatic stress without yielding. If this hypothesis can be accepted a single experimental test gives the complete yield locus for this material. The yield locus is different for foams (and other plastic materials) because there is an influence of the hydrostatic stress component [2–3]. A single experimental test is not enough to characterize the material behaviour completely. As a consequence a series of different tests are necessary in order to cover the widest possible range of combinations of deviatoric and hydrostatic stress components.

2

Experimental tests

Within the 6th Framework Programme European project APROSYS [4], a complete characterisation of aluminium foam [5–8] was carried out at Politecnico di Torino. The experimental tests were performed in the Laboratory of the 2nd Faculty of Engineering, in Vercelli. In this paper the hydrostatic compression and the hydro-compression tests are reported. σdev

σA

Hydro-compression tests area Uniaxial compression

σR

Hydro-compression Hydrostatic compression

3

β = atan(r) 1

Figure 1:

σR

σR σR σA

p = σ A /σ B

σ dev = σ A − σ R σ hyd =

σhyd

σ A + 2σ R

3 r = σ dev / σ hyd

Hydrostatic and hydro-compression tests area in the deviatorichydrostatic stress components space.

Hydrostatic compression tests were performed first to assess the behaviour of the tested aluminium foam in a pure hydrostatic stress condition. Then hydroWIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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compression tests were carried out changing the ratio p in order to obtain different σhyd - σdev combinations. As shown in fig. 1, combining in different ways the axial stress σA and the radial stress σR it is possible to investigate the highlighted area of the σhyd - σdev plane. The limits of this area are the pure hydrostatic test (σA = σR) and the pure compression test (σR = 0). 2.1 Testing device Performing hydrostatic and hydro-compression tests on a metal foam is a challenging task. The foam specimen have to be completely separated from the fluid used to load it; for this reason it was covered with a latex sheath in order to avoid seepage of the fluid in the specimen. The cover have to be enough strong to not break during the test and, at the same time, it have to be very light and thin in order to not change the test results. Furthermore in the hydro-compression tests it is necessary to control the axial and the radial pressure separately. For all this reasons a testing device (fig. 2) was designed and built expressly to perform these tests. The test chamber, containing the foam specimen, was filled with a fluid (mixture of water and glycol) and mounted directly on the fixtures of a general purpose hydraulic testing machine DARTEC HA100 controlled with a DARTEC 9600 electronic unit.

Figure 2:

Schematic of the loading system.

In the hydrostatic tests the specimen was put on the vertical rod connected to the testing machine. This rod could move inside the test chamber in the axial direction changing the volume of the chamber and, consequently, the pressure of the fluid inside the chamber, i.e. the pressure on the specimen. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

252 High Performance Structures and Materials III In the hydro-compression tests the moving rod connected to the hydraulic testing machine pressed the specimen against a fixed rod inside the chamber generating the axial stress σA on the specimen. Meantime, a double-effect pneumatic piston, was connected to the lateral surface of the test chamber. The rod of this piston could move inside the test chamber in the radial direction changing the volume of the chamber and, consequently, the pressure inside the chamber, generating the radial stress σR on the specimen. In both tests the electronic unit performed the test control and the data acquisition, while a PC equipped with a National Instruments acquisition board was used to acquire the load, stroke and pressure. In the hydro-compression tests the PC carried out the control of the radial pressure σR (pressure inside the test chamber) by moving the piston rod, whose displacement was measured through a potentiometer. The load applied though the rod connected to the testing machine was measured with a 100 kN, class A, strain-gage load cell, while the stroke of the rod was measured by means of a LVDT transducer connected to the hydraulic actuator. The pressure inside the chamber was measured with a 350 bar strain-gauge pressure transducer applied to the chamber. 2.2 Samples The samples were cylinders cut from blocks, made of aluminium foam, produced by the Fraunhofer-Institute for Applied Materials Research (IFAM) in Bremen (D). These blocks were obtained through a powder metallurgical process for preparing foamed metals. According to this process, commercial powders are mixed with small quantities of a powdered foaming agent by means of conventional techniques. The mixture is compacted to a semi-finished product of low porosity by applying compaction techniques such as extrusion or coextrusion. The result of the compaction process is a foamable semi-finished product that can be worked into sheets, profiles, etc. by applying conventional deformation techniques. During a final heat treatment at temperatures above the melting point of the corresponding alloy, the material expands and develops its highly porous, closed-cell structure. Three blocks (308 mm × 125 mm × 45 mm), identified as B2, B3 and B4, were cut in seven sub-blocks each to obtain the samples. These prismatic square section sub-blocks were turned with a CNC lathe so as to get circular cylindrical specimens with a nominal diameter and height of 41 mm. As in the quasi-static compression tests (not reported in this paper) an anisotropic behaviour of the specimens was found, the specimens were machined in two different ways. In the ones obtained from block B2 the axial direction was parallel to the topbottom direction of the block (fig. 3). On the contrary, the specimens from blocks B3 and B4 the axial direction was parallel to the transversal direction of the block (fig. 3). In the foaming process the top-bottom direction is the first foaming direction (direction for which the boundary planes of the block were reached first), while the transverse and the longitudinal directions were the second and the third foaming direction respectively.

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Top-bottom

253

Specimens from block B2 Specimens from blocks B3 and B4

Longitudinal Transversal

Figure 3:

Position of the axis of the cylindrical specimens with respect to the block directions.

The properties of the specimens used for hydrostatic compression and hydrocompression tests are reported in the following tables. Table 1:

Properties of the specimens for hydrostatic compression tests. DIMENSIONS (mm)

SPECIMEN CODE

Diameter

Height

DENSITY (kg/dm³)

AXIAL DIRECTION

H-B2-1

41.2

41.8

0.66

Top-Bottom

H-B2-2

40.8

39.7

0.66

Top-Bottom

H-B3-1

40.8

45.5

0.64

Transversal

H-B3-2

40.8

45.3

0.37

Transversal

H-B4-1

40.8

41.2

0.27

Transversal

H-B4-2

41.0

41.0

0.36

Transversal

H-B4-3

41.0

40.7

0.40

Transversal

Table 2:

Properties of the specimens for hydro-compression tests. DIMENSIONS (mm)

SPECIMEN CODE

Diameter

Height

DENSITY (kg/dm³)

SPECIMEN AXIAL DIRECTION

HC-B2-1

40.6

40.5

0.40

Top-Bottom

HC-B2-2

40.8

39.2

0.34

Top-Bottom

HC-B2-3

40.6

40.2

0.57

Top-Bottom

HC-B3-1

40.8

45.3

0.54

Transversal

HC-B3-2

40.8

45.3

0.47

Transversal

HC-B3-3

40.8

45.0

0.38

Transversal

HC-B4-1

40.8

45.4

0.60

Transversal

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254 High Performance Structures and Materials III

3

Experimental results

3.1 Hydrostatic compression tests The results of the hydrostatic compression tests are reported in fig. 4. As it is possible to notice, the results are strongly dependent from the density which is the apparent density, the average value within the volume of the sample. Since there is a large scatter in size of the cells, cell thickness, and cell distribution the apparent density is affected by a very large scatter; as a consequence the test results are rather scattered. However a general behaviour can be detected. The elastic modulus, the yield stress and the slope of the stress-strain curve after yielding, all increase with increasing density.

Hydrostatic stress (MPa)

30

H-B2-1 0.66 H-B2-2 0.66 H-B3-1 0.64 H-B3-2 0.37 H-B4-1 0.27 H-B4-2 0.36 H-B4-3 0.4

25 20 15 10 5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Volumetric strain (-)

Figure 4:

Hydrostatic compression tests: stress-strain curves

Furthermore, the tested foam showed an anisotropic behaviour, which can be detected by examining the deformed shapes of some specimens after the tests (fig 5). In specimen H-B3-1 the directions of the cross section are the longitudinal and the top-bottom directions of the base block. Because the topbottom direction is weaker than the longitudinal one, the cross section of the deformed specimen is elliptical (fig 5.(a)). This didn’t occur in the H-B2-2 specimen for which the directions of the cross section are the longitudinal and the transversal directions of the base block, which have approximately the same strength. As a consequence, despite the slightly lower density, the yield stress of specimen H-B3-1 is higher than the one of the H-B2-1 and H-B2-2 specimens. The hydrostatic yield stress for the specimens with a transversal axial direction was evaluated as intersection of the two lines interpolating the first part (elastic) and the second part (plateau) of the experimental stress-strain curve. An equation, which relates the hydrostatic yield stress to the density, was found through an interpolation of these experimental data. σyield-hyd = 13.38·ρ1.773 (1) WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Density ρ in the previous equation is expressed in kg/m³ and stress in MPa.

(a) Figure 5:

(b)

Cross section of H-B3-1 (a) and H-B2-2 (b) specimen after the tests.

In fig. 6 hydrostatic tests are compared with uniaxial quasi-static uniaxial tests (not reported in this paper) for two different densities. As it can be noticed the yield stress is higher for the uniaxial test because in this loading condition the specimen material is not constrained in the radial direction and therefore it can move laterally before the axial yield. Furthermore, the densification rate is greater for the hydrostatic loading condition.

Hydrostatic stress (MPa)

80 S-B1-1 0.36 Uniaxial

70 60

H-B4-2 0.36 Hydrostatic

50

S-B1-2 0.64 Uniaxial

40

H-B3-1 0.64 Hydrostatic

30 20 10 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Volumetric strain

Figure 6:

Comparison of uniaxial quasi-static tests and hydrostatic tests.

3.2 Hydro-compression tests As explained above, one of the main aims of the experimental tests was evaluating the failure locus of the foam as shown in fig. 1. For this reason each hydro-compression test was performed applying a proportional load, i.e. a fixed ratio σR / σA, as shown in the following table. In addition two hydrostatic tests,

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256 High Performance Structures and Materials III on specimen H-B3-1 and H-B3-2 (reported in the previous section) were carried out using the testing device for the hydro-compression tests with a ratio p = 1. Table 3:

Hydro-compression tests settings.

SPECIMEN

Density (kg/m3)

p = σR / σA

r = σdev / σhyd

HC-B2-1

0.40

p = 0.658

r = 0.445

HC-B2-2

0.34

p = 0.366

r = 1.111

HC-B2-3

0.57

p = 0.658

r = 0.445

HC-B3-1

0.54

p = 0.658

r = 0.445

HC-B3-2

0.47

p = 0.366

r = 1.111

HC-B3-3

0.38

p = 0.658

r = 0.445

HC-B4-1

0.60

p = 0.366

r = 1.111

H-B3-1

0.64

p = 1 (hydrostatic)

r = 0°

H-B3-2

0.37

p = 1 (hydrostatic)

r = 0°

Examining the experimental data (axial and radial stress-strain curves), it was possible to notice a different behaviour in the axial and radial directions. This behaviour is shown clearly in the fig. 7, where the results of the hydrostatic test performed on specimen H-B3-2 are reported. This test was performed with the experimental device for the hydro-compression tests so it was possible to evaluate the axial and the radial characteristics separately. Radial strain (-) 0

0.05

0.1

0.15

0.2 8

Axial stress Radial stress

7 6

7 6

5

5

4

4

3

3

2

2

1

1

0

Radial stress (MPa)

Axial stress (MPa)

8

0

0

0.05

0.1

0.15

0.2

Axial strain (-)

Figure 7:

Hydrostatic test (p = 1) on specimen H-B3-2.

In fig. 7, it is possible to notice that, even if the load ratio p = 1 was kept at a constant level during the test, the axial and the radial yield stress were quite WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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different. This means that the ratio between the axial yield stress and the radial yield stress was different from the ratio p at which the experimental test was carried out. This event is rather clear in the hydrostatic tests, while in the hydrocompression tests it occurred only some times and not in a so evident way. However, due to this phenomenon, it was not possible to use the experimental data directly to obtain the failure locus because it was not clear which one of the yield stresses should have been chosen. In order to overcome this problem the following procedure was used. First of all, the hydrostatic stress-volumetric strain curve was calculated from the axial and radial stress-strain curves through the following equations:

σhyd = (σA + 2 σR) / 3, εvol = εA + 2 εR

(2)

Then the yield stress for each test was evaluated considering the hydrostatic stress-volumetric strain curve. In order to remove the effect of the density on the results, each yield stress, calculated as described above, was divided by the theoretical yield stress evaluated through eqn (1). Therefore a normalised yield σyield-adim stress was obtained Finally, the normalised deviatoric and hydrostatic stress components were obtained combining the following equation:

σR = p σA, σhyd = (σA + 2 σR) / 3, σdev = | σA – σR |

(3)

These components are plotted in the σdev – σhyd (fig. 8) in order to obtain the failure locus. 1.8 Uni-axial compression test

ηdev normalised

1.6 1.4 1.2

HC-B4-1

1 HC-B3-2

0.8 0.6

HC-B3-1

HC-B3-3

H-B3-2

H-B3-1

0.4 0.2 0 0

0.25

0.5

0.75

σhyd normalised

Figure 8:

Failure locus.

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1

1.25

258 High Performance Structures and Materials III

4 Summary and conclusions Performing hydrostatic and hydro-compression tests on a metal foam proved to be a challenging and rather complicated task. The insulation of the specimen from the fluid used to load it was particularly difficult; indeed in many tests the cover used to protect the specimen from seepage of the fluid broke and the fluid penetrated into the specimen causing an early end of the test. For this reason many tests were carried out in order to set up the testing device and procedure; in this paper only the most important results are reported. To sum up, the mechanical behaviour of aluminium foam was investigated through a series of experimental tests performed within the European Project APROSYS. As for foams the yielding is not independent from the hydrostatic stress component, a single experimental test is not enough to characterize the material behaviour completely. Therefore pure hydrostatic compression and hydro-compression tests were carried out through a testing device expressly developed. From the results of these tests it has been possible to obtain the yield locus of the aluminium foam in the deviatoric-hydrostatic stress components space.

Acknowledgements The financial support of the European Commission by means of the APROSYS project and Mr. Dirk Lehmhus of Fraunhofer-Institute for Applied Materials Research (IFAM) are gratefully acknowledged.

References [1] Gibson, L.J, Ashby, M.F., Cellular solids: structure and properties (Second Edition), Cambridge University Press, 1997. [2] Collins, J.A., Failure of materials in mechanical design, John Wiley & Sons, 1980. [3] Khan, A.S., Huang, S., Continuum theory of plasticity, John Wiley & Sons, 1995. [4] Integrated Project on Advanced PROtection SYStems (APROSYS). On line: www.aprosys.com. [5] Ashby, M.F., Evans, A., Fleck, N.A., Gibson, L.J., Hutchinson, J.W., Wadley, H.N.G., Metal foams – A Design Guide, Butterworth Heinemann, 2000. [6] Ehlers, W., Mullerschon, H., Klar, O., On the behaviour of Aluminium Foams under Uniaxial and Multiaxial Loading, Verlag Mit, 1999. [7] Deshpande, V.S., Fleck, N.A., Isotropic constitutive models for metallic foams, Journal of the Mechanics and Physics of Solids, Pergamon, 1999. [8] Deshpande, V.S., Fleck, N.A., Multiaxial yield behaviour of polymer foams, Acta Materialia, Pergamon, 2001.

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Characterisation of the high strain rate properties of Advanced High Strength Steels J. Van Slycken1, P. Verleysen1, J. Degrieck1 & J. Bouquerel2 1

Department of Mechanical Construction and Production, Faculty of Engineering, Ghent University, Belgium 2 Department of Metallurgy and Materials Science, Faculty of Engineering, Ghent University, Belgium

Abstract In the automotive industry a lot of energy is put into the development of lightweight auto body structures that are able to outperform the classic structures. For these purposes tremendous advances have been made in the field of multi-phase steels. Complex steel grades have been developed with exceptional mechanical properties: they combine high strength values (yield strength, tensile strength, etc.) with an excellent ductility. TRIP steels (TRansformation Induced Plasticity steels) show these properties pre-eminently. To guarantee a controlled dissipation of the energy released during a crash, knowledge and understanding of the impact-dynamic material properties is essential. In this paper the results are presented of an extensive experimental program to investigate the strain rate dependent mechanical properties of different TRIP steels. The influence of different alloying types (Al, Si, SiAl, etc.) on the static and dynamic stress-strain behaviour is investigated. A split Hopkinson tensile bar set-up was used for the experiments. Microstructural observation techniques such as different optical methods, SEM and XRD were used to reveal the mechanisms governing the observed high strain rate behaviour. From the results it is clear that the excellent mechanical properties are not only preserved at higher strain rates, but still improve. The influence of the alloying elements is comparable in the static and dynamic case: aluminium tends to increase the elongation level of the material, whereas silicon improves the stress that is achieved. Keywords: high strain rate behaviour, split Hopkinson tensile bar, TRIP steel, alloying elements. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06026

260 High Performance Structures and Materials III

1

Introduction

Engineered steels provide automotive designers and manufacturers with the unique option to combine lightweight design with the traditional advantages of steel: low cost and eco-efficiency. Under the impulse of several steel-auto partnerships, including the Ultra Light Steel Auto Body (ULSAB) program, new types of high-strength steel, called Advanced High-Strength Steels (AHSS) (Bleck [1]), are engineered to complete the traditional steel grades. In the ULSAB-Advanced Vehicle Concept (AVC) program, the need to reduce the added mass which is required to satisfy future safety mandates presents the opportunity to apply these newer types of high-strength steels in the design of an efficient lightweight body structure. Members of the AHSS family include Dual Phase (DP), Transformation Induced Plasticity (TRIP), Complex Phase (CP) and Martensite Steels (MS). The low alloy TRIP steels show high strength values in combination with an excellent deformability, making them the material of choice for impactabsorbing structural parts of auto-bodies (Bleck [1]). TRIP steel has a microstructure of soft ferrite (α) grains with bainite (αΒ) and retained austenite (γ) (figure 1). The retained austenite transforms into martensite (α’) during deformation. The hard martensite delays the onset of necking leading to high total elongation values and high crash energy absorption. TRIP steels can therefore be engineered or tailored to provide excellent formability for manufacturing complex parts. In addition, these steels can be designed into the automotive body structure to offer excellent crash energy absorption.

Figure 1:

SEM-micrograph showing the multiphase microstructure of a nondeformed CMnAl-TRIP steel (magnification: x 5000).

The need for more optimized crashworthiness analysis in the automotive industry makes high strain rate tensile testing of sheet steels very important. It is well known that steels display positive strain rate performance, i.e. at the higher rates of strain which are typically associated with crash events, steels have higher WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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strengths and consequently higher crash energy absorption. Different types of testing techniques have been used to generate data under these dynamic conditions, each serving a specific range of strain rates. One of the most commonly used setups is a split Hopkinson bar setup. Mostly, a pressure apparatus is used to obtain the dynamic parameters (Davies and Hunter [2]), a tensile setup is however preferred for the testing of sheet metals. Stress-strain curves at strain rates varying from 500 to 2000 s-1 can be obtained with the tensile setup developed at Ghent University. The behaviour of steels during high strain rate loading is the result of the interaction between two opposing processes: strain rate hardening and thermal softening, which is due to adiabatic heating during deformation. In the case of TRIP steels, thermal softening also affects the γ-α’ transformation: the increased temperature reduces the transformation rate. This paper presents the results of an extensive experimental program, which is set up to assess the dynamic mechanical behaviour of TRIP steels. Special attention is paid to the influence of the different alloying elements in TRIP steels on the stability of the austenite phase and thus on the transformation in the material.

2

Materials and methods

2.1 Materials A key parameter for the TRIP effect is the stability of the meta-stable austenite which is mainly determined by the austenite particle size and the composition, especially the carbon content is an important parameter (Itami et al. [3]). Alloying elements such as silicon, aluminium and phosphor in TRIP steels principally added to inhibit carbide precipitation during the second stage isothermal holding temperature in the production process of TRIP steels - also significantly influence the thermodynamic stability of the austenite phase. The influence of the alloying elements is assessed by the study of four different TRIP steel grades: CMnAl-, CMnSi-, CMnSiAl- and CMnSiAlP-TRIP. Specific care was taken to keep the same carbon content for each steel grade. Table 1 lists the composition of the low alloy TRIP steels. Table 1:

Chemical composition of the investigated TRIP steel grades (in weight percent).

Steel grade CMnAl-TRIP CMnSi-TRIP CMnSiAl-TRIP CMnSiAlP-TRIP

C 0.24 0.25 0.25 0.20

Mn 1.61 1.67 1.70 1.56

Al 1.54 0.78 0.69 0.29

Si 0.091 1.28 0.55 0.38

P 0.006 0.012 0.011 0.012

Silicon strengthens the ferrite phase considerably, but high Si-contents results in an adherent FeO.SiO2 oxide layer on the sheet surface, which generates surface defects on the hot rolled sheet and which is difficult to remove by WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

262 High Performance Structures and Materials III pickling. Moreover, due to this oxide layer CMnSi-TRIP steels are complicated to galvanize and thus to process in continuous galvanizing lines. Therefore, Al has been used to replace the silicon in TRIP steels. Al increases the ductility significantly because it has a lower solid solution strengthening potential for the ferrite phase (Girault et al. [4]). 2.2 Experimental testing procedure A Split Hopkinson Tensile Bar (SHTB) apparatus is used to characterize the dynamic properties of the investigated materials. It mainly consists of two bars: an input and an output bar between which a specimen of the test material is attached (figure 2). A tensile wave is produced by an impactor that is accelerated towards the anvil of the input bar. After the impact this incident wave travels along the input bar towards the specimen where it interacts with the sample and is partly reflected back into the input bar. The other part, the transmitted wave, travels along the output bar. The strain histories of the different waves (incident, reflected and transmitted wave, respectively denoted as εi(t), εr(t), εt(t)), are recorded by means of strain gauges mounted on both bars. By adjusting the impact speed of the impactor, the strain rate can be varied.

Figure 2:

Schematic representation of the Split Hopkinson Tensile Bar (SHTB) set-up.

According to the one-dimensional wave theory and the assumption of a uniaxial and homogeneous stress and strain in the specimen, the stress, strain and strain rate in the specimen can be written as follows (Kolsky [5]): σ(t)=

EbAb U -U 2C ε t ( t ) , ε ( t ) = ob ib =- b As Ls Ls

t

∫ ε ( τ ) dτ , r

0

Vob -Vib 2Cb (1) =εr ( t ) Ls Ls where, As and Ls are the cross-sectional area and the length of the testing region of the specimen, respectively. Cb is the one-dimensional elastic stress wave velocity in the input/output bar material, Ab is the cross-section area and Eb is Young’s modulus of the input/output bar. Uib and Uob are the displacements of the specimen/input bar interface and the specimen/output bar interface, respectively. Vib and Vob are the corresponding interface velocities. ε ( t ) =

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3

263

Results

3.1 Influence of the alloying elements Figure 3 shows a comparison of the engineering stress-strain curves of the different investigated steel grades at a strain rate of ~1600 s-1. The four steel grades exhibit a remarkable uniform elongation despite their high strength levels. This is due to the occurrence of the strain-induced transformation and thus, to the TRIP effect. The difference in their respective strength and elongation levels can be attributed to their chemical composition. The differences in dynamic flow stress are fully consistent with the ones in the static case: silicon addition increases the strength considerably, whereas aluminium has little effect on the strength level (Girault et al. [4]) and the CMnSiAl-TRIP steel shows an intermediate behaviour. The addition of phosphor on the other hand, has limited influence on the behaviour at the early stages of deformation. When the strain increases, the strain hardening of the CMnSiAl-TRIP steel is more important than of the CMnSiAlP-TRIP, which reaches maximum stress and uniform elongation at a lower level of deformation.

Figure 3:

Comparison of the engineering stress-strain curves of the investigated steel grades at high strain rate (~1600 s-1).

3.2 Strain hardening To investigate the hardening behaviour, the strain hardening coefficient or nvalue, appearing in the Hollomon stress-strain relation of eqn (2) (Stout and Follansbee [6]) is calculated as a function of strain for the different stress-strain curves.

σ = Kε n

p

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(2)

264 High Performance Structures and Materials III where σ is the true stress, εp the true plastic strain and K is a material constant. A strain window of 0.05 in which the strain hardening coefficient is averaged out, is used to minimize the effect of oscillations in the stress-strain curves. In figure 4 the evolution of the strain hardening coefficient as a function of the true strain is represented for the TRIP steels. In the early stages of deformation, the strain hardening is highest for the CMnSiAl (P)-TRIP steels, followed by the CMnSiTRIP. The TRIP steel with high aluminium content shows the lowest strain hardening in the beginning. The hardening of CMnAlSiP-TRIP begins to stabilize and in the region between 7 and 14% of true strain the hardening behaviour of CMnAl, CMnSi and CMnAlSiP is similar. After approximately 14% of true strain, the strain hardening coefficient of the CMnSi-TRIP steel begins to decrease. The decrease of the strain hardening of the CMnAl- and the CMnSiAl (P)-TRIP steels begins later during the deformation. The CMnAl-TRIP steel shows a slighter decrease in strain hardening at the end of the deformation.

Figure 4:

Comparison of the strain hardening coefficients or n-value of the investigated TRIP steel grades in function of the true strain at a strain rate of ~1200 s-1.

When looking at the uniform elongation, CMnAl-TRIP steel shows the highest n-value, whereas CMnSi-TRIP steel shows the lowest. Figure 5 displays the strain hardening rate (dσ/dε) for the investigated steel grades during dynamic deformation. The strain hardening rate of ferrite is known to increase with silicon additions and to be insignificantly affected by the presence of aluminium. Accordingly, this behaviour is seen in dynamic conditions during the early stages of deformation. CMnSi-TRIP shows the highest strain hardening rate, whereas CMnAl-TRIP has the lowest values. CMnSiAl shows an intermediate behaviour. After approximately 7% of true strain, the strain hardening rate of the CMnSiTRIP and the CMnSiAl (P)-TRIP steels become similar. The behaviour of the CMnAl-TRIP steel shows a more constant evolution during deformation, whereas the other steel grades have a steeper decrease in strain hardening rate. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Figure 5:

265

Comparison of the strain hardening rate dσ/dε of the investigated TRIP steel grades in function of the true strain at a strain rate of ~1200 s-1.

3.3 Influence of the strain rate In figure 6 the tensile strength in function of the strain rate is presented for the CMnAl-, CMnSi- and the CMnSiAl-TRIP steel grades. The influence of the strain rate is material dependent, but positive for both the CMnSi-TRIP steel and the CMnAl-TRIP steels: stress levels rise as the strain rate increases whereas for the CMnSiAl-TRIP this influence is limited. Quite some scatter can be noticed for the tensile strength values for the CMnSi-TRIP steel.

Figure 6:

Tensile strength as a function of strain rate for the CMnAl-, CMnSi and the CMnSiAl-TRIP steel grades.

Different parameters can be used to evaluate the crash-resistance performance of steels. Since, for most applications, the material rarely deforms up to fracture, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

266 High Performance Structures and Materials III the yield stress and energy absorbed by the material at certain levels of deformation gives valuable information. In figure 7 the energy dissipated by the CMnAl-, CMnSi- and the CMnSiAl-TRIP steel at 10% strain is given as a function of the strain rate. From this figure it is clear that all steel grades exhibit a positive strain rate dependency. The CMnSi-TRIP steel shows the highest energy absorption values, the other two steel grades have similar energy absorption levels.

Figure 7:

Energy absorption until 10% deformation as a function of strain rate for the CMnAl-, CMnSi and the CMnSiAl-TRIP steel grades.

In dynamic conditions the strain rate has limited influence on the material properties. If these dynamic properties are compared to properties after static deformation on the other hand, an important difference can be observed. Table 2 shows the mechanical properties of the CMnAl-TRIP steel after static (strained at a constant strain rate of 10-4 s-1) and after dynamic deformation. An important increase in both yield and tensile strength can be noticed when comparing static to dynamic conditions. The uniform elongation however shows limited changes, as well as the energy dissipation until 10% deformation. Table 2:

Static 704 s-1 1246 s-1 1898 s-1

Mechanical properties of CMnAl-TRIP steel in static and dynamic conditions. Upper yield strength, MPa

Tensile strength, MPa

Uniform elongation, -

506 622 646 672

689 786 800 814

0,265 0,264 0,301 0,252

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Energy dissipation at 10% deformation, 106 J/m3 55,91 59,16 61,39 64,99

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267

This can be explained when looking at the X-ray diffraction (XRD) patterns in order to reveal the mechanisms governing the complex mechanical behaviour. In figure 8 XRD patterns of non-deformed specimens are compared with patterns of specimens after static and dynamic tests. The measurements are performed on a Siemens D5000 diffractometer. Only the ferrite (b.c.c.) and austenite (f.c.c.) peaks are considered in the measurements. Diffractograms were obtained in the 19°-40° 2θ-range using a filtered molybdenum Kα radiation.

Figure 8:

XRD pattern for the CMnAl-TRIP steel, 2θ-range between 19° and 40°.

The diffractogram of the non-strained sample presents an initial amount of 12% of retained austenite. After both static and dynamic tensile tests, no (220)γ austenite peaks, except the (200)γ one, are present. In the dynamic tests, no austenite was expected because of the importance of the adiabatic heating during deformation. The temperature in the specimen is expected to reach, at the end of the test, about 90-100°C. The temperature increase inhibits the austenite to martensite transformation kinetic (Samek et al. [7]).

4

Conclusions

Results are presented of an extensive study of the strain rate dependent behaviour of TRIP steels. These materials combine high strength with high ductility and offer therefore an excellent crash energy absorption potential. Split Hopkinson tensile bar tests are performed to obtain the stress-strain curves at higher strain rates. The influence of the strain rate on the behaviour is especially important when comparing static with dynamic testing conditions. The amount of austenite that is transformed to martensite and the adiabatic heating during dynamic deformation plays an important role. Special attention is paid to the influence of alloying elements such as Al, Si and P on the dynamic behaviour of TRIP steels. As in the static case, silicon contributes to a significant solid solution strengthening of the ferrite matrix. TRIP steels with high silicon content WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

268 High Performance Structures and Materials III show therefore high strength levels. TRIP steels mainly alloyed with aluminium on the other hand exhibit lower strength values but higher elongation levels. The work hardening of the latter steel grade is more constant during deformation, whereas TRIP steels with high silicon content show higher work hardening. These properties can further be used to develop several material models. The strain rate dependent behaviour cannot be described in a general way and various types of constitutive relations have been proposed. The validation of these models can be used for crashworthiness analysis in the automotive industry.

References [1]

[2] [3] [4]

[5] [6]

[7]

Bleck, W., Using the TRIP effect - the dawn of a promising group of cold formable steels. Proceedings of the International Conference on TRIPAided High Strength Ferrous Alloys, ed. B. C. De Cooman, pp. 13-24, 2002. Davies, E. D. H. & Hunter, S. C., The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. J. Mech. Phys. Solids, 11, pp. 155-179, 1963. Itami, A., Takahashi, M. & Ushioda, K., Plastic Stability of Retained Austenite in the Cold-Rolled 0.14-Percent-C-1.9-Percent-Si-1.7-PercentMn Sheet Steel. Isij International, 35(9), pp. 1121-1127, 1995. Girault, E., Mertens, A., Jacques, P., Houbaert, Y., Verlinden, B. & Van Humbeeck, J., Comparison of the effects of silicon and aluminium on the tensile behaviour of multiphase TRIP-assisted steels. Scripta Materialia, 44, pp. 885-892, 2001. Kolsky, H., An investigation of mechanical properties of materials at very high strain rates of loading. Proc Phys Soc Lond Sec B, 62, pp. 676-700, 1949. Stout, M. G. & Follansbee, P. S., Strain rate sensitivity, strain hardening, and yield behavior of 304L stainless steel. Journal of Engineering Materials and Technology - Transactions of the ASME, 108(4), pp. 344353, 1986. Samek, L., De Cooman, B. C., Van Slycken, J., Verleysen, P. & Degrieck, J., Physical Metallurgy of Multi-Phase Steel for Improved Passenger Car Crash-Worthiness. Steel Research International, 75(11), pp. 716-723, 2004.

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269

Evaluation of bond strength in Roller Compacted Concrete under various normal pressures M. Madhkhan & A. Arasteh Department of Civil Engineering, Isfahan University of Technology, Iran

Abstract The construction of dams using Roller Compacted Concrete (RCC) is a relatively new technology, which has rapidly developed in recent years. To increase the safety factor in designing the various horizontal construction joints in RCC dams, an experimental research using interlayer cement grout has been carried out to improve the bond strength. A device with the ability of exerting the normal pressure in the direct shear method has been designed and used. 144 tests were carried out to assess the effect of interlayer cement grout along with exerting the normal pressure. The tests include different proportions of water to cement of the interlayer grout (0.5, 0.75, 1) and different values of normal pressure (0, 5, 10, 15 kg/cm 2 ) in the ages of 7, 28 and 90 days. The results indicate that the use of interlayer cement grout in the range of the normal pressure exerted gives higher bond strength compared to the state of no grout. By reducing the ratio of water to cement of the grout, the bond strength increases. Keywords: Roller Compacted Concrete, horizontal construction joint, interlayer cement grout, bond strength, normal pressure.

1

Introduction

The RCC dams are built in relatively thin layers, with a thickness of approximately 30 centimeters placed, spread and compacted over each other; therefore a large number of horizontal joints are formed in the dam. The quality of the concrete at the joint surface is of great importance to ensure the integration of layers. But in practice, this surface has a lower quality compared to the concrete mass; and a lower tensile and shear strength and a higher permeability is expected [1]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06027

270 High Performance Structures and Materials III The shear strength is a function of cohesion and angle of internal friction. The minimum shear properties occur at the horizontal construction joints between the lifts of RCC. Generally, the shear strength of the joint is dependent upon the amount of cement used in the RCC mixture, grading and quality of aggregates, amount of compaction of each layer, existing conditions of each layer surface (including weather conditions, construction conditions, and etc), layer age, improvement of layer surface by adding bedding mixture and index of maturity or curing. A large range of the bond strength depends on the RCC mixture and above named factors. As a general range, the cohesion can be taken between 0.1 and 2.5 MPa and the angle of friction between 30 and 65 degrees [2]. Although the normal pressure varies from one point to another in dam height, but the experimental researches and the bond (shear) strength determination are mostly carried out in the absence of normal pressure. The shear strength parameters stated in the technical data, are based on coring studies and joint tests on specimens taken from the dam built or being built. In this research, the influence of different normal pressures on the bond (shear) strength of RCC construction joints is studied using common laboratory specimens and by using interlayer cement grout.

2

RCC materials

2.1 Cement The heat generation control is of great importance in selecting the cement type. Heat generation is typically controlled by using pozzolans and slags, therefore Portland cements including Type Ι, Π and V can be used [3]. In the present research, exports cement Type Ι of the Isfahan cement factory was used with the chemical analysis given in Table 1. The quantities of C 3 S, C 2 S, C 3 A, C 4 AF and insoluble residue in this cement are respectively 46.13, 27.38, 9.61, 9.98 and 0.44%. Table 1:

Chemical analysis of the cement used in the RCC mix design (%).

SiO 2

Al 2 O 3

Fe 2 O 3

CaO

MgO

SO 3

Na 2 O

K2 O

21.68

5.72

3.28

63.53

1.75

1.63

0.2

0.52

2.2 Pozzolans Ninety percent of the dams built until 1998 contain some kind of additional material such as slag or pozzolan, and in only 10 percent of the cases, the cement has been used individually to make RCC [3]. To replace a portion of the cement by pozzolan (25%) a natural pozzolan powder was used in the RCC mix design. The chemical analysis is given in Table 2. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Table 2:

271

Chemical analysis of natural pozzolan (%).

SiO 2

Al 2 O 3

Fe 2 O 3

CaO

MgO

SO 3

Na 2 O

K2 O

57.35

18.2

5

6.23

1.45

1.93

3.5

1.72

2.3 Aggregates Considering that the aggregates fill 75 to 85% of the RCC volume [4], Their properties are an effective factor on fresh and hardened RCC. The gravel is crushed river aggregates with a maximum size of 25.4 millimeters in accordance with ASTM C33, and the sand is natural sand with 4.2% of the material passing No. 200 sieve (non-plastic) in accordance with ASTM C33. The gravel/sand ratio in the RCC mix design is 55 to 45. The grading of the coarse and fine aggregates is given in Table 3. Table 3:

Grading of the aggregates (Gravel and sand) in the RCC mix design.

Coarse Agg.

Fine Agg.

Agg. size (mm)

137

3

12.5

9.5

4.75

2.36

1.18

Passing Percent

92.3

52.1

28.2

1.3

0.0

0.0

Agg. size (mm)

4.75

2.36

1.18

0.6

0.3

0.15

Passing Percent

96.5

87.1

66.9

42.8

27.7

12.2

Table 4:

Water

19.0

Mix design specifications (in one cubic meter). Weight of mix material (kg/m3) Aggregates Cementitious material Gravel Sand Cement Pozzolan 1161 950 139 35

Tests program

The above materials, cubic RCC specimens with 15x15x15 cm dimensions were made using the optimum mix design [5] given in Table 4. The specimens were made in two layers and each layer was compacted for 15 seconds using the electric vibrating hammer (ASTM C1435). The specimens are cured and protected by common laboratory methods to carry out shear strength tests at the ages of 7, 28 and 90 days. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

272 High Performance Structures and Materials III The tests carried out include different proportions of water to cement of interlayer grout (0.5, 0.75, 1) and different values of normal pressure (0, 5, 10, 2

and 15 kg/cm ). 3.1 Test method for determining the bond strength Considering that the construction joint can be referred to as an imposed crack (failure), for determination of the RCC bond strength according to previous researches [6], a device with a similar mechanism to the soil direct shear apparatus was designed and used. The RCC specimen was placed in the frame (figure1). The frame consists of a fixed plate and a moving plate. The normal pressure is exerted according to the mechanical theory of high strength bolts and by tightening the bolts on the moving plate. The device was then placed under the pressure jack (with the pressure being exerted in the same direction as the contraction joint). Finally, two sliding plates were placed tangent to the joint and on opposite sides of it. By exerting the pressure on the jack, shear failure would occur at the joint (figure 2).

Figure 1:

4

Device for exerting normal pressure to the RCC specimen.

Analysis of results

4.1 Bond strength at different ages In Table 5, the average results of the bond strength determination at the ages of 7, 28 and 90 days (each test done at least 3 times) are given with the specifications of the specimen (interlayer grout) and also test along (4 values of normal pressure). The variations of bond strength versus normal pressure for different values of proportion of the interlayer grout are shown in figures 3, 4 and 5 (for different WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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273

ages of specimens). Any test where no grout is used has the least bond strength, and where the w/c ratio is equal to 0.5 the most bond strength is obtained. By decreasing the w/c ratio, the bond strength increases. This matter can be observed more significantly in the diagrams of figure 4 (at the age of 28).

Figure 2: Table 5:

Exertion of normal and shear forces.

Average results of RCC bond strength at the ages of 7, 28 and 90 days. 2

Age (days)

7

28

90

w/c ratio

No grout 1 0.75 0.5 No grout 1 0.75 0.5 No grout 1 0.75 0.5

Normal pressure σ= 0 2 kg/cm 16 17.2 17.6 19.4 21.8 23.6 23.3 24 25.4 25.3 27.4 28.5

Bond strength [τ] (kg/cm ) Normal Normal Normal pressure pressure pressure σ= 5 σ= 10 σ= 15 2 2 2 kg/cm kg/cm kg/cm 20.9 22.2 26.7 19 24.45 25.9 23 24.15 28.3 20.3 25.2 28.6 23.1 30.4 31.1 25.8 31.9 33.2 29.8 30.1 35.4 30.8 32.1 35.4 27.1 34.6 35.1 30.9 35.6 36.1 28.4 35.0 37.9 28.8 33.5 39.2

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Bond Strength (kg /cm 2)

274 High Performance Structures and Materials III 30 28 26 24 22 20 18 16 14 12 10

no grout

R2 = 0.9419

w /c of grout=1

R2 = 0.945

w /c of grout=0.75

w /c of grout=0.5

2

R = 0.9496

no grout

R2 = 0.9595

w /c of grout=1 w /c of grout=0.75

0

5 10 15 Norm al Pressure(kg/cm 2)

Figure 3:

20

w /c of grout=0.5

Bond strength versus normal pressure diagram at the age of 7 days, comparing different proportions of w/c.

Bond Strength (kg /cm 2)

38 36

no grout

R2 = 0.9099

34 32

w /c of grout=1

2

R = 0.9147

30

w /c of grout=0.75

w /c of grout=0.5

28 26

no grout

R2 = 0.8853

24

w /c of grout=1

R2 = 0.9386

22

w /c of grout=0.75

20 0

Figure 4:

5 10 15 Norm al Pressure(kg/cm 2)

20

w /c of grout=0.5

Bond strength versus normal pressure diagram at the age of 28 days, comparing different proportions of w/c.

The RCC shear strength parameters which include the interlayer cohesion (C) and the angle of interlayer friction (ϕ) have been derived for each age and each proportion of w/c. The results are given in table 6. 4.2 Effect of specimen age on the bond strength For all normal pressure values and proportions of w/c, on average, the 7-day bond strength is approximately 77.7% of the 28-day bond strength. Also, the 90WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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day bond strength has an increase of 10.1% compared to the 28-day bond strength (figure 6). 40

R2 = 0.9328

B o n d Stren g th (kg /cm 2)

38

no grout

36

w /c of grout=1

34

w /c of grout=0.75

R2 = 0.8959

32

w /c of grout=0.5

R2 = 0.8868

30 28

no grout

R2 = 0.9007

26

w /c of grout=1

24

w /c of grout=0.75

22 0

Figure 5:

5 10 15 Normal Pressure(kg/cm2)

Bond strength versus normal pressure diagram at the age of 90 days, comparing different proportions of w/c. Table 6:

Specimen Age (days) 7

28

90

w /c of grout=0.5

20

RCC shear strength parameters (C, ϕ).

w/c ratio No grout 0.5 0.75 1 No grout 0.5 0.75 1 No grout 0.5 0.75 1

Interlayer Cohesion, kg/cm 16.44 18.51 18.28 16.87 21.32 25.25 24.16 23.39 25.06 26.98 26.46 26.41

2

Interlayer friction angle 33.7 32.96 33.58 32.41 35.14 35.4 36.2 34.9 36.2 36.4 37.3 36.6

4.3 Effect of w/c ratio of interlayer grout and specimen age on shear strength parameters The interlayer cohesion is intact with the bond strength, resulting that the maximum and minimum cohesion occur at w/c=0.5 and state of no grout used, respectively (figure 7). The angle of interlayer friction has no specific trend, and WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

276 High Performance Structures and Materials III

Bond strength (kg/cm2)

its maximum and minimum values occur at w/c=0.75 and w/c=1.0, respectively (figure 8). 34 32 30 28

w /c=-

26

w /c=1

24

w /c=0.75

22

w /c=0.5

20 0

10

20

30

40

50

60

70

80

90

100

Age(days)

Figure 6:

Bond strength versus age curves for different proportions of w/c.

Cohession (kg/cm2)

30 28 26 24 22

w/c=-

20

w/c=1

18

w/c=0.75

16

w/c=0.5

14 0

10

20

30

40

50

60

70

80

90

100

Age(days)

Figure 7:

5

Cohesion versus age curves for different proportions of w/c.

Conclusions

In this research, a device with the ability of exerting the normal pressure was designed to assess the RCC bond strength. The analysis of the overall results obtained from the optimum mix design, different w/c ratios (0.5, 0.75, and 1) and 2

different values of normal pressure (0, 5, 10 and 15 kg/cm ) at the ages of 7, 28 and 90 days gives the following conclusions: The Mohr coulomb equation [τ = C + σ tan ϕ] has regressed the bond strength versus normal pressure for each proportion of w/c and also for the state of no grout used, along with the high correlation coefficients derived from above. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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277

Friction(angle)

38

36

34

w /c=w /c=1

32

w /c=0.75 w /c=0.5

30 0

Figure 8:

10

20

30

40 50 60 Age (days )

70

80

90

100

Angle of friction versus age curves for different proportions of w/c.

The minimum bond strength was obtained in the tests where no grout was used, and where the interlayer grout is used, the bond strength would decrease whilst the w/c ratio is increased. The variation of the interlayer cohesion including all proportions of w/c and also the state of no grout used is intact with the variation of the bond strength, whereas the angle of interlayer friction does not follow any special pattern.

References [1] [2] [3] [4] [5] [6]

Roller Compacted Mass Concrete, American Concrete Institute, ACI 207.5R, 1999. Schrader, E.k. Shear Strength and Lift Joint quality of RCC International Journal on Hydropower & Dams, v6, n1, 2002, p 46-55. Dunstan, M.R.H Latest Development in RCC Dams, Proceedings of International Symposium on Roller Compacted Concrete Dams, Chengdu, China, April 21-25, 1999. Guide for Selecting Proportions for No-Slump Concrete, American Concrete Institute, ACI 211.3R, 1997. Ramazanian pour, A. A & Hassankhani, A., “Optimum Mix Design of Roller Compacted Concrete”, Proceeding of 6th International Conference of Civil Engineering, Isfahan, IRAN, 2003. Bogat, H. & RAHGOZAR, R., “Effect of Polymer Grouts on RCC dams bond strength”, Proceeding of 6th International Conference of Civil Engineering, Isfahan, IRAN, 2003.

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High performance fibres and the mechanical attributes of cut resistant structures made therewith S. Rebouillat & B. Steffenino DuPont International S.A., Geneva, P.O. Box 50, Geneva, Switzerland

Abstract High performance materials are rarely used as 100% compound, therefore most of the time when reference is made to these categories of materials it is as a matter of fact a reference to a system comprising a well-combined ingredients exhibiting together high performances in use. Fibres, micro-fibres and now nanofibres are definitely part of the high performance reinforcing ingredients, which in a system largely contributes to the performance in use of the designed system. High performance fibres, and more specifically aramids for their outstanding heat and mechanical properties, although most often used as reinforcing compound, are also used more uniquely and directly in situations where mechanical and thermal protections are required. It is therefore useful as a preliminary grounding to review these types of materials in terms of their properties. Then their attributes and potential contribution alone or in a system become more specific and can be extrapolated more easily. The mechanical performance and more specifically the cut resistance of articles made of high performance materials are generally measured against several norm specifications. Those are not designed to predict the “real” in use performance of the considered protective articles. This gap is even larger when the cut resistance of the compounding fibre increases. Although relationships exist between the various cut performance levels obtained with the ASTM, ISO and EN norms (respectively American, International and European standards), there is a need for a more fundamental understanding and interpretation of the data. The ASTM and ISO approaches remain the most suitable methods for that purpose. Various parameters, such as the high performance material and fibre composition, the effect of kinetic such as the cutting speed and the speed profile, the force distributions, the artefacts generated by various surface coatings are presented and discussed aiming at improving the interpretations of the data generated through the norm procedures. Therewith, significant progress in the understanding of the cut fundamentals and the norm harmonization are made available to the users of high performance fibres, such as KEVLAR®. Keywords: high performance materials and fibres, textile structures, coated textile materials, mechanical protection, cut mechanism, cut fundamentals, KEVLAR®. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06028

280 High Performance Structures and Materials III

1

Introduction

Aromatic polyamides came as breakthrough materials in commercial applications as early as 1961, with the market launch of the meta-aramid fibre Nomex®, which opened new horizons in the field of thermal and electrical insulation. A much higher tenacity and modulus fibre was developed and commercialised, also by DuPont, under the trade name KEVLAR® in 1971. Scientists, in the fields of liquid crystals, polymers, rheology and fibre processing, as well as, process and system engineers spent several years prior and during the early stage of its market introduction to establish the basics and fundamental understanding necessary to take full advantage of this new class of high performance materials. The outstanding potential of these materials derived mostly from the anisotropy of their superimposed substructures presenting pleated, crystalline, fibrillar and skin-core characteristics. In this work we will focus on the main high performance fibres and the applications of the commercial aramids. Basic papers outlining the engineering of such products, especially their surface modifications, are available and provide a good basis for a scientific background [1–7]. We will study in more detail the protective solutions obtained from high performance fibres. A gentle force of 0.01 N applied on a razor blade, such as the one used by the hairdresser, represents more than 1000 bar or 104 (N/cm2), there is no doubt that the performance of an effective cut protective barrier has to be a high performance. In the professional activities the awareness and the prevention regarding hand injuries have substantially contributed to contain the injury level, nonetheless the hand protection remains an area where the room for improvement is substantial. Thanks to the personal protective equipment, such as gloves, made of high performance fibres, the situation is improving on a worldwide basis. General guidance about hand protection is provided in the ANSI/ISEA 105-2000 [8]. Norms, developed in 1987 in Europe and in 1997 in the USA, have played an important role in classifying cut resistant materials. Glove performance can now reach the upper range of the cut scale level defined by certain norms with a substantial dexterity enhancement. Norm levels are nowadays considered more as entry criteria in the proposition of a given hand protection than as a truly predictive tool to anticipate the real cut risk where experience and expert judgment prevail. The fundamental understanding of the cut mechanism also plays an essential role to complement this approach. The test equipment, which are described in the various norms relative to cut protection, such as ASTM, EN and ISO (respectively American, European and International standards), are good tools to develop a basis for the cut fundamental understanding. The work presented in this paper provides and illustrates a number of situations where cut fundamentals can be link with the properties of high performance fibres. There is no attempt to extrapolate the results to a real risk encountered in the field, therefore this study cannot constitute a basis for risk analysis but can serve WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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as guidance towards an improved understanding of factors which may influence the cut performance of protective equipment.

2

High performance fibres

2.1 Main high performance fibres The following designation was adopted in 1974 by the United States Federal Trade Commission to describe the aromatic polyamide based fibres under the generic term aramid: “a manufactured fibre in which the fibre forming substance is a long chain synthetic polyamide in which at least 85% of the amide (- CO - NH -) linkages are attached directly to two aromatic rings”. The earliest representative of this class is poly-m-phenyleneisophthalamide, which was commercialised by DuPont in 1967 as Nomex® aramid fibre. Its chemical formula is:

N

N

H

H

O

O

C

C

n

Nomex®

The discovery in 1965 of high tenacity, high modulus fibres from liquid crystalline solutions of synthetic para-aromatic polyamides led to the commercial production of KEVLAR® aramid fibre by DuPont Co in 1971[9]; the corresponding chemical formula is given below:

N

N

H

H

O

O

C

C

KEVLAR®

n

KEVLAR® fibres are poly (p-phenylene terephthalamide) (PPTA) [10]. Another para-aramid, Twaron®, similar to KEVLAR®, and an aromatic copolyamide appeared on the market towards the end of the 80s. Teijin commercialised the copolyamide Technora® fibre. For the sake of completeness, Monsanto in the 1970s developed, based on an aromatic polyamide-hydrazine composition, an aromatic copolyamide fibre under the code X500 which almost reached the market. PBO (commercialised under the trade name Zylon®) and PIPD, often referred as to M5, are other high performance fibres sufficiently close to the aramids to be mentioned here [1]. 2.2 Spinning fibres Production of fibres initially involves heating the spinning polymeric solution up to a suitable processing temperature, which is of the order of 80oC for the highly WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

282 High Performance Structures and Materials III concentrated solutions in 100% (water-free) [11] sulphuric acid. The solution state corresponds to a nematic liquid crystalline phase. The concentration limit for the polymer in spinning solution is 20 wt %. Polymer spinning solutions are extruded through spinning holes and subjected to elongational stretch across a small air gap, illustrated in fig. 1. The spinning holes fulfill an important function. Under shear, the crystal domains become elongated and orientated in the direction of the deformation [12–14]. Once in the air gap, elongational stretching takes place. This is effected by making the velocity of the fibre as it leaves the coagulating bath higher than the velocity of the polymer as it emerges from the spinning holes. This ratio is often referred to as the draw ratio, which can be fine-tuned to obtain higher tenacities and moduli with lower elongations and denier. The resulting stretch in the air gap further perfects the respective alignment of the liquid crystal domains.

Figure 1: Schematical representation of the extrusion of the liquid crystalline solution in the dry-jet wet spinning process. Overall a higher polymer orientation in the coagulation medium corresponds to higher mechanical properties of the fibre. Because of the slower relaxation time of these liquid crystal systems, the high as-spun fibre orientation can be attained and retained via coagulation with cold water [15]. Essentially, the crystallinity and orientation of the solution are translated to the fibre. These factors allow the production of high strength, high modulus, as spun fibres. 2.3 Structures and properties Aramid fibres have unique properties that set them apart from other fibres. Aramid fibre tensile strength and modulus are significantly higher than earlier WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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organic fibres and fibre elongation is lower. Aramid fibres can be woven on fabric looms more easily than brittle fibres such as glass, carbon or ceramic. They also exhibit inherent resistance to organic solvents, fuels, lubricants and exposure to flame. The superimposed structures, such as the crystallites, the fibrils and the skincore boundaries, are definitely unique attributes, which can be partially tailored through the fibre process engineering. 2.4 Mechanical properties Typical stress-strain curves of different KEVLAR® fibres are provided in fig. 2(a), which clearly outlines the modulus increase from KEVLAR® 29 to KEVLAR® 149. There are differences between KEVLAR® 49 and KEVLAR® 29. The respective moduli, brought by various spinning conditions and posttreatments performed on para-aramid precursors are generally considered as intermediate between those of graphite, boron and glass fibres. The linear stressstrain behaviour of para-aramids is special compared to most man-made fibres which tensile behaviours are depicted in fig. 2(b).

Figure 2:

Typical stress/strain curves of (a) KEVLAR® fibres and (b) other commercially representative industrial yarns.

Creep is measured either by the length variation under tension or by the stress decrease at constant gage length. Para-aramids, which exhibit little creep, differ significantly from other highly oriented polymeric fibres, such as HMPE fibres, which can break after several days under intermediate load due to their high creep properties associated with a stress slip of molecules compared to a structure tightening in the case of para-aramids. The temperature, the load relative to the fibre ultimate strength, the water content and other parameters affect creep.

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284 High Performance Structures and Materials III 2.5 Applications The present section takes into consideration a selected range of applications of the aramids with an attempt to underline some of the system engineering aspects. This approach may provide not only a better understanding of the major reasons for these fibres to contribute to the best balanced performance of the system, but also may stimulate new ideas in the way these fibres are used in the considered system or how they could be used in a new one. As an example, the understanding of how a para-aramid fibre can be shaped into an optimised fabric pattern to resist to a fragment impact, can provoke new ideas regarding the engineering of these fibres to offer cut and heat protection in other sectors where the kinetics may be much closer to static conditions. This system approach, led to the development of some of the aramid fibre applications to be outlined in this section illustrated in table 1 [11]. Table 1:

Aramid market segments and key attributes.

The associated fig. 3 provides a concrete illustration of the dynamic network established since the inception of the aramids. This network outlines the dynamic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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integration of aramids in advanced and modern technologies with a constant adaptation to the new challenges associated with various generations of product. There are end-use market segments, which have been quite rethought because of the aramids, such as the asbestos replacement by para-aramid pulps. There are other areas, which will continue to evolve because of the ever-growing stringent requirements for energy saving in transportation for example.

Figure 3:

Applications of para-aramid fibres.

The aramid contribution in this sector is outstanding. Clearly, communications, including transportation as well as transmission, leisure and sports, life protection and, health and safety in general, have been tremendously improved and adapted to modern technologies in part because of the aramids.

3

High performance fibres for protective solutions

The cut performance of articles made of high performance materials is generally measured against several norm specifications [16, 17]. Those are not designed to predict the “real” in use performance of the considered protective articles. This gap is even larger when the cut resistance of the compounding fibre increases. Although relationships exist between the various cut performance levels obtained with the ASTM, ISO and EN norms, there is a need for a more fundamental understanding and interpretation of the data. The ASTM and ISO approaches remain the most suitable methods for cut level exceeding 3. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 4(a): Test equipments (top) with schematic representation (bottom) of the force application, (EN388). 3.1 Cut testing methods EN, ISO, ASTM The first standardized cut test method was developed in France in 1987 (prEN388 [18], hereafter called EN) followed by a method developed in 1997 by the American Society for Testing Material, (ASTM- F1790-97 [19], hereafter called ASTM) and its most recent international adaptation via the International WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Standardisation Organization (ISO-13997 [20], hereafter called ISO). General guidance about hand protection is summarized in the ANSI/ISEA 105-2000 [8], which relates to theses various norms. Each testing procedure is described in the norm documents, [18–20], mentioned above. The three related test equipments are depicted in figs. 4 (a), 4(b) with a schematic representation of the blade orientation, the load force application and the blade displacement. There are similarities between the ASTM and ISO procedures and equipments, but there are fundamentals differences between those two and the EN equipment and test method.

Figure 4(b): Test equipments (top) with schematic representation (bottom) of the force application, blade displacement and blade orientation, (ASTM/ ISO). The EN test method is based on the measurement of the substrate cut resistance when submitted to the cutting effect generated by the rotation of the circular blade on the sample tested, which is under a relatively moderate pressure WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

288 High Performance Structures and Materials III provided by a 5 N load force applied on the blade. The number of revolutions of the blade to cause the cut-through of the specimen will be used to calculate the cut index and therewith the cut performance. A cotton fabric is used as the reference material. The ASTM and ISO methods on the opposite are based on the measurement of the load necessary to provoke the cut-through of the sample after a sliding distance of the rectangular blade of 20 mm (most recently adopted harmonized distance). The corresponding load can reach, for knitted samples made of high performance fibres, the equivalent of 1000 to 3500 g, i.e. 10 to 35 N, which is 2 to 7 times higher than the load force applied on the circular blade in the case of the EN method. This is a difference worth highlighting. 3.2 Experimental part The design of the experiments was selected in order to clarify and better understand some of the aspects, which were partly introduced in the materials and methods section. For the EN test method, the blade cutting edge degradation during the course of the testing of high cut performance materials, hereafter HCP materials, has been illustrated and studied by conducting tests on a “stop and go” basis, i.e. changing the blade during the test every 1, 2, 3, and 5 cycles as illustrated in table 2. Both 100% KEVLAR® and 5.5 % steel reinforced para-aramid knitted structures were used. Table 2: EN test method: the blade cutting edge degradation profile during the course of the testing of high cut performance materials.

The analysis of the influence of PVC dots, melted and cured onto the knitted surface to enhance the grip properties of the fabric or the gloves, was performed using the EN and the ASTM procedures. This further illustrates some of the specific features of the two methods. Fig. 5 provides the resulting data outlining NL1, which is the load, in g, provoking the cut through of the specimen for a blade motion of 25.4 mm. The motion of the blade on the sample tested has been studied in terms of speed using the ASTM and ISO blade calibration material, i.e. a 1.6 mm neoprene layer, and 100% para-aramid felt, which characteristics are indicated in fig. 6(a) and 6(b). As depicted in fig. 7 a load cell, linked to a computer, was placed in the balanced arm holding the blade in order to estimate the force opposing the blade motion during the ASTM testing. Therewith the ratio of this resisting force to the applied load force has been calculated for various materials, such as neoprene, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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100% KEVLAR® knitted gloves and another HCP material. Results are summarized in fig. 8

Figure 5:

PVC dots influence on the EN (top table) and the ASTM results (top table and curves).

Figure 6(a): Blade speed influence on the cut performance of the calibration material.

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Figure 6(b): Blade speed influence on the cut performance of a 100% paraaramid textile.

Figure 7:

CPPT (ASTM) equipped with a compressive load cell.

Finally, a series of experiments was run to estimate the similitude between cutting and sliding process. In fig. 12 a sliding experimental assembly is depicted and was made in order to estimate the sliding force. A large part of the force applied onto a cutting device can be dissipated via the sliding friction, associated sometime with the compression and the relaxation of the materials under cut. Therefore it is important to relate the cut forces measured during cut with the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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more easily accessible sliding and friction forces. It is not our intention to pretend that a direct liaison can be established between these two physical phenomena but, in view of the lack of published data on the subject matter, it is valuable to bring another piece of comprehension to this scientific matter of relevant value to personal protection. During an accidental cut the sliding attributes of the personal protective equipment are determinant.

Figure 8:

F2 / F1 ratio for various materials submitted to ASTM testing.

Figure 9: Various knife cutting edge profiles and recommended uses.

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Figure 10:

A proposed cut mechanism sequence: “Densification-CompressionCut-Relaxation-Densification...”.

3.3 Results and discussion The cut mechanism and related fundamentals are rarely described in the literature. The large diversity of situations where cuts are involved and the rather large number of physical phenomena superimposed in a cutting operation, render the scientific approach further more difficult. There are some qualitative reflection points, which may guide the novice and help the reader through this section. Materials, which are generally considered as cut resistant materials, are also exhibiting: · a good tear resistance · a reasonable puncture resistance towards sharp but not too thin perforators · a certain resistance to abrasion or ability to fibrillate In the cutting process itself one may consider the following as useful considerations: · the amount of material per unit volume, which the cutting threat has to go through · the deformability of this material and the tendency for this material to maintain a high density under pressure still keeping its flexibility · the mobility of the fibres on which the blade is sliding; a revolving material being more difficult to cut. · the blade dulling effect of the material being cut. Glass is a good example.

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·

the eventual surface or bulk phase changes occurring during the cut either due to local heating under friction or heating. . the water or liquid retention and absorbency of the protective barrier. Fibres with skin-core structures are generally more protective. The thickness of the skin is a positive attribute. Para-aramid fibres are cut resistant. Rather brittle materials tend to be less effective. On the other hand ductility can be a good attribute. Steel and glass tend to support these two statements. As exhibited in fig. 9 the design of blade profiles, leaving apart the steel selection and processing, remains an art. For example what are the criteria taken into account to make an effective knife cutting edge to cut tomatoes versus bread versus meat versus cheese, versus wood, versus textile, etc. When one observes a few cutting edges, it seems, as depicted in fig. 10, that the sequence of -1- material blockage-densification and compression followed by -2- the effective sliding and cut of the previously compressed materials and -3the relaxation and separation of the “already cut material” is a sequence which may be taken into account to better understand the selection of a large proportion of cutting edge profiles. That densification-cut-relaxation sequence repeats itself along the blade as the cutting operation proceeds. The threshold compression ratio at which cut can progressively take place depends on the fibre mobility, the deformation/cut speed, the local humidity as well. 3.3.1 Blade degradation during EN testing Using the EN norm, the degradation effect of the materials on the blade cutting edge was already demonstrated by Lara [21] with a series of test performed on fibreglass reinforced glove samples. According to this study, a fraction of cycle – a cycle being represented by one blade full rotation counter-clockwise and its reverse rotation clockwise - was initially sufficient to cut the cotton calibration fabric. The same blade, after running for 50 revolution cycles on the HCP glasscontaining material, then cut the reference cotton fabric after 25 cycles compared to 0.8 cycles initially. This shows that less than a cycle was necessary to dull the blade. Glass fibres are known for their dulling effect. A major concern shall be outlined regarding products, which have a dulling effect. Shall they be considered as effective high cut performance materials? A partial answer is established by the fact that the ASTM/ISO norms are more and more used for these types of materials. These norms being less sensitive to the dulling effect since one spot of the blade is in contact with the sample to cut only once. The EN norm recent revisions suggest the use of the ISO testing in such cases. This is a significant step and improvement. 3.3.2 Speed effect on cut resistance In which proportion the blade speed can affect one material more than another. For example figs. 6(a), 6(b) compares the speed effect, on a neoprene rubber layer and a KEVLAR® sample stamped out of a knitted glove, for a speed variation ranging from 14 to 28 mm/s. Such a variation affects much more the neoprene load-to-blade displacement curve than the load-to-blade displacement relationship of the para-aramid WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

294 High Performance Structures and Materials III samples. This may raise some concern about the 2.5 mm/s versus the 110 mm/s blade displacement speed respectively selected for the ISO and the EN procedures. The speed selected for the ASTM approach, i.e. 14 mm/s, and the highest speed of 28 mm/s selected in the present study lies in between the ISO and the EN selected values. Should those values be reconsidered and harmonized? This question goes beyond the scope of this study. Nonetheless, some additional data will be discussed later on and may converge with this observation providing some elements of explanation. One can already underline that in a textile structure, the fibres are relatively free to move, especially when the textile is knitted which adds bulk to the structure. This mobility can compensate for the blade displacement speed variation since the fibre elements provoke an interface slippage. Since these standards have been put together mostly for the testing of hand protective equipment mostly made of HCP fibres such as para-aramids, the results shown in fig. 6(b) tend to support the rather limited effect of speed within the range tested. Although the emergence of dipped supported gloves may provoke some more questioning of the speed effect since a dipped or coated glove is no more than a textile structure covered by a rubber layer. 3.3.3 Coating/printing effect on the cut resistance of high performance textile materials The testing of gloves with PVC dots is reported in fig. 5. For the dotted side, the level 3 given by the EN cut testing is not consistent with the 770 g load necessary to provoke the cut through after a 25 mm blade displacement using ASTM method. One would expect at least a load of 1100 g, i.e. 1.5 folds, to reach an EN level 3. Furthermore, the non-coated side of the glove, i.e. without PVC dots, reach an inferior level 1versus level 3 for the dotted side. This does not seem to make sense in terms of cut performance. For the ASTM analysis, one observes that, for the non-dotted side, the necessary load to cut through after a 25 mm blade displacement is 820 g. This is more in line with the expected results since the coating is not a cut resistant material and is a spot by spot treatment. Overall in this specific case, the EN testing largely over-estimates the real cut performance. The EN testing does not appear to reflect the real cut performance of the material. Since it tends to over-estimate it, precautions shall be taken regarding the interpretation of the results. In terms of ASTM or ISO testing, if the polymeric layer deposited on the textile substrate is continuous and interlocked into this substrate, the cut performance is likely to be less than the substrate alone since the fibres mobility is restricted. This is not the observation made from fig. 5 since the PVC dotting is discontinuous. The blade dulling, the speed effect and the interference of non-cut performing polymeric coating have been discussed and may find some additional explanations in the study, which follows. 3.3.4 Forces involved during cutting/sliding process The apparatus designed to measure the cut performance as per the ASTM procedure is shown in fig. 4 along with a schematic diagram, which provides the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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blade orientation and displacement, and force application point. This force is generated by weights placed on a plate mounted on a lever arm assembly in a way that doubles the resulting force applied to the specimen. The test apparatus also consists of a motor-driven balanced arm, which holds the blade. In order to measure the co-linear force component resisting to the displacement of the blade, a 50 N (or 200N) load cell, type S2 from HBM Germany, was installed in the arm holding the blade as per the diagram of fig. 7. This load cell was connected to a computer interface, module MP55 from HBM

G

Germany. The resisting force, − F2 , was recorded during the cutting process

G

using the CATMAN® software from HBM Germany. The normal force, F1 , applied onto the blade edge is assumed to be directly proportional to the double weight force value, which is an approximation since during the displacement, and due to the thickness of the sample, the blade cutting edge does not remain strictly perpendicular to the sample contacting point. This is depicted in fig. 7.

G

G

Fig. 8 provides various F2 / F1 ratio.

G

For the neoprene calibration material the resisting force, − F2 , is about 2.2

G

times superior to the normal applied force, F1 . The resisting force found in this study is almost equal to the one, identified as a frictional force, reported by Massé et al. [22] in his study of the basic principles used in the development of a new cut-test machine now specified in the ISO test method. Therefore, our findings, although conducted independently on a different testing apparatus, ASTM versus ISO, are fully in line with the work conducted by Massé et al. [22] at the IRSST, Québec Canada. This is a good demonstration of the interchangeableness of the two equipments now both recommended in the ASTM norm.

G

G

Fig. 8 also provides the F2 / F1 ratio for a KEVLAR® knitted structure and a ceramic coated para-aramid woven fabric. The two ratios are more or less ¼ of the neoprene, measured ratio which may bring a question regarding the suitability this rubber material in the ASTM and ISO norms as a blade sharpness normalizing factor. Furthermore, the amount of friction generated to cut trough neoprene may explain why a dotted knitted structure exhibits an artificially high cut EN performance. The necessity to use the blade cutting edge only once during a test is essential in a situation where abrasion and resulting friction of the surface of the tested material are high. The apparatus designed to measure the sliding forces is shown in fig. 11 along with a schematic diagram, very close to the fig. 7. In this case the resisting force is generated in part by the weight placed onto the specimen but also in part by the specimen surface characteristics inclusive of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

296 High Performance Structures and Materials III its deformability under pressure. The resisting force is measured during the linear displacement at a constant controlled speed equal to 4.2 mm/s.

Figure 11:

Figure 12:

Sliding test: (side view) with schematic representation (bottom) of the force application, specimen displacement on calibrated engineered surface.

F2 / F1 ratio for various in case of cut testing and sliding tests.

Figure 12 outlines the similitude of the forces involved in the case of cutting versus the ones involved in the case of sliding. An amazing and revealing level of concordance can be observed, which means that one could definitely relates, to a certain extend, the surface wear and friction with the cut mechanism, which is by far more complex. This goes beyond the scope of this study, although WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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during an accidental cut the sliding attributes of the personal protective equipment are determinant. High molecular weight polyethylene textile structures are known to change their sliding and friction properties with moderate temperature as well as under significant abrasion and wear while other materials such as para-aramids tend to fibrillate bringing somehow and to a certain extend additional grip sometime to the detriment of appearance.

4

Conclusion

High performance materials based on high performance fibres significantly contribute to the emergence of new technologies under development today by fibre producers as well as converters and end-users relying on the unique attributes of para-aramids such as KEVLAR® in diversifying applications. Thermal and mechanical attributes reviewed, measured and partly discussed in the previous part of this work are contributing to better protection. Their “scientific contain” remains to be further explored. Several tests methods, based on their equivalent standard procedures, were reviewed beyond their normal application frames. In term of harmonization, the considered test conditions reinforce the appropriateness of the convergence towards a single norm, still allowing for multiple choices at the equipment and procedure levels to best represent the suitability of the material versus the expected cut protection level. From this study one can appreciate as an improvement of the EN norm the possibility to test as per ASTM or ISO norm procedure especially when abrasive materials beyond level 3 are tested. Similarly, the interchangeability of the ASTM and the ISO apparatus appears more and more useful as the fundamental understanding of test condition limits increases. Several situations studied in the present work support this aspect. In terms of evolution, the emergence of high cut performance materials reinforced with glass or steel fibres, the market progression of coated protective equipments, the improvement of the test variability, and the substitution of the calibration material seems to be good focus points for reflection in the near term. Several conclusions drawn in the course of this study may serve as a basis. The dulling effect of glass fibres towards the blade cutting edge can be misleading in terms of cut performance analysis. Similarly the artificial cut contribution of polymeric dots deposited on the textile substrate can also cause difficulties in terms of the cut performance interpretation. Those two specific situations are avoided when the blade cutting edge is in contact with the sample only once during the testing. ASTM or ISO are therefore more favourable in such cases. A large amount of work remains to be done to fully integrate all the components of the identified improvements in a global understanding of how the cut mechanism and testing can evolve to be better predictive tools. Reference [16] provides more insight into this study.

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298 High Performance Structures and Materials III The present study has brought, hopefully, a refreshing scientific description of the state of the art knowledge of high performance fibres and related protective solutions with an effort to stimulate the desire to go beyond the knowledge and applications of today. There are properties of the aramids, a high performance family of fibres, which can still be optimised via the system approach. These materials have the potential to participate to the solutions of tomorrow.

Acknowledgements The authors wish to express their gratitude to André Miret-Casas for his meticulous technical assistance and enthusiasm. Portia Yarborough, chairman of the F23 ASTM committee brought valuable comments in her review. DuPont fibre production centre in UK was instrumental to the independent evaluation and analysis of the materials tested. Special thanks are also due to André Courgey from the European Automotive and Transportation Institute in France; his contribution to the wear and friction study is especially recognised.

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[22]

299

P. Yarborough, C. N. Nelson, Performance of Protective Clothing, Global Needs and Emerging Markets, ASTM International, STP1462, 2005. Tejani, N., Blocker, R, Schiffelbein, P., and Rivet, E., “Cut Protection Performance Test for Measuring Cut Resistance of Materials Used in Protective Clothing,” Performance of Protective Clothing: Sixth Volume, ASTM STP 1273, Jeffrey O. Stull and Arthur D. Schwope, Eds., American Society for Testing and Materials, 1997. BS EN388, 1994, Protective gloves against mechanical risks ASTM F1790-97, Standard test method for measuring cut resistance of materials used in protective clothing ISO13997, ISO / TC 94/SC 13/WG 5 N 173, Protective clothingMechanical properties- Determination of resistance to cutting by sharp objects Lara, J., Turcot, D., Daigle, R, and Payot, F., “Comparison of Two Methods to Evaluate the Resistance of Protective Gloves to Cutting by Sharp Blades,” Performance of Protective Clothing: Fifth Volume, ASTM STP 1237, James S. Johnson and S. Z. Mansdorf, Eds., American Society for Testing and Materials, 1996. Massé, S., Lara, J., Sirard, C., and Daigle, R., “Basic Principles Used in the Development of a New Cut-Test Apparatus for Standardization,” Performance of Protective Clothing: Sixth Volume, ASTM STP 1273, Jeffrey O. Stull and Arthur D. Schwope, Eds., American Society for Testing and Materials, 1997.

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The study of surface oxidation of tin(II) fluoride and chloride fluoride materials by Mössbauer spectroscopy: to oxidize or not to oxidize, that is the question G. Dénès, E. Laou, M. C. Madamba & A. Muntasar Laboratory of Solid State Chemistry and Mössbauer Spectroscopy, Department of Chemistry and Biochemistry, Concordia University, Montréal, Canada

Abstract Experimental methods designed to study the bulk of materials do not necessarily detect the changes taking place at the surface of the crystallites. For example, divalent tin-containing materials appear to be stable at ambient conditions in air, provided they are not hygroscopic, X-ray powder diffraction shows only the peaks of the expected tin(II) phase. However, we have observed that the Mössbauer spectrum of polycrystalline samples contain, in addition to the expected tin(II) peak(s), a small peak at 0 mm s relative to CaSnO3 at ambient conditions, that can be attributed only to tin(IV) coordinated by oxygen. A detailed study of this phenomenon has shown that Mössbauer spectroscopy is quite sensitive for detecting thin layers of oxide at the surface of crystallites of tin(II). This phenomenon has been exploited for the study of spontaneous oxidation of various tin(II) fluoride and chloride-containing materials, some of these fluorides being the highest performance fluoride-ion conductors known to date. It was observed that passivation is quite efficient in the fluorides, and in the chloride fluorides that have all their tin(II) covalently bonded. On the other hand, the materials containing a mixture of covalently bonded tin(II) and the Sn2+ stannous ion namely, the Ba1-xSnxCl1+yF1-y solid solution, show a higher rate of oxidation, which is highly dependent on the method of preparation and the composition parameters, x and y. Keywords: passivation, oxidation, disordered phases, fluorite-type structure, Mössbauer spectroscopy, X-ray diffraction, ionic conductivity, BaClF. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06029

302 High Performance Structures and Materials III

1

Introduction

The MF2 fluorides of large divalent metals (M = Ca, Sr, Ba & Pb) have the well known fluorite type structure, with a cubic unit-cell, space group Fm3m, and a cubic coordination of the metal ion (fig. 1) [1]. This contrasts with SnF2 tin(II) fluoride (stannous fluoride) that has a molecular structure; it is made of Sn4F8 tetramers and the Sn-F bonds are strongly covalent [2]. When SnF2 and MF2 are combined together, some new materials are produced, some of which are ordered, others disordered. The MSnF4 compounds have a …M M Sn Sn… order parallel to the c axis of the unit-cell (fig. 1) [3]. Tin and lead belong to group 14, and therefore they have four valence electrons, and in the +2 suboxidation state, the 5s2 (Sn) or 6s2 (Pb) electrons are unused and form a nonbonding electron pair, also called a lone pair. When bonding is ionic, the orbitals are not hybridized and the lone pair is located on the ns native orbital that has spherical symmetry, unless it is distorted by polarization, and a quite regular coordination is observed, and the lone pair is said to be non-stereoactive since it does not modify the coordination. In fluorite-type β−PbF2, lead has a cubic coordination, therefore its lone pair is located on the native 6s orbital and bonding is ionic, like in BaF2 (fig. 1). In tetragonal MSnF4, the coordination of M is a distortion of the MF8 cube of the fluorite structure, with in addition, four longer interactions, to form an overall MF4F’4F”4 where the bond length increases as follows: M-F (from M-F-M bridges) < M-F’ (from M-Feq-Sn bridges) < M-F” (from M-Fax-Sn bridges), where Feq and Fax form equatorial (4) and axial (1) bonds with tin, respectively (fig. 1). For the sake of clarity, the BaF” bonds are not shown on figure 1. The lead coordination in β−PbF2, is still ionic since Sr2+ and Ba2+ take the same coordination in SrSnF4 and in BaSnF4, respectively. In contrast, tin forms covalent bonding in MSnF4, with a very short axial Sn-F” bond, and the lone pair is located on a hybrid orbital, in transposition to F” (fig. 1). The tin lone pair is said to be stereoactive since it changes drastically the tin stereochemistry (lower coordination number, highly distorted coordination). A consequence of the M/Sn order and of the stereoactivity of the tin lone pair is that the lone pairs cluster in sheets that are cleavage planes, which make the structure very highly two-dimensional. Examples of disordered structures are PbSn4F10, the M1-xSnxF2 solid solution (M = Ca and Pb), and µγMSnF4 (M = Ba and Pb) obtained by ball-milling. The disordered phases have an Fm3m cubic unit-cell like the fluorite-type MF2, and their diffraction pattern shows no peak splitting or additional low angle peak that would indicate the presence of a lattice distortion or a superstructure. Therefore, Sn and the other metal M are fully disordered on the metal ion site of the MF2 structure (fig. 2). The M bonding is ionic and the coordination is cubic with some disordered local distortions in the neighborhood of tin, while tin bind to only one of the faces of the F8 cubes, with its stereoactive lone pair pointing towards the center of the cube, and thus its bonding is covalent. BaClF crystallizes in the PbClF structure (space group P4/nmm). Its structure is also a tetragonal distortion of the fluoritetype BaF2, with layers of F and Cl ordered parallel to the c axis of the unit-cell, making it a layered structure, in contrast with the three-dimensional cubic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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structure of BaF2 (fig. 1). We have recently found two methods for substituting substantial amounts of Ba by Sn in BaClF, and also some F by Cl (y>0), or some Cl by F (y<0), making it a doubly disordered Ba1−xSnxCl1+yF1-y solid solution. Xray diffraction shows that Ba and Sn are fully disordered on the Ba site of the BaClF structure. In addition, while the two anionic sites remain ordered, for y>0 the y Cl in excess are disordered with the remaining (1–y) F on the F site, whereas for y<0, the (–y) excess F are disordered with the remaining (1+y) Cl on the Cl site. However, the electronic situation at tin cannot be assessed by X-ray diffraction. All the tin(II)-containing fluorides studied here are high performance fluoride ion conductors (about three orders of magnitude higher than the conductivity of the corresponding MF2).

Figure 1:

Projection of the crystal structure of BaSnF4, BaF2 and BaClF.

All these materials appear stable in air at ambient conditions, and X-ray diffraction shows no sign of oxidation to tin(IV). The present study shows that tin-119 Mössbauer spectroscopy is an excellent method for detecting minute amounts of tin(IV) oxide in tin(II) compounds, and that all materials show at the minimum some surface oxidation, and some passivate very well while other do so less efficiently.

2

Experimental

Fluoride materials synthesis was carried out either in aqueous medium or at high temperature under dry conditions. Tetragonal α-PbSnF4 was obtained by precipitation on addition of an aqueous solution of lead(II) nitrate to a solution of SnF2. Ca1-xSnxF2 precipitated when a solution of calcium nitrate was added to a solution of SnF2 at high Ca/Sn ratio, or by leaching, on stirring, CaSn2F6 in a large amount of water. The synthesis of Pb1-xSnxF2 and BaSnF4 was carried out under nitrogen in sealed copper tubes heated at 500 oC and quenched. Barium WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

304 High Performance Structures and Materials III tin(II) chloride fluorides were prepared by reaction of barium nitrate and tin(II) fluoride in aqueous solutions versus the X = BaCl2/(SnF2 + BaCl2) molar ratio. Three new materials were obtained: BaSn2Cl2F4 (0.25<X<0.35), BaSnClF3.0.8H2O (0.50<X<0.80), and a material that has the diffraction pattern of BaClF (X>0.87). Chemical analysis showed that it is a Ba1-xSnxCl1+yF1-y solid solution (x = 0.15, y = 0.11). A very similar “BaClF phase” was obtained by synthesis in dry conditions when mixtures calculated according to eqn (1) were heated three days under nitrogen in sealed copper tubes at temperatures varying from 350 to 800 oC. 1 − 2x − y 2

Figure 2:

BaF2 + xSnF2 +

1+ y 2

BaCl2 → Ba1 − x Sn xCl1 + y F1 − y

(1)

Model of a disordered fluorite type M1-xSnxF2 structure.

X-ray powder diffraction was carried out on a PW-1050/25 Philips diffractometer automated with the SIE RAY 122® system from Diffraction Technology, using the Kα radiation of copper (λKα1 = 1.54051 Å) and a monochromator. Mössbauer spectra were recorded in a TN7200® multichannel analyzer from Tracor Northern, using an Elscint driving system, a Harshaw (Tl)NaI detector and a 10 mCi CaSnO3 γ-ray source. All chemical isomer shifts were referenced relative to a standard CaSnO3 absorber. All spectra were collected at ambient temperature and fitted using the GMFP software. Samples were studied when young (freshly prepared) and old (after several years storage in air at ambient temperature). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Transmission

1 0 .9 6 0 .9 2

(a )

0 .8 8

Transmission

0 .9 9 5 0 .9 7 5 0 .9 5 5

(b )

0 .9 3 5 -8

-6

-4

-2

0

2

4

6

8

Ve l o c i t y ( m m /s )

Figure 3:

3

Ambient temperature 119Sn Mössbauer of BaSnF4 prepared by the dry method at 500 oC for 2 hours: (a) young sample, (b) sample aged for 69 months at ambient conditions.

Results and discussion

3.1 Oxidation of the fluorides Various ordered and disordered tin(II)-containing fluorides were checked for oxidation, using X-ray diffraction and Mössbauer spectroscopy. Figure 3 shows the spectrum of BaSnF4 when freshly prepared and after 69 months of age. Both spectra have a large quadrupole doublet in the 2-4 mm/s velocity range, which is characteristic of the presence of a stereoactive lone pair on tin, and confirms that bonding is covalent (fig. 1). In addition, a weak peak is observed at ca. 0 mm/s. The peaks positions are given relative to a tin(IV) oxide standard, CaSnO3, and therefore, the peak at 0 mm/s in BaSnF4 shows some tin(IV) oxide is contained in the ample, even when freshly prepared. This can be assigned to Sn(II) to Sn(IV) oxidation of the surface of the solid particles by atmospheric oxygen. After aging 69 months in air at ambient conditions, the tin(IV) signal at ca. 0 mm/s has not increased significantly, therefore no much further oxidation occurred during the nearly 6 years of aging. It is therefore established that there is an initial very rapid surface oxidation of the particles that creates a passivating layer, which protects the bulk of the particles against further oxidation. X-ray diffraction does not show any peak foreign to BaSnF4, and more particularly no WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

306 High Performance Structures and Materials III SnO2 peak. This suggests that either the amount of SnO2 is too small to be detected by diffraction, or the oxide is amorphous, or the particles are too small to diffract. This establishes the superiority of Mössbauer spectroscopy to detect small amounts of oxide in tin(II) halides. The relative intensity of Mössbauer peaks is a function of both the amount of each kind of tin and their recoil-free fraction fa. The recoil-free fraction is the fraction of 119Sn nuclei that absorb γ−rays without dissipating the recoil energy through phonons (lattice vibrations). The stronger the lattice, the higher the recoil-free fraction, and the stronger the Mössbauer peaks at equal amount of tin. Since Sn(IV) forms very strong bonds to oxygen in the highly tight rutile structure of SnO2, it has a high recoil-free fraction and gives a disproportionably high signal, in comparison with small recoil-free fraction tin(II) halides, that have much weaker bonds. It results that the fraction of SnO2 in a tin(II) halide is more than ten times smaller than its relative signal. Therefore, the amount of SnO2 in BaSnF4 is much smaller than the intensity of it signal suggests. Disordered systems, even very poor in tin, like the M1-xSnxF2 at small tin content (x = 0.27 for M = Ca, x = 0.10 for M = Pb) also show no SnO2 diffraction peak (fig. 4Ab), and the tin(IV) Mössbauer peak is barely detectable (fig. 4Cb), showing excellent passivation.

(A) Figure 4:

(B)

(C)

(A) X-ray powder diffraction pattern of CaF2 and Ca1-xSnxF2 (x = 0.27), (B) X-ray powder diffraction pattern of BaClF and Ba1−xSnxCl1+yF1-y (x = 0.15, y = 0.094), (C) 119Sn ambient temperature Mössbauer spectrum of Ba1-xSnxCl1+yF1-y (x = 0.15, y = 0.094) and Ca1-xSnxF2 (x = 0.27).

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3.2 Oxidation of the chloride fluorides The stoichiometric chloride fluorides, BaSn2Cl2F4 and BaSnClF3.0.8H2O have a behavior similar to that of the fluorides. Freshly prepared samples have a tin(IV) oxide peak, and therefore there is a rapid initial surface oxidation (fig. 5), and it also does not show any significant increase after aging, therefore efficient passivation occurs. X-ray diffraction does not detect the oxide. The stronger tin(IV) Mössbauer signal of BaSn2Cl2F4, in comparison to BaSnClF3.0.8H2O, does not mean that the former compound contains more SnO2. It could just mean that tin(II) in BaSn2Cl2F4 has a lower recoil-free fraction than in BaSnClF3.0.8H2O. Very low temperature spectra would be required to determine the reason for their unequal tin(IV) peak intensity.

Transmission

1

(a) 0.93

0.86

1

Transmission

0.97

(b)

0.94 0.91 0.88 0.85 -8

-6

-4

-2

0

2

4

6

8

Ve locity (mm/s)

Figure 5:

Ambient temperature 119Sn Mössbauer spectrum of: (a) BaSn2Cl2F4 and (b) BaSnClF3.0.8H2O.

The behavior of the non-stoichiometric Ba1-xSnxCl1+yF1-y is much more complex. The Mössbauer spectrum of the precipitated solid solution is made of a single tin(II) line at ca. 4 mm/s, characteristic of ionic bonding, and of a tin(IV) oxide peak at ca. 0 mm/s (figs. 4Cb & 6). Like in the materials above, no SnO2 could be detected by X-ray diffraction (fig. 4Bb). Contrary to the fluorides and the stoichiometric chloride fluorides, a significant increase of the intensity of the tin(IV) oxide Mössbauer peak takes place on aging for 4 years. Therefore, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

308 High Performance Structures and Materials III although the material is still quite well protected against oxidation, it is not fully passivated.

1

Transmission

0.99 0.98 0.97 0.96 0.95

(a) (d)

(AM-234, X = 0.860, aged for 2.5 m)

0.94 0.93

-8

-6

-4

-2

0

2

4

6

8

Velocity (mm/s)

Transmission

0 .9 8 5

0 .9 6 5

( A M - 2 3 4 , X = 0 .8 6 0 , a g e d f o r 4 8 .0 m )

0 .9 4 5

(b)

0 .9 2 5 -8

-6

-4

-2

0

2

4

6

8

V e l o c i t y i n ( m m /s )

Figure 6:

Ambient temperature 119Sn Mössbauer spectrum of the precipitated Ba1−xSnxCl1+yF1-y versus age: (a) 2.5 month old, (b) 48.0 months old.

The rate of oxidation of the Ba1-xSnxCl1+yF1-y solid solution prepared by high temperature direct reactions in dry conditions is highly dependent on the x and y composition parameters (fig. 7). At low tin content (low x), freshly prepared samples have a tin(IV) oxide Mössbauer peak quite stronger than the tin(II) peak (fig. 7a1). This shows that initial resistance to oxidation is much weaker than in the low tin content fluoride solid solutions. For example, the tin(IV) peak of Pb1-xSnxF2 at low tin content (x = 0.10) is barely detectable. After less than three years aging, the tin(IV) peak of figure 7a1 has much increased and the tin(II) peak is now a minority, therefore passivation is much poorer. When the tin content increases to x = 0.121 at y = 0 (Cl/F = 1), the

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tin(IV) peak is weaker than the tin(II) peak, however, it is stronger than that of stoichiometric chloride fluorides. (AM-1180, x = 0.05, y = 0.10)

(AM-1180, x = 0.05, y = 0.10)

1

0.999

0.994 Transmission

Transmission

0.995

0.99

0.985

0.984

(a1)

(a2)

0.98

0.979

(AM-315, x = 0.121, y = 0.000)

1

AM-315, x = 0.121, y = 0.00

1

0.99

0.99

0.98

0.98

Transmission

Transmission

0.989

0.97 0.96

(b1)

0.95

0.97 0.96

(b2)

0.95

0.94

0.94

(AM-1155, x = 0.225, y = 0.2, RT aged)

1

(AM-1155, x = 0.225, y = 0.200) 0.998

Transmission

Transmission

0.995

0.988

(c 1)

0.99

(c 2)

0.985

0.98

0.978 -8

-6

-4

-2

0

2

4

6

8

-8

-6

-4

Velocity (mm/s)

Figure 7:

119

-2

0

2

4

6

8

Velocity (mm/s)

Sn Mössbauer spectra of Ba1-xSnxCl1+yF1-y at RT, prepared by direct reactions under dry conditions (freshly prepared and aged): (a1 & a2) AM-1180, x = 0.050, y = 0.10 [freshly prepared (a1) and aged for 32 months (a2)], (b1 & b2): AM-315, x = 0.121, y = 0.000 [freshly prepared (b1) and aged for 36.7 months (b2)], (c1 & c2): AM-1155, x = 0.225, y = 0.200 [freshly prepared (c1) and aged 31.0 months (c2)].

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310 High Performance Structures and Materials III Upon aging to three years, the tin(IV) peak dominates slightly. This shows that both the oxidation is slower at higher tin content, and passivation is somewhat better, although not great. Near the upper limit of the solid solution in tin (x = 0.225) and at high chlorine content (y = 0.200), the tin(II) signal predominates by far, however, it is also very weak, which shows that its recoilfree fraction is extremely small. After aging two and a half years, the tin(IV) signal is twice as strong as the tin(II) signal, therefore initial oxidation was not very large, however, passivation was not very good. These results can be understood, at least in part, from the bonding model of tin in the solid solution. At low x, Sn is diluted in the solid and there is little chance of a Sn being neighbor to another Sn. The coordination of an isolated Sn is mostly determined by the size of the Ba site it occupies. Since Ba2+ is quite larger than Sn2+, the latter is loose in its site, i.e. its bonds are weak, resulting in a weak recoil-free fraction and fast oxidation (figs. 7a1 & 7a2). At higher tin content, the rest of the solid network has to accept adjustments to satisfy the tin bonding requirements, which allows for stronger bonds, hence a higher recoilfree fraction and some improvement of passivation (figs. 7b1 & 7b2). However, at high x and high y, the excess chlorine, probably clusters around tin than around barium because there is more room around the smaller tin. However, since Sn-Cl bonds are weaker than Sn-F bonds, it results in a decrease of recoilfree fraction and passivating power (figs. 7c1 & 7c2).

4

Conclusion

Mössbauer spectroscopy was an essential method for understanding the complex bonding of tin(II) and its oxidation to tin(IV) in the fluorides and chloride fluorides.

Acknowledgements This work was made possible by the support of Concordia University and the Natural Science and Engineering Research Council of Canada. Grateful thanks are also due to the Procter and Gamble Co. (Mason, Ohio) for supporting our Mössbauer laboratory.

References [1] [2] [3]

Rayner-Canham, G. & Overton, T., Descriptive Inorganic Chemistry, 3rd ed., Freeman, New York, pp. 78-80, 2002. Dénès, G., Pannetier, J., Lucas, J. & Le Marouille, J.Y., About SnF2 stannous fluoride. I. Crystallochemistry of α-SnF2. J. Solid State Chem., 30(3), pp. 335-343, 1979. Birchall, T., Dénès, G., Ruebenbauer, K. & Pannetier G., A neutron diffraction and 119Sn Mössbauer study of PbSnF4 and BaSnF4. Hyperf. 29, pp. 1331-1334, 1986. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Study of double ionic disorder (cationic and anionic) and disorder of two kinds of tin(II) (ionic and covalent) within the same material: the incredibly complex Ba1-xSnxCl1+yF1-y solid solution and its study by Mössbauer spectroscopy S. Boufas2 G. Dénès1, J. Kochuparampil1, H. Merazig2 & A. Muntasar1 1

Laboratory of Solid State Chemistry and Mössbauer Spectroscopy, Department of Chemistry and Biochemistry, Concordia University, Montréal, Canada 2 Laboratoire de Chimie Moléculaire, du Contrôle de l’Environnement et des Mesures Physico-Chimiques, Département de Chimie, Université Mentouri, Constantine, Algeria

Abstract A doubly disordered Ba1-xSnxCl1+yF1-y solid solution has been prepared. X-ray diffraction shows that Ba and Sn are fully disordered on the Ba site of the BaClF structure. In addition, while the two anionic sites remain ordered, for y>0 the y Cl in excess are disordered with the remaining (1-y) F on the F site, whereas for y<0, the (-y) excess F are disordered with the remaining (1+y) Cl on the Cl site. 119 Sn Mössbauer spectroscopy showed the Ba1-xSnxCl1+yF1-y solid solution is a truly unique material. Indeed, in addition to the rarity of being a doubly disordered solid solution over a wide range of chemical composition, it also exhibits a variable mixture of ionic tin(II) (Sn2+ stannous ions, with a non-stereoactive lone pair) and covalently bonded tin(II) (with a stereoactive lone pair). A new material, Ba2SnCl6, has been prepared and its crystal structure solved. It contains octahedrally coordinated Sn2+, with a perfectly non-stereoactive lone pair, for the first time in a chloride. This makes the lone pair a potential charge carrier. Keywords: disordered phases, BaClF structure, Mössbauer spectroscopy, X-ray diffraction, mixed conductivity. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06030

312 High Performance Structures and Materials III

1

Introduction

BaClF crystallizes in the PbClF structure. Its structure is a tetragonal distortion of the fluorite-type BaF2, with alternating layers of F and Cl ordered parallel to the c axis of the unit-cell (fig. 1), making it an anisotropic structure, in contrast with the isotropic cubic structure of BaF2 (fig. 1) [1]. The Ba2+ ions are bonded to four fluoride ions at the corners of a square, like half of its cubic coordination in BaF2. An apical Ba-Cl bond is located in trans-position relative to the fluorine square, with in addition four chloride ions at the corners of a square parallel to the F4 square and rotated by 45o parallel to (a,b). It results that the barium coordination in BaClF is a BaF4ClCl’4 monocapped square antiprism, in contrast with the BaF8 cube found in BaF2. The BaClF structure is the result of …F Cl… order of the fluorite structure parallel to the c axis of the unit-cell. An alternate tetragonal distortion of the fluorite structure, also with order parallel to the c axis is the BaSnF4 structure, with the metals being ordered (…Ba Ba Sn Sn…). The structure of BaSnF4 is very strongly two-dimensional (fig. 1). In the present work, the reaction of barium chloride and tin(II) fluoride in aqueous solutions, at a high BaCl2/SnF2 ratio, gives a precipitate that appears at first glance, from its X-ray diffraction pattern, to be BaClF. A similar phase, although not completely identical, was obtained by high temperature reaction of a mixture of BaF2, BaCl2 and SnF2 in dry conditions. The detail investigation of the material obtained, particularly the study of tin by Mössbauer spectroscopy, is presented here. In addition, the first perfectly ionic tin(II) in a chloride fluoride has been discovered in a new compound, Ba2SnCl6.

Figure 1:

2

Projection of the crystal structure of BaSnF4, BaF2 and BaClF.

Experimental

The “BaClF phase” precipitate was obtained when aqueous solutions of barium chloride and tin(II) fluoride were mixed, at a X = BaCl2/(SnF2 + BaCl2) molar WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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ratio above 0.85, when barium chloride was added to tin(II) fluoride (Ba→Sn), or the other way around (Sn→Ba). Synthesis under dry conditions was performed when mixtures calculated according to eqn (1) were heated for three days under nitrogen in sealed copper tubes at temperatures varying from 350 to 800 oC. 1 − 2x − y 2

BaF2 + xSnF2 +

1+ y 2

BaCl2 → Ba1 − x Sn xCl1 + y F1 − y

(1)

Crystals of Ba2SnCl6 were collected from a batch obtained by slow evaporation of an aqueous solution of BaCl2.2H2O and SnCl2.2H2O. Chemical analysis was carried out by use of a Perkin Elmer 503 atomic absorption spectrometer (Ba and Sn), an Orion Research Laboratories 94-09 fluoride ion selective electrode (F), and by argentometry according to the Fajans titration method (Cl). The analytical methods required substantial adjustments in order to eliminate interferences created by the presence of tin and fluorine [2]. X-ray powder diffraction was carried out on a PW-1050/25 Philips diffractometer automated with the SIE RAY 122® system from Diffraction Technology, using the Kα radiation of copper (λKα1 = 1.54051 Å) and a monochromator. Identification to known phases was done by Search-Match® using the µPDSM software from Feinn-Marquat and the JCPDS PDF1 database from the ICDD. Mössbauer spectra were recorded in a TN7200® multichannel analyzer from Tracor Northern, using an Elscint driving system, a Harshaw (Tl)NaI detector and a 10 mCi CaSnO3 γ-ray source from Ritverc. All chemical isomer shifts were referenced relative to a standard CaSnO3 absorber. The spectra were fitted using the GMFP5 software.

3

Results and discussion

The X-ray powder diffraction pattern of the precipitate (fig. 2b) has the same set of peaks as BaClF (fig. 2c), at the same positions, and therefore, it seems that the product obtained is simply BaClF. This seemed confirmed by Search-Match® that resulted in BaClF being the only reasonable match. Although the match was reasonably good (Similarity Index SI>150 and all 15 peaks matched), a close examination of the peak position revealed a small, but significant shift, shown by arrows on figure 3a, relative to unsubstituted BaClF (fig. 3b). The shift of all peaks is consistent with a 0.91% decrease of the a parameter and a 0.97% increase of c, while the unit-cell volume decreases by 0.86% and the c/a tetragonal distortion increases by 1.89%. The change of unit-cell parameters suggests a change of chemical composition. A logical change was the substitution of a fraction of Ba by Sn, to give non-stoichiometric Ba1-xSnxClF. Chemical analysis was performed to determine the tin molar fraction x, and it showed that the compound is also non-stoichiometric for the anions, such that the formula is best written as Ba1-xSnxCl1+yF1-y (x = 0.15, y = 0.11), to show it is a substituted BaClF phase, as shown by X-ray diffraction.

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(a) 220

212 104

211 202

103

003 112 200

102

110

101

Intensity

001

002

314 High Performance Structures and Materials III

Intensity

(b)

(c) 5

10

15

20

25

30

35

40

45

50

55

60

Deg. 2 theta

Figure 2:

X-ray diffraction of (a) Ba1-xSnxCl1+yF1-y prepared by the dry method (x = 0.150, y = 0.050, 43 hours at 500 oC), (b) precipitated Ba1-xSnxCl1+yF1-y (X = 0.923), (c) BaClF precipitated by addition of HF(aq) to a solution of BaCl2.2H2O.

Preparations carried out by the dry method yielded a material that also gives the BaClF diffraction pattern (fig. 2a). A wide double non-stoichiometric solid solution was obtained: 0≤x≤0.25, -0.15≤y≤0.25. The inversion of the relative intensities of the (002) and (110) peaks could be in part due to the change of chemical composition. It is likely that it is mostly due to preferred orientation favoring the (00l) peaks in the latter, i.e. the material tends to cleave parallel to (a,b). This is consistent with the crystal structure (fig. 1): the long Ba-Cl bonds (Cl atoms at the corners of a square) are the weak links. In addition, figure 3 shows that the peak shift in the solid solution prepared by the dry method is negligible, and it was found it changes very little with the composition parameters x and y. The different unit-cell parameters of the solid solution prepared by the two methods shows that something differentiates them from one another, although both are doubly disordered BaClF structures. For these

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110

110

101

(c)

(a)

20

002

101

Intensity

(b)

110

002

101

Intensity

002

disordered phases, only average Ba/Sn values can be obtained by diffraction alone. Could the difference be due to the way tin binds to the BaClF network?

25

30

Deg. 2 theta

Figure 3:

Enlargement of a portion of the X-ray powder diffraction pattern of: (a) Ba1-xSnxCl1+yF1-y prepared by the dry method (x = 0.150, y = 0.050, 43 hours at 500 oC), (b) BaClF precipitated by addition of HF(aq) to a solution of BaCl2.2H2O, (c) precipitated Ba1-xSnxCl1+yF1y (X = 0.923).

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316 High Performance Structures and Materials III

(b)

Figure 4:

Electrical monopolar and electrical quadrupolar interactions generated by different kinds of divalent tin, and their effect on the Mössbauer spectrum: (a) unhybridized lone pair (e.g. semiconducting CsSnBr3), (b) hybridized lone pair (e.g. BaSnF4), (c) unhybridized lone pair in the case of Ba1−xSnxCl1+yF1-y discovered in the present work.

A selective probing of the tin sites, that ignores barium, is required. Mössbauer spectroscopy of the 119Sn nuclide probes the tin sites only, regardless of crystallinity or order/disorder. Figure 4 shows the expected Mössbauer spectrum for the modes of bonding of divalent tin. Tin belongs to group 14 and period 5 of the periodic table of the elements, and therefore it has 4 valence electrons. It results that it forms two stable oxidation states, +2 and +4. In the suboxidation state +2, tin has a non-bonded electron pair (lone pair). Two types of bonding can be found in halides: ionic bonding (most iodides and some bromides) or covalent bonding (all fluorides and chlorides, and most bromides). When bonding is ionic, the tin lone pair is located on the 5s native orbital (nonstereoactive lone pair). Figure 4a shows that in the case of ionic bonding in an undistorted coordination, the 5s orbital is perfectly spherical and there is no WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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electric field gradient (efg) at the Sn2+ site. It results no splitting of the nuclear levels, and therefore only one nuclear transition takes place (1/2 →3/2), and a single Mössbauer absorption line is observed. This contrasts with the case of covalent bonding, where the tin lone pair is located on one of the hybrid orbitals (stereoactive lone pair). This lone pair is highly axial and therefore generates a large efg, (Vzz)val. In addition, the stereoactive lone pair distorts very strongly the tin polyhedron of coordination, which generates another term of the efg, (Vzz)latt. The overall effect is the splitting of the first excited state which results in two nuclear transitions, and the Mössbauer spectrum is a doublet, with a large quadrupole splitting ∆ (fig. 4b). In addition, the higher the valence s electron density acting at tin, the higher the isomer shift δ, i.e. the line shift relative to a CaSnO3 standard. As can be seen from figure 4, this makes the isomer shift of Sn2+, 4.0-4.1 mm/s (5s electron density concentrated around the tin atom) higher than that for Sn(II)cov, 2.5-3.5 mm/s (5s electron density pulled away in hybrid orbitals). The ambient temperature spectrum of precipitated Ba1-xSnxCl1+yF1-y is a single line at 4.07 mm/s (fig. 5a1) which shows that tin is present in the form of Sn2+ stannous ions. For the Ba1-xSnxCl1+yF1-y solid solution prepared in dry conditions, the Mössbauer spectrum is a function of the x and y composition parameters. For a high tin content (x = 0.225), the spectrum is also a single line at ca. 4 mm/s if Cl/F>1 (y = 0.25), and therefore bonding is ionic (fig. 5b1). For the same x value and Cl/F<1 (y = -0.15), the tin(II) region of the spectrum is a very highly asymmetric doublet, which is actually due to the sum of a doublet and a single line. Since for fluorides, the high velocity line of a doublet is at about 4 mm/s, it adds to the single line of ionic tin to makes a spectrum that looks like a strongly asymmetric doublet. Therefore, at high tin content, and when there is more fluorine than chlorine, the Ba1-xSnxCl1+yF1-y solid solution contains a mixture of Sn2+ ions and covalently bonded tin(II). The line at ca. 0 mm/s in the spectra of figures 5c1 and 5c2 is due to a tin(IV) impurity generated by surface oxidation of the particles. This is common in tin(II) compounds. The ambient temperature spectra have a highly unequal intensity, and strangely enough, the poorest in tin is the strongest (fig. 5a1). In addition, the spectrum of figure 5b1 is much weaker than that of figure 5c1, even though they contain exactly the same amount of tin. The line intensity is a function of the recoilless fraction fa, which is itself a function of bond strength. All the Mössbauer observations make it possible to build a model of tin bonding in Ba1-xSnxCl1+yF1-y solid solution: (i) at low tin concentration (small x), like in the precipitated solid solution, due to dilution tin is isolated from other tin atoms, the solid network is determined by the size of the Ba2+, Cl- and F- ions, and since tin occupies a barium site, it takes the same mode of bonding, i.e. ionic (figs. 6a); (ii) at high tin content (x = 0.25), the tin concentration is high enough to have some tin clustering, and in an excess of fluorine (y = -0.15)each tin atom is connected by bridging fluorine, and this result in covalently bonded tin with a stereoactive lone pair (fig. 6b). The sample of figure 5c1 contains a mixture of the two models of figures 6a and 6b. However, at the same tin concentration (x = 0.25), in the presence of an excess of chlorine (y = 0.25), the excess Cl is likely WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

318 High Performance Structures and Materials III to coordinate tin where there is more room than around barium, due to the smallest size of tin, and this probably creates a situation similar to anhydrous SnCl2. Mössbauer spectroscopy has shown that ionic bond is present in SnCl2. This is probably the reason for the single Sn2+ line of figure 5b1. The weaker SnCl bonds, compared to Sn-F bonds, are the reason for the very weak lattice (extremely weak line intensity at ambient temperature). 1

0.93

0.98

Transmission

Transmission

0.98

(X = 0.896, Ba ---> Sn, T = RT) 0.96

X= 0.896, Ba->Sn, T = RT

0.88

X= 0.896, Ba->Sn, T = 17K

(X = 0.896, Ba ---> Sn, T = 17 K)

(a)

(a1) (a )

0.94

(a(b) 2)

(a

0.83

0.78 -8

-6

-4

-2

0

2

4

6

8

Velocity (mm/s)

1

1

Transmission

0.96 T r a n s m is s io n

0.995

0.99

(b1)

(x = 0.225, 0.225, y = 0.25, T x= y = 0.250, T = RT

(b1)

0.92

x= 0.225,

(x y == 0.225, 0.250,y = 0.25, T = 17 K) T = 17K

0.88 0.84 0.8 -8

0.985

(b(b22))

-6

-4

-2

0

2

4

6

8

Velocity (mm/s)

1

1

0.995 T r a n sm issio n

0 .9 5

0.99 0.985

0 .9

x =x= 0.225, y = - 0.15, T = RT 0.225,

y = -0.150, T = RT

0.98 0.975

(c1)

(x =x= 0.225, y = - 0.15, T= 17 0.225, K)

y = -0.150, T = RT

0 .8 5

(c1)

(c2)

(c2)

0 .8

0.97 -8

-6

-4

-2

0

2

4

6

8

-8

-6

-4

Velocity (mm/s)

Figure 5:

119

-2

0

2

4

6

8

V e lo c it y ( mm/ s )

Sn Mössbauer spectrum at 293 K (a1, b1, c1) and 17 K (a2, b2, c2) of samples: (a1 & a2) AM-1273 (precipitated at ambient conditions, X = 0.896, Ba→Sn), (b1 & b2) AM-1270 (dry method, x = 0.225, y = 0.250), (c1 & c2) AM-1148 (dry method, x = 0.225, y = -0.150).

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(a)

(b)

Figure 6:

Bonding model in Ba1-xSnxCl1+yF1-y: (a) ionic bonding for isolated Sn2+ ions at low x, (b) covalent bonding to Sn(II) in Sn clusters at higher x.

On cooling to 17 K, thermal motion is frozen and therefore the Mössbauer spectra are stronger. Since weak and strong lattices have their thermal motion frozen, the difference between them is very small at low temperature, and weak lattices benefit more from cooling (fig. 5b1). The reason for some of the difference between the precipitated solid solution and that prepared in dry conditions is unclear at this point. However, our current research under progress has shown that there is a phase transition on heating the precipitates, to give a high temperature phase metastable at ambient temperature, that is similar to the samples prepared in dry conditions at ambient conditions. The theoretical spectrum for purely ionic tin(II), i.e. the stannous Sn2+ ion, is a single line at high velocity (ca. 4.1 mm/s), shown on figure 4a. However, SnCl2 has an orthorhombic structure, and therefore the tin site is distorted, which results in a small efg due to possible polarization of the lone pair and site distortion, therefore one can expect a moderate broadening of the line. The same WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

320 High Performance Structures and Materials III applies to ionic tin in Ba1-xSnxCl1+yF1-y, where the presence of a mixture of fluorine and chlorine around tin also contributes to the efg (fig. 4c). Therefore, no strictly purely ionic tin(II) had been obtained before in a chloride or in a fluoride. Now, this has been done: crystals of Ba2SnCl6 were obtained and its crystal structure has been solved. The unit-cell is cubic, and tin is located in an undistorted octahedral site (fig. 7). This is the first case of a perfectly spherical, i.e. completely non-stereoactive, lone pair, in a tin(II) chloride. Efforts are now focusing on producing a sufficient quantity in order to collect its Mössbauer spectrum. It should have the narrowest possible line for Sn2+, and give, for the first time, the true isomer shift of purely ionic tin in a chloride matrix. The full crystal structure of Ba2SnCl6 will be published elsewhere.

Figure 7:

4

Crystal structure of Ba2SnCl6.

Conclusion

The combination of Mössbauer spectroscopy with X-ray powder diffraction has made possible a deep understanding of tin bonding in the Ba1-xSnxCl1+yF1-y solid solution. For the first time, a doubly disordered solid solution, with Ba2+ ions, Sn2+ ions and covalently bonded tin(II) disordered on the same site has been obtained. In addition, this is the first time that ionic tin(II) and covalently bonded tin(II) have been reported to be present in the same sample, and also to be varied by changing the stoichiometry.

Acknowledgements This work was made possible by the support of Concordia University, the Natural Science and Engineering Research Council of Canada and the université Mentouri. Grateful thanks are also due to the Procter and Gamble Co. (Mason, Ohio) for supporting our Mössbauer laboratory. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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References [1] [2]

Flahaut, J., Les structures de type PbFCl (EOI) et type anti-Fe2As (C38) des composés ternaires à deux anions MXY. J. Solid State Chem., 9, pp. 124-131, 1974. Muntasar, A., Preparation, characterization and properties of novel materials in the BaCl2/SnF2 system, PhD thesis, Concodia University, Montreal, pp. 39-89, 2002.

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Study on fatigue and energy-dissipation properties of nanolayered Cu/Nb thin films Y.-C. Wang, T. Hoechbauer, J. G. Swadener, T. Darling, A. Misra, R. Hoagland & M. Nastasi Centre for Integrated Nanotechnologies Group, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, USA

Abstract Energy dissipation and fatigue properties of nano-layered thin films are less well studied than bulk properties. Existing experimental methods for studying energy dissipation properties, typically using magnetic interaction as a driving force at different frequencies and a laser-based deformation measurement system, are difficult to apply to two-dimensional materials. We propose a novel experimental method to perform dynamic testing on thin-film materials by driving a cantilever specimen at its fixed end with a bimorph piezoelectric actuator and monitoring the displacements of the specimen and the actuator with a fibre-optic system. Upon vibration, the specimen is greatly affected by its inertia, and behaves as a cantilever beam under base excitation in translation. At resonance, this method resembles the vibrating reed method conventionally used in the viscoelasticity community. The loss tangent is obtained from both the width of a resonance peak and a free-decay process. As for fatigue measurement, we implement a control algorithm into LabView to maintain maximum displacement of the specimen during the course of the experiment. The fatigue S-N curves are obtained. Keywords: nanolayered thin films, fatigue, loss tangent.

1

Introduction

Materials consisting of nano-scale microstructures are of scientific and industrial interest due to their unusual mechanical properties. It has been shown that nanolayered multilayers exhibit ultra-high yield strength [1]. Transmission electron microscopy (TEM) studies on cold-rolled Cu/Nb multi-layers showed that the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06031

324 High Performance Structures and Materials III films exhibit large plastic deformation without the formation of dislocation cell structures [2]. Based on the high tensile strength and lack of dislocation cell structure formation during monotonic loading, we hypothesize that the fatigue strength and failure mechanisms in nanolayered metals may differ significantly from bulk metals. However, the fatigue and energy-dissipation properties of selfsupported nano-layered thin films have not been well studied. Experimental methods for measuring mechanical fatigue properties of materials have been well documented in many textbooks [3, 4]. In recent years, new methods are continuously under development, such as the methods for the fatigue investigations in gigacycle regime [5]. Similarly, there is a need to develop methods to evaluate fatigue properties of novel nanoscale thin film materials. We develop a method to perform fatigue tests on self-supported films via oscillating cantilever beams at their resonant frequencies. This method is superior to traditional methods, such as the tension-tension and rotating beam fatigue methods, in that high stress and high cycle fatigue can be accomplished simultaneously with fewer alignment problems. Our method applies fully reversed stress to specimens, i.e. the load ratio σmin /σmax = -1. Furthermore, our resonant frequency device also facilitates measurement of linear viscoelastic damping.

Figure 1:

2

Schematic of the resonant frequency device for the measurement of fatigue and energy-dissipation properties of self-supported thin films.

Experimental

The schematic of our resonant frequency setup [6] is shown in Figure 1. A bimorph piezoelectric plate-like actuator (Moran Electro Ceramics, Fairfield, NJ, U.S.A.) with the dimensions of 15 mm in length, 6 mm in width and 0.64 mm in thickness was used to generate excitation at the support of a cantilever beam specimen. The bimorph actuator was clamped into the foundation with proper electrical connections, and driven by a function generator (Agilent 33250A, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Agilent Technologies, Inc., Loveland, Colorado, U.S.A.). The free length of the piezo actuator was 10mm. The multilayer specimens were mounted onto the top of the piezo actuator with a set of aluminium clamp. The weight of the aluminium clamp was 235.4 mg. The deflection of the specimen and aluminium clamp was monitored by a fibre-optic measurement system (MTI-2000 FotonicTM sensor, MTI Instruments, Inc., Albany, NY, U.S.A.) with the probes labelled Fibre 1 and Fibre 2 in the figure. The location of Fibre 1 was chosen to be close to the fixed end of the specimen to avoid signal overload due to large deflection. Lock-in amplifiers (Model SR830, Stanford Research Systems, Inc., Sunnyvale, CA, U.S.A.) were connected with the fibre-optic probes to reduce noise in the displacement signals. Experiments were performed under moderate vacuum conditions typically about 50 mtorr to reduce air damping and achieve a reasonably high Q factor. Since our primary interest is to observe the initiation of fatigue cracks, not crack propagation, we define fatigue failure when a 10 Hertz reduction in specimen’s resonant frequency is observed. Cu/Nb multilayers, with individual layer thickness of 40 nm and consisting of 500 bilayers of Cu and Nb with a total thickness of 40 µm, were prepared by magnetron sputtering deposition on silicon wafers that were cut about a half way through thickness by a wafer saw into a pattern of 2 mm long and 1 mm (or 0.5 mm) wide rectangular strips on the wafer surface. A typical cross-sectional TEM image of the Cu/Nb multi-layers is shown in Figure 2 along with selected area diffraction pattern. Note the polycrystalline structure of the layer with in-plane grain size on the order of the layer thickness. The multi-layer exhibited a strong Kurdjumov-Sachs orientation relationship: {111}Cu // {110} Nb, and <110>Cu // <111>Nb. The parallel {111}Cu and{110} Nb planes formed the interface.

Figure 2:

3

Cross-sectional TEM micrograph (left) and the corresponding selected area diffraction pattern (right) of a sputter-deposited Cu/Nb multilayer with 40 nm individual layer thickness.

Results and discussion

It is known that the Cu and Nb films manufactured by the magnetron sputtering technique respectively have <111> and <110> textures parallel to their growth direction (i.e. normal to layer surfaces). However, their in-plane properties are WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

326 High Performance Structures and Materials III composed of poly-crystals in random directions. The in-plane Young’s modulus (E) of a material along a direction can be calculated as follows [7].

1 l 4 m 4 n 4 2m 2 n 2 2l 2 n 2 2m 2 l 2 = + + + + + , F1 F2 F3 E E1 E2 E3

(1)

where 2 1 C22C33 − C23 1 C11C 33 − C132 1 C11C22 − C122 = , = , = , E1 E2 E3 ∆ ∆ ∆

2 2 (C12C13 − C11C23 ) 1 = + ∆ C44 , F1 2 2 (C13C23 − C33C12 ) 1 = + F2 ∆ C55 ,

2 2 (C12 C23 − C22C13 ) 1 = + C66 , F3 ∆ C11 C12 C13 ∆ = C12 C22 C23 C13 C23 C33 . Here Cij represents elastic constants and ( , m, n) is the direction cosine. For single crystal Cu, C11 = 169.8 GPa, C12 = 122.6 GPa and C44 = 75.3 GPa [8]. The calculated principal Young’s moduli are E1 = E2 = E3 = 66.99 GPa. The Young modulus along <111> is E<111> = 191.2 GPa. As for single crystal Nb, C11 = 245.6 GPa, C12 = 138.7 GPa, C44 = 29.3 GPa [8]. It is found that E1 = E2 = E3 = 145.48 GPa and E<110> = 93.2 GPa and E<111> = 83.2 GPa. It can be seen from Figure 3 (a), the theoretical results of the in-plane Young’s modulus of single crystalline Cu and Nb, that Cu in (111) plane is transversely isotropic but Nb in (110) is direction-dependent. A simple uniform homogenization for Nb along different directions results in an averaged in-plane Young’s modulus about 100 GPa. Since our multilayers have equal volume fraction for Cu and Nb throughout thickness, we adopt the Voigt homogenization procedure, and obtain the homogenised in-plane Young’s modulus to be 115 GPa. Figure 3 (b) shows comparison between experimental and calculated inplane Young’s modulus. Good agreement is achieved. A typical SEM picture of the fatigue crack observed on the Cu/Nb multilayers before complete fracture is shown in Figure 4 (a). The length of the sample was 7.1 mm, and the crack caused an about 60 Hz reduction in resonant frequency. This sample was cycled at the stress amplitude 800 MPa for 8500 cycles. The position of the crack is very near the aluminium clamp. The bright spots are the cyanoacrylate glue used to pre-mount the specimen on the piezo WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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actuator before mounting the aluminium clamp and fatigue testing. Fatigue damage such as extrusions and intrusions are not observed around the crack on the surface where maximum stress was exerted. A SEM image of annealed Cu fatigued by the resonant frequency device is shown in Figure 4 (b). This sample was fatigue tested at a stress after several thousand cycles. It can be seen that two cracks were initiated at the edges of specimen, and propagated inwards. The bright regions along the cracks are slip-related intrusions and extrusions. 160

4800

Ave. E 111 = 131.4 GPa for Cu Ave. E 110 = 100 GPa for Nb

140

Resonant frequency, Hz

Directional E, GPa

Cu in (111)

120 Nb in (110)

100

Theoretical calculations based on E = 115 GPa

4000

80

3200 2400 1600 800

<100>

60 -100

Figure 3:

-50

0 θ, deg

50

100

(a)

0

2

8

10

(b)

(a) Theoretical calculation of directional in-plane Young’s modulus of the Cu and Nb with the texture of <111> and <110> directions normal to the layer surface, respectively, and (b) experimental results of resonant frequencies versus sample lengths.

(a) Figure 4:

4 6 Length, mm

(b)

SEM images of (a) fatigued 40 nm Cu/Nb multilayers with a total thickness of 40 µm and (b) fatigued annealed 40 µm-thick Cu, clamped at the top of the image.

The S-N curves of several materials, quenched and tempered (550˚F) 4340 steel, the 40 nm Cu/Nb multilayers, mild steel, 6061-T6 and annealed Cu, are shown in Figure 5. The nanolayered Cu/Nb exhibits a fatigue limit at about 450 MPa. It can be seen that the high fatigue endurance of the 40 nm Cu/Nb multilayers correlates well with its high tensile strength of around 1400 MPa. We obtain the ratio of fatigue endurance limit to ultimate tensile strength to be about 0.35 for the 40 nm Cu/Nb multilayers [9]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

328 High Performance Structures and Materials III

4340 Steel Quenched and Tempered (550ūF)

Maximum Stress, MPa

1000

40 nm CuNb multilayers

Al 6061-T6

100 Annealed Cu

100

10

2

10

4

10

6

10

8

10

10

N

Figure 5:

S-N curves (maximum stress vs. number of cycles to failure curves). Symbols are defined as follows. Circles: 4340 steel. Diamonds: 40nm Cu/nb multilayers. Triangles: mild steel. Up-side down triangles: annealed Cu.

The deformation mechanism of the fatigue of the 40 nm Cu/Nb multilayers is not completely understood. Since the formation of dislocation cell structures is unlikely due to the nanoscale thickness of an individual layer between two adjacent interfaces, extrusion or intrusion is not observed on the surface subjected to maximum stress. This is consistent with our experimental findings. However, concerning detailed deformation mechanisms, specifically how the nanolayered thin films accumulate damage during cycling loading that eventually leads to fatigue crack initiation require further TEM study, which will be presented in a future article. As for linear viscoelastic damping measurement in terms of loss tangent (i.e. tan δ), two approaches are adopted. From a resonant peak, one can use the FWHM method to calculate tan δ, as follows [10].

tan δ =

∆ν

ν0 3

.

(2)

Here ν0 is the resonant frequency of the specimen and ∆ν the frequency span at half maximum intensity. The other approach is to extract loss tangent from a free decay process, as follows.

tan δ =

1 A ln 0 . kπ Ak

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(3)

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Here k represents number of peaks in a time interval; A0 and Ak are amplitudes respectively corresponding to the beginning and end of the time interval. Figure 6 (a) and (b) show the results of our damping measurements on the 40 nm Cu/Nb multilayers. From resonant peaks, we obtain the loss tangent of the nanolayered thin films, which is approximately 3.8 x 10-3 (thick solid line). After 20 million cycles of excitation under a small driving force (dotted line), tan δ is 5.3x10-3. The shift in resonant frequency, about 3 Hz, is unlikely due to fatigue of the specimen. This resonant frequency shift may be system related, and it is the reason why the 10-Hertz rule is adopted for determination of fatigue failure. We remark that the measured loss tangent for the nanoscale films appears higher than expected. This may be due to defects inherently from the magnetron sputter deposition. Further experiments are underway to determine possible parasitic damping from the support of the cantilever or air damping. 6

Cycled

2

-3

tan δ = 3.8x10 (virgin)

15

tan δ = 5.3x10 -3 (cycled) 10 5

0 -2 -4

L=11.3 mm

0

-6

-5 380

Figure 6:

4

4

Virgin

20

Amplitude, volts

Specimen displacement amplitude, µm

25

-8

385

390 Frequency, Hz

395

400

tan δ = 3.3 x 10 -3 50

100

150

(a)

200

Time, ms

250

300

(b)

(a) Measured tan delta from resonant frequency peaks of a 6.5-mm long specimen, and (b) loss tangent from the free decay method of an 11.3 mm long specimen. Both results were obtained under a pressure of 20 mtorr.

Conclusions

The resonant frequency method was constructed and shown to be efficient for mechanical fatigue and energy-dissipation investigations of self-supported thinfilm materials. Cu/Nb multilayers with 40 nm individual layer thickness exhibit over an order of magnitude greater fatigue strength compared to bulk Cu. No slip band intrusions and extrusions were observed near the fracture surface. The ratio of the fatigue endurance limit to ultimate tensile strength was around 0.35, consistent with similar empirical scaling observed in other materials. Loss tangent of the nanoscale films was about 4 x 10-3. However, potential parasitic damping from the cantilever supports requires further study.

Acknowledgements This research was supported, in part, by the Department of Energy, Office of Science, Office of Basic Energy Sciences. YCW acknowledges support from the LANL Director’s postdoctoral fellowship. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

330 High Performance Structures and Materials III

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Clemens, B.M., Kung, H., & Barnett, S.A. (1999). Structure and strength of multilayers. MRS Bulletin 24(2), 20-26. Misra, A., Zhang, X., Hammon, D., & Hoagland, R.G. (2005b). Work hardening in rolled nanolayered metallic composites. Acta Materialia 53, 221-226. Frost, N.E., Marsh, K.I., & Pook, L.P. (1974). Metal Fatigue. Clarendon Press, Oxford, UK. Suresh, S. (1998). Fatigue of Materials. Cambridge University Press, Cambridge, UK. Bathias, C., & Paris, P.C. (2005). Gigacycle fatigue in mechanical practice. Marcel Dekker, New York. Wang, Y. C., Hoechbauer, T., Swadener, J. G., Misra, A., Hoagland, R. G., Nastasi, M., “Mechanical fatigue measurement via a vibrating cantilever beam for self-supported thin solid films”, submitted (2005). Love, A.E.H. (1944). A Treatise on the Mathematical Theory of Elasticity. Dover Publications, New York. Hirth, J. P. and Lothe, J. (1982). Theory of Dislocations, Wiley, New York. Wang, Y. C., Misra, A. and Hoagland, R. G., “Fatigue properties of nanoscale Cu/Nb multilayers”, submitted (2005). Lakes, R.S. (2004). Viscoelastic measurement techniques. Review of Scientific Instruments 75(4), 797-810.

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Advances in computational modeling through the use of higher-level microstructure characterization M. Groeber1, M. Uchic2, D. Dimiduk2, Y. Bhandari3 & S. Ghosh3 1

Department of Materials Engineering, The Ohio State University, USA Air Force Research Laboratory, MLLMD, Wright-Patterson AFB, USA 3 Department of Mechanical Engineering, The Ohio State University, USA 2

Abstract In this paper, orientation maps of consecutive serial sections are collected in an automated manner by a Focused Ion Beam–Scanning Electron Microscope (FIB-SEM) outfitted with an EBSD system. Micro-Imager, a program developed in this work, uses the 2D EBSD maps to define microstructural features such as grains and grain boundaries. Parameters used to characterize microstructure are also calculated by Micro-Imager for every section. The statistical measurements of each section are compared to assess variability in the microstructure. The 2D sections are reconstructed into a volume by Micro-Imager3D, another program developed in this work. Statistics analogous to those measured in 2D are calculated and compared to the expected distributions predicted by the 2D measurements coupled with stereology. As a result, quantitative descriptions of microstructure are made and improvements over conventional methods are yielded. Information about individual constituents allows correlations between distributions to be derived. The correlations drawn allow models to account for aspects of microstructure that have classically been overlooked. The resulting 3D grain structure serves as a realistic model microstructure. Keywords: quantitative characterization, equivalent representation, modelling.

1

Introduction

The ability to characterize microstructure is an important tool for materials scientists and computational modelers, because it allows one to predict the capability of a material for a given application. For example, it is well known WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06032

332 High Performance Structures and Materials III that the grain size of a material has a strong effect on mechanical properties; therefore an accurate measure of the grain size distribution is desirable to predict material performance. Classic methods for characterizing microstructure usually involve viewing an image from a sectioned surface, where the area of interest would be mechanically polished [1]. Stereology or other methods can be used to interpolate three-dimensional (3D) statistics from the 2D microstructural images. However, there are some microstructural parameters that cannot be inferred from 2D sections [2]. In addition, many stereological parameters yield only average values to describe microstructural features. Recognizing the fact that many properties (especially those associated with failure) require extreme values of the microstructure [3], it is evident that characterizing the full distribution of these features may be more appropriate for some predictive models [4,5]. This need to more completely characterize microstructure has led to more direct methods, such as serial sectioning, that allow one to obtain true 3D microstructural data [6,7]. For this study, a Focused Ion Beam (FIB) is used to serial-section a specimen and an EBSD system is used to obtain an orientation map of each section. This method shows great potential because the dual beam FIB-SEM microscope can be automated to perform this analysis without user interaction. Post-processing of the orientation data is performed using a program developed in this study called Micro-Imager. Micro-Imager automatically defines grains and grain boundary segments using the 2D EBSD maps, and calculates statistics of the microstructure based on this information. MicroImager characterizes the microstructure more completely; because it calculates full distributions of parameters in both the 2D sections and the 3D reconstructed volume. This study focuses on the task of developing refined data collection and also improving statistical analysis techniques to increase predictive capabilities. This methodology may be particularly useful for generating a host of statistical microstructural correlations that enable the construction of truly representative material microstructures as input for modeling and simulation programs.

2

Brief overview of the FIB-EBSD serial sectioning process

The serial sectioning experiment in the Dual Beam FIB-SEM is comprised of moving the sample repeatedly between two microscope stage positions– the “sectioning” position and the “ion imaging/EBSD analysis” position. Image recognition is critical for precise alignment at the two stage positions, as the microscope stage is only accurate to within few microns when moving between the two stage positions. The image recognition of the DB235 ensures a consistent slice thickness, which is important factor for reconstruction of the serialsectioning data into a 3D volume and is one of the distinct advantages of this automated process. The key advantage of acquiring the orientation information is it allows unsupervised segmentation of grains, which is difficult using only secondary electron or even ion images. A more detailed explanation of the serial sectioning process can be found in [8].

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3

333

3D orientation data reconstruction and grain segmentation and measurement by Micro-Imager

3.1 Grain segmentation with Micro-Imager Micro-Imager is a newly developed software program that automatically locates and approximates the complex grain boundary positions using a series of line segments and an error-per-unit-length algorithm. This process corrects artifacts that are produced when collecting EBSD data as well as simplifies the structure for subsequent mesh generation for computational modeling. Grain boundary detection consists of comparing orientations of neighboring data points. The midpoint between two data points is tagged as a boundary point if there is a misorientation of more than 4 degrees. This process is completed for all data points, and then special boundary points are located. Special points are the points that three or more grains share. The initial approximation of the grain boundaries is simply the connection of these special points. Micro-Imager allows the user to adjust the value of the acceptable error-perunit-length. As the user lowers the acceptable value, the approximated boundaries will have an error-per-unit-length that is intolerable. The specialpoint connection will be changed into two equal-length segments that share an endpoint on a defined boundary point and have the same bounding special points as the initial segment’s endpoints. A schematic of the evolving grain boundary approximation is shown in Fig. 1. Low tolerances closer represent the actual data, but do not simplify the structure. Higher tolerances simplify the structure, but it is important not to oversimplify and make erroneous microstructures. A more detailed description of the grain segmentation process is discussed in [8].

Figure 1:

Schematic showing the evolution of the grain boundary approximation in Micro-Imager. Here t(x) represents the true grain boundary shape and the line segment(s) labeled a(x) represent the approximated boundary.

3.2 Measurement of microstructural parameters by Micro-Imager Once Micro-Imager has created a representative microstructure, it has the ability to extract statistical information about the grain morphology. In the initial phase WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

334 High Performance Structures and Materials III of this work, five parameters were selected for measurement. These parameters are characterized in 2D, and corresponding 3D measurements have also been calculated for some of these parameters. The five parameters measured in 2D are: misorientation, number of neighboring grains, number of grain boundary edges, grain boundary perimeter, and grain area. Grain volume and number of neighboring grains were also measured in 3D.

4

Statistical analysis

4.1 Micro-Imager analysis versus stereology In this work, measurements of two microstructural parameters by Micro-Imager are compared to the same parameters measured by conventional stereology to validate the accuracy of Micro-Imager calculations. The two parameters are grain area and grain boundary surface-area-per-volume.

(A)

(B) Figure 2:

Comparison of parameter measurements from Micro-Imager and Fovea Pro, where thickness means the sectioning depth. (A) Average grain area and (B) Average grain boundary-surface-areaper-volume.

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The measurement of grain area in Micro-Imager is straightforward. The area of each grain is simply the number of data points originally assigned to that grain multiplied by the area associated with each data point. The calculation also includes the error calculated during grain boundary approximation discussed previously. The stereological measurement of grain area was performed using Adobe Photoshop with the image analysis toolkit Fovea Pro. The grain area was determined using the lineal mean intercept measurement in Fovea Pro. Figure 2a shows a comparison of average grain area determined by MicroImager and Fovea Pro for section. To calculate the grain area using the lineal mean intercept method, grains were considered to be spherical, i.e. the grain area was assumed to be that of a circle with a diameter equal to the lineal mean intercept. There are two curves for the grain area as calculated by Fovea Pro; the higher curve includes a multiplicative factor that accounts for the error in the lineal intercept method for circular grains in a 2D image [10]. Figure 2b shows a comparison of surface area-per-volume measurements. Micro-Imager calculates this value by adding the length of all the grain boundary segments for a grain and dividing by the total area of the grain. A stereological factor is used to translate the value of boundary-length-per-area to boundarysurface-area-per-grain-volume [9]. The calculation using Fovea Pro counts the total number of intersections of a grid of cycloids with the grain boundaries. The number of intersections is divided by the length of the cycloids. This value is multiplied by a stereological factor to convert the intersections-per-length to boundary-surface-area-per-volume [9]. The radius of the cycloids will affect the results, because as the radius approaches zero the intersections approach exactly the grain boundary lines and the cycloids fill the full area, yielding the same value as the length-per-area determined by Micro-Imager. It can be seen in Fig. 2b that measurements in Fovea Pro converge to the measurement by MicroImager. This is one advantage of Micro-Imager; that some microstructural parameters can be calculated in a manner that is not sensitive to the measurement method, in contrast to traditional stereological techniques.

(A) Figure 3:

(B)

Probability Density Functions generated by Micro-Imager from a single 2D section. (A) grain diameter (B) misorientation angle. The value of grain diameter is calculated as that of a circle with the same area as the grain.

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336 High Performance Structures and Materials III 4.2 Probability density functions In addition to calculating the mean value of the parameters, the full distributions have been plotted and fitted to determine probability density functions (PDFs), which provide a much more complete characterization. Figure 3 shows the PDFs of grain size and misorientation angle. Probability density functions are useful in creating statistically equivalent microstructures, because they clearly describe the likelihood of a value of a parameter to occur. 4.3 Parameter variation between slices Figure 4 is a 3D surface plot constructed from the series of individual probability density functions. This plot does not substitute for a PDF from a 3D reconstruction, but it can easily display the variability of a parameter through the thickness of the serial sectioning data set. That is, if the PDFs vary markedly, then one 2D section may not be sufficient to describe a particular microstructural feature.

Figure 4:

PDFs of grain area shown collectively as a surface plot in 3D to highlight variation from section-to-section.

Two additional applications of this methodology to examine parameter variation are shown in Figs. 5 and 6. In Fig. 5, a histogram of the grain area is plotted, which was constructed by taking the average value of the bin across all the 2D sections. The error bars denote the maximum and minimum values for each bin across the entire set. The figure illustrates that there can be significant deviations in the distribution of grain size across the 2D images—even for sizes which are at or near the mean. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 5:

Histogram of grain area that is constructed from the averaged bin values for each size range for all of the sections. The error bars show the maximum and minimum values for each size throughout the entire data set.

Figure 6:

Plot showing the values for average grain area and maximum grain area for each slice.

Figure 6 tracks the average grain area and the maximum grain area on every slice. One can observe that the average grain area appears essentially constant for every slice, but there is noticeable variation in the maximum. The data shown in this section further illustrates the potential need for more detailed microstructural analysis than simply the average of a parameter. Often, the average value for a particular parameter appears constant from slice-to-slice, but can vary markedly in the extreme values. If only the average value of a feature WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

338 High Performance Structures and Materials III is needed for property prediction, then a single section of sufficient size may be adequate. However, if knowledge of the extreme values of a micro-constituent is needed, then the information provided by only one or two sections from a material cannot confidently describe the rest of the microstructure. Furthermore, there are some parameters that can only be calculated from a true 3D microstructural environment [2].

Figure 7:

Plot of correlation between grain volume and number of neighboring grains in 3D. Plot shows that small can grains cluster, but large ones tend not to.

Figure 8:

Plot showing the change in the distribution of neighboring grains as grain size increases.

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4.4 Correlations between microstructural parameters In addition to representing distributions of individual microstructural parameters, it may be important for many properties to consider the correlation between multiple parameter distributions. A grouping of similarly oriented grains or a clustering of extremely large or small grains can greatly affect local responses. Figure 7 displays the correlation between grain size and the number of neighbors of the grain. It can be seen in Fig. 7 that small grains can have many neighbors, implying clustering; however, larger grains do not appear to cluster. Accounting for relationships between parameters yields a more accurate description of microstructure. Figure 8 shows distributions of the number of neighbors for various grain sizes. It is clear that as the grain size increases the distributions shift to larger numbers of neighbors as well as broaden. Parameter correlations can yield microstructure models that may produce more realistic results for locally driven properties.

Figure 9:

Plot comparing the unbiased distribution of grain volumes from the reconstructed microstructure and the average extrapolated distribution of grain volumes from all the 2D sections.

4.5 Comparison of 2D and 3D measurements The two parameters correlated in Fig. 7, grain volume and number of neighbors, are computed in an unbiased manner from the reconstructed volume. These unbiased measurements can be compared to extrapolated 2D measurements. Comparisons of the two measurements will allow for the evaluation of the extrapolation technique. Figure 9 shows a plot of the unbiased distribution of grain volumes from the reconstructed microstructure along with the average extrapolated distribution of grain volumes from all the 2D sections. Figure 9 WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

340 High Performance Structures and Materials III shows that the extrapolated distribution differs noticeably from the actual 3D distribution. The extrapolation technique used in this comparison, which assumed the grains to be shaped as Kelvin polyhedra, over-predicts the number of very small grains and the size of the largest grains. As a result, the average grain volume for the two distributions is nearly the same, but the extremes of the distributions are quite different. Parameter measurements on the reconstructed volume provide unbiased descriptions of the grain morphology. Comparisons such as that in Fig. 9 will show where classic stereological relations breakdown. In addition, there are anomalies in the reconstructed volume that maybe deviate significantly from classic assumptions of microstructure. For example, the average number of neighbor grains in the reconstructed microstructure may agree with theoretical calculations, but theoretical calculations greatly under-predict the maximum number of neighbor grains.

5 Conclusion Material characterization is a key in the prediction of material function. The ability to understand the microstructure is the first step in being able to optimize parameters that inevitably lead to superior materials. This study has introduced both experimental and computational procedures that enable new techniques to the forefront of microstructural analysis. This paper presents a snapshot of the representation and analysis methodology as it exists currently. The focus of this work is to continue to improve the representation of microstructures to create realistic computational models for prediction of material capabilities.

References [1] [2] [3] [4] [5] [6]

[7]

Samuels LE. Metallographic Polishing by Mechanical Methods. Metals Park, OH: American Society for Metals; 1982. DeHoff R T. Quantitative serial sectioning analysis: preview. J. Microscopy 1983; 131: 259-263. Ghosh S and Moorthy S. Particle Fracture Simulation in Non-Uniform Microstructures of Metal-Matrix Composites. Acta Materialia 1998; 46: 965-982. Kurzydlowski K J, Ralph B, Bucki J J, and Garbacz A. The grain boundary character distribution effect on the flow stress of polycrystals. Materials Science and Engineering A 1995; 205: 127-132. Kurzydlowski K J. On the dependence of the flow stress on the grain size distribution in polycrystals. Scripta Metallurgica 1990; 24: 879-884. Li M, Ghosh S, Richmond O, Weiland H, and Rouns T N. Three dimensional characterization and modeling of particle reinforced MMCs, Part I: Quantitative description of microstructural morphology. Material Science and Engineering A 1999; A265: 153-173. Li M, Ghosh S, Richmond O, Weiland H, and Rouns T N. Three dimensional characterization and modeling of particle reinforced MMCs, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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[8]

[9] [10]

341

Part II: Damage characterization. Material Science and Engineering A 1999; A266: 221-240. Groeber M, Haley B K, Uchic M D, Dimiduk D M, and Ghosh S. Towards 3D Reconstruction and Characterization of Polycrystalline Microstructures Using a FIB-SEM System. Submitted to Materials Characterization 2005. Russ J C and Dehoff R T. Practical Stereology. New York: Kluwer Academic/Plenum; 2000. Thompson A W. Calculation of True Volume Grain Diameter. Metallography 5 1972; 366-369.

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Deformation of aluminum alloys AD-1, AMg-6 and D-16 at dynamic compression and temperatures of 25–250oC V. A. Pushkov, S. A. Novikov, V. A. Sinitsyn, I. N. Govorunov & O. N. Ignatova Russian Federal Nuclear Center – VNIIEF, Russia

Abstract Using the Kol’sky method, the authors studied dynamic diagrams of uniaxial compression of aluminum alloys AD-1, AMg-6, and D-16. These alloys are applied in aviation, space technologies, shipbuilding and many other industries. Alloys similar to the above mentioned alloys have been studied abroad as well (for example, aluminum alloys 1100-0, 5182, 6061-T6); however in general data on dynamic deformation of the mentioned aluminum alloys have been obtained only at normal temperature. In this work, we performed systematized research of the aluminum alloys at strain rates of 200–1400s-1 and temperatures of 25– 250oɋ. Keywords: Kol’sky method, compression, temperature, strain rates, aluminum alloys, temperature and velocity dependencies V-0.2, model of strength.

1

Introduction

Study of dynamic mechanical properties of structural materials is an urgent problem for many areas of science and engineering, and VNIIEF experts are actively involved into efforts for solving this problem. The Kol’sky method [1] is one of the reliable methods for study of dynamic mechanical properties of materials at strain rates of 102–104s-1. In its classical form, it is intended to study dynamic diagrams of uniaxial compression V-H. The method falls into the category of tests with the parameter ȑ=dH/dt=const, i.e. with constant strain rate. The method essence is quasistatic loading of a sample placed between two steel bars by passing and reflected waves of stresses. The bars are in elastic state, while the sample undergoes elastic-plastic deformation. Various techniques of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06033

344 High Performance Structures and Materials III dynamic tests are presently based on the Kol’sky method principles. These techniques are used in tests with extension, torsion, torsion simultaneously with compression or extension, crack-resistance tests, tests for study of the Baushinger’s effect [2, 3].

2

Research results

Using the Kol’sky method, the authors studied dynamic diagrams of uniaxial compression of aluminum alloys AD-1, AMg-6, and D-16. These alloys are applied in aviation, space technologies, shipbuilding, and many other industries. Available data on these alloys are mostly data obtained at static velocities of loading [4]. There are some data obtained under dynamic loading, but only for normal temperature [5−8]. Alloys similar to the above mentioned alloys have been studied abroad as well (for example, aluminum alloys 1100-0, 5182, 6061T6), strain diagrams have been obtained for them as well, but they are also for normal temperature [9−11]. In this work, we performed systematized researches of the aluminum alloys at strain rates of 200–1400s-1 and temperatures of 25−250oС. Samples in as-delivered condition having sizes ∅8×8mm were studied. In each experiment, diagrams of dynamic compression were obtained, and yield strengths σ-0.2 were determined. The diagrams were plotted in the “stress intensity – strain intensity” coordinates. Recalculation was performed using the dependencies taking account for change of the effective Poisson’s ratio in the area of elastic-plastic transition, eqns (1) and (2):

σi=σ/[1-µ′⋅ln(1+ε)]2

(1)

εi=2⋅(1+µ′)⋅ln(1+ε)/3

(2)

In eqns (1) and (2) µ′=0.5-0.5⋅σ⋅(1-2⋅µ)/(Е⋅ε). Fig 1 shows averaged diagrams σiεi of the alloys. To make the figure simpler, the diagrams are given for each material at temperatures of 25 and 250oС (the diagrams at 150oС are located in the middle). Experiments revealed that strain hardenings of alloys D-16 and AMg-6 were approximately the same, and it was higher than that of alloy AD-1. D-16 is the hardest, and AMg-6 and AD-1 are located in the decreasing order (see Fig 1). The yield strengths temperature dependencies σ-0.2=f(T) and velocity dependencies σ-0.2=f(έ) are shown in Fig 2 and Fig 3. The same as in Fig 1, the dependencies in Fig 3 are given at temperatures of 25 and 250oС. These dependencies have generally the linear character. It should be noted that the velocity dependence σ-0.2 does not evidently result from curves 7, 8 (for AMg-6) and curves 11, 12 (for AD-1) from Fig 1. However, in Fig 3, where results of each experiment are pointed, one can see, nevertheless, the weak dependence of σ-0.2 on έ for the mentioned materials. It follows from Fig 2 and Fig 3 that the strongest change of σ-0.2 occurs in D-16 with growth of Т and έ, and the weakest change occurs in AD-1. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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σi, MPa

345

1 5 2

400

4

3 7

6 8

9 200 10

11 12

0 0

0.1

0.2

εi

Figure 1: Averaged diagrams of dynamic compression of aluminum alloys. D-16: 1–Т=25oС, έ =800–1100s-1; 2-Т=25oС, έ =310-500s-1; 3-Т=250oС, έ =900–1050s-1; 4-Т=250oС, έ =280–550s-1; AMg-6: 5– Т=25oС, έ=1100–1300s-1; 6-Т=25oС, έ=250–550s-1; 7-Т=250oС, έ=1200–1400s-1; 8-Т=250oС, έ =190–530s-1; AD-1: 9–Т=25oС, έ =800–1200s-1; 10-Т=25oС, έ =200–370s-1; 11-Т=250oС, έ =920– 1100s-1; 12-Т=250oС, έ =330–600s-1. So, for alloys D-16, AMg-6, AD-1 in the considered temperature-velocity conditions of loading, we observe in this or that extent the drop of σ-0.2 with temperature growth, and the increase of σ-0.2 with strain rate growth. It is characteristic of many other materials and alloys [11]. The obtained data are in agreement with data of the other authors. For instance, it is mentioned for D-16 in [7] that the yield strength of the alloy is σ+0.2=290–300 MPa at Т=20oС and έ ∼500s-1 (extension). In [11], for alloy 6061-Т6 (that is an analog of D-16), there is σ-0.2=300–310 MPa at Т=20oС and έ =900s-1 (compression) from the diagrams σ-ε. It is noted in [5] for AMg-6 that at Т=20oС and έ =270s-1 and 2000s-1, values of the yield strengths are 205 MPa and 230 MPa, respectively. It is characteristic that these data confirm the velocity dependence of σ-0.2 for AMg-6 in [5]. In [7, 8], the authors mention that σ+0.2=170–175 MPa at Т=20oС and έ ∼500s-1 (extension). At the same time, it is noted in [7, 8] that the yield strength σ+0.2 is WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

346 High Performance Structures and Materials III actually unchanged, when έ is growing from static values to dynamic values (up to 103s-1). This difference from our data can be caused by different histories of AMg-6 deformation. Time of loading growth in sample in our experiments was ∼ 70µs, and it was ∼5µs in [7, 8]. However, this problem requires more comprehensive study.

σ-0.2, MPa

300

1

200

2 3 4

100 0

5 6 0

100

Т, 0С

200

Figure 2: Temperature dependencies of σ-0.2 of aluminum alloys. D-16: 1– έ=800–1100s-1; 2–έ =280–550s-1; AMg-6: 3–έ =1020-1400s-1; 4– έ=190–620s-1; AD-1: 5–έ =800–1200s-1; 6–έ =200–600s-1. σ0,2, MPa 1 300

2 3

200

4

100 0

5 6 0

400

800

1200

έ, s-1

Figure 3: Velocity dependencies of σ-0.2 of aluminum alloys. D-16: 1–Т=25oС; 2–Т=250oС; AMg-6: 3–Т=25oС; 4–Т=250oС; AD-1: 5–Т=25oС; 6– Т=250oС. No similar data have been revealed for AD-1 in Russian literature. In foreign literature, Lindholm and Yeakley [10], for alloy 1100-0 (that is an analog of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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AD-1) tested by the Kol’sky method, increase of strength properties is noted with strain rate growth.

3

Modeling

To create a more reliable model of behavior of aluminum alloys, additional tests were performed with dynamic extension and normal temperature. Basing on experimental results, we developed the phenomenological model of strength [13] for description of our obtained dynamic diagrams of compression and extension of aluminum alloys AMg-6, AD-1, D16 at strain rates of 102–103 s-1 and temperatures of 25–250оС (298–523 K). The model content is the following. If medium is elastic-plastic, stress intensity σi (yield strength at uniaxial p compression and extension) can be presented as function of three variables εi ,

εip , Т characterizing its stress-strain state, eqn (3): σ i = σ i (ε ip , εip , T )

(3)

p p Here εi - intensity of plastic strains, εi - intensity of rate of plastic strains, Т -

current temperature. Stress intensity σi can be presented as product of simple functions, where each function depends only one variable:

σ i = A ⋅ f 1 ⋅ (ε iP )⋅ f 2 (εiP )⋅ f 3 (T )

(4)

In eqn (4), f1 (εip ) describes strain hardening, f 2 (εip ) – influence of rate of plastic strain, and f 3 (Т ) – thermal softening. In the expanded form, the constitutive equation of dynamic deformation is the following eqn (5): m   εip    σi = A⋅ 1+ a⋅ (ε ) ⋅ 1+b⋅ ln p  ⋅ 1−T k   εi0   

(

p n i

)

(

)

(5)

Here А, а, b, m, n, k are constant values, which can be determined using experimental data; εiop =1⋅s-1– normalizing value of εip , Т

k

=

Тr , Тm– melting Тm

temperature; Tr – current temperature. For aluminum alloys, steel and copper tested by the SHPB method, Table 1 presents values of constant coefficients and exponents selected basing on available experimental data. The area of application of the suggested model is limited by the area of experimentally obtained characteristics. It is equal to:

εip ≤0.20 ; εip ≤1,4⋅103 1/s ; Т≤Тm

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348 High Performance Structures and Materials III Table 1: Material AMg-6 AD-1 D-16

Values of constant coefficients and exponents, eqn (5).

process compression extension compression extension compression extension

А, MPa 200 130 43.5 30 240 220

a 2.45 2.40 2.8 2.8 2.28 2.28

n 0.4 0.4 0.335 0.330 0.28 0.28

b 5⋅10-4 5⋅10-4 2⋅10-4 2⋅10-4 1.2⋅10-4 1.2⋅10-4

m 2.55 2.50 3.6 3.6 3.3 3.3

k 2 2 2 2 2 2

Table 2: Experimental and calculated values of conventional yield strength at compression of aluminum alloys AD-1, AMg-6 and D-16 at various strain rates and initial temperatures of samples. Material

Т,oK 298

AMg-6

423 523 298

AD-1

423 523 298

D-16

423 523

έ ,s-1

σ-0.2 aver, GPa

250–550

0.187±0.12

1100–1300 360–620 1020–1300 190–530 1200–1400 200–370

0.210±0.08 0.169±0.09 0.190±0.14 0.159±0.12 0.175 0.055±0.04

0.201 0.155 0.169 0.170 0.177 0.058

800–1200 230–290 900–1200 330–600 920–1100

0.067±0.03 0.049±0.03 0.053 ±0.03 0.045±0.02 0.049±0.02

0.065 0.051 0.056 0.043 0.049

310–500

0.310±0.09

0.320

800–1100 280–520 800–1100 280–550 900–1050

0.338±0.1 0.263±0.14 0.288±0.045 0.245±0.09 0.265±0.11

0.340 0.280 0.300 0.245 0.263

test

σ-0.2 aver, GPa calculation 0.189

Tables 2 and 3 present values of conventional yield strength at compression (σ-0.2) and extension (σ+0.2), which are experimental and calculated by (5) for aluminum alloys AD-1, AMg-6 and D-16 at various strain rates and initial temperature of samples. And for example, Fig 4 shows σ-ε diagrams of AMg-6 compression and extension, which are experimental and calculated by eqn (5). One can see in Table 2, Table 3, and Fig 4 that the suggested elastic-plastic model is in satisfactory agreement with the experimental data within test error. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 3:

4

349

Experimental and calculated values of conventional yield strength at extension of aluminum alloys AD-1, AMg-6 and D-16 at various strain rates and normal temperature of samples.

Material

Т,oK

AMg-6

298

AD-1

298

D-16

298

έ, s-1

σ+0.2 aver, GPa test

σ+0.2 aver, GPa calculation

640–800 1200–1450 210–370 910–1100 230–370 650–1130

0.141±0.05 0.156±0.08 0.040±0.08 0.043±0.08 0.285±0.05 0.303±0.014

0.147 0.152 0.040 0.045 0.288 0.300

Conclusion

Studies of dynamic diagrams of uniaxial compression of alloys AMg-6, AD-1, D-16 were performed at strain rates έ =200–1400 s-1 and temperatures Т=25– 250oС. Temperature and velocity dependencies of yield strengths σ-0.2 were obtained. Among the tested alloys, D-16 has the highest strength. The strongest change of σ-0.2 with growth of Т and έ is revealed for D-16, and the weakest – for AD-1. Additional tests were performed with dynamic extension and normal temperature.

Figure 4: Experimental (different form points) and calculated (solid and dotted lines) σ-ε diagram of compression and extension of aluminum alloy AMg-6 at strain rate of 103 s-1. 1–Т0=298oК (compression), 2– Т0=423oК (compression), 3–Т0=523oК (compression), 4–Т0=298oК (extension). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

350 High Performance Structures and Materials III Basing on experimental results, we developed the phenomenological model of strength for description of our obtained dynamic diagrams of compression and extension of aluminum alloys. The presented results can be helpful for conduction of various strength calculations, as well as for prediction of material behavior under dynamic loading [13].

References [1] [2] [3] [4] [5] [6]

[7] [8] [9]

[10] [11] [12] [13]

Kol’sky H., An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading. Proceedings of the Physical Society, Section B, 62, pp. 676-700, 1949. Muzychenko V.P., Kashenko S.I. and Guskov V.A., On application of split Hopkinson bar method. Plant laboratory (Rus.) 1, pp. 58-66, 1986. Novikov S.A., Pushkov V.A., Sinitsyn V.A. and Tsoi P.A., Study of the Baushinger’s effect at dynamic loading. Appl. Mech. and Tech. Phys. (Rus.), 4, pp. 163-169, 1995. Shalin R.E. Aviation materials. Reference book, v.4, part 1, ONTI-VIAM, Moscow, pp. 180-185, 1982. Bol’shakov A.P., Novikov S.A. and Sinitsyn V.A., Study of the tension and compression uniaxial dynamic diagrams of copper and alloy AMg6. Problems of strength (Rus.), 10, pp. 87-88, 1979. Glouschenkov V., Novobratsky R. and Bourmistrov A., Influence of the spread in values of aluminum alloy dynamic properties upon the final results of magnetic-pulse strain, Proc. of the DYMAT-91, Physics Eds., Association DYMAT: Strasbourg, pp. C3/331-C3/334, 1991. Stepanov G.V., Astanin V.V. and Romanenko V.I., Study of properties of the aluminum alloys at dynamic loading. Problems of strength (Rus.), 2, pp. 59-63, 1983. Popov N.N., Ivanov A.G., Strekin V.P. and Barinov V.M., Study of AMg6 dynamical behaviour. Problems of strength (Rus.), 12, pp. 50-54, 1981. Higashi K., Mukai T. and Kaizu K., The microstructural evolution during deformation under several strain rates in a commercial 5182 aluminum alloy, Proc. of the DYMAT-91, Physics Eds., Association DYMAT: Strasbourg, pp. C3/347-C3/352, 1991. Lindholm U.S., Yeakley L.M., High strain rate testing: tension and compression. Experimental Mechanics, 8(1), pp. 1-9, 1968. Mayden S., Green S., Compressive strain-rate tests on six selected materials at strain rate from 10-3 to104 in/in/sec. Applied Mechanics, Ser. Е, v. 33(3), pp. 20-30, 1966. Novikov S.A., Pushkov V.A., Dynamic crack-resistance of В4С-based materials, Proc. of the Conf. on Urgent Problems of Protection and Safety, Publ. House of SIC “Spetsmaterials”: St. Petersburg, pp. 4-8, 1999. Glushak B.L., Ignatova O.N., On modeling of dynamic deformation diagrams. Questions of Atomic Science and Technique (Rus.), 2, pp. 45-49, 1998.

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351

X-ray residual stress measurements on plasma sprayed molybdenum coatings K. Hirukawa1, K. Akita2, S. Tobe3, T. A. Stolarski4 & S. Ohya2 1

Research Division in Engineering, Graduate School of Musashi Institute of Technology, Japan 2 Department of Mechanical Systems Engineering, Musashi Institute of Technology, Japan 3 Department of Mechanical Engineering, Ashikaga Institute of Technology, Japan 4 School of Engineering and Design, Brunel University, UK

Abstract Plasma sprayed molybdenum coatings have been applied to friction parts in the automotive field because of their high wear resistance. Clarification concerning the significance of mechanical properties and the residual stress state of the coating is important to improve the performance of the coating. However there are some difficulties in the measurement of mechanical properties of the coating, especially Young’s modulus, because it has complex structures including lamella, pores, micro cracks and so on. Young’s modulus is required to determine the x-ray stress constant for x-ray residual stress measurements. Strain gauges are often used to measure the strains of a component. If glue is applied to thermal spray coatings to attach a strain gauge, it might be that it penetrates into the pores and becomes solidified, hence incorrect strains will be measured. In this research, firstly, Young’s modulus and x-ray stress constant, K, for molybdenum thermal spray coating were determined experimentally by four-point bending tests. The effects of quick-drying glue on strain measurement using a strain gauge were investigated. Secondly, the residual stresses of the coatings were measured using x-ray diffraction. Six types of substrates with a different thermal expansion coefficient (TEC) were prepared. The materials used, in ascending order of TEC, were Super Invar, molybdenum, SCM440, SK5, SUS304 and A6063. Molybdenum powder was sprayed on the substrates in air with various thicknesses. The following results were obtained. (1) The effect of quick-drying glue on the mechanical strain measurement on the sprayed coating was negligible. (2) The Young’s modulus of the coating was lower than that of the commercial molybdenum sheet. (3) Linear strain response against applied mechanical loads was observed in the case of the polished coating surface. (4) The x-ray stress constant K of the coating and the bulk were almost the same. (5) Residual stresses on the coating surfaces were of tensile type on all substrates used in this study. (6) Tensile residual stresses of the coatings increased with the decreased thermal expansion coefficient of substrate. Keywords: x-ray stress measurement, residual stress, plasma spraying, molybdenum coating, thermal expansion coefficient, quick-drying glue, strain response. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06034

352 High Performance Structures and Materials III

1

Introduction

Thermal sprayed molybdenum coating is one of the surface modification applications improving the wear resistance. Mo coating also has high heat resistance so that Mo coatings are applied to the engines and sliding parts of automobiles [1]. The effect of controlled residual stress on wear resistance has been reported [2]. Evaluation of the residual stress is essential to improve the enabled functions. In this study, firstly, the effects of quick-drying glue on mechanical Young’s modulus measurement using strain gauge were considered. Secondly, the effects of thermal expansion coefficient (TEC) of substrates on residual stress were investigated.

2

Experiment procedure

2.1 Specimen preparation Materials used as substrates, in ascending order of TEC in air, were Super Invar, molybdenum, SCM440, SK5, SUS304 and A6063. Molybdenum powder was used as a spraying material. TEC of all materials used are shown in table 1. A condition “substrate TEC < coating TEC” can be provided when Super Invar is used as substrate in the case of molybdenum coating despite it has low TEC. Two types of substrate were used for mechanical Young’s modulus measurement and residual stress measurement. The former was 100 × 15 × 3mm, the later was 15 × 15 × 3mm. Table 1:

Thermal expansion coefficient of materials at the room temperature . α , ×10-6.

Super Molybdenum SCM440 SK5 SUS304 A6063 Invar ASTM AISI Molybdenum AISI 4140 SK50H AISI 304 ISO AlMg0.5Si F1684 JIS

0.5

4.9

11.8

11.8

17.3

24.3

2.2 Spraying condition For Young’s modulus measurement, molybdenum was sprayed with the thickness of 300µm by the Atmospheric Plasma Spraying system. The spraying condition is shown in table 2. There were two surface conditions. One is assprayed surface and another is polished with waterproof abrasive paper. For residual stress measurement, six types of substrate were sprayed with various thicknesses. Spraying was applied with the different number of passes.

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High Performance Structures and Materials III

Table 2:

Plasma spraying conditions.

Spraying material

Mo powder

Method

APS

Current

800 A

Voltage

26 V

Spraying distance Maximum thickness

3

353

100 mm 300 µm

Experimental procedure

3.1 Measurements of mechanical Young’s modulus E Loads were applied to coating-substrate composite beam by four-point bending. Tensile or compressive stress was generated at coating surface. The range of applying loads was determined in the elastic region (within -1600 × 10-6 strain) determined by Acoustic Emission test [3]. In this paper, the elastic region of the coatings was defined below AE increasing point. In measuring mechanical strain using the strain gauge, it is said that quickdrying glue penetrates into the pores and solidifies thus making strain measurement ambiguous. In our study, strain gauges were attached using quickdrying glue; Young’s modulus of the glue after solidified was 3.2 GPa. With reference to Young’s modulus, the coating was measured with various conditions of coating surfaces and substrate thicknesses. All the strain gauges were held for 24 hours after bonding. Both strains of substrate and coating were measured at the same time. Young’s modulus of coating, EC, was calculated by the following equation (1)

ε  tC AS ES −  C +1 EC Z S εS  EC = ε C   +1 Z S − tC AC ε S 

(1)

from the theory of composite beam. Where t is the thickness, A is the crosssection area and Z is the geometrical moment of area, respectively. Two suffixes C and S mean coating and substrate. Coating surfaces are postulated as uniaxial stress. Furthermore the coating was broken by bending just after cooling with liquid nitrogen. The fractured surface was observed by a Scanning Electron Microscope to clarify the penetration depth of quick-drying glue. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

354 High Performance Structures and Materials III 3.2 Measurements of x-ray stress constant K Tensile and compressive stress was applied to coating surfaces within the elastic region described previously. Loads were applied gradually and slope of 2θ-sin2ψ diagram [4] M was determined in each case under the condition shown in table 3. 2θ-sin2ψ diagrams were obtained by averaged diffraction angle in three times continuous measurements. X-ray stress constant K was calculated with M-σapp diagram. The equations for K and M are as follows.

K= Table 3:

σ app M

,

M=

∂ (2θψ ) ∂ (sin2ψ)

(2)

Conditions of x-ray diffraction.

Characteristic x-ray X-ray wavelength Diffraction plane Diffraction angle 2θ Kβ filter Peak determination Stress determination sin2ψ range, step Detecter Tube voltage, current Peak count Diameter of incident x-ray

V-Kα 0.250483 nm Mo211 154.267 deg Ti Center of FWHM 2θ -sin2ψ method 0.0 - 0.6 , 0.1step PSPC 30 kV, 8 mA 2048 φ 3 mm

3.3 Residual stress measurements by x-ray diffraction A flattened particle had a thickness of about 4µm and x-ray penetration depth was about 0.9µm. This means that the lattice strain of extreme upper part of the coating surface could be measured in this study. Residual stresses were measured on the principle of the Ω-goniometer method

4

Results and discussion

4.1 Penetration depth of quick-drying glue, and Young’s modulus Fig.1 shows the cross-sectional secondary electron images of fractured surface of the free-standing coating. The chemical composition of quick-drying glue provides white image when exposed to the electron beam irradiation. The right image indicates that the glue penetration depth was about 40µm. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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355

Young’s modulus of the coating was measured using three specimens with different conditions. Tensile and compressive stresses were applied to as-sprayed or polished surface, and thicknesses of substrates were 3mm and 5mm. As a result measured EC was about 235 GPa on an average. EC seems to be lower than that of bulk molybdenum (323 GPa [5]) because of the porosity of the coating. Then Young’s modulus of quick-drying glue was 3.2 GPa. The penetration depth of quick-drying glue was about 10% of the coating thickness, and the Young’s modulus of the glue was about 1% of the coating. It means that the effect of quick-drying glue on mechanical strain measurement on the sprayed coating was negligible. Coating surface

Coating surface

50µm

As-sprayed

Increment of slope of 2T-sin2\ diagram ¨M , deg

Figure 1:

50µm

Glue applied

Cross-sectional secondary electron images of fractured surface of the free-standing coating.

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -400 -300 -200 -100 -0.1 0 -0.2 Bulk : Bulk molybdenum -0.3 Sub3 : Substrate t=3mm -0.4 Sub5 : Substrate t=5mm Tens. : Tension -0.5 Comp. : Compression -0.6 -0.7

Bulk Tens.

-607 MPa/deg

Bulk Comp.

-547 MPa/deg

Sub3 Tens.

-536 MPa/deg

Sub3 Comp.

-630 MPa/deg

Sub5 Tens.

-523 MPa/deg

Sub5 Comp.

-530 MPa/deg

100

200

300

400

Mechanical applied stress Vapp , MPa Figure 2:

The relationship between slope of 2θ-sin2ψ diagram M and mechanical applied stress σapp.

4.2 X-ray stress constant K In the case of as-sprayed surface slope M was not changed regularly. This is the same result with a previous study [6] so that details will be skipped. Fig.2 shows WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

356 High Performance Structures and Materials III the relationship between M and σapp on polished coating specimens. These specimens had different thicknesses of substrates as t=3mm (Sub3) and t=5mm (Sub5). Bulk molybdenum (Bulk, t=3mm) was also tested. Increment of slope M was evaluated differently depending on applied stress, tension or compression. Applied stresses were calculated based on Young’s modulus of coating, EC. Calculated K values are written in explanatory note in Fig.2. Considering that the results of bulk molybdenum are criterion, K of thermal sprayed molybdenum coating were almost the same with that, and independent to experimental conditions. This means that the strain response was improved because of polishing the coating surface. To conclude the averaged K value -563 MPa/deg would be used for residual stress determination.

Surface residual stress Vr , MPa

200

9 passes 4 passes 2 passes 1 pass 0.5 pass

150

B 100

50

A

0 0

5

10

15

20

25

Thermal expansion coefficient of substrate D, ×10-6 Figure 3:

The relationship between TEC of substrates α and surface residual stress σr.

4.3 The relationship between TEC and residual stress Fig.3 shows the relationships between TEC of substrates and surface residual stress of coatings. This result includes the changes of scanning passes of the spraying gun. Residual stresses of all specimens were tension. Here, the melting point of molybdenum is about 2893 K [7]. Therefore the temperature difference was about 2273 K when the molten particles impacted the pre-heated substrate. As a result the temperature difference caused significant shrinkage when the coating began to cool. Therefore the tensile residual stress was generated. Except for two plots signed A and B in fig.3, residual stresses tend to increase with decreasing the TEC of substrate. It suggested that the shrinkage of substrate contributed to that of lamella in cooling process so-called the bimetal. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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357

Conclusion

(1) The effect of quick-drying glue on mechanical strain measurement on the sprayed coating was negligible. (2) The Young’s modulus of the coating was lower than that of the bulk molybdenum. (3) Linear strain response against applied mechanical loads was observed in the case of polished coating surface. (4) X-ray stress constant K of the coating and the bulk were almost the same. (5) Residual stresses on the coating surfaces were of tensile type on all substrates used in this study. (6) Tensile residual stresses of the coatings increased with decreasing the thermal expansion coefficient of substrate.

References [1] [2] [3] [4] [5] [6] [7]

Sulzer Metco, Automotive Solutions Kit, p.8, 2002. XIE C X and OUYANG M-L, Wear, 137, No.2, p.159, 1990. K. Akita, S. Tobe, Journal of the Society of Materials Science, Japan, p.741, 2004. Standard method of X-ray stress measurement for steel, The Society of Materials Science Japan, p.71, 2002. Technical Data “Modulus of Elasticity of Metals”, The Japan Society of Mechanical Engineers, p.209, 2001. S. Takahashi, I. Uchibayashi, H. Misawa and H. Suzuki, 34th Symposium on X-ray Studies on Mechanical Behavior of Materials, The Society of Materials Science Japan, p.61, 1998. E. Yajima, R. Ichikawa, K. Furusawa, T. Miyazaki, T. Kozakai, Y. Nishino, Mechanical and Metallic Materials for Young Engineers, Maruzen CO., LTD, p.343, 2002.

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359

Compressive residual stress generation process by laser peening without pre-coating H. Tanaka1, K. Akita2, Y. Sano3 & S. Ohya2 1

Graduate Student, Musashi Institute of Technology, Tokyo, Japan Department of Mechanical Systems Engineering, Musashi Institute of Technology, Tokyo, Japan 3 Power and Industrial Systems Research and Development Center, Toshiba Corporation, Yokohama, Japan 2

Abstract Laser peening without pre-coating has been applied to water-immersed specimens of high tensile strength steel. In order to understand the generation process of compressive residual stress, specimens with various laser irradiation patterns, i.e., single spot, line scanning and area scanning, were prepared. Detailed distributions of residual stress on the specimens were measured using synchrotron radiation. Large tensile residual stresses, which might be caused by the thermal effect of the laser pulse, were observed in the center region on the single spot. It decreased towards the edge of the spot, and changed to compression around the spot. The compression became larger with the increasing pulse numbers irradiated on the same spot. In the line scanning, a tensile residual stress was observed in the final spot of the line, which decreased and changed to compression as the distance from the final spot increased. The residual stress on the area scanning was compression as a whole. It was estimated that the compressive residual stress in the area scanning would be generated from the overlapping effect of the compressive field made around each laser spot. The residual stress generated by laser peening without pre-coating is considered to be the superimposition of the tensile and the compressive components due to thermal effect and plastic deformation, respectively. Keywords: laser peening, surface residual stress, single pulse, line irradiation, laser pulse density, x-ray stress measurement, synchrotron diffraction.

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360 High Performance Structures and Materials III

1

Introduction

Laser peening is one of surface treatment technologies to improve fatigue strength and resistance to stress corrosion cracking (SCC) [1, 2]. Laser peening utilizes a mechanical interaction between the surface of material and plasma caused by the irradiation of a nanoseconds-order laser pulse. Laser peening introduces compressive residual stress in surface layer (more than 1 mm in depth) and, therefore, has been applied to reactor core shrouds of nuclear power plants to prevent SCC [3]. Our laser peening method is able to generate the compressive residual stress on the material surface without any ablative coating [4], and the laser power is relatively low. Therefore the treatment process is simple and the laser pulses can be led easily through optical fibre. However, the residual stress generation mechanism is not simple because the thin surface layer of material is exposed to the high temperature plasma and it can cause a tensile residual stress on the component. Mukai et al. reported that the surface residual stress changes to the compression side when the number of laser pulses irradiated per unit area was increased. Recently, microscopic distributions of residual stress on a single laser spot were measured using synchrotron radiation source [5, 6, 7]. As a result, the tensile residual stress was observed on the surface of the center of the single laser spot. The phenomenon that the residual stress changes from tension at the first pulse to compression in the area irradiation is considered to be a key factor to clarify the residual stress generation mechanism. In this study, laser pulses were irradiated on the same spot to investigate the effect of overlapping of laser pulses. A line scanning specimen was also prepared to investigate the effect of offset over lapping of laser pulses. Microscopic distributions of residual stress were measured on the surface of these specimens to discuss the residual stress generation mechanism.

2

Fundamental mechanism of residual stress generation by laser peening

Fig. 1 shows the fundamental process of laser peening. Laser beam focused by a lens is irradiated on a specimen surface in water. The material surface layer becomes plasma if the power density of the laser pulse at the material surface exceeds the threshold of the ablation of the material. The inertia of water prevents the expansion of the plasma, and the energy of the plasma concentrates in a narrow space. The resulting plasma pressure becomes 10～100 times higher than that in air and it reaches 1 to 10 GPa [8]. The shock wave is generated by this pressure, and it propagates in the material. Plastic deformation occurs near the surface of the material due to the dynamic stress of the shock wave, and the material is hardened through the process. The compressive residual stress is generated by the elastic restraint from the un-deformed region. The generation of the compressive residual stress has been explained basically by this mechanism, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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361

since the depth profile of the residual stress by laser peening is well reproduced by elastic-plastic dynamic simulations. Laser pulse Lens Plasma

Water

Compression

Test sample During laser irradiation

Figure 1:

3

After laser irradiation

Fundamental process of laser peening.

Experimental procedure

3.1 Material and specimens Material used in this study is high tensile strength steel, HT1000. Table 1 and Table 2 show the chemical compositions and the mechanical properties, respectively. Fig.2 shows the micrograph of the HT1000 surface. The grain size is around 10 µm or less, so it is suitable for the specimen of microscopic X-ray stress measurements. The dimensions of the specimens are 20 mm × 80 mm with the thickness of 15 mm and 28 mm × 30 mm with the thickness of 17 mm. To minimize the effect of residual stress by machining, the electrolytic polishing was applied on the surface of the specimens prior to laser irradiation. Table 1:

C

Si

Mn

0.13

0.25

0.92

Table 2:

Chemical compositions of HT1000.

P

S

0.011 0.001

Mo

Nb

V

Cr

0.39

0.02

0.04

0.87

Mechanical properties of HT1000.

Yield strength [MPa] Tensile strength [MPa] 965 1055

Elongation [%] 23

3.2 Laser peening conditions The schematic of experimental setup for laser peening is shown in Fig.3. The fundamental wave of a Q-switched Nd:YAG laser (λ = 1.06 µm) was frequencydoubled to a water-penetrable wave (λ = 532 nm) by a second harmonic generator with a nonlinear optical crystal. The laser beam was focused by a lens WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

362 High Performance Structures and Materials III and irradiated on a specimen in a water jacket through an optical window. The diameter of the laser spot was about 1 mm. The specimen was fixed on a specimen holder and driven to x- and y-directions in the water jacket during laser irradiation.

30 µm

Figure 2:

Micrograph of HT1000 specimen surface. Mirror Frequency-doubled Nd:YAG laser X-Y table Test sample Lens

Mirror Window Water jacket

Figure 3:

1 pulse 4 pulses 10 pulses 40 pulses

15 20

Experimental setup of laser peening.

6

27

18

18

80

Figure 4:

Scheme of pulse laser irradiation on the same spot on HT1000 specimen.

Fig.4 illustrates the single spot specimen where laser pulses irradiated at the same point 1, 4, 10, and 40 times. The line and area scanning were performed as WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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363

shown in Fig.5 (a) and (b). The lines with pulse densities of 1, 5, 10 and 100 /mm were prepared in the line scanning, changing the scanning speed. In the area scanning, an area of 10 × 10 mm2 is irradiated by scanning laser pulses twodimensionally, as shown in Fig.5 (b). The coverage was 800% which was defined in the same way in shot peening. The laser irradiation conditions are summarized in Table 3. 17

17

(D)100 /mm (C)10 /mm

28

(B) 5 /mm

28

Laser irradiated area (10mm×10mm) Y X

(A) 1 /mm

Figure 5:

30

30

(a) Front surface.

(b) Back surface.

Laser pulses irradiation scheme (Numerals in (a) mean the number of laser pulses per unit length. Arrows mean the laser scanning direction). Table 3:

Laser peening conditions on HT1000 specimens.

Material Pulse energy Spot diameter Pulse duration Coverage Pulses irradiated per unit length Single pulse

HT1000 215 mJ 1.0 mm 8 ns 800% ( Fig,5 (b) ) 1 /mm (A) 5 /mm (B) 10 /mm (C) 100 /mm (D) 1, 4, 10 and 40 pulses

3.3 Residual stress measurement The microscopic residual stress distributions near single laser pulse spot, laser irradiated line and laser irradiated area were measured on the HT1000 specimens using the synchrotron radiation source at the beam line 3A of Photon Factory, KEK, Tsukuba, Japan. Synchrotron radiation is suitable for the microscopic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

364 High Performance Structures and Materials III residual stress measurement required in this research because the intensity of the radiation is much higher than that of the typical laboratory X-ray. The wavelength used was 0.228 nm. α-Fe211 diffraction was measured. The X-ray irradiated area was φ 0.2 mm or 0.5 mm in diameter. Residual stress was derived by the sin2ψ method. X-ray stress constant, K, is assumed to be the mean value of the calculated ones using Reuss and Voigt models and the elastic compliance of a single crystal of α-Fe. The X-ray stress measurement conditions are summarized in Table 4. Table 4:

Conditions of X-ray stress measurement.

X-ray source Material Diffraction plane Wavelength Diffraction angle Detector Irradiated area X-ray stress constant, K

4

Synchrotron radiation KEK, PF BL3A HT1000 α- Fe211 0.228 [nm] 154 [deg] PSPC φ 0.2 [mm] or φ 0.5 [mm] -353 [MPa/deg]

Experimental results

4.1 Surface residual stress distributions on the overlapped laser spot Fig. 6 shows the schematic illustration of the measuring positions of surface residual stress in and around the overlapped laser spot. The X-ray irradiated area was 0.2 mm in diameter.

Y X

Laser spot φ 1.0mm : Stress measured positions,φ 0.2 mm

Figure 6:

Schematic illustration of residual stress measurement positions in and around a single pulse laser irradiated spot (see Fig. 7).

Fig. 7 shows the surface residual stress distributions along X-axis of Fig. 6. Large tensile residual stresses of about 600-800 MPa were observed in all the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

365

laser spots with 1 to 40 pulses. The tensile residual stresses decreased toward the edge of laser spot, and changed to compression in the outside of the spot. The compression was about -200 MPa in the single pulse spot. When the number of laser pulses increased, the compression increased at 0.3mm outside from the edge of laser spot. The maximum compressive residual stress reached to -600 MPa for 40 pulsed spot.

Residual stress, σX [MPa]

Laser spot 1000 800 600 400 200 0 -200 -400 -600 -800

1 pulse 4 pulses 10 pulses 40 pulses

0

Figure 7:

0.5

1

Distance from spot center, X [mm]

1.5

Surface residual stress distribution in and around the laser spots irradiated 1, 4, 10 and 40 pulses at the same position. Laser scanning direction 1 pulse/mm (A)

Final spot

5 pulses/mm (B) 10 pulses/mm (C)

100 pulses/mm (D)

X : Stress measured positions,φ 0.5 mm

Figure 8:

Schematic illustration of residual stress measurement positions on the center line of laser irradiated line.

4.2 Surface residual stress distributions in laser irradiated line and area The interaction of adjoined laser spots was examined using the line-scanned specimen. Fig.8 shows the positions of residual stress measurement on the laser WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

366 High Performance Structures and Materials III irradiated line. Fig.9 shows the surface residual stress distributions along the lines. The horizontal axis of the figure shows the distance from the center of the final laser spot center of each line. The X-ray irradiated area was 0.5 mm in diameter. The tensile residual stresses were observed near the final spot for the cases A, B and C. The tensions decreased with increasing the distance from the final spot. The tensile residual stress was decreased with increasing the number of laser pulses per unit length, and it changed to the compression side in case of D with the 100 pulses/mm irradiation density. On the area-scanned specimen with the coverage of 800%, the surface residual stress was compression with about -150 MPa. Final spot Residual stress, σX [MPa]

800

1 pulse/mm (A) 5 pulses/mm (B) 10 pulses/mm (C) 100 pulses/mm (D)

600 400 200 0 -200 -400 0

0.5

1

1.5

2

2.5

Distance from edge of irradiated line, X [mm] Figure 9:

5

Surface residual stress distributions along the center of laser irradiated line.

Discussion

The residual stress at the center of the single spot was tensile even if laser pulses were irradiated 40 times at the same position (Fig. 7). Meanwhile when the laser pulses were scanned in a line as shown in Fig. 9, tensile residual stresses were rapidly decreased with increasing distance from the final spot. These facts mean the overlapping of laser pulses at the same position dose not generate compression at the center of laser spot, on the other hand the off-center overlapping of laser pulses decreases the tension as shown for the lines A, B and C in Fig. 9. The reason of this phenomenon is considered that the compressive residual stress region around laser spot overlaps with tension region at the spot center of the adjoining spot by off-center overlapping. When the irradiation density was increased to 100 pulses/mm (Line D), the residual stresses changed to compression. However, although the irradiation density per unit length of the area irradiation was 3.2 pulses/mm, the residual stress was compression. These facts mean that an off-center overlapping of laser scanning lines is also important to generate higher compressive residual stresses. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

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Our laser peening process does not use any pre-coating. Therefore, the material surface is exposed to high temperature plasma, and the resulting surface residual stress depends mainly on the balance of the compressive stress component generated by cold working due to high-pressure laser plasma and the tensile stress component generated by rapid cooling due to thermal effect. Thermally affected depth is only several micrometers [9] and the effect seems to be reset pulse by pulse. However, the plastic deformation caused by cold working can be accumulated [10]. Therefore, the tensile component might be constant and the compressive component increases when the laser pulse density increases. Thus, the whole stress level is changed to the compression side, when the number of laser pulses increases, as shown in Fig. 10. Residual stress

σth

Tension 1

Laser pulse density

σr = σth + σsw

Compression

σsw σr: Surface residual stress on laser peened area σth: Tensile stress component generated by thermal effect σsw: Compressive stress component generated by compressive residual stress around laser spot

Figure 10:

6

accumulation

of

Schematic of the residual stress generation mechanism on laser peening without pre-coating.

Conclusion

Overlapped single laser pulse, line irradiation and area irradiation were performed on high tensile strength steel, HT1000 and the residual stress distributions near the irradiated spot, line and area were measured using a synchrotron radiation source. The results obtained through the experiments are summarized as follows: (1) Residual stress was tension in the single spot even if 40 laser pulses were irradiated at the same position. Compressive residual stress was observed in the outside of the spot. (2) Off-center overlapping of laser pulses generates the compressive residual stress on the surface of HT1000 because the compressive stress region at the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

368 High Performance Structures and Materials III outside of laser spot would change the tensile residual stress at the spot center to compression.

Acknowledgement This work has been performed under the approval of the Photon Factory Program Advisory Committee of the High-Energy Accelerator Research Organization, Japan (Proposal No. 2003G032).

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10]

Y. Wakabayashi, K. Masaki, Y. Ochi, T. Matsumura, Y. Sano, T. Kubo, Japan Society Of Mechanical Engineers M&M2003 pp.283-284 (2003). P. Peyre, C. Braham, J. Lédion, L. Berthe and R. Fabbro, Journal of Materials Engineering and Performance, 9, pp.656-662 (2000). Y. Sano, et al, Proc. of the 7th Int. Symp., JWS, pp.439-444 (2001). Y. Sano, M. Kimura, M. Obata, N. Mukai, A. Sudo and S. Shima, 6th International Conference on Nuclear Engineering (ICONE-6236), 1998. Y. Yoshioka, Proc. 38th Symposium on X-ray Studies on Mechanical Behaviour of Materials, 83 (2002). K. Akita, Y. Sano, T. Kubo, Y. Yoshioka and H. Suzuki, Int. Conf. On Advanced Technology in Experimental Mechanics 2003 (ATEM’03), (2003). K. Akita, H. Tanaka, Y. Sano and S. Ohya, Material Science Forum, 490491, pp.370-375 (2005). Y. Sano, N. Mukai, K. Okazaki and M. Obata, Nuclear Instruments and Methods in Physics Research B pp.432-436 (1997). Y. Sano, M. Kimura, K. Sato, M. Obata, A. Sudo, Y. Hamamoto, S. Shima, Y. Ichikawa, H. Yamazaki, M. Naruse, S. Hida, T. Watanabe, Y. Oono, 8th International Conference on Nuclear Engineering (ICONE8441), 2000. Y. Sano, N. Mukai, M. Yoda, K. Ogawa and N. Suezono, Materials Science Research International, Special Technical Publication-2, pp.453458 (2001).

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369

The scale effect of roughness in contact problems S. Mezghani1, A. Jourani1 & H. Zahouani2,3 1

Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS, 5513, France 2 Ecole Nationale d’Ingénieurs de St Etienne, France 3 Ecole Centrale de Lyon, France

Abstract In this paper we have used the Hölder exponent to characterise the scale of roughness and to study the scale effect of high spatial frequencies on elastic contact between solids. The mathematical approach shows that the Hölder exponent of roughness is a sophisticated tool for modelling realistic surface roughness at different scales of observation. The incidence of Hölder exponent on the prediction of pressure and displacement is studied in elastic contact between a smooth plane and rough surface. Keywords: Holder exponent, scale effect, elastic contact.

1

Introduction

Surface topography plays an important role in a multitude of physical and tribological phenomena such as contact mechanics, friction, adhesion, wear, wettability, lubrication, etc. Surface topography causes discrete contact points, when two rough nominally flat surfaces are brought together, the real area of contact is the accumulation of the area of the individual contact points. For most metals at normal loads, this will be only a small percentage of the apparent contact area. Typical models of surface deformation are either elastic, plastic or mixed elastic-plastic, and can be represented schematically as a function of surface topography and material constants ƒ(σh, σs, Rc , E, H) with σh the root mean square of height, σs the root mean square of summits, Rc the mean radius of summit curvature, E and H are respectively the Young’s modulus and the hardness of the solid which are the intrinsic parameters of the solid. On the other WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06036

370 High Performance Structures and Materials III hand the principal statistical parameters of roughness, such as variance of height, slope and curvature are unfortunately not always independent of length scales. The definition of slope or summit of surface roughness systematically involves a high wave vector cut off due to the measurement process itself and not to the surface characteristics. Consequently, instruments with different resolutions and scan lengths yield different values of these statistical parameters for the same surface. Therefore it is very important to characterize rough surfaces by intrinsic parameters which are independent of the sampling length or area. Hölder exponent which indicates the scale of perturbation of surface topography has been widely used in recent years [1-5], this is due in part to the observations that fractal geometry can reflect an intrinsic property of random phenomena and can be applied to characterize surface topography and contact mechanics. Majundar and Bhushan [5,7] explored these applications using the theory developed by Mandelbrot [6] concerning the cumulative distribution of −

D

islands on the earth’s surface relief, which follows the power law N ≈ a 2 , where N is the total number of islands whose area larger than a, and D is the fractal dimension of its coast line ( 0 < D < 1 ). In this work we have used the Hölder exponent to characterize the scale of roughness and to model the effect of roughness scale in contact mechanics. In the second part we have studied, through experiments, the mechanism of asperity deformation using two numerical approaches based on geomorphologic characterization and fractal geometry using the concept developed by Mandelbrot.

2

Elastic model of rough contact

The majority of the methods used to determine the relationship between normal load and displacements, except finite elements methods, are based on the assumption of elastic solid, homogeneous and semi - infinite The basic assumptions necessary to use the continuous equations of elasticity: The slopes of roughness are weak; The contact area is small in front of other dimensions. With zero loads, surfaces of the two bodies in the immediate neighbourhoods of the point of contact can be comparable with paraboloids of equations: z1 = A1x2 + A2xy + A3y2

,

z2 = B1x2 + B2xy + B3y2

(1)

Each surface is characterized at the point of contact by its principal radii of curvature R1 and R'1, R2 and R'2 is the angle which forms between them. Constant A and B depend on the size of the principal curves of surfaces and

 1 1 1 1  . + + +  R1 R '1 R2 R '2 

the angle γ with A + B = 

By choosing an orientation of axes X and Y, X1 and Y1, the terms in xy become negligible. The expressions of the equation (1) become: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

z1 =

371

 1 1 2 1 2 1 2 y  x + y , z2 = − ' x22 + ' 1 '' 1 '' 2  R R 2 R1 2 R1 2 2 2   2

(2)

The geometry of the contact between two unspecified solids is thus brought back to the problem of the contact between two paraboloids:

h( x, y ) = Ax 2 + By 2 =

1 2 1 2 x + y 2 R1 2 R2

(3)

The equations of linear elasticity make it possible to connect displacement wi(x, y) of a M(x point, y) of the surface of a semi elastic body infinite, with the pressure p(x', y') in Me (x', y') pertaining to the surface of contact, by means of the following integral equation of Boussinesq:

1 − νi 2 wi ( x , y ) = πEequ

∫∫Ac

p( x ', y ')dx ' dy '

( x − x ')

2

+ ( y − y ')

(4)

2

with Ac the surface of contact

S1

Z1

δ S2

Figure 1:

w2(x,y)

w(x,y) Z2

Contact region Région du Région ducontact contact

w1(x,y)

General diagram of the areas of contact between two solids.

So if one calls w(x, y), the difference between the M1(x point, y, z1) of the surface of the solid 1 (S1) and the M2(x point, y, z2) of the surface of solid 2 (S2), the geometry enables us to write the following relation:

w( x, y ) = w1 ( x, y ) + w2 ( x, y ) = δ −

x2 y2 − 2 R1 2 R2

(5)

where w1(x, y) and w2(x, y) are relative displacements on the two solids, R1et R2 the principal radii of curvature of the two solids. The determination of displacement in a point of surface due to the field of pressure is carried out by the superposition of the influence of all the efforts. Displacement at the point ij can then be written in the form: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

372 High Performance Structures and Materials III

wij = Cijkl pkl

(6)

with C ijkl the matrix of the coefficients of influence.

P(x,y)

P(x,y)>0 P(x,y)=0

Figure 2:

Pressures field at the contacts points between the two bodies.

The problem is thus to find pressures, such as pij > 0 , such as

C ijkl p kl + h( x, y ) − δ = 0

(7)

If one consider the pressure as an input signal and displacement like a response of exit by the following relation to a dimension (Johnson 1985): w( x) = h( x) ⊗ p( x) (8) This equation is well defined in the space field in the form of a product of convolution: If the pressure applied is form p( x) = B0 sin( 2πx / λ ) (9) The obtained displacement has the following expression (Johnson 1985)

w( x) =

(1 − ν 2 )λ B0 sin( 2πx / λ ) + cst πE

(10)

what gives like transfer function of the system in the field of the space frequencies:

H (ω ) =

2(1 − ν 2 ) Eω

(11)

and in the field of the space frequencies one has like response

W (ω ) = H (ω ) P(ω ) =

2(1 − ν 2 ) P(ω ) ω E

(12)

In addition, by using the properties of the Fourier Transform one can write that the derivative of displacement can be written in Fourier domain as

W (ω ) = iωW (ω ) by replacing in the equation of the pressure one can write: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(13)

High Performance Structures and Materials III

P(ω ) =

F (ω ) =

373

E W (ω ) i 2(1 − ν 2 )

(14)

E W (ω ) i 2(1 − ν 2 )

(15)

These last relations are important characteristics of contact problems. They imply that the contact pressure and slope of displacement as well as the load and displacement are two pairs of parameters which have similar frequency composition. The passage in the spectral field makes it possible to solve this last equation, while subjecting the functions a two-dimensional transform of Fourier. What makes it possible to write a relation between the spectrum of the pressures and the spectrum of displacements as [1, 2 ,3] : W (ν x , ν y ) =

2 2(1 − ν ) P (ν x , ν y )

(16)

ν

E

2 2 where ν = ν x + ν y represent a space frequency. The determination of the field of pressure p(x, y) is determined by opposite transform of Fourier for each level of bringing together δ :

-1  pδ ( x, y ) = TF 

E

2  2(1 − ν )



ν W (ν x ,ν y )



(17)

The spectrum of pressure is thus spread out between the low and high frequencies:

vx =

1 1 1 1 , vy = mm-1 , ν hx = mm-1 ,ν hy = Lx Ly 2∆x 2∆y

with Lx, Ly , the size of the surface, ∆x , ∆y the sampling steps in the x and y directions, which initially define rough surface before contact. For normal macroscopic imposed load F, one numerically brings closer two surfaces and to solve the equation for each level of bringing together δ. For each normal position δ of the rigid plan, one determines contact pressures p(x, y) by using the direct and inverse Fourier transform of the contact equations. Convergence is reached when the imposed effort is equal to the sum of the local efforts calculated for each iteration: F = ∑ ∑ pij ( x, y ) ∆x∆y i, j WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(18)

374 High Performance Structures and Materials III

3

Hölder exponent of rough surfaces

As the details of roughness: Z(x) depends on the length scale, we assume each realization of Z(x) to be a continuous, but non-differentiable function. It means that the presence of any small roughness elements may prevent us from reaching a satisfactory limit of

(Z ( x + ∆) − Z ( x) ) as ∆ → 0 ∆

(19)

with ∆ the sampling step of roughness measurement A simple way to obtain this behaviour for a function Z(x) is to assume that the increment of Z(x) is related to ∆ by the self-affinity relation:

Z ( x + ∆) − Z ( x) ∝ ∆H , 0 < H < 1 ∆ → 0

(20)

lim ∆ →0 ∆H −1 , only exists if H=1. The derivative of Z, which is proportional to the limit, for 0
(Z ( x + ∆) − Z ( x) )2

∝ ∆2 H 0
(21)

This relation can be generalized to dimensions E > 1 in the following way

(Z ( x + ∆) − Z ( x) )2

∝ ∆

2H

, 0 < H < 1, ∆ → 0

where x represents a point in E-dimensional Euclidean space RE and

(22) stands

for the usual norm in this space. This is not a new condition; it is proven for all

∆ values by the fractional Brownian motions.

4

The spectral exponent

From the previous relation it appears that the functions Z(x+∆)-Z(x) and

( Z ( x + s∆ ) − Z ( x)) sH

(23)

Considered as functions of ∆, have the same law of probability. If we assume that Z(0) = 0, then function Z(x) verifies the statistical self- affinity relation

Z ( sx ) = s H ( Z ( x ) WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(24)

High Performance Structures and Materials III

It means that the two functions Z(x) and Y ( x ) =

375

Z ( sx ) are statistically sH

indistinguishable. Voss (1985) showed that the fractal dimension is related to parameter H by the relation D = E+1-H where E is the dimension of the Euclidean space RE. These functions, which correspond to a non-stationary random process with stationary increments, present many other interesting properties. For instance their power spectrum S(ω) presents a well-defined decrease following the power law

S (ω ) =

1

ω

β

=

1

ω

β = 2H + E

,

2H +E

(25)

The power spectrum of roughness verifies another self-affinity relation

S (kω ) =

1 k

2H +E

S (ω )

(26)

If one makes the analogy with the relation between the spectrum of pressure and the spectrum of the slopes of displacements which vary in the same space frequency band, one can notice that the law of power spectrum of pressure depends on the Hölder exponent as following

P (ω ) =

E W (ω ) i 2(1 − ν 2 )

P (kω ) =

1 k

2H +E

P(ω )

(27) (28)

This relation, shows well the effect of the exhibitor of Hölder on the spectrum of pressure, and consequently on the local singularities of pressures and displacements which will induce local plasticity.

5

The effect of the Hölder exponent of roughness on elastic contact

In order to study the influence of the Hölder exponent of roughness on the parameters of elastic contact: pressure, displacement and real contact area, we propose to generate three rough surfaces parameterised in roughness by their Hölder exponent. Each of those surfaces is the sum of a periodic surface (sinusoidal) and a random rough surface generated by the random displacement model [14-15] with different roughness Hölder exponent H=0.1, H=0.5, H=0.8 (figure 3). Those surfaces obtained (figure 4 {a1, a2, a3}) were put in contact under a pressure of 400 MPa with a rigid plan. Pressure fields are calculated for each of these three rough surfaces as using the model of the elastic contact (figure 4). We notice that the pressure field gotten is as much more discontinuous than the surface is rougher. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

376 High Performance Structures and Materials III

(a) Figure 3:

S1

S2

S3

Figure 4:

(b)

(c)

Three examples of simulation by the algorithm of the midpoint method with random successive additions (a) H=0.1 (b) H=0.5 (c) H=0.8.

(a1)

(b1)

(c1)

(a2)

(b2)

(c2)

(a3)

(b3)

(c3)

Roughness surfaces: (a1) H=0.1, (a2) H=0.5, (a3) H=0.8; Pressure field: (b1) H=0.1, (b2) H=0.5, (b3) H=0.8; deformed surfaces: (c1) H=0.1, (c2) H=0.5, (c3) H=0.8; {Applied pressure 400 MPa}.

The following table (Table 1) regroups the whole of contact parameters below (Bearing surface area and mean contact pressure) of the three different rough surfaces. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 1:

377

Contact parameters for various roughness surfaces.

Surface roughness S1 (H=0.1) S2 (H=0.5) S3 (H=0.8)

Bearing area (%) 11.66 27.78 28.60

Mean Pressure (GPa) 3.40 1.41 1.38

These results (Table 1) show that the roughest surface (S1) which has the lower Holder exponent is the more resistant in contact. Indeed, it presents the lowest bearing area surface (11.66%) and the most important mean pressure (3.40 GPa). This proves the narrow relation between punctual pressure field’s singularity and the Holder exponent indicator of roughness.

6

Conclusion

The incidence of the Hölder exponent of roughness is important in contact mechanics, wettability and lubrication of rough material, where the knowledge of the surface bearing area, the developed area or void volume is directly related to the scale of observation. In contact mechanics there are two incidences. The first is related to the size of the spot area and surface bearing area, the second is connected to the evolution of the void and material volume during contact as a function of observation scale. The mathematical modelling study shows that the Hölder exponent of roughness is a sophisticated tool for modelling realistic surface roughness at different scale of observation. It can be used as the input data in contact mechanic modelling. This scale parameter can be used as an indicator of the real length, area and volume of rough material, independently of the scale of observation. This scale parameter is very promising to simulate the behaviour of rough morphology in static contact.

References [1] [2] [3] [4] [5] [6] [7]

Johnson K.L., N – J Mech. Phys. Solids – 1970. Ju Y., Farris T.N., Spectral analysis of two dimensional contact problems, Transactions of the ASME, Vol 118, pp 320-329, 1996. Liu Sh., Wang Q., Liu G., A versatile method of discrete convolution and FFT for contact analyses. Wear 243 pp. 101-111, 2000. Gagnepin J.J and Roques-Carmes C., Fractal approach to twodimensional and three-dimensional surface roughness, Wear, Vol 109, pp.119-114, 1986. Majunder A. and Bhushan B., Role of fractal geometry in roughness characterisation and contact mechanics of surfaces, ASME J. Tribology, Vol. 112, pp.205-216, 1990. Majunder A. and Tien C.L, Fractal characterisation and simulation of rough surfaces, Wear, Vol 136, pp. pp. 313-327, 1990. Majunder A. and Bhushan B., Fractal model of elastic-plastic contact between rough surfaces, J. Tribology, Vol 13, pp.1-11. 1991. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

378 High Performance Structures and Materials III [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17]

Lopez J., Hansali G., Zahouani H., Lebossé J.C., Mathia T., 3D Fractalbased characterisation for engineered surface topography, Int.J.Mach.Tools Manufact, Vol 35, pp.211-217, 1995. Mandelbrot B., Stochastic models for the Earth’s relief, the shape and the fractal dimension of coastlines, and the number-area rule for islands, Proceedings of the National Academy of Science, Vol 72, pp.3825-3838, 1975. Richardson L.F., The problem of contiguity: An appendix of statistics of deadly quarrels, General Systems Yearbook 6, pp.687-689, 1961. Mandelbrot B., How long is the coast of Britain - Statistical self-similarity and fractional dimension, Science, Vol 155, pp.636-638, 1967. Brown S.R. and Scholz C.H., Broad bandwidth study of the topography of natural rock surfaces, J. Geophys. Res., 90, pp. 12575-12582. , 1985. Felder J., Fractals, Plenum Press, New York, 1988. Mandelbrot B. and Van Ness J.W, Fractional Brownian motions, fractional noises, and applications, SIAM Review 10, 422-437, 1968. Russ John C., Fractal surfaces, Plenum Press, New York, 1944. Saupe D. and Peitgen H.O., The science of fractal images. SpringerVerlag pp.82-91, 1988. Zahouani H., Vargiolu R., Fractal models of surface topography and contact mechanics, Vol28, N° 4-8, pp 517 – 534, 1998.

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Section 5 High performance concretes

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381

Copper slag as fine aggregate for high performance concrete K. S. Al Jabri Department of Civil and Architectural Engineering, Sultan Qaboos University, Sultanate of Oman

Abstract This paper presents results from an experimental investigation carried out to study the potential use of copper slag as fine aggregate on the strength of both normal and high strength concrete. Concrete mixtures were prepared using different proportions of copper slag as partial and full replacement of fine aggregate. The percentage of copper slag added by weight ranged between 10100% of sand used in concrete. For each concrete mixture, six 150mmx150mmx150mm cubes, three 300mmx150mm dia. cylinders and three 100mmx100mmx500mm prisms were cast. Density, compressive, tensile and flexural strengths were determined at 28-day of curing. Cube compressive strength was also determined at 7-day of curing. Results demonstrated that there is general an increase in the density and workability of both normal and high strength concretes as copper slag quantity increases. Also results showed that the compressive strength of concrete is generally improved, compared with the control mix, with the increase of copper slag up to a certain copper slag content beyond which the strength generally reduces. Mixes with large copper slag percentage showed signs of bleeding and segregation due to the significant increase of workability. Keywords: copper slag, fine aggregate, concrete, strength, workability, density.

1

Introduction

Aggregates are considered one of the main constituents of concrete since they occupy more than 70% of the concrete matrix. In many countries there is scarcity of natural aggregates that are suitable for construction whereas in other countries the consumption of aggregates has been increased, in recent years, due WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06037

382 High Performance Structures and Materials III to the increase demand by the construction industry. In order to reduce dependence on natural aggregates as the main source of aggregates in concrete, artificially manufactured aggregates and artificial aggregates from industrial wastes provide an alternative for the construction industry. Therefore utilisation of aggregates from industrial wastes can be alternative to the natural and artificial aggregates. Copper slag is an industrial by-product material produced from the process of manufacturing copper. It has been estimated that approximately 24.6 million ton of slag is generated from world copper industry [1,2]. In the Sultanate of Oman, approximately 60,000 tons of copper slag is produced every year by Oman Mining Company. Although the majority of the produced copper slag is used in the sand blasting industry and in the manufacturing of abrasive tools, the remainder is disposed without any further reuse or reclamation. There are many mechanical and chemical characteristics that qualify copper slag to be used in the concrete as partial replacement of Portland cement or a substitute of aggregates. For example, copper slag has number of favourable mechanical properties to be used as an aggregate such as excellent soundness characteristics, good abrasion resistance and good stability. Also, copper slag exhibits pozzolanic properties since it contains a low content of CaO, and it can possesses cementitious properties as CaO content increases or under activation of NaOH. Use of copper slag in concrete industry as replacement of cement or/and fine aggregates can have the benefit of reducing the costs of disposal and minimising environmental pollution [1,2]. Little research work has been conducted to investigate the potential use of copper slag as fine aggregate on the properties of normal concrete. Akihiko and Takashi [3] investigated the use of slag from copper smelting as a fine aggregate in concrete. From mortar strength tests with a cement/slag/water ratio of 1/2/0.55, the ball milled slag gave a higher strength. The effects of using several types of slag on mortar and concrete reaction, reinforcing steel corrosion, abrasion, workability and slump, shrinkage, and freezing and thawing characteristics were examined. The effect copper slag as replacement of fine aggregates on the strength, setting time and durability of concrete mixtures was also studied by Toshiki et al. [4]. Recently, Demirboğa and Gül [5] used blast furnace slag aggregates to produce high strength concrete. Different water/cement ratios were used to determine the 7- and 28-day compressive strength and other properties. Results showed that compressive strength of blast furnace slag aggregate concretes were approximately 60-80% higher than control concretes containing no slag for different water/cement ratios. It was concluded that blast furnace slag aggregate, in combination with other supplementary cementitious materials, could be utilised in making high strength concretes.

2

Research objectives

The main objective of this study was to investigate the use of copper slag as partial and/or full replacement for fine aggregate in normal and high strength concrete mixtures. The following were specific tasks: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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(1) Evaluate the effect of copper slag addition on the workability and density of normal and high strength concretes (2) Conduct compressive, tensile and flexural strength testing on concrete mixtures

3

Materials

3.1 Cement The cement used in this study was ordinary portland cement (OPC) purchased from Oman Cement Company. This cement is the most widely used one in the construction industry in Oman. 3.2 Coarse and fine aggregates Coarse aggregates (i.e. 20 mm and 10 mm) and fine sand were purchased from a nearby crusher in Al-Khoudh area, which are typically the same materials used in normal concrete mixtures. The gradation test conducted on aggregates showed that they met specifications requirements. 3.3 Copper slag Copper slag is a by-product material produced from the process of manufacturing copper. As the copper settles down in the smelter, it has a higher density, impurities stay in the top layer and then are transported to a water basin with a low temperature for solidification. The end product is a solid, hard material that goes to the crusher for further processing. Copper slag used in this work was brought from Oman Mining Company, which produces an annual average of 60,000 tons. 3.4 Silica fume The silica fume used in the production of high strength concrete was supplied and added to the mix in a powder form (Elkem Emsac 500s). 3.5 Superplasticizer In order to improve the workability of high strength concrete, superplasticizer in the form of a polynaphthalene sulphonate-based admixture (conplast SP430) was used. This had 40% active solids in solution.

4

Laboratory testing program

4.1 Mix design and sample preparation The mix proportions chosen for this study are given in Table 1 for both normal strength concrete (NSC) and high strength concrete (HSC). The constituents were weighed in separate buckets. The materials were mixed in a rotating pan in accordance with ASTM C192-98. The overall mixing time was about 4 minutes. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

384 High Performance Structures and Materials III The mixes were compacted using vibrating table. The slump of the fresh concrete was determined to ensure that it would be within the design value and to study the effect of copper slag replacement on the workability of concrete. The specimens were demoulded after 24 hours, cured in water and then tested at room temperature at the required age. Table 1:

Mix proportions and water-to-cement (w/c) ratios for normal and high strength concretes.

Type Cement NSC HSC

416 400

Mix proportions (kg/m3) Silica Sand 10mm 20mm fume Agg. Agg. 721 338 790 44 710 1190 -

w/c Wat er 207 140

ratio

SP l/m3

0.5 0.35

7.9

To determine the unconfined compressive strength, six cubes (150mmx150mmx150mm) were cast for each mix and water-to-binder ratio, and three samples were tested after 7 and 28 days of curing. Three 150 mm diameter x 300 mm long cylinders were prepared for each mix in order to determine the 28-day tensile strength of concrete. Also, to determine the flexural strength (modulus of rupture) for each mix, three 100mmx100mmx500mm prisms were cast and tested after 28 days of curing. 4.2 Testing procedure After curing, the following tests were carried out on the concrete specimens: • 7- and 28-day cube compressive strength test was conducted in accordance with BS 1881: Part 116 using a loading rate of 2.5 kN/s; • 28-day cylinder tensile (splitting) strength test was done in accordance with ASTM C496-96 using a loading rate of 2 kN/s; and • 28-day flexural strength test was conducted in accordance with ASTM C78-94 using a simple beam with third point loading at a loading rate of 0.2 kN/s. All tests were conducted using a DARTEC compression machine.

5

Test results and discussion

The properties of normal and high strength concretes in terms of density, slump, compressive strength, tensile strength and modulus of rupture are shown in Tables 2 and 3, respectively. The effect of copper slag addition as replacement of fine aggregate on the density of normal and high strength concretes is shown in Fig. 1. It can be seen from Fig. 1 that there is general increase in the density of both normal and high strength concretes with the increase of copper slag replacement in the concrete. The density of concrete was increased by almost 5% for both concrete types. This is mainly due to the higher specific gravity of copper slag which was 3.4 compared to fine aggregate which has a specific gravity of 2.8. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 2:

Properties of normal concrete at 7- and 28-days of curing.

Mix No.

Mix Type

1

Control (100% S) 10%CS + 90S 20%CS + 80%S 30%CS + 70%S 40%CS + 60%S 50%CS + 50%S 70%CS + 30%S 90%CS + 10%S 100%CS + 0%S

2 3 4 5 6 7 8 9

385

Strength (MPa) (Fcu)+ (Ft)+

Density (kg/m3)

Slump (mm)

(Fcu)*

2524

50

33.2

44

3

7.7

2515

90

37.1

44.9

3.5

7.2

2540

90

39.6

48.5

3.7

7.2

2558

75

37

48.9

3.2

6.9

2550

145

36.2

48.1

3.8

6.5

2560

180

39.5

53.1

4.1

7.3

2601

185

37.2

46.6

3.6

6.3

2597

200

37.9

50.1

3.6

7.2

2653

210

38

45

3.4

5.9

(Fcr)+

S= Sand

CS= Copper Slag, *= Cured at 7 days

Table 3:

Properties of high strength concrete at 7- and 28-days of curing.

Mix No.

Mix proportions

1

Control (100% S) 10%CS 90S 20%CS 80%S 40%CS 60%S 50%CS 50%S 60%CS 40%S 20%CS 80%S 100%CS 0%S

2 3 4 5 6 7 8

S= Sand

+ + + + + + +

+= Cured at 28 days.

Density (kg/m3)

Slump (mm)

(Fcu)*

Strength (MPa) (Fcu)+ (Ft)+

(Fcr)+

2568

28

76.9

93.9

5.4

14.6

2530

28

79.6

99.8

5.2

13.6

2588

50

74.5

95.3

6.2

12.4

2586

125

64.8

79.6

4.6

10.8

2625

115

77.8

96.8

6.1

12.9

2658

128

69.0

83.0

4.8

11.1

2673

102

63.8

79.0

4.7

10.3

2700

150

63.4

82.0

4.4

10.1

CS= Copper Slag, *= Cured at 7-days += Cured at 28-days.

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386 High Performance Structures and Materials III Density (kg/m3) 2700

Normal concrete High strength concrete

2650 2600 2550 2500 2450 2400 1

2

3

4

5

6

7

8

9

M ix No.

Figure 1:

Effect of copper slag addition on the density of normal and high strength concrete

Slump (mm) 250

Normal concrete High strength concrete

200 150 100 50 0 1

2

3

4

5

6

7

8

9

M ix No.

Figure 2:

Effect of copper slag addition on the workability of concrete.

The workability of concrete was assessed based on the measured slump of fresh concrete as shown in Fig. 2 for different proportions of copper slag. It is clear from Fig. 2 that the workability of concrete increases significantly with the increase of copper slag in concrete mixes. For the control mix (i.e. mix 1), the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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measured slump was 50mm and 28mm for normal and high strength concretes respectively whereas for mixes 8 and 9, with 100% replacement of copper slag for both NSC and HSC respectively, the measured slump was 210mm and 150mm for both types of concrete. This suggests that copper slag has much less absorption characteristics than sand which explains the considerable increase in workability as copper slag quantity increases. This increase in workability can have beneficial effect on concrete in the sense that concrete mixes with low water-to-cement ratios can be produced which can have good workability, greater strength and improved durability than conventional concrete. However, it should be noted that mixes with high contents of copper slag showed signs of bleeding and segregation of which can have detrimental effects on concrete performance.

Compressive Strength (MPa) 60 50 40 30

7-days

20

28-days

10 0 1

2

3

4

5

6

7

8

9

Mix No. Figure 3:

Average cube compressive strength of normal concrete at 7 and 28days of curing.

The average unconfined 7- and 28-days compressive strengths are shown in Figs. 3 and 4 for NSC and HSC, respectively. Results from both Figures show that the compressive strength of concrete is generally increases as copper slag quantity increases up to 50% addition of copper slag beyond which the compressive strength is slightly decreased due to the significant increase in the workability causing reduction in the strength. High water content in the mix causes the particles of the constituents to separate leaving pores in the hardened concrete which consequently reducing the strength. For normal concrete, Mix 6 (50% sand and 50% copper slag) yielded the highest compressive strength value of 53 MPa whereas the control mix gave a compressive strength value of 44 MPa. Mix 9 (with 100% copper slag replacement) yielded a compressive strength of 45 MPa which is comparable with the control mix (Table 2). For high strength concrete, the highest compressive strength was achieved by Mix 2 with 10% replacement of copper slag. The highest compressive strength was WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

388 High Performance Structures and Materials III 99.8 MPa compared with 93.9 MPa for the control mix. However, the mix with 100% replacement of copper slag (Mix 8) gave the lowest compressive strength value of approximately 82 MPa which is almost 12.5% lower that the control mix (Table 3). Tables 2 and 3 indicate that the tensile strength of concrete and the modulus of rupture showed a similar behaviour to the compressive strength results. Compressive Strength (MPa) 100 90 80 70 60 50 40 30 20 10 0

7-days 28-days

1

2

3

4

5

6

7

8

Mix No.

Figure 4:

6

Average cube compressive strength of high strength concrete at 7 and 28-days of curing.

Conclusions

This paper presented the results of a research study on the effect of using copper slag as a substitute for fine aggregate in normal and high strength concrete mixes. Normal and high strength concrete mixes were prepared with different proportions of copper slag. Concrete mixes were tested for workability, density, compressive strength, tensile strength and flexural strength. Results showed that the increase in the copper slag content causes an increase in both workability and density of normal and high strength concretes. Results also showed that the compressive strength of concrete is generally improved, compared with the control mix, with the increase of copper slag up to a certain percentage beyond which the strength was slightly reduced. Mixes with large copper slag substitution suffered from bleeding and segregation due to the increase in workability. Generally it is concluded that copper slag can be used as fine aggregate to produce concrete with good strength.

References [1]

Shi, C., and Qian, J.; High Performance Cementing Materials from Industrial Slags – A Review. Resources, Conservation and Recycling, 29:195-207, 2000. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

[2] [3] [4] [5]

389

Gorai, P., Jana, R. K., and Premchand. Characteristics and utilisation of copper slag – a review. Resources, Conservation and Recycling; 39:299313; 2003. Akihiko, Y., and Takashi, Y. Study of Utilisation of Copper Slag As Fine Aggregate for Concrete (in Japanese). Ashikaya Kogyo Daigaku Kenkyu Shuroku; 23:79-85, 1996. Toshiki A, Osamu K, Kenji S. Concrete with copper slag as fine aggregate (in Japanese). J. Soc Mater Sci Jpn; 49:1097-102, 2000. Demirboğa, R., and Gül, R., Production of High Strength Concrete by Use of Industrial by-products, Building and Environment, in press, 2006.

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Application of FRC in tunnel reinforcement P. Procházka Czech Technical University in Prague, Czech Republic

Abstract Fibers in concrete should primarily restrain the material from cracks during the curing process and dehydration. For standard evaluation of the influence of the fibers, the three or four point test is studied, both experimentally and numerically on a straight beam. In tunnel lining the situation is slightly different. While the three or four point test leads to the assessment of the structure being only bended, in tunnel reinforcement not only this but also bending with normal force has to be observed. Moreover, the curing is affected by the non-uniform distribution of moistening, as from the side of the surrounding rock the measure of wet is principally higher than on the face where ventilation speeds up the dehydration. These two assumptions are considered and treated experimentally and numerically. Because a real tunnel lining is too large, a scale model is used and, consequently, more samples can be studied. Similarity rules and an appropriate calibration have to be obeyed. This paper is principally focused on the theory of possible coupled modeling of behavior of the one-sided moistened tunnel lining.

1

Introduction

Recently, the author published several papers on behavior of steel FRC with Dramix type fibers in underground structures, especially in tunnel linings, [1]. One-sided moistened concrete beams have been studied from the mechanical point of view. Scale models were prepared and studied. The process, which we deal with in this paper, starts with coupled modeling targeted to description of one-sided moistened beams. Then arches are modeled and the mechanical properties are taken from the straight beam behavior. From the numerical point of view a linear system of equations for unconstrained problem has to be solved and when restrictions are imposed on stresses a nonlinear problem is to be observed. The solution of such a problem can be WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06038

392 High Performance Structures and Materials III formulated in terms of series including eigenparameters, which simulate measure of different moistening at different layers in the structure. The application of the proposed problem is suitable for structures from composite materials, for laminated or layered structures, in tunnel construction, especially in NATM, etc. In order to approximate the real structure of tunnel lining, in our study a laminated hollow cylinder will be investigated from theoretical standpoint. Uniform distribution of eigenparameters is considered in each layer.

2

Experiments with a beam

The behavior of one-sided moistened FRC was studied on straight beam, since curvilinear beam could be modeled in a cumbersome way. The samples were prepared in an enameled basin, the tap water in which was pored, which was permanently added to hold the water level at a proper elevation. The wetted medium surrounding the sample simulated real situation on site. Consequently, the sample was prepared from two kinds of layers: the first layer was built up from high porously liquid-absorbing mass. This mass was separated from the concrete (the sample itself) by a filter paper to ensure convenient removal of the concrete slender element from the equipment for preparing the samples. As a consequence of this structure the lower side of the samples was permanently highly moistened. This was also the aim of the experimental study The samples were built up by auxiliary timber frame; its thickness along three sides had the value of the right layer of the samples. The fourth side was free in the beginning of the layers creation. Then the first layer of the fiber reinforced concrete was constructed and the rectangular frame was completed by gluing the last part (the shape of it corresponded to the height of the layer). This stage of building the experimental equipment is depicted in Fig. 1, where also wire holders for sticking next molding are seen to create the following layers of the concrete. In this figure still no water is pored, and the three remaining moldings inside the form are still not yet put inside the form to create the first concrete layer. In order to create the thicknesses of the layers as accurately as possible, the moldings were put into the form along all sides of the equipment, so that the room for the first and the subsequent layers was exactly defined. In other words, at each stage of layers creation, one planar frame was built up with the same thickness of moldings (as desired). A plateau with the set up of fibers was drawn in Corel Draw on hardened paper of 200 g/m2 In order to hold the cold temperature, which could cause temporary sticking of fibers to the paper, the temperature of the paper with the lay-up of fibers was dropped by putting the system in a freezer. Frozen at 32 degrees of Celsius every paper was three times sprinkled to assure the adhesion of fibers to the paper. The number of fibers was calculated in such a way that the fiber volume ratio was 1 percent. In each layer are 270 fibers in 11 interfaces of concrete layers (6 and 5 fibers are repeated in rows). The thicknesses of the layers are 6 + 4 +...+ 4 + 2 mm.

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The concrete mixture was made of Knauf BP-8, the stones of which larger than 0.8 mm were removed by filtering. For 28 days in room temperature the slender beam was dried in such a way that from the upper side a fan two times a day for 2 minutes was used from a distance of 2 m. From below steady delivery of water was assured. The final stage is seen in Fig. 2. The water was poured in the basin immediately after the first layer of concrete was put in the frame. The sample was not fully in a directly contact with the water, but along the interface of the sample with the plateau the moistening of the lower layer was ensured. The dimensions of the whole sample element were: 150 x 250 x 48 mm3. This sample was cut into beams of a cross section 2.8 x 4.8 mm2, which were loaded by a concentrated load in the middle of span of the beams. The ratio of thickness to span was about 1/3. From that point of view, the mechanical behavior was closer to a plate.

Figure 1:

Experimental equipment with the basin and frame at the commencing stage.

Then a comparative study provided a time-dependent distribution of strains in the beam during the bending test. The one-sided moistened beam during the curing process and standard fiber reinforced beam are compared in both experimental studies, static and time dependent. Because a non-reinforced beam has very poor bearing capacity, it is not considered in these comparative studies. A standard cement mixture has modulus of elasticity E = 20 GPa and Poisson’s ratio v = 0.16. Steel fibers possess modulus of elasticity E = 197 GPa and Poisson’s ratio v = 0.3. Regularly distributed Dramix-type fibers are used. A typical bending test with one concentrated load in the middle of 128 mm span of beams was applied. The beams were loaded on the moistened side to worsen the situation. It was expected that the pore pressure would expand the layers on the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

394 High Performance Structures and Materials III moistened side, while in the dry side the pore pressure slowly disappears. This caused a sort of prestress at the moistened side.

Figure 2:

Final stage of the sample element after construction.

The aim of this study was to discover how the situation would worsen during the bending test comparing these samples with that without loading by the moistening. In Fig. 3, a relation applied force - deflection is depicted. Two cases of the beams are compared: moistened and external moistening free plates are considered. Both types of beams were cured 28 days and then the experiments were accomplished. From Fig.3 one can see a sudden drop at the value of loading about 1.8 kN in the case of moistened beam and about 2.6 kN in the case of the wet-free beam. These drops differ also in deflections, which were in the first case 0.18 mm and 0.22 mm in the second case. Sudden opening of a crack below the concentrated load causes the drops of loading. After increase of loading peaks are reached. While the externally moistened beam attains 3.8 kN, the moistening free beam 4.5 kN. After this, the beams still bear, but their bearing capacity lowers and softening of the aggregate occurs. This is caused by a pulling of fibers out of the concrete matrix. When the destruction of the beams is attained, it can be seen that no steel fiber were disconnected, fibers were only pulled out of the concrete. The different behaviors of both beams are caused by pore pressure, which is stored in cured beams. In mathematical models this property is very easy to describe. Pore pressure is a special case of eigenstrain tensor, µ , or eigenstress tensor, λ , which both are expressed in the generalized Hooke’s law as: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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σ = L(ε − µ ) or σ = Lε + λ , where σ is the stress tensor and ε is the strain tensor. L is the elastic material stiffness matrix. Another type of experiment was carried out on these beams. A time dependent behavior of both types of beams is observed by bending with the same disposition as in the previous experiments, see Fig. 4.

Figure 3:

Figure 4:

Deflection - concentrated load relation.

Time dependent behavior of the beams (creep).

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396 High Performance Structures and Materials III From the picture it is seen that the behavior of both types of beams is not different that much. Only in the neighborhood of the peak values the differences are larger than for the lesser values of the concentrated load. From both examples we have a conclusion that both bearing capacity and time dependent deflections are larger in moistened beams. Nevertheless, the differences are not that much decisive that the steel fiber reinforced concrete has to be removed from the production for the type of structures, such as underground works (tunnels, retaining walls), foundations, floor decking, etc.

3

Numerical description of a cylindrical segment

In this section we concentrate our attention on a special type of one-sided moistened FRC structure, which can be used in design of underground works, foundation engineering, floor decking of industry halls, sport halls, etc. The mechanical behavior of the moistened arches was compared with the fiberreinforced straight beams. The reinforcement of both types of beams was put into structures regularly (uniformly) in a very careful way, as described in the above text.

Figure 5:

Geometry and disposition of the beam under study.

It was shown that the bearing capacity of the moistened structure is less than that of beams created in the standard way. Also the creep seems to be more important and larger in the case of moistened structure.

4

Relations on a single layer

The clamped arch can be solved by Fourier method: the unknowns and the loading are split into series in one direction and in the remaining direction the ordinary differential equation is solved. The problem of clamped arch is divided into solution of simply supported arch and the rotation at clamped edges is annihilated by applying external bending moment. The solution of the latter case can be found in [2]. The first problem is a generalization of the problem from [3]. Results from similar cases can be seen in [4]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Introducing cylindrical coordinate system 0rφz, components of the displacement vector are considered: u = u r , v = uϕ , w = u z , where u r is the displacement in the radial direction and uϕ is the displacement in the circumferential (hoop) direction and w = u z is the displacement in the axial direction. The cylindrically orthotropic structure is a particular example of a cylindrically anisotropic solid with the constitutive equations written as (in this section the indices describing number of the layer are omitted):  σr   σϕ σ  z  σ rϕ   σ ϕz σ  rz

  Lrr     L rϕ  L  =  rz   0     0   0  

L rϕ Lϕϕ Lϕz 0 0 0

Lrz Lϕz L zz 0 0 0

0 0 0

G rϕ 0 0

0 0 0 0

Gϕz 0

 ε r − µ r     ε ϕr − µ ϕr   ε zr − µ zr     2(ε rϕ − µ rϕ )   2(ε ϕz − µ ϕz )    G rz  2(ε rz − µ rz )  0 0 0 0 0

(1)

where the stress tensor σ was related with the strain tensor ε and the eigenstrain tensor µ through the material stiffness matrix L, and Lij , G rϕ , Gϕz , G rz are the stiffness coefficients. Nine stiffness coefficients and six eigenstrains describe this kind of anisotropy. Cylindrical orthotropy is characterized by the fact that properties in the tangential, radial and axial directions are distinct; in other words, the material is orthotropic in a Cartesian system which is located at any point within the structure, with the three axes pointing in the axial, tangential and radial directions respectively. If Lrr > Lϕϕ then the material is called radially orthotropic, and if Lϕϕ > L rr then it is called circumferentially orthotropic. In the special case of transversally isotropic solid which may represent the average structure of fiber reinforced material, nine independent material constants in (1) are related as: Lrr = Lϕϕ ,

Lrz = Lϕz ,

G rz = Gϕz ,

Lrr - Lrϕ = 2G rϕ

In what follows we concentrate our attention on 2D case, assuming a long enough cylinder. Then the number of displacement components is reduced to u = u r , v = u ϕ , w = u z , and three stresses remain non-zero: σ r , σ ϕ , τ rϕ . The cylindrical coordinate system reduces to polar one. Recall basic geometrical relations:

εr =

∂u , ∂r

εϕ =

1 ∂v u + , r ∂ϕ r

ε rϕ =

1 ∂u ∂v v + − r ∂ϕ ∂r r

(2)

If we adopt the assumption σ z = 0 and isotropic material is considered, Hooke’s law takes the form: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

398 High Performance Structures and Materials III ε r − µr =

1 (σ r − νσ φ ), E

ε φ − µφ =

1 (σ φ − νσ r ), E

ε rφ − µ rφ =

1 σ rφ (3) G

where ν is Poisson’s ratio. Substituting the last equations in the geometrical equations yields the inverse relations of Hooke’s law for expressing stresses in the terms of displacements:   1 ∂v u  ∂u  − µ r  + Lrφ  + − µφ  σ r = Lrr ( ε r − µ r ) + Lrφ ( ε φ − µφ ) = Lrr  r  ∂r    r ∂φ   1 ∂v u  ∂u  − µr  + − µφ  + Lφφ  σ φ = Lrφ  ∂ ∂ r φ r r      1 ∂u  ∂v v σ rφ = G ε rφ − µ rφ = G  + − − µ rφ  r φ r r ∂ ∂  

(

)

(4)

(5)

Two equations of equilibrium provide relations among three components of stress tensor:  ∂σ r 1  ∂σ rφ + σ r − σ φ  = 0, +  r  ∂φ ∂r 

 1  ∂σ φ + 2σ rφ  = 0 +  r  ∂φ ∂r 

∂σ rφ

(6)

Expanding two components of displacements into sine and cosine series and denoting the k-th term by U r( k ) and U φ( k ) yields

u r( k ) (r , ϕ ) = U r( k ) (r ) cos 2kϕ , uϕ( k ) (r , ϕ ) = U ϕ( k ) (r ) sin 2kϕ ,

(7)

u z( k ) = 0 where U r( k ) ≡ U ( k ) and U ϕ( k ) ≡ V ( k ) are unknown functions of r which need to be determined from the equations of equilibrium. In particular, the substitution of (1) in the stress-strain relations gives stresses which, when substituted into the equations of equilibrium in cylindrical coordinates, provide the following equations for evaluation of U ( k ) and V ( k ) . Similarly the eigenparameters will be expanded into series with the k-th components:

µ r( k ) (r , ϕ ) = R ( k ) (r ) cos 2kϕ , µφ( k ) (r , φ ) = F ( k ) (r ) cos 2kϕ , µ r(φk ) (r , ϕ ) = M ( k ) (r ) sin 2kϕ WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(8)

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Components of strains, (2), and stresses, (6), are then expressed as (identification (k) of the number in the series is dropped out and the prime denotes derivative by r):

ε r( k ) = U ' cos 2kϕ , εϕ( k ) =

1 (2kV + U ) cos 2kϕ , ε r(ϕk ) =  − 2k U + V '− 1 V  sin 2kϕ r r   r (9)





1

σ r( k ) =  Lrr (U '− R ) + Lrϕ (2kV + U − F ) cos 2ϕ , r   (k )

σϕ

1   =  Lrϕ (2kV + U − F ) + Lϕ (U '− R)  cos 2ϕ , r  

 2k

1



σ r(φk ) = G − U + V '− V − M  sin2φ r  r 

(10)

(11)

Substituting (10), (11) to (6) and using the last four relations, two equations for unknown amplitudes of displacements read as:

ν

(2kV '+U '− F ') + r 2k (1 − ν )  2k V  1 +  − U + V '− − M  − [2kV + U − F ] = 0 r r r   r

U ' '− R' '+(1 − ν )(U '− R' ) +

M ν F 3 8k 2 + ν − 1 4k + ν 4k ( ν − 3) V ' ' − V '+ V− U '+ U − M '+ + R '+ 2 = 0 2 r r r R 1− ν r (1 − ν ) r

The last two equations lead to the solution, which defines the displacements on each lamina. Four integration constants are determined from interfacial conditions between adjacent layers. The angle determined from the above approach is eliminated by introducing moments at the end points to get conditions for clamped segment. The influence of such a moment to the segment is taken from [1], where the approach is described in details.

5

Conclusions

In this paper the procedure for calculating a segment of one-sided moistened tunnel lining from FRC. Information from straight one-sided moistened beams has been transferred to the segment and then the numerical procedure is suggested. Since it allows us to apply semianalytical (Fourier) solution, the procedure is relatively easy. First simply supported segment is considered, then using the approach from [2] for involvement of given moment at the end points, the clamped edge can be simulated. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

400 High Performance Structures and Materials III

Acknowledgment This paper was prepared under financial support of GA AV ČR, project No. IAA 2119402.

References [1]

[2] [3] [4]

Procházka, P., Starikov, N.: Optimal Slope of Dramix Type Fibers in Reinforced Concrete, 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, 30 May - 03 June 2005, Brazil. Lekhnitski, S.G. Anisotropic plates. Gordon and Breach Sci. Publ. NYC London – Paris, 1968. Chen, T., Dvorak, G.J. and Benveniste, Y. Stress fields in composites reinforced by coated cylindrically orthotropic fibers. Mechanics of Materials 9 (1990) 17-32. Sokolnikoff, I.S. Mathematical theory of elasticity. 2nd edition, McGrawHill, New York, 1996.

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Innovative procedure to produce high performance pretensioned concrete girders combining high strength concrete and normal or special concrete types C. Vázquez-Herrero & F. Martínez-Abella Department of Construction Engineering, School of Civil Engineering, University of La Coruña, Spain

Abstract As is well known, prefabrication procedures allow significant reductions in construction time. In many countries, precast concrete bridges are customary and made up of pretensioned prestressed girders. Both the compressive and tensile strength of the concrete mix must be enough at transfer (which takes place typically two or three days after concrete casting) in order to avoid excessive prestress losses, and undesirable cracking. Many attempts have been made to increase the spectrum of the usual high performance concretes to other concrete types (i.e. normal strength concrete, lightweight concrete and recycled aggregate concrete). Why are these special concrete types not used in the current production of large pretensioned concrete girders? Technical problems have been observed at the critical zones: development length, geometrical discontinuities, supports, etc. A new technique has been developed by the authors to produce hybrid pretensioned concrete elements, combining the usual high strength concretes in the critical zones with other concrete types in the rest of the elements. Keywords: prestressed concrete, precast concrete, pre-tensioning, pretensioned, pretensioned, high performance, high strength, prestressed, girders, beams.

1

Introduction

As is well known, the prefabrication procedures allow significant reductions in the construction time. In many countries precast concrete bridges are customary, made up of pretensioned prestressed girders deck. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06039

402 High Performance Structures and Materials III In post-tensioned structures, the transfer of prestress is done through mechanical anchorage. However, in pretensioned structures strand force is transmitted to concrete by bond, which is essential to: •

•

Transfer the prestressing force to the beam during the service life. Transfer length is defined (after Weerasekera [1]) as the distance over which the strand should be bonded to concrete in order to develop the effective prestress stress (fpe) in the strand (see Figure 1) . Guarantee the anchorage of the prestressing force. The flexural bond length is defined as the distance beyond the transfer length required to achieve the prestress design stress (fps) during the service life (see Figure 1). The development length is the sum of the transfer length and the flexural bond length.

Accordingly two different zones are determined along these elements: the external zones, where the bearing capacity is lower than the design capacity (development length) and the inner zone, where prestress is fully effective. Both the compressive and tensile strength of the concrete mix must be high enough at transfer (which takes place typically two or three days after concrete casting) in order to avoid excessive prestress losses, and undesirable cracking. The stresses created by bond in concrete are so high that the use of high strength concrete is usually required to prevent cracking along the transfer length.

Figure 1:

Transfer length and development length.

Many attempts were made to increase the spectrum of the conventional high performance concretes to other concrete types (i.e. moderate strength concrete, lightweight concrete and recycled aggregate concrete). The use of lightweight concrete having similar strength compared to the current high strength concrete, would allow either the reduction of the self weight of these elements while maintaining their span length, or the increase of the span length while maintaining the self weight. On the other hand, a reduction of the requirements of the resistant properties of concrete would imply a significant cost reduction. The use of normal strength WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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concrete is desirable, instead of the current high strength concrete. However this possibility must be carefully studied, in order not to affect the performance of the structure during the service life. Environmental regulations are hardening in developed countries. The extraction of aggregates is being penalized. In the future, a considerable increase of the cost of natural aggregates will be experienced. Consequently, the applicability of recycled aggregate concrete to the prefabrication processes must be studied. Why are these special concrete types not used in the current production of large pretensioned concrete girders? In our opinion it is due to their high cost and eventually to the technical problems at the critical zones: development length, geometrical discontinuities, supports, etc. Nowadays, the bearing capacity of the pretensioned concrete elements is limited by the feasibility of introducing the maximum prestress in the cross section due to the restrictions of minimum center-to-center strand distance as well as on minimum cover. Several attempts to use high strength lightweight concrete instead of the current normal weight concrete, with the same design criteria have failed due to longitudinal cracking along the ends of the girders (Vázquez-Herrero and Martínez-Abella [2]). These cracks are generated by the high stresses in the boundary between strands and surrounding concrete: •

•

Normal stresses between strand and the surrounding concrete, caused by the recovery of the strand diameter after transfer. As a result transverse tensile stresses appear in the surrounding concrete (bursting stresses) along the whole length of the pretensioned concrete elements. Bond stresses along the transfer length. The resulting transverse stresses which appear in the surrounding concrete are named splitting stresses.

Longitudinal cracking along the pretensioned concrete elements is unacceptable, because it nullifies the protection against corrosion of the strand provided by concrete cover. So it is necessary to prevent this kind of damage. The dispositions that have been usually considered in order to reduce the risk of longitudinal cracking are the following: •

•

Methods to enhance the confinement of the strands. With this aim transverse reinforcement is disposed, either through stirrups or spiral reinforcement. This procedure does not prevent cracking, but it certainly reduces crack width, and prevents losing the confining effect. These methods imply a considerable increase of operational costs. Methods to reduce bond between active reinforcement and concrete by different means, i.e.: reduction of the strand surface roughness, use of lubricating substances, use of partially debonded tendons, etc. An adverse effect is collateral to these measures: the increase of the development length means a reduction of the bearing capacity of the girders at this zone. Besides, it is difficult to exert and control the rate of bond reduction desired.

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404 High Performance Structures and Materials III In this paper an alternative procedure to produce pretensiond concrete elements, which allows the use of a normal strength concrete is presented. This is explained in the following section.

2

A procedure to produce hybrid pretensioned concrete elements

The scope of the procedure is to produce pretensioned concrete elements with what will be named objective concrete, which can be either a normal strength concrete or a special type of concrete (i.e. lightweight concrete, recycled aggregate concrete, fiber reinforced concrete, etc.). The critical zones of the structure are determined by : • •

The bending moments along the structure determine the situation of the zones where the prestressing force is not constant, which means the existence of bond stresses. The stress concentration zones, such as supports, holes, zones where the geometry changes abruptly,...

The objective concrete will be used in most of the concrete girders, but in the critical zones, which will be executed using a high strength concrete mix that has proved a good performance. The adequate performance of the objective concrete in the non critical zones must be checked experimentally, in order to determine the adequacy of the material for this use. The production procedure is similar to current girders’, but for the different phases, which will be determined by the different critical zones. The boundary surface between the different concrete types can be accomplished by the use of any kind of formwork, or by the natural slope adopted by concrete.

4

1

3

2

Figure 2: Hybrid pretensioned concrete girder. Figure 2 presents an example of a simply supported beam, which will take part of an isostatic bridge with precast girders deck. The depth of the critical zone (see flag 1 in Figure 2) is determined by the upper strand level, adding the convenient cover. The critical zones are delimited by the development length of the lower tendons (see flag 2 in Figure 2). The adopted angle (see flag 3 in Figure 2) of the boundary must be fixed to completely define the critical zone. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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The reinforcement required to transmit the horizontal component of the shear stresses (see flag 4 in Figure 2) must be conveniently anchored at both sides of the interphase. The rest of the element is cast with the objective concrete.

3

A procedure to assess the performance of hybrid pretensioned concrete elements

In order to study the feasibility of producing hybrid pretensioned concrete beams, the folowing procedure is proposed: 1.

2.

3.

Initially, several hybrid pretensioned concrete prisms must be produced and tested (see Figure 3). The critical zones have to be performed using a high strength concrete mix, and the rest using the objective concrete. These elements must be monitored for at least several months after transfer, to detect eventual cracking or unexpected growth of the transfer length. If the behavior of the prisms is considered satisfactory, the second stage consists of producing at plant several hybrid pretensioned girders, these elements have to be monitored, and some of them tested to failure (see Figure 4) to estimate their bearing capacity and development length. As a final stage of the design process, a prototype structure will be designed, built and monitored for several years. Only if this final stage is successful, current production of the hybrid pretensioned concrete elements can be attempted.

Figure 3:

Pretensioning bench and hybrid pretensioned concrete prism.

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406 High Performance Structures and Materials III

Figure 4:

4

Flexural test on a pretensioned concrete girder.

The behavior of hybrid pretensioned elements of lightweight concrete-high strength concrete

An experimental research was planned to study the feasibility of producing pretensioned concrete elements with lightweight concrete (LC, see Table 1). The proposed LC mix contains expanded clay as lightweight aggregate. Previously an experimental research had been undertaken to study the feasibility of lightweight concrete pretensioned elements, but longitudinal cracking appeared at the transfer length regions of these elements some time after transfer (VázquezHerrero and Martínez-Abella [2]). The pretensioned concrete prisms (square cross section 105 mm wide, 3750 mm long) were produced at the School of Engineering Laboratory, in a 4 m pretensioning bench designed for this research (see Figures 3 and 5). This part of the experimental research has been the subject of the undergraduate research project of Félix Sañudo-Herrera [3]. The critical zones (1500 mm long from the end of the elements) were cast with a high strength concrete (RC, see Table 1), and the inner part (750 mm long) was cast afterwards with the lightweight concrete mix. Table 1:

Prop. UNITS

Concrete specific weight KN/m3

Properties of the concrete mixes.

Average Average Average Average Average Average concrete direct concrete direct split tensile modulus of cylinder tensile cylinder tensile strength, at elasticity, at strength, at strength, at strength, at strength, at 2 days 2 days 2 days 2 days 28 days 28 days MPa MPa MPa MPa MPa MPa

RC

22

50

3.2

30441

3.3

67

>3.3

LC

17

48

2.7

19939

2.7

56

2.9

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Strand type was Y1860 S7 (1860 MPa), with a diameter of 15.2 mm (0.6”). Strand was tensioned until a stress of 75% of the minimum ultimate stress was attained. From the stressing process to the prestress force release 48 hours elapsed. Transfer was performed through gradually by flame heating. The prisms were continuously monitored during the transfer process and afterwards. Transfer length was measured periodically after transfer, and its values kept steady within the high strength concrete critical zones. The prisms did not show any visible cracking for several months, but after 3-5 months longitudinal cracks appeared along the lightweight concrete inner part of all the hybrid specimens. Cracks progressed throughout the whole lightweight concrete inner part. Consequently, the tested lightweight concrete mix used is not able to withstand the bursting stresses resulting from the strand diameter recovery which takes place after transfer due to the Poisson effect. Although the performance of the hybrid prisms is adequate for the first several months after transfer, durability cannot be assessed throughout the whole service life of the structure.

Figure 5:

5

View of the pretensioning bench and the prism during testing.

Conclusions

After the experimental program carried out on hybrid pretensioned concrete elements, the following conclusions have been established: •

A procedure has been developed to design and produce hybrid pretensioned concrete elements (girders, beams, hollow core plates, etc.) using a given objective concrete in most parts of the element. The critical zones will be performed using a concrete mix that has shown a good performance (i.e. high strength concrete). Special attention must be paid to the design of the boundary surface between both concrete mixes. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

408 High Performance Structures and Materials III •

•

•

An objective high strength lightweight concrete was tested, by means of producing hybrid pretensioned concrete prisms. This lightweight concrete mix is not adequate for the production of hybrid pretensioned concrete girders, using 15.2 mm diameter strand, with a cover of 50 mm, measured from the strand center. The prism test has shown a good procedure to detect potential problems of the objective concrete, for a given strand size and cover. In our opinion this first stage is very convenient to study the feasibility of the proposed objective concrete mixes. In our opinion further research should be done using a stronger lightweight concrete aggregate, strong enough to restrain cracks propagation. Eventually the use of fiber reinforced concrete, either lightweight or normal weight concrete, is another way to increase concrete tensile strength.

Further research will be done at the University of La Coruña dealing with different objective concrete types (i.e. normal weight concrete of moderate strength, fiber reinforced concrete and recycled aggregate concrete) in order to widen the scope of pretensioned concrete elements.

Acknowledgements This research project has been sponsored by the cement and concrete firm Corporación Noroeste, and by the projects XUGA22801A97 and CICYTMAT20001-0765 (Ministry of Science and Technology, Spain). The authors appreciate the contribution of the engineers Mr. Humberto Vázquez, Mr. Arturo Martínez, Mr. Marcos Fernández and Mr. Juan Rabuñal, and of Professor Dr. Pablo Rodríguez-Vellando for the revision of this text.

References [1] Weerasekera, I.R., Transfer and Flexural Bond in Pretensioned Prestressed Concrete, PhD Thesis, UMI: Michigan, 1991. [2] Vázquez-Herrero, C., Martínez-Abella, F., Bond Properties of High Strength Pretensioned Lightweight and Normal Weight Concrete, Proceedings of Bond in Concrete, Budapest, pp. 791-798, 2002. [3] Sañudo-Herrera. F., Experimental study of the behavior of hybrid pretensioned concrete elements using high strength concrete and lightweight concrete, Undergraduate Research Project (in Spanish), University of La Coruña, 2003.

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Properties of high performance concrete: the effect of cracks E. Mňahončáková, M. Jiřičková & R. Černý Department of Structural Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic

Abstract The moisture transport and storage parameters of high performance concrete belong to the most critical parameters in designing and using complex reliability based models for service life prediction of concrete structures. However, mostly it is supposed that the material is compact without any significant cracks, which is not always true. In this paper, the effect of thermally induced cracks on the basic moisture transport and storage properties is analyzed. Experimental results show that the presence of cracks in hardened high performance concrete mixtures dramatically changes all measured hygric parameters. Also, the effects of both microsilica and the size and presence of aggregates on moisture transport and storage parameters are very important. The liquid water transport is found to be affected by cracks, microsilica and aggregates in a much more significant way than water vapour transport. Keywords: high performance concrete, moisture transport and storage properties, cracks.

1

Introduction

The progressive building materials development improves their quality and utility value. In order to achieve better material properties, besides the new technologies also specific admixtures are employed. The accurate admixtures dosage and technological conditions observation can improve the material parameters like strength and durability. Most frequently, the mineral admixtures, for example pozzolana, fly ash or flue cinder, are used in concrete production. Fine siliceous materials usually called silica fume belong to relatively newly applied materials improving concrete quality. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06040

410 High Performance Structures and Materials III Silica fume can influence the hydration process on the one hand physically, ultra fine elements fill space between hydrating cement grains, on the other one chemically; they can react together with calcium hydroxide to form qualitatively different binding products. It leads to lower porosity, it changes the micro-pores distribution, the cementing agent is compacting and the strength, insolubility and corrosion durability of final product are improved. Despite the gradual shift in the HPC design and application philosophy towards a generally recognized necessity to measure a wider scale of HPC parameters the mechanical properties still remain the far most frequent parameters investigated in the research work being done on HPC. Water and water vapor transport and storage properties of HPC were not yet in the center of interest of most researchers until now although they possess a very high predicative potential concerning the HPC quality. In this paper, we analyse the influence of cracks on moisture transport and storage parameters of two different types of concrete, namely the high performance concrete C90/105 containing microsilica and the same material C60/75 without microsilica.

2

Materials

Basic tested material was high performance concrete C90/105 containing silica fume suspension, microsilica, denoted as BI in subsequent text. For the sake of comparison, the same concrete mixture C60/75 (denoted as BII) was prepared without microsilica. In order to analyse the effect of aggregates on moisture transport and storage parameters, two other mixtures based on BI and BII were prepared. The first one (denoted as BBI and BBII) was cement mortar without the 8-16 mm aggregate fraction. The second one was cement paste denoted as PI and PII. Table 1 presents the composition of the studied cement mixtures. Table 1:

Composition of studied cement mixtures.

Composition in g ** SiO2 Aggregates Woer- *** CEM I Lentan suspen- 0-4 ment 4-8 8-16 52.5 R VZ 33 sion mm mm mm FM 794 BI 480 72 664 207 995 7.74 2.58 BII 470 0 668 209 1001 5.17 2.35 BBI 480 72 664 1202 0 7.74 2.58 BBII 470 0 668 1210 0 5.17 2.35 PI 2346 352 0 0 0 38 12.67 PII 2348 0 0 0 0 26 12 * SiO2 suspension – water suspension consisting of 88-95% of SiO2 and amounts of calcium oxide, magnesium oxide and nitrogen oxide. ** FM 794 – plasticizer on the basis of polycarboxylateether. *** Lentan VZ 33 – hydration retarder on the saccharose basis. Type of concrete

*

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w/c 0.36 0.33 0.38 0.35 0.31 0.34 small

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Each mixture was cast into the forms, after one day the samples were removed from the forms and then varnished using curing solution. After the hardening period of 28 days, the samples were cut to required dimensions. These samples were then heated up to 600°C in an oven to reach cracks, see Figure 1.

Figure 1:

The HPC BI sample after thermal loading.

3 Experimental methods 3.1 Basic material parameters As fundamental physical material characteristics, bulk density ρb [kgm-3], vacuum saturation moisture content wsat [kgm-3], porosity [Vol.-%] and matrix density ρm [kg m-3] were determined. They were obtained using the gravimetric method and the vacuum saturation method. The vacuum saturation moisture content was calculated according to the equation

wsat = ρ w

msat − m0 = ρ wψ , msat − ma

(1)

where ρw is the water density [kg m-3], m0, msat and ma are the mass of dry sample, water- saturated sample and mass of the immersed water - saturated sample [kg], respectively, and ψ is the open porosity, which is defined as the ratio of the volume of open pores in material to its total volume. Matrix density was calculated as

ρ mat =

m0 , V (1 −ψ )

(2)

where V is sample volume [m3]. The measurement of basic parameters took place in a conditioned laboratory at the temperature of 22±1 ˚C and 25-30% relative humidity. Each result represents the average value from three to five measured values. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

412 High Performance Structures and Materials III 3.2 Water vapour and water transport parameters Two versions of the common cup method were employed in the measurements of the water vapor diffusion coefficient [1]. In the first one the sealed cup containing silica gel (5% relative humidity) was placed in a controlled climatic chamber with 97% relative humidity and weighed periodically. In the second one the cup containing the saturated solution of K2SO4 (97% relative humidity) was placed in the 25% relative humidity environment. The measurements were done at 20°C in a period of two weeks. The steady state values of mass gain or loss were determined by linear regression for the last five readings. The water vapor diffusion coefficient D [m2s-1] was calculated from the measured data according to the equation

D=

∆m ⋅ d ⋅ R ⋅ T S ⋅ τ ⋅ M ⋅ ∆p p

,

(3)

where ∆m the amount of water vapor diffused through the sample [kg], d the sample thickness [m], S the specimen surface [m2], τ the period of time corresponding to the transport of mass of water vapor ∆m [s], ∆pp the difference between partial water vapor pressure in the air under and above specific specimen surface [Pa], R the universal gas constant, M the molar mass of water, T the absolute temperature [K]. On the basis of the diffusion coefficient D, the water vapor diffusion resistance factor µ was determined:

µ=

Da D

,

(4)

where Da is the diffusion coefficient of water vapor in the air. The water sorptivity was measured using a standard experimental setup. The specimen was water and vapor-proof insulated on four edges and the face side was immersed 1-2 mm in the water, constant water level in tank was achieved by a Mariott bottle with two capillary tubes. One of them, inside diameter 2 mm, was ducked under the water level, second one, inside diameter 5 mm, was above water level. The automatic balance allowed recording the increase of mass. The water absorption coefficient A [kgm-2s-1/2] was then calculated using the formula

i = A⋅ t ,

(5)

where i is the cumulative water absorption [kg/m2], t is the time from the beginning of the suction experiment. The water absorption coefficient was then employed for the calculation of the apparent moisture diffusivity in the form [2]

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2

 A  ,  κ app ≈   wc − w0 

(6)

where wc is the saturated moisture content [kgm-3] and w0 the initial moisture content [kgm-3]. In the experimental work, the following samples were used: water vapor diffusion coefficient – 10 cylinders with the diameter 105 mm and thickness 20 mm, water sorptivity - 5 specimens 50 x 50 x 20 mm. 3.3 Sorption isotherms The water adsorption and desorption in a porous material are based on van der Waals forces between the surface of the porous matrix and water molecules. The dry material mass increases after a contact with moist air because of gradual bonding of water molecules from the air to the pore walls, in the case of adsorption. Desorption is reversed physical phenomenon, the initial state is capillary saturated sample. At the moment of achieving the equilibrium state between the water vapour pressure in the moist material and in the surrounding air this process is stopped. The mass of samples was measured in specified periods of time until steady state value of mass was achieved. Then, the moisture content by mass was calculated according to the equation um =

mw − m0 m0

,

(7)

where mw is the mass of wet sample and m0 is the mass of dry sample [kg]. The samples were placed into the desiccators with different salt solutions to simulate different values of relative humidity, see Table 2. Table 2:

Relative humidity over saturated salt solutions.

Temperature/Relative humidity [-] 20°C 23°C 25°C LiCl 0.111 0.111 MgCl2.6H2O 0.330 0.329 0.328 NaNO2 0.654 0.643 NaCl 0.755 0.754 0.753 NH4Cl 0.792 0.788 0.786 KCl 0.851 0.845 0.843 K2SO4 0.979 0.976 Salt

In the measurements of adsorption and desorption isotherms, for each relative humidity 3-10 test specimens of each material with overall dimensions of 40 x 40 x 10 mm were prepared. The measurements were done at 23±1°C. The initial WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

414 High Performance Structures and Materials III states were dry samples in the case of adsorption, and capillary saturated samples in the case of desorption.

4 Experimental results and discussion The results of basic parameter measurements are presented in Table 3. Table 3:

Basic material parameters.

Without cracks Type of mixture

ρb

ρmat

wsat

With cracks ψ

ρb

ρmat

[kgm-3]

[%]

[kgm-3]

wsat

ψ [%]

BI

2423

2760

122.0 12.20 2391

2856

162.2 16.30

BII

2388

2647

95.1

9.80 2350

2713

133.2 13.40

BBI

2240

2652

150.6 15.60 2147

2627

182.0 18.30

BBII

2210

2578

140.5 14.30 2205

2669

174.8 17.40

PI

1988

2809

248.4 29.20 1942

3068

375.8 36.70

PII

1991

2779

285.2 28.30 2019

3088

355.7 34.60

The highest bulk density exhibited the hardened HPC mixture BI. The second highest had the BII mixture without microsilica. Nearly the same values had mixtures denoted BBI and BBII, cement pastes had the lowest values. The bulk density of cement mixtures with cracks was lower then of the same without cracks. The matrix densities differed only in the range of about 5% and increased about 10% in the case of mixtures with cracks compared to mixtures without cracks. The highest vacuum saturated moisture content had the cement pastes PI and PII, where the difference was within the error range, and increased about 30 % in the case of mixtures with cracks compared to mixtures without cracks. The most porous materials were cement pastes PI and PII, the presence of microsilica and admixtures did not have a significant effect. The porosity increased from 3 to 6% in the case of mixtures with cracks compared to mixtures without cracks. The results of measurements of water and water vapour transport parameters of the studied hardened cement mixtures are presented in Tables 4 and 5. The water vapour diffusion coefficient was affected by the addition of microsilica only for cement paste PI in a remarkable way where it decreased three to four times. However, for the mixture BII its effect was within the error range of the measuring method. The cracks caused dramatic increase of the water vapour diffusion coefficient in all cases. The effect of microsilica on the liquid moisture transport parameters was more pronounced than on water vapour transport. The water sorptivity for the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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basic mixture BI decreased five times due to the microsilica addition, and for the cement paste PI by about 30%. The moisture diffusivity of mixtures with microsilica decreased in a corresponding way. In the case of cement mixtures with cracks the moisture diffusivity increased one order of magnitude compared to mixtures without cracks. Table 4:

Water vapour transport properties of cement mixtures. Without cracks

Type of mixture

BI BII BBI BBII PI PII

-6

D [10 m s ] 9725% RH 1.06 1.10 0.287 0.288 0.136 0.357

Table 5: Type of mixture BI BII BBI BBII PI PII

2 -1

597% RH 0.397 0.375 0.132 0.151 0.079 0.291

µ [-] 9725% RH 22.5 20.9 80.3 81.4 170.7 65.4

597% RH 59.3 61.4 175.4 152.1 295.9 81.7

With cracks D [10-6 m2sµ [-] 1 ] 97597525% 97% 25% 97% RH RH RH RH 1.48 1.55 15.8 15.4 2.22 1.76 10.4 13.2 1.73 1.96 13.8 12.3 2.11 2.23 11.0 10.6 2.29 3.20 10.2 7.2 2.22 1.76 10.4 13.2

Water transport properties of cement mixtures.

Without cracks A [kg m-2s-1/2] κapp [m2s-1] 3.63E-03 8.9E-10 1.80E-02 3.3E-09 1.76E-02 1.5E-08 4.04E-02 8.3E-08 2.36E-02 7.0E-09 3.52E-02 1.8E-08

With cracks A [kg m-2s-1/2] κapp [m2s-1] 4.84E-02 9.2E-08 4.23E-02 1.1E-07 8.08E-02 2.5E-07 6.21E-02 2.0E-07 2.77E-01 5.5E-07 1.59E-01 2.0E-07

The effect of aggregate presence was found to be quite different for liquid water transport than for water vapour transport. The water sorptivity was for the basic hardened HPC mixture BI several times lower than for the cement pastes. On the other hand, the water vapour diffusion coefficient of cement pastes was always lower than for the basic hardened HPC mixtures. The 5-97% RH values of the water vapour diffusion coefficient were always lower than the 97-25% RH values, which is in accordance with the previous measurements on many other materials including concretes and cement pastes. Figure 2a-f shows the measured adsorption (lower curves) and desorption (upper curves) isotherms. We can see that cement pastes had higher water adsorption capacity than the basic hardened HPC mixtures. This is an expected behaviour due to absence of the inert aggregates in cement pastes. The presence of microsilica slightly increased the water adsorption capacity of basic hardened WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

416 High Performance Structures and Materials III HPC mixture but for cement pastes an opposite behaviour was observed. Therefore, any clear effect of microsilica on water adsorption capacity was not evidenced. The sorption curves are almost the same for the mixtures without and with cracks. 0.04

0.06

BII without cracks BII with cracks

moisture content [kg kg-1]

moisture content [kg kg-1]

BI without cracks BI with cracks 0.04

0.02

0

0.02

0 0

0.2

0.4

0.6

0.8

1

0

0.2

relative humidity [-]

0.4

(a)

0.8

1

BBII without cracks moisture content [kg kg-1]

moisture content [kg kg-1]

1

(b)

BBI without cracks BBI with cracks 0.04

0.02

0

BBII with cracks 0.04

0.02

0 0

0.2

0.4

0.6

0.8

1

0

0.2

relative humidity [-]

0.4

0.6

relative humidity [-]

(c)

(d) 0.14

0.12

0.12 moisture content [kg kg-1]

0.1 moisture content [kg kg-1]

0.8

0.06

0.06

0.08 0.06 0.04 PI without cracks

0.02

PI with cracks

0

0.1 0.08 0.06 0.04 PII without cracks

0.02

PII with cracks

0 0

0.2

0.4

0.6

0.8

1

0

0.2

relative humidity [-]

(e)

Figure 2:

5

0.6

relative humidity [-]

0.4

0.6

0.8

1

relative humidity [-]

(f)

Sorption isotherm of mixture (a): BI, (b): BII, (c): BBI, (d): BBII, (e): PI and (f) PII.

Conclusions

The experimental work performed in this paper has shown that the presence of cracks in hardened HPC mixtures dramatically changed all measured parameters. Also the effect of both microsilica and aggregates on the moisture transport and storage parameters was very important. The liquid water transport was found to be affected by cracks, microsilica and aggregates in a much more significant way than water vapour transport.

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Acknowledgement This research has been supported by the Ministry of Education of Czech Republic, under grant No. MSM: 6840770003.

References [1]

[2]

S. Roels, J. Carmeliet, H. Hens, O. Adan, H. Brocken, R. Černý, Z. Pavlík, C. Hall, K. Kumaran, L. Pel, R. Plagge, Interlaboratory Comparison of Hygric Properties of Porous Building Materials. Journal of Thermal Envelope and Building Science 27(2004), 307-325. R. Černý, J. Poděbradská, J. Drchalová, Water and water vapor penetration through coatings. Journal of Thermal Envelope and Building Science 26(2002), 165-177.

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Seismic upgrading of square and rectangular RC columns using FRP wrapping M. A. N. Abdel-Mooty1, M. E. Issa2, H. M. Farag3 & M. A. Bitar4 1

Department of Construction Engineering, American University in Cairo, Egypt 2 Department of Structural Engineering, Cairo University, Egypt 3 Department of Structural Engineering, Military Technical College, Egypt 4 Department of Structural Engineering, Military Technical College, Egypt

Abstract This paper presents the results of an experimental study to evaluate the effectiveness of using FRP wrapping for repair and seismic upgrading of square and rectangular RC columns in buildings. The factors affecting the performance of FRP wrapping in rectangular columns under the action of both axial and combined axial and lateral loads are considered in this paper. These factors include the rectangularity ratio of the column cross section, the thickness of the FRP jacket, and the use of carbon verses glass FRP for column jacketing. Techniques to improve the performance of strengthening rectangular columns were also proposed and evaluated in the paper. Such techniques include rounding the sharp edges of the columns, and transferring square into circular columns using mortar. A total of fourteen half-scale reinforced concrete columns, divided into two groups, are tested in this research. The first group of columns consists of three square columns and three rectangular columns which are tested under axial loads. The second group consists of four square columns and four rectangular columns tested under combined axial and lateral loads. Keywords: seismic upgrading, column strengthening, CFRP and GFRP.

1

Introduction

The use of FRP wrapping for repair, strengthening, and seismic upgrading of columns has gained increasing attention in recent years [1–3]. FRP wrapping was proposed for increasing the ductility of column under axial and axialWIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06041

420 High Performance Structures and Materials III flexural loading through confinement [4–8], improving insufficient shear strength [9], and insufficient lap splice length [10]. Due to uniform confinement, FRP wrapping has been proven effective for strengthening circular columns [4, 10]. Other studies showed the improvement in seismic capacity of square columns wrapped with FRP jacket [7, 8]. It was recommended that the sharp corners of the column must be rounded to avoid premature failure of the FRP jacket due to stress concentration at the corners. Studies conducted on wrapping rectangular columns showed inferior performance to that of circular and square columns [5, 11]. The work presented in this paper describes the first phase of a two-phase research program addressing the performance of RC rectangular columns strengthened using FRP wrapping jacket. The first phase is an experimental study to evaluate the effectiveness of using FRP wrapping for repair and seismic upgrading of square and rectangular RC columns in buildings. The second phase is a theoretical investigation to develop a numerical model capable to accurately predict the actual behaviour of the repair techniques. The factors affecting the performance of FRP wrapping in rectangular columns under the action of both axial and combined axial and lateral loads are considered in this paper. These factors include the rectangularity ratio of the column cross section, the thickness of the FRP jacket, and the use of carbon verses glass FRP for column jacketing. Techniques to improve the performance of strengthening rectangular columns are also proposed and evaluated in the paper. Such techniques include rounding the sharp edges of the columns and transferring square into circular columns using mortar. A total of fourteen half-scale reinforced concrete columns, divided into two groups, are tested in this research. The first group of columns consists of three square columns and three rectangular columns which are tested under axial loads. The second group consists of four square columns and four rectangular columns tested under combined axial and lateral loads.

2

Experimental program and setup

The test program in this research aims at studying the various parameters affecting the behaviour of reinforced concrete rectangular columns strengthened using FRP wrapping under both axial and seismic forces. These parameters include the rectangularity ratio of the column cross section, the thickness of the FRP jacket, and the use of carbon verses glass FRP for column jacketing. A total of fourteen half-scale reinforced concrete columns, divided into two groups, are tested in this research. The first group of columns, Group I, consists of three square columns; CS1, CS2 and CS3; and three rectangular columns; CR1, CR2 and CR3; which are tested under increasing axial loads to failure. The second group, Group II, consists of four square columns; CS4, CS5, CS6 and CS7; and four rectangular columns; CR4, CR5, CR6 and CR7; tested under combined axial and lateral loads. Constant axial loads, 350 kN for square columns and 700 kN for rectangular columns, were applied to Group II columns during increasing cyclic lateral loading up to failure. This load level is approximately 30% of the axial load carrying capacity of the tested columns. The details of the test specimens and program are summarized in Table 1. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 1: Testing load and set-up Group I: Axial load

Summary of test program and setup. Specimen crosssection and reinforcement

Specimen No.

Square Columns

CS1

Control

200×200 mm

CS2

CFRP

CS3

CFRP curving sides

CR1

Control

CR2

CFRP

CR3

CFRP

Square Columns

CS4

Control

200×200 mm

CS5

GFRP

CS6

CFRP

CS7

2-layers CFRP

CR4

Control

CR5

GFRP

CR6

CFRP

CR7

2-layers CFRP

φ 6/150 mm

FRP 1500 mm

4φ12 Rectangular Col.

Designation

400×200 mm φ 6/150 mm

6φ12 Group II: Lateral cyclic loading with constant axial load

φ 6/150 mm

4φ12

1600 mm

FRP Rectangular Col. 750 mm

400×200 mm φ 6/150 mm

400 mm

Table 2:

6φ12

Physical and mechanical properties of GFRP and CFRP.

Property Tensile strength in fibre direction (MPa) Elongation at breaking (%) Tensile modulus (GPa) Nominal laminate thickness (mm) Fabric width (mm)

GFRP 575 2.2 26.1 0.17 300

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CFRP 965 1.33 73 0.13 350

421

422 High Performance Structures and Materials III The average concrete strength for the test specimens after 28 days from casting is 32.5 MPa. Mild steel of diameter 6 mm was used for stirrups while high grade steel of diameter 12 mm was used for longitudinal reinforcement. Two types of fibre reinforced polymers were used for jacket wrapping of the column specimens namely Glass FRP and Carbon FRP laminates. The physical and mechanical properties for both GFRP and CFRP are displayed in Table 2. Rigid frame Jack and load cell

Steel cap

LVDT LVDT & load cell

Steel cap

a) axial load

b) axial and lateral cyclic load

Figure 1:

Schematic diagram for test set-up.

a) Control Specimen CR1 during testing Figure 2:

Axial load testing and failure of control specimen CR1.

a) CR6 during testing Figure 3:

b) Failure of CR1

b) Failure of CR6

Lateral cyclic load testing and failure of specimen CS7.

Group II columns were cast integrally with a 1000×400×400 mm footing for square columns and a 1200×400×400 mm footing for rectangular columns. The footing had six bars of diameter 12 mm as top and bottom reinforcement with WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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stirrups 10 mm diameter spaced at 10 cm. The footing dimensions and reinforcement were chosen such that its deformations would not affect the measurements in the test zone. All columns of Group II were tested under constant axial load and increasing cyclic load to failure with columns CS4 and CR4 used as reference. FRP wrapping of columns was applied within the potential plastic hinge zone of the column for a length of 750 mm from the footing top face. Columns CS5 and CR5 were wrapped by one layer of GFRP, Columns CS6 and CR6 were strengthened by one layer of CFRP, and Columns CS7 and CR7 were wrapped by two layers of CFRP. The installation of the FRP jacket necessitates surface treatment. Sandpaper and sand blasting were used for concrete surface preparation before applying the epoxy resin used for installing the FRP wrapping. The sharp corners of the rectangular columns were rounded to avoid premature failure of the jacket due to stress concentration at sharp corners. Epoxy was applied uniformly on the entire face of the concrete column as well as on the FRP laminate. The fabrics were then compressed to the concrete surface with a roller. The thickness of the epoxy layer was controlled to be about 1.2 mm. The specimens were treated at room temperature for at least 24 hours before testing. Group I specimens were instrumented with 2 LVDT’s at the mid-height to measure out of plane deformation. Strain gauges were attached to longitudinal reinforcements and the stirrups at the column mid-height to measure longitudinal and transverse strain during loading. Surface strain gauges were also mounted to the concrete surface and to the FRP jacket. Those measurements will be used for verification of the detailed numerical models developed in the second phase of this research. As for Group II specimens, LVDT’s are used to measure the lateral displacement at the top and mid-height of the columns during testing. Furthermore, strain gauges are used to measure vertical and lateral strain on reinforcement and concrete and FRP surfaces near the column base. The measured load, displacements, and strains at the various locations were fed into the data acquisition system and recorded for further processing and analysis. Figure 1 shows schematic diagrams of the test set-up for Group I and Group II specimens. Figures 2 and 3 show sample specimens during testing and at failure. Avreage stress at failure (N/mm2)

60

50

40

CR2 CS1 CS2 CS3

30

20

10

0 0

0.5

1

1.5

2

2.5

Strain measured on longitudinal steel RFT(%)

Figure 4:

Variation of axial strain with axial stress for square columns.

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424 High Performance Structures and Materials III

Avreage stress at failure (N/mm2)

60

50

40

CR2 CS1 CS2 CS3

30

20

10

0 0

0.5

1

1.5

2

2.5

Transverse strain measured on stirrups (%)

Figure 5:

Variation of transverse strain with axial stress for square columns. Avreage stress at failure (N/mm2)

60 CR1 CR2 CR3

(CR1)" (CR2)"

50

(CR3)"

40 30 20 10 0 0

Figure 6:

0.5

1

1.5

2

2.5

Strain measured on longitudinal steel RFT (%)

Variation of axial strain with axial stress for rectangular columns.

Avreage stress at failure (N/mm2)

60 50 CR1 (CR1)

40

(CR2)" CR2 CR3 (CR3)"

30 20 10 0 0

0.5

1

1.5

2

2.5

Transverse strain measured on stirrups (%)

Figure 7:

Variation of transverse strain with axial stress for rectangular columns.

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425

Analysis of test results

Group I specimens are tested under increasing axial load to failure. The square column control specimen formed inclined cracks at an average stress of 30.2 N/mm2 followed by fast progressive failure in the form of falling off the concrete cover and buckling of longitudinal reinforcement. As for specimen CS2, and CS3, the confinement provided by the CFRP jacket delayed the failure to higher stresses values, 48.2 N/mm2 for CS2 and 48.9 N/mm2 for CS3. At failure the CFRP jacket was ruptured due to hoop tensile stresses after which buckling of steel longitudinal bars occurred. The rectangular specimens showed similar behaviour up to failure except that the effect of confinement due to the CFRP jacket was less pronounced. Figures 4–7 show the variation of axial strain and lateral strain with axial stress for square and rectangular column specimens respectively. Figure 8 compares the average axial stress at failure for axially loaded specimens. FRP confinement creases the ultimate capacity of axially loaded square columns by approximately 50%. The confinement effect was less pronounced for rectangular columns where the increase in the ultimate capacity was about 21%. Transferring square column to circular one slightly increase the average failure stress by 1.5%, however the overall load capacity of the column is increased by 5.2% from 193 kN to 203 kN due to increasing the cross section. Average stress at failure (N/mm2)

60

48.25 N/mm2

50

48.88 N/mm2

38.875 N/mm2

40

35.625 N/mm2 30.75 N/mm2

30.2 N/mm2 30

20

10

0

CS1

Figure 8:

CS2

CS3

CR1

CR2

CR3

Average axial stress at failure for axially loaded columns.

Group II specimens were tested under constant axial load (30% of the axial load capacity) and increasing cyclic load to failure. Figure 9 shows the loaddisplacement hysteresis loops for specimen CR6 during testing. All specimens in this group showed similar behaviour. As load increased cracking started first at the base, and then more cracks started to develop within approximately 450 mm from the base. With further increasing load, crack opened and concrete in compression crushed forming plastic hinge. Figure 10 shows the envelope for lateral load variation with lateral displacement for square specimens. The variation of lateral load with strain measured on steel stirrups for square specimens is displayed in Figure 11. The load-displacement envelope for rectangular columns is shown in Figure 12. The ultimate lateral load increased by 11%, 22%, and 44% for square column wrapped by GFRP, CFRP and two WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

426 High Performance Structures and Materials III CFRP layers, respectively. Similar improvements were observed for rectangular columns where lateral load increased by 12.5%, 25%, and 33.3% for rectangular column wrapped by GFRP, CFRP and two CFRP layers, respectively. Figure 13 compares the lateral load capacity for cyclically loaded columns. 400 300

Lateral load (kN)

200 100 0 -40

-30

-20

-10

0

10

20

30

40

-100 -200 -300 -400

Displacement (mm)

Figure 9:

Load-displacement hysteresis loops for specimen CR6. 150

Lateral load (kN)

100

CS4 CS5 CS6 CS7

50

0 -50

-40

-30

-20

-10

0

10

20

30

40

50

-50

-100

-150

Lateral displacement (mm)

Figure 10:

Lateral load vs. lateral displacement envelop for square columns. 150

Lateral load (kN)

100

CS4 CS5 CS6 CS7

50

0 0

0.5

1

1.5

2

2.5

3

3.5

4

-50

-100

-150

Transverse strain on stirrups (%)

Figure 11: Lateral load vs. lateral strain on steel stirrups for square columns. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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400 300

CR4 CR5 CR6 CR7

Lateral load (kN)

200 100 0 -40

-30

-20

-10

0

10

20

30

40

-100 -200 -300 -400

Lateral displacement (mm)

Figure 12: Lateral load vs. lateral displacement envelope for rectangular columns. 350 300 kN

Maximum lateral load (kN)

300

320 kN

270 kN 240 kN

250

200 130 kN

150

100

90 kN

100 kN

110 kN

50

0

CS4

CS5

CS6

CS7

CR4

CR5

CR6

CR7

Figure 13: Ultimate lateral load for cyclic loaded specimens.

4 Summary and conclusions The performance of fourteen RC square and rectangular columns strengthened using FRP wrapping jacket under axial loading and increasing lateral cyclic loading was experimentally evaluated in this paper. The loads, displacements and strains during testing were recorded for further numerical study. It was found that strengthening RC columns using FRP wrapping is much more efficient in square columns than in rectangular columns. Failure stresses increased due to FRP wrapping by 50% in square columns and 21% in rectangular columns. Transferring square columns to circular columns to improve FRP confinement increases the ultimate stress by only 1.5%. Group II columns subjected to combined axial and lateral loads were wrapped with FRP in the plastic hinge zone. FRP wrapping delayed concrete cracking, and increase ductility. The rupture of the FRP wrap occurs gradually with increasing load giving more warning before failure. The ultimate lateral load is increased by 11%, 22%, and 44% for square column wrapped by GFRP, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

428 High Performance Structures and Materials III CFRP and two layers of CFRP respectively. Similar improvements in the lateral load capacity were observed for rectangular columns (12.5%, 25%, and 33.3%).

References [1] Lehman, D.E., Gookin, S.E., Nacamuli, A.M. & Moehle, J.P, Repair of earthquake-damaged bridge columns. ACI Structural journal, 98(2), pp. 233-242, 2001. [2] Mosallam, A.S. (ed.). Innovative System for Seismic Repair and Rehabilitation of Structures – Design and Application, Proc. (SRRS2), Technomic Publishing co., Inc., 2000. [3] Tan, K.H. (ed.). Fibre-Reinforced Polymer Reinforcement for Concrete Structures, Proc. of the Sixth International Symposium on FRP Reinforcement for Concrete Structures, Singapore, 8-10 July, 2003. [4] Fam, A.Z. & Rizkalla, S.H., Confinement model for axially loaded concrete confined by circular fiber-reinforced polymer tubes. ACI Structural journal, 98(4), pp. 451-461, 2001. [5] Wang, Y.C. & Restrepo, J.I., Investigation of concentrically loaded reinforced concrete columns confined with glass fibre-reinforced polymer jackets. ACI Structural journal, 98(3), pp. 377-385, 2001. [6] Chaallal, O. & Shahawy, M., Performance of fiber-reinforced polymerwrapped reinforcement concrete column under combined axial-flexural loading. ACI Structural journal, 97(4), pp. 659-668, 2000. [7] Mahmoud, Kh., Fouad, E., Ramadan, M.O. & Abd-Elalim, A., Behaviour of axially loaded square RC column confined with sandwich FRP wraps. Proc. International Conference: Future Vision and Challenges for Urban Development, Paper SG96F, Cairo, 20-22 December 2004. [8] Allam, H.M., Strengthening of square columns by a New Technique. Proc. International Conference: Future Vision and Challenges for Urban Development, paper SG167F, Cairo, 20-22 December 2004. [9] Haroun, M. A. & Elsanadedy, H.M., Seismic retrofit of shear-deficient reinforced concrete bridge columns by advanced composite jackets. Proc. Structural Composite for Infrastructure Applications, Aswan, Egypt, 2002. [10] Haroun, M.A., Mosallam, A.S., Feng, M.Q. & Elsanadedy, H.M., Experimental Investigation of Seismic Repair and Retrofit of bridge columns by composite jackets. Journal of Reinforced Plastics and Composites, 22(14), pp. 1243-1268, 2003. [11] Hosny A., Shahin, H. Abdelrahman, A., & El-Afandy, T., Uniaxial tests on rectangular columns strengthened with CFRP. Proc. Structural Composite for Infrastructure Applications, Aswan, Egypt, 17-20 Dec., 2002.

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Arch bridge made of reactive powder concrete D. Cizmar 1, D. Mestrovic2 & J. Radic2 1 2

Ekonerg Institute, Zagreb, Croatia Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia

Abstract Concrete is the most commonly used building material. It is used for buildings, industrial structures, bridges and dams. Every day the concrete is being improved, to achieve better characteristics, reduce price and to be environmental acceptable. In the introduction an historical overview of concrete is given – beginning in the 1950s when the compression strength was 40 N/mm2. Today this is the Industry standard. Concrete with compression strength greater than 40 N/mm2 is called high performance concrete (HPC). The first HPS concretes were made in 1960. Around 1990, Reactive Powder Concrete (RPC) started to appear. Their strengths go up to 800 N/mm2. Today, HPC is limited to bridges, and RPC is still very rarely used. In this work we present a method for making a RPC composite material with a compression strength of up to 170 N/mm2. Preparation and testing of material is performed in the laboratory of the Faculty of Civil Engineering in Zagreb. In addition to the mechanical properties, the durability parameters are also tested. Detailed concrete mix proportions are given in the article. The possibility that RPC could be used to construct an arch bridge over Bakar strait (Croatia) is analyzed. The bridge would have arch span of 432 m. Due to very high compression strength of RPC the span length may be increased, therefore reducing the total construction cost. RPC is also used to increase the resistance to freeze-thaw cycles, increase abrasion resistance, and reduce chloride permeability. Many large Adriatic bridges have experienced durability problems that are primarily the result of using standard concrete. Each of these durability enhancements provided by RPC decrease maintenance costs and lengthens the service life of a structure, which is vital for bridges in the Adriatic region. Keywords: arch bridge, reactive powder concrete (RPC), construction technology.

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430 High Performance Structures and Materials III

1

Introduction

Cement is around 12 millions years old. Reactions between limestone and oil shale during spontaneous combustion occurred in Israel to form natural deposits of cement compounds. The deposits were first described by Israeli geologists between 1960 and 1970. In 3000 years BC, the Egyptians used mud mixed with straw to bind dried bricks. They also used gypsum mortars and mortars of lime in the construction of the pyramids. Chinese used cementitious materials to hold bamboo together in their boats and in the Great Wall of China. The Romans used pozzolana cement from Pozzuoli, Italy near Mt. Vesuvius to build the Appian Way, Roman baths, the Coliseum and Pantheon in Rome, and the Pont du Gard aqueduct in southern France. They used lime as a cementitious material. Pliny reported a mortar mixture of 1 part lime to 4 parts sand. Vitruvius reported a 2 parts pozzolana to 1 part lime mixture. Animal fat, milk, and blood were used as admixtures (substances added to cement to improve properties).

Figure 1:

Maxentius basilica.

Fig. 1 shows the Maxentius basilica built in the 4th Century. These materials were not used again until the beginning of the 19th Century. In 1824, Joseph Aspdin of England invented Portland cement by burning finely ground chalk with finely divided clay in a lime kiln until carbon dioxide was driven off. The sintered product was then ground and he called it Portland cement named after the high-quality building stones quarried at Portland, England. Portland cement started the era of modern composition cements. In 1828, I. K. Brunel made the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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first significant engineering application of Portland cement, using it to fill a breach in the Thames Tunnel. In 1867, Joseph Monier of France reinforced William Wand’s (USA) flower pots with wire ushering in the idea of iron reinforcing bars (re-bar). In 1889, the first concrete reinforced bridge was built. Around 1950, concrete with a compression strength of 40 N/mm2 was made. In the 1960’s, High Performance Concrete (HPC) was made. HPC is the name given to a class of materials that exhibits properties superior to those of conventional concrete. The superiority may lie in one or more of several attributes, such as strength, stiffness, freeze-thaw durability, or resistance to chemical attack. The properties are selected on the basis of the requirements of the particular application. Its compression strength ranges from the 50 N/mm2 to 100 N/mm2. In the 1970’s, fiber reinforcement in concrete was introduced. During the 1980’s, superplasticizers were introduced as admixtures. Around 1990, Reactive Powder Concrete (RPC) appeared. Their compression strength goes up to 800 N/mm2. While HPC is increasingly used for bridges, RPC is still very rarely used.

2

Components for RPC mixture

2.1 Introduction This article presents the possibility of making RPC concrete with compression strengths up to 200 N/mm2. Four different mixtures are analyzed. The first is mixture of hybrid micro-fiber concrete; the others are composed of only one type of fibers. 2.2 Cement As the class of cement increases the compression strength increases. For this mixture we have selected the Portland cement PC 55 with no mineral ingredients. 2.3 Fine aggregate Aggregate that is used for the making of this mixture is quartz aggregate. Two fractions of this material are used: one with soil size of 0.125–0.25 mm, other with 0.25–0.5 mm, This effectively means that maximal soil size is 0.5 mm. 2.4 Steel fibers Two different types of steel fibers are used (shorter and longer fibers). Shorter fibers are 13 ± 2 mm long, diameter is 0.2 ± 0.02 mm. Minimal tensile strength is 2600 MPa. Longer fibers have curved ends; their length is 40 ± 3 mm and diameter 0.5 ± 0.02 mm. Minimal tensile strength is 2600 Mpa. In both cases, the high tensile fibers are used to achieve necessary ductility. 2.5 Silica fume Silica fume is a pozzolanic additive, with specific area of 20 m2/g. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

432 High Performance Structures and Materials III Other parameters are: - density - sieve residue 45 µm - pH value 8.44

2.23 g/cm3 5.6%

2.6 Superplasticizer Superplasticizer is primary used to decrease the participation ratio of water in the concrete mixture. Chosen superplasticizer is based on policarbon-silate. It is a brown fluid, dissolves in water and doesn’t contain chlorides. Other parameters are: - density 1.064 kg/dm3 - pH value 7 - alkalinity 0.31 % - viscosity on 20˚C 134 mPas Table 1:

Ingredients per m3 of RPC.

Mixture number

M1

M2

M3

M4

Steel fibers (kg/m3) SF1 (40/0.5) SF2 (13/0.2)

76 190

228

228

234

720

955

720

980

230

239

230

303

123 1112

105 945

123 1111

105 965

Superplasticizer (kg/m3)

30

35

31

40

Water (l/m3)

190

215

190

209

24˚C 5% 2.41 kg/m3 140 mm

25˚C 5% 2.35 kg/m3 250 mm

24.5˚C 5% 2.36 kg/m3 190 mm

26˚C 5% 2.306kg/m3 220 mm

Cement (kg/m3) Fine aggregate (kg/m3) Quartz sand 0.125–0.25 0.25–0.5 (kg/m3)

Concrete properties temperature pores density consistency

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2.7 Compatibility of cement and superplasticizer The compatibility of PC 55 cement and superplasticizer Glenium ACE 30 is one of the important requirements for achieving a good RPC mixture. Compatibility is measured by a change in the consistency of the concrete. To achieve proper treatment of concrete it must retain its characteristics for approximately 1.5 h for site application and 0.5 h for prefabrication.

3

Experimental results

Compression and tensile strength measurement were conducted on four specimens with a prismatic shape (40 x 40 x 160 mm). The specimens were 28 days old. First, the tensile strength was measured, then the compression strength. Table 2 presents the experimental results for each mixture. Fig. 2 shows the specimens after the tests. Table 2:

Flexure strength (mean) (MPa)

Compression strength (mean) (MPa)

M1 M2

46.9 42.8

132.0 155.6

M3 M4

42.8 48.8

153.3 174.8

Figure 2:

4

Experimental results.

Specimens after testing.

Gas permeability test

Gas permeability is tested according to Croatian regulations EN 993-4. Specimens were cylindrical, the diameter and length were 50 mm. They were taken from a 10 x 10 x 50 cm prism (first mixture). The specimens were put in WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

434 High Performance Structures and Materials III dry chamber until constant mass was achieved. The specimens were than cooled to room temperature, polished and coated with epoxy. A pressure difference of 3 bars wasn’t detected, which means that gas permeability should have been very low (none was detected).

5

Capillary water test

A capillary water test is conducted according to Croatian regulations HRN.U.M8.300:1985 in the laboratory of the Civil engineering faculty in Zagreb on specimens 28 days old. The diameter of the specimen was 15 cm and hadd a height of 10 cm. Before test the specimens are dried on 105˚C and then left for 2 days in laboratory. Sealing putty is applied on one side. Specimens are then weighed. The capillary water test was performed in intervals of 1, 3, 5, 10, 15, and 30 minutes after immersing in water, and then after 1, 2, 3, 4, 5, 6, and 24 hours. Results show the linear proportion between the capillarity water and square root of time. The height of capillarity water could not be determined because it didn’t cross the area sealed with sealing putty. The starting absorption capacity for ordinary concrete is 0.25 ml/m2/s after 10 min, 0.17 ml/m2/s after 30 min, and 0.10 ml/m2/s after 1 hour. These results show (table 3) that RPC has very little water permeability. Its absorption after 10 minutes is much lesser than that of ordinary concrete after an hour. Table 3: Capillary water test. Time 0 min 1 min 3 min 5 min 10 min 15 min 30 min 1h 2h 3h 4h 5h 6h 24 h

A sample (kg/m2/s) 0 0.000613 0.00029 0.00022 0.000122 0.000089 0.0000526 0.0000314 0.0000182 0.0000133 0.0000105 0.00000888 0.00000451 0.00000242

B sample (kg/m2/s) 0 0.000754 0.000306 0.000212 0.000122 0.0000896 0.0000518 0.0000310 0.0000181 0.0000133 0.0000107 0.00000888 0.00000451 0.00000251

C sample (kg/m2/s) 0 0.00066 0.000291 0.000198 0.000117 0.0000864 0.0000511 0.0000302 0.0000176 0.0000131 0.0000103 0.00000864 0.00000439 0.00000241

6 Bridge over Bakar strait We will now explore the possibility that a future arch bridge over Bakar strait could be made of RPC. By building this bridge the highway would be 7 km WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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shorter. The bridge arch span would be 423 m, vertical clearance is 72 m. Bridge has 22 piers (spans are 30+20x38+30 m). Fig. 3 represents the longitudinal section of the bridge. A cross section of the bridge is shown on fig. 4. The cross section is constant and aerodynamic. With the exception of the foundations, abutments, and highest piers, the whole bridge will be build from pre-cast segments 3.8 m long. To achieve symmetric load bearing structure will be supported from both sides. Segments that weight 718 kN are being composed and pre-stressed in pressurising station behind the abutments and then pushed. Computer simulation of the Bakar bridge is presented in fig 5.

Figure 3:

Figure 4:

Figure 5:

Longitudinal section of the bridge.

Cross section of the bridge.

Computer simulation of the bridge.

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Basic design

Calculations are made for stresses during usage, strains during construction, stresses and strains of bearing structure during construction and usage. With the exception of dead and traffic load these calculations cover temperature load, creep, lateral wind, and earthquake load. Table 4 shows maximal stresses for each part of the bridge. Road slab and transverse beams are the only parts of the structure where tensile stresses are about 15 MPa. Due to very high stresses, passive reinforcement with ropes will be used. In transverse beams internal prestressing with addition of passive reinforcement will be used. This conception is not very often with RPC and in future phases of the project it should be carefully considered. Table 4: Part of structure Road slab Transverse beam Highest pier Arch

8

Maximal stresses.

Maximal compression (Mpa) 15.30 10.24 15.36 67.35

Maximal tension (Mpa) 15.30 15.50 6.92 10.61

Conclusion

In this article we have presented the possibility that a future arch bridge over the Bakar strait (Croatia) could be made from RPC. Due to the very high compression strength of RPC it can be used for large spans. RPC also has superb durability parameters as well as a high abrasion resistance and reduced chloride permeability. This makes RPC an ideal material for bridges on the Adriatic cost because durability problems are primary related to the fact that the reinforcement’s protective layer is rapidly being destroyed. Winds that reach up to 250 km/h drift large amounts of chlorides that destroy the bridge structure, primary the arch and the columns. Each of these durability enhancements provided by RPC decrease maintenance costs and lengthen the service life of a structure, which is vital for bridges in the Adriatic region.

References [1] Candrlic, V., Concrete arch bridge over Bakar straits. Proceedings for Conference of Croatian builders, eds. V. Simovic, Cavtat, pp. 358-364, 2001. [2] Jagar, A., High performance concrete, Faculty of civil engineering: Zagreb, 2003.

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[3] Mrakovcic, S., Precast arch bridges made of reactive of powder concrete, Faculty of civil engineering: Zagreb, 2001. [4] Edward Nawy, G., Fundamentals of high-performance concrete, John Wiley & Sons: New York, 2001.

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Elastic and inelastic seismic response comparison of reinforced concrete buildings with normal resistance concrete and with high resistance concrete J. A. Avila1, 2 & D. Rivera2 1

Institute of Engineering, National University of Mexico (UNAM), Mexico 2 Faculty of Engineering, National University of Mexico (UNAM), Mexico

Abstract The elastic and inelastic seismic response of 3, 9, 17 and 25 levels of reinforced concrete buildings (offices), located in the soft soil of Mexico City, with normal resistance concrete (f´c= 250 kg/cm2) and with high resistance concrete (f´c= 700 kg/cm2) are compared. The design with normal and high resistance concrete is made with the RDF-93 and RDF-04 codes respectively. The design results (transversal sections dimensions, vibration periods, lateral displacements, reinforcement steel, etc) are compared, after making a spectral modal dynamic analysis, as well as the non-linear responses (lateral displacements, global and local ductility demands, global distribution of plastic hinges, etc) from the step-by-step dynamic analysis with the SCT-EW record of the 1985 earthquakes.

1

Introduction

The elastic and inelastic behavior of 3, 9, 17 and 25 levels reinforced concrete buildings (offices) is compared; it is designed with the RDF-93 Code [1] for normal resistance concretes and with the RDF-2004 Code [2] for high resistance concretes. The service (the ratios of relative lateral displacement do not exceed of 0.012 times the story high) and failure (given resistances to satisfy the seismic behavior factor Q= 3 requirements) limit states are satisfied according to the Complementary Techniques Norms in both Codes. The structures are designed WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06043

440 High Performance Structures and Materials III for the soft soil conditions (zone III for the RDF-93 and zone IIIb for the RDF2004) in Mexico City. For the design, the spectral modal dynamic seismic analysis is used, considering the elastic-lineal three-dimensional behavior; the vibration periods, maximum lateral displacements, ratios of relative lateral displacements to story high, shear forces, mechanics design elements (internal actions), and longitudinal and transversal reinforcements are compared. Based on these designs, the non-linear response is later determined making step-by-step dynamic analysis in time history, using the SCT-EW record representative soft soil and the bigger damages zone during the 1985 September earthquake in Mexico City. The maximum local ductility in beams and columns are calculated, as well as the global ductility and the tendencies that the failure mechanics develop. Finally, conclusions and recommendations to follow in the practical design of the kind of structures are presented, regarding both concrete resistance type. 1.1 High resistance concretes Some of the high resistance concrete mechanical properties are different from the conventional concrete. Due to this some doubts about if the actual structural design procedures for conventional concretes are also used in high resistances ones. In the Techniques Norms of RDF-2004 it is allowed to use high resistance concretes with values of f´c until 700 kg/cm2, except the structures designed with a seismic behavior factor Q= 4 and members facing flexion-compression that take part of frames resisting more than 50% of the seismic actions and which design axial load (Pu) is bigger than 0.2 PRO, where PRO is the design resistant axial load; in this cases only concretes with f´c until 550 kg/cm2 (55 MPa) can be used. This Norms propose to use the following equation in order to obtain the elasticity modulus in high resistance concretes: Ec= 7700 (f´c)1/2 + 163000 (kg/cm2) or Ec= 2400 (f´c)1/2 + 16300 (MPa) Fig. 1 shows some curves stress-deformation for different values of f´c. Some of the main parameters to consider in high resistance concretes are the following: a) workability, b) permeability, c) volumetric changes, d) durability in useful life. For design consideration purposes, flexion and axial load, many of the design procedures used for the conventional concretes have been applied and in most of the cases good results have been obtained. The maximum unitary deformation value in the concrete εcu= 0.003 adopted in conventional concrete design, has also adequate, but less conservative for high resistance concretes. The equivalent rectangular block of compression stress is proposed, but its application in high resistance concrete members facing to flexion and flexure-compression has been questioned because in the Norms a superior limit of f´c from which such block is invalid has not been specified. The concrete ductility is going to diminish as long as the f´c value increases. The transversal reinforcement presence increases the resistance and ductility of a high resistance concrete column, but in less magnitude than the normal concrete due WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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to the tendency of lateral deformations to be considerably smaller, which produces a less effective confinement. If the axial load is bigger than 0.40 PRO the use of high resistance steel for the transversal reinforcement is recommended, trying to avoid the reinforcement congestion in the structure nodes. The confinement level is directly proportional to the concrete resistance and does not consider the axial load level in the structural element. Therefore in members where the f´c resistance is high, we have a high transversal steel percentage and in consequence construction problems. Despite of the augmentation in concrete costs as its resistance increases, the total cost of the structure will be less, because the concrete resistance augmentation causes an important reduction in the structural elements dimensions as well as in the reinforcement areas.

Figure 1:

2

Curves stress-deformation for several concretes.

Structures description

Two kinds of structures are considered: A case (normal concrete with f´c= 250 kg/cm2 and B case (high resistance concrete with f´c= 700 kg/cm2). The elasticity modulus for both cases is Ec= 14000 (f´c)1/2 (221,359 kg/cm2) and Ec= 7700 (f´c)1/2 + 163000 (366,723 kg/cm2), respectively. The concrete is class 1 with volumetric weight γc= 2400 kg/cm2 and ν= 0.2; the reinforced steel used has a yield stress fy= 4200 kg/cm2. Figs. 2 and 3 show the type plant and transversal cuts of 3 and 9 level buildings; fig. 4 shows the type plant of 17 and 25 level buildings. Also, the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

442 High Performance Structures and Materials III structural characteristics of each building are presented. The floor system is concrete slab type of 10 cm thickness. 2.1 Design criteria

800

800

800

800

800

250

800

250

The gravitational loads effects and those of second order (P-∆) are included in the analysis. For A case the seismic design spectra of soft soil is considered (Zone III) of the RDF-93 and the Zone IIIb spectra of the RDF-2004 for the B case, considering the seismic behavior factor Q= 3 (see fig. 5). The reinforcement steel areas of the structural elements design (beams and columns) is made with the last mechanical elements obtained from the structural analysis for the critical load combination; it is according to the general requirements and ductile frames of the Concrete Norms. These designs include the load factor and the resistance reduction factor.

Figure 2:

SECONDARY BEAMS VIGAS SECUNDARIAS

(CENTIMETERS)

Type plant and three-dimensional 3 level model.

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High Performance Structures and Materials III 1

2 800

3 800

4 800

800

A

800

B

800

C

800

D

E VIGAS SECUNDARIAS SECONDARY BEAMS

(CENTIMETERS)

astic responses ROOF 3.50

al8 sections and reinforcement areas 3.50

s7 of the3.50 structural elements of the transversal sections in B cas is6 reduction is more important in columns, so that the resistanc mpression3.50in the concrete has a bigger influence in the behavior o h5igh axial loads. In the majority of B case elements, the require areas w3.50 ere smaller; only the some external axes columns, th 4 esult similar in both cases. 3.50 3

3.50 s of vibration ntal period 2 3.50 res the fund amental periods of vibration of analyzed A and B cas 1 A case structures tend to be less rigid. 7.50

STREET LEVEL

daLSmental periods of vibration comparison, 3, 9, 17 and 25 leve 3.75 dels (A and B cases) BAS

(METERS)

3.00

CIM

Figure 3:

Type plant and transversal cut, 9 level model.

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444 High Performance Structures and Materials III 2

1

3

800

4

800

800

800

A

800

B

800

C

D VIGAS SECUNDARIAS

SECONDARY BEAMS (CENTIMETERS)

Figure 4:

Type plant, 17 and 25 level models.

Design spectra, zones III and III-b 0.50

Sa/ g

0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

1

2

3

4

5

6

Ti [seconds]

RDF-93,Q= 1

Figure 5:

3

RDF-04, Q= 1

RDF-93, Q=3

RDF-04, Q=3

Design spectra RDF-93 and RDF-2004, Q= 1 and Q= 3.

Design elastic responses

3.1 Transversal sections and reinforcement areas The dimensions of the structural elements of the transversal sections in B case are smaller. This reduction is more important in columns, so that the resistance increase to compression in the concrete has a bigger influence in the behaviour of elements with high axial loads. In the majority of B case elements, the required reinforcement areas were smaller; only the some external axes columns, the reinforcement result similar in both cases. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3.2 Fundamental periods of vibration Table 1 compares the fundamental periods of vibration of analysed A and B case buildings. The A case structures tend to be less rigid. Table 1:

Fundamental periods of vibration comparison 3, 9, 17 and 25 level models (A and B cases).

MODEL 3 9 17 25

Fundamental period of vibration [seconds] X Y θ 0.80 0.80 0.69 0.70 0.74 0.63 1.49 1.45 1.07 1.44 1.41 1.09 1.91 1.91 1.38 1.84 1.84 1.42 2.10 2.10 1.39 1.88 1.88 1.38

CASE A B A B A B A B

Fig. 6 has the September 19th 1985 earthquake SCT-EW accelerogram, used later in the inelastic analysis. Fig. 7 shows the elastic and inelastic response spectra of this record (φ = 5%) and of RDF-2004, as well as the fundamental periods of vibration location (X direction) of all the structures, A and B cases. When the level number increases, the difference between both cases tends to be bigger. 0.20 0.15 0.10 0.05 Sa/g

a/ g

0.00 -0.05 -0.10 -0.15 -0.20 0

5

10

15

20

25

30

35

40

45

50

tiempo [s]

Time (seconds)

Figure 6:

th

September 19 1985 earthquake SCT-EW accelerogram.

3.3 Total maximum lateral displacements In both design cases very similar responses are presented, so that, if the A case structures result less rigid than those of B case is because they were designed with the RDF-93 with a maximum spectral level of 0.40, in comparison with the 0.45 value for the RDF-2004. 3.4 Ratios of relative lateral displacement to story high (drifts) The 3 and 9 level buildings present less drifts because in these the resistance conditions ruled, therefore it was necessary to increase the dimensions in some WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

446 High Performance Structures and Materials III of the structural elements regarding to the required in the service limit state, checking not to exceed the 0.012 permissible limit. 1.20

1.20

1.10

1.10 1.00

1.00

A case

B case

0.90

0.80

0.80

0.70

0.70 Sa/g

Sa/g

0.90

0.60

0.60

0.50

0.50

0.40

0.40

0.30

0.30

0.20

0.20 0.10

0.10

0.00

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

Period (seconds) duct=1

duct=2

Figure 7:

4

duct=3

duct=4

RDF-04 Q=1

2.0

2.5

3.0

3.5

4.0

Periodo [s] Period (seconds)

Periodo [s]

RDF-04 Q=3

duct=1

duct=2

duct=3

duct=4

RDF-04 Q=1

RDF-04 Q=3

Fundamental periods of vibration location regarding the design spectra of the RDF-2004 and of response spectra SCT-EW, A and B cases.

Inelastic step-by-step seismic analysis responses

4.1 Maximum lateral displacements and global ductility demands Generally the inelastic response was less than the elastic, especially in high buildings (17 and 25 levels), because in these there was bigger energy dissipation; in the 3 level building both responses were very similar. Table 2 presents the global ductility maximum demands (µG) developed in each structure; excepting the 3 level building, the mG values are bigger in A case buildings with a bigger energy dissipation than those in B case; it is important to notice that in any case the µG was bigger than the design value of Q= 3. 4.2 Ratios of relative lateral displacement to story high (drifts) The inelastic response diminishes considerably from the elastic, until reaching permissible level values; the A case responses tend to be bigger. Table 2:

MODEL 3 9 17 25

Global ductility demands (mG) 3, 9, 17 and 25 level models (A and B cases). AXE A 3 A C A B A B

∆max [cm] 5.48 5.64 41.12 43.06 51.00 48.98 77.26 76.73

Global ductility demand (µG) A case B case ∆y [cm] ∆max [cm] ∆y [cm] µG 4.82 1.14 5.27 3.74 5.42 1.04 5.90 3.15 17.40 2.36 32.30 13.80 16.70 2.58 29.10 14.90 22.40 2.28 39.88 19.85 18.50 2.65 38.36 18.84 29.30 2.64 45.18 20.61 30.70 2.50 44.10 28.56

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µG 1.41 1.91 2.34 1.96 2.01 2.04 2.19 1.54

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4.3 Ratios of basal shear force-roof lateral displacement Table 3 compares the maximum values of the roof lateral displacement and the basal shear force, elastic and inelastic behavior, of the analyzed structures (3, 9, 17 and 25 levels), A and B cases. Excepting the 3 level building, the ratios of basal shear force-roof lateral displacement result bigger in B case. Table 3: MODEL

Ratios of basal shear force-roof lateral displacement.

AXE

Roof lateral displacements and maximum basal shear forces A case B case V [t] ∆ [cm] ∆ [cm] 5.47 161 5.68 5.45 156 5.27 5.61 60 5.89 5.64 57 5.90 49.30 859 27.90 41.12 382 30.93 44.80 706 28.11 43.05 406 29.11 149.00 2710 93.14 51.00 582 44.81 148.00 2610 92.00 48.97 566 46.58 143.00 3920 107.84 77.26 1261 55.56 142.00 3830 107.57 76.73 1269 55.39

CASE Elas. Inel. Elas. Inel. Elas. Inel. Elas. Inel. Elas. Inel. Elas. Inel. Elas. Inel. Elás. Inel.

A 3 3 A 9 C A 17 B A 25 B Level

V [t] 203 164 56 52 525 331 518 329 1792 424 1757 407 3277 921 3229 910

Level

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

-8

-6

-4

-2

0

2

4

Ductility maximum demand A case

Figure 8:

6

8

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Ductility maximum demand

B case

A case

B case

Local ductility maximum demands (µL) in beams, 17 and 25 level buildings, A and B cases.

4.4 Local ductility maximum demands (µL) in beams and columns and global distribution of plastic hinges The 3 level building does not present yield in columns. The beams in all buildings have very similar responses, independently of the case type; the beams maximum demands are rarely bigger in A case. The µL maximum values in beams as in columns are obtained in the 9 level building, but inside the admissible limits from the practical point of view. Fig. 8 compares the local WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

448 High Performance Structures and Materials III ductility maximum demands developed in the beams of the 17 and 25 level buildings central axes, A and B cases, for illustrative purposes. In all buildings, A and B cases, global distribution of plastic hinges presents a general tendency towards the failure mechanism known as “strong column-weak beam” type. This means that plastic hinges are in the most part of the beams and only in some columns, which is according to the actual design philosophy of RDF-93 and RDF-2004 codes.

5

Conclusions

Use of high resistance concretes (f´c > 400 kg/cm2) in reinforced concrete columns improves in a great deal its axial load capacity; therefore in front of actions where compression high loads predominate its behavior is very adequate; nevertheless, in front of less magnitude loads or even tensions its use results almost no efficient because the resistance is given mainly by the reinforcement steel. The most part of the energy dissipation is in the beams, no matter what type of structure it is. The A case structures present a bigger amplitude in the hysteretic cycles of the ratios of basal shear force-roof lateral displacement, with bigger dissipated energy amount for inelastic deformations. The maximum values of local ductility demands are found inside the permissible by the RDF-93 and RDF-2004. The inelastic response is considerably reduced regarding to the elastic one, except in the 3 level building, where the elastic and inelastic responses tend to be practically the same. In every analyzed case, reviewing the shear forces history, the corresponding resistance is never reached; this is, there is always a resistance reservation, which is bigger in B case buildings, so that for this case a bigger quantity of transversal reinforcement by confinement was required because the design conditions for transversal reinforcement must be more strict as the f´c value increases.

References [1] [2]

Diario Oficial de la Federación, Reglamento de Construcciones del Distrito Federal, 1993. Gaceta Oficial del Gobierno de la ciudad de México, Reglamento de Construcciones del Distrito Federal, 2004.

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The strength effects of synthetic zeolites on properties of high performance concrete P. Frontera1, S. Marchese1, F. Crea1, R. Aiello1 & J. B. Nagy2 1

Department of Chemical Engineering and Materials, University of Calabria 2 Laboratoire de RMN, Facultés Universitaires Notre-Dame de la Paix

Abstract The influence of several kinds of synthetic zeolite as mineral admixtures on the workability and performance strength of mortars cement and concrete has been investigated. In the first series of experiments zeolites have been used to replace 10% of cement in the preparation of mortars. The second series of experiments have regarded the preparation of high performance concrete. In order to evaluate the influence of alkali cations, two different forms of zeolite A have been used with one sodium (NaA or 4A) and the other with calcium (CaA or 5A) and also to evaluate the filling effects of particles zeolites 4A and 5A with different crystals sizes have been used. Furthermore we have used three types of cement with a different content of clinker: CEM I 42.5R, CEM II/A-S 42.5 R and CEM III/A-S 42.5 N. Every prepared sample has been water-cured at 20°C for 2, 7 and 28 days. The results of samples (mortars and concrete) made with zeolites were compared with those obtained with silica fume, usual fine material used in the preparation of high performance strength concrete. Keywords: admixtures, zeolites, mortars, concrete, compressive strength.

1

Introduction

High strength concrete exhibits the rheological, mechanical and durability properties better than those of conventional concrete. The use of high concrete results in many advantages, such as reduction in beam and column sizes and increase in the building height. Moreover high strength concrete can perform much better in extreme and adverse climatic condition, reducing maintenance and repair costs. Silica fume is among one of the most recent pozzolanic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06044

450 High Performance Structures and Materials III materials currently used in high performance concrete production, this material is a by-product of the manufacture of silicon of various silicon alloys. The use of silica fume is very expensive, because it is light and has a low bulk density, which may cause difficult in transporting and handling. The goal of this research was to study production and utilization of alternative materials that is suitable for use in concrete industries as a substitute for the expensive silica fume. The material examined has been the synthetic zeolites (LTA, FAU), with different content of alkali cation (Na+, Ca++) and micronic particles size. The structure of the natural and synthetic zeolitic materials is characterized by a large number of channels and cavities which exhibit a high surface area. The natural zeolites are hydrated alumino-silicate minerals containing alkaline and alkaline-earth metals, formed by the alteration of volcanic ash which is mainly an amorphous, siliceous material. They are considered as materials with high pozzolanic activity for their open structure and chemistry [1-5]. The pozzolanic activity of zeolites depends on the presence of reactive SiO2 and Al2O3 which react with the Ca(OH)2 liberated during the hydratation of cement and converting it into C-S-H gels and aluminates. As a result, the microstructure of hardened cement concrete is improved and the concrete becomes more impervious. In fact, a work of Chan et al., found that the use of natural zeolites as admixtures in concrete concurs to make high performance concrete (HPC concrete with high strength over 80 MPa), that usually are obtained with admixtures such as silica and fly ash [5]. In literature numerous reports exist on the use of natural zeolites in the concrete [7-11], little work instead consider the use of synthetic zeolites in cement industry [12-14]. This is probably linked with the wide availability of natural zeolites that economically concur being more favourable for large scale applications. However, the limit in the use of natural zeolites is in the fact that they have variable chemical compositions depending on their origin, and in addition, variable amounts of impurities can be present. Another limitation is that the natural zeolitic material before being used as additives in concretes must be finely crushed. For this reason we use the synthetic zeolites in form of powders, with fixed chemical compositions, which have also a low cost. In this work the effect of synthetic zeolite A has been examined, in particular the zeolite 4A and 5A, which differ for the presence of Ca++ cations. Moreover, crystals with different sizes have been used in order to evaluate the filling effect of the particles.

2

Experimental

2.1 Materials - Portland cement CEM I 32.5 R with content of clinker >95% (UNI EN 197/1). - Portland- slag cement CEM II/A-S 42.5 R with content of clinker between 80 and 94% and content of slag ranging between 6-20%( (UNI EN 197/1). - Blastfurnace cement CEM III /A-S 42.5 N with content of clinker between 35 and 94% and slag ranging between 6 and 65 % (UNI EN 197/1). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Four different synthetic zeolites were prepared: two of which having small particle size: - zeolite 4A with size of crystals ranging about 1-5 µm (NaAL) (Fig.1). - zeolite 4A with size of crystals ranging about 0.2-1 µm (NaAS) (Fig.2). - zeolite 5A with size of crystals ranging about 1-5 µm (CaAL) (Fig.1). - zeolite 5A with size of crystals ranging about 0.2-1 µm (CaAS) (Fig.2). Silica Fume (SF) (trade name MAPEPLAST SF) produced and supplied by MAPEI (Italy). Coarse Aggregate rounded with 25 mm maximum diameter in accordance with UNI 8520-2. Fine Aggregate is standard sand in accordance with UNI EN 933-3. The Superfludifiant used was acrylic polymers without formaldehyde with 30.5 solid content (trade name Dynamon SP1) produced and supplied by MAPEI (Italy). 2.2 Preparation of zeolites The small crystals of zeolites 4A were prepared starting from the following initial reaction mixture: 2.37Na2O 1Al2O3 1.90 SiO2 62 H2O The synthesis procedure was the following: sodium aluminate was added to a solution of sodium hydroxide solution and after the homogenisation the silica source was added. The solution was stirred for 15 min at room temperature and then submitted to hydrothermal treatment, in static conditions, at 25°C for about 7 days. The large crystals of zeolites 4A were prepared starting from system having the composition: 3.2 Na2O 1Al2O3 1.90 SiO2 95 H2O. The reagents admixture was made as described above. Successively, the solution obtained has been submitted to hydrothermal treatment at 95 °C for 2 h, in stirred conditions. After the hydrothermal treatment every samples are recovered, washed with distilled water, dried at 110°C for 12 hours and successively equilibrated at room temperature before X-ray and microscope analysis. The X Ray analysis confirms the presence of only zeolite A in all samples obtained. Through microscope analysis it is possible to evaluate the different dimensions of crystals obtained (Fig.1, Fig.2). The zeolites 5A were prepared by ionic exchange of zeolite 4A, 250g of 4A have been contacted with 1liter of a 1M aqueous solution of calcium nitrate at room temperature. This exchange procedure was applied three times. The efficiency of exchange has been checked by EDAX (Energy Dispersive X-Ray Analysis), the zeolite 5A obtained has a ratio of ionic content (Ca++/ (Ca+++ Na+)) of about 0.9. 2.3 Preparation of mortars Cement mortars were prepared for determination of compressive strength of the mixtures. Zeolites were used as direct replacement of cement on a weight to weight basis at the levels of 10%. The sand/cement ratio was 3 and the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

452 High Performance Structures and Materials III water/solid ratio was 0.5. To obtain the cement mortars all solid and water was mixed for 5 minutes and then cast in prismatic moulds. The specimens were demolded after 24 hours of moist-curing at 23°C. Companion specimens were placed in water at 20°C after demolding. Compressive strength was determined at designated ages.

Figure 1:

A) SEM micrographs of the small crystal of zeolite A (NaAS, CaAS) B) SEM micrographs of the large crystal of zeolite A (NaAL, CaAL).

2.4 Preparation of concrete The concrete mixtures were prepared at water to cementitious material ratio 0.35. Zeolite was used in addition of cement on a weight to weight basis at the levels of 10%. Also the concrete mixtures containing silica fume were prepared at a water-to cementitious ratio of 0.35. Cube specimens of concrete were cast in 100mmx100mmx100mm standard moulds, respectively. The specimens were removed from the moulds after 1 day and were then cured in water at 20°C until the age of strength testing. The mix proportions of concrete prepared are shown in Table 1. Table 1:

Mixture proportions of concrete.

Mix Proportion (kg/m3) W/C Zeolite water Cement 0,35

50

175

500

Fine aggregate 693

Course Aggregate 1040

Superfluidifiant 5

2.5 Characterization and testing The compressive strength of the concrete and mortars were tested at the ages of 2, 7 and 28 days using a machine (MATEST in accordance with UNI 6686-1,2) the loading rate of 1000 N/mm2sec. Moreover, in the case of preparation of concrete the workability has been measured, following the prescription of UNI 9418, we have executed the slump test on fresh concrete. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3 Results and discussions 3.1 Properties of mortars The results of the compressive strength test of the mortars are shown in Table 2. Table 2:

Compressive strength of mortars. Compressive strength (MPa)

Cement

Zeolites

2 days

7 days

28 days

CEM I

-NaAS NaAL CaAS CaAL SF -NaAS NaAL CaAS CaAL SF -NaAS NaAL CaAS CaAL SF

18.9 17.7 21.9 26.1 25.9 20.3 22.5 20.6 28.0 26.8 30.0 21.1 15.7 15.7 20.4 20.1 22.3 13.5

29.2 21.2 30.1 34.5 33.4 29.5 32.6 28.5 34.3 35.7 37.2 28.6 23.5 20.9 26.7 30.4 32.6 22.9

34.0 25.3 31.3 36.2 39.1 36.4 42.8 39.9 37.7 42.7 42.8 40.5 42.5 23.9 30.5 38.0 40.8 39.0

CEM II/A-S

CEM III/A

The replacement of 10% of CEM I, with most of zeolites, generally improves the compressive strength of the mortars cement, both at early ages and longer. Only the zeolites NaAS and NaAL make worse the compressive strength of the mortars made with CEM I. The mortars prepared with CEM I and Zeolite CaAS and CaAL exhibit compressive strength alike and higher, respectively, than the mortars with CEM I and silica fume. In the case of CEM II/A-S, the replacement of 10% of this type of cement in the preparation of mortars with all zeolites and also with silica fume does not improve significantly the compressive strength at later ages as 28 days, but at early ages it is possible to remark an increase of compressive strength. For the CEM III/A-S, that is a cement with a high content of fly ash with a low rate of hardening, the replacement of 10% of zeolites NaAL, CaAL, CaAS improves the compressive strength at early ages, concurring to transform a cement of slow hardening in to a rapid hardening. The increase in the strength of mortars due to the replacement of zeolites CaAS, CaAL and silica fume can be attributed to the improved aggregate-matrix WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

454 High Performance Structures and Materials III bond associated with formation of a less porous transition zone and a better interlock between the paste and sand. Furthermore, the zeolitic materials with calcium ions play an important role in improving the fine aggregate-paste bond through the formation of more calcium hydrates as showed by microscope analysis (Fig. 2).

Figure 2:

SEM micrograph of fibrous C-S-H (Calcium Silicate Hydrate) in the mortars made with zeolite CaA.

3.2 Properties of fresh concrete The concrete mixes were tested in the fresh state for the workability. The standard test measured the slump of fresh concrete and the results are summarized in figure 3. 300

slum p [m m ]

250 200

CEM I

150

CEM II

100

CEM III

50 0 --

Figure 3:

NaAL CaAS CaAL

SF

Workability of fresh concrete prepared with cements with and without 10% of various admixture.

According to these results, most of the concrete made with zeolites CaAs and CaAL, had a high slump and were workable. In particular, the fresh concrete made with zeolites are more workable than that made with silica fume. This is WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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probably caused by the presence of amount of water in zeolitic materials (20 weight % of zeolites). The concretes made did not show tendencies to segregate or bleed much, because of the high total cementitious contents, including zeolites or silica fume and low water to cementitious ratio. 3.3 Properties of hardened concrete

Compressive strength [MPa]

For investigating the strength development of different concrete mixtures, the results of compressive tests are plotted in Figures 4-6.

Figure 5:

60 50 40 30 20 10 0

0

10

20

30

Days

Compressive strength of concretes prepared with CEM I with and without additions: (■)CEM I, (y) NaAL, (S) CaAS, (T) CaAL, () SF.

Compressive strength [MPa]

Figure 4:

70

70 60 50 40 30 20 10 0

0

10

20

30

Days

Compressive strength of concretes prepared with CEM II/A-S with and without additions: (■)CEM II/A-S, (y) NaAL, (S) CaAS, (T) CaAL, () SF.

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Compressive strength [MPa]

456 High Performance Structures and Materials III

Figure 6:

70 60 50 40 30 20 10 0

0

10

20

30

Days

Compressive strength of concrete prepared with CEM III/A-S with and without additions (■)CEM III/A-S, (y) NaAL, (S) CaAS, (T) CaAL, () SF.

The concrete made with zeolites CEM I and both CAS and CAL exhibits comparable values of compressive strength at 2, 7 and 28 days. They do not have relevant differences caused by the size of the particles, probably due to the fact that at a ratio a/c= 0.35 the pozzolanic effects of zeolites are more important than the effects of particles packing. Also it is possible to notice at early ages a more increment of compressive strength for samples prepared with zeolites instead of silica fume, caused by accelerating effects of zeolites. For the concrete made with CEM II/A-S the addition of zeolites does not improve significantly the compressive strength at 28 days, the greater increment is observed by addition of silica fume. It is probably due to more finesse of CEM II/A-S, that involves a greater mechanical resistance of the samples without admixtures, in fact at 28 days we have obtained compressive strength a little advanced with the fine additions as silica fume. The concretes made with CEM III/A-S and zeolites CaA, exhibit an increment of resistance with respect to concrete without admixture of about 9 %. The greater increments are obtained at the early ages, caused by the alkali content of zeolites that accelerate the process of activation of the fly ash. As expected, in agreement with the results of mortars, the strength of the concretes made with zeolite NaA, was less than all the other mixes at all ages thus reflecting the fact that the sodium content influences negatively the development of hardening properties of cement concrete. The increase in strength in systems of concrete containing zeolite CaAS and CaAL can be explained in a way very similar to the strength increase in mortars mixes. The aggregate-matrix bond improvement induced by zeolites materials is probably the result of a combined filler and pozzolanic effect. The filler effect leads to reduction in porosity of transition zone and provides a dense microstructure and WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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thus increases the strength of the system. The pozzolanic effects help the formation of bonds between the densely packed particles in the transition zone through the pozzolanic reaction with the calcium hydroxide liberated during the hydration of Portland cement to form extra binding calcium silicate hydrates which lead to further increase in the strength.

4 Results and discussions Based on the results of this study it can be concluded that some artificially made zeolites can improve the strength of mortars in a way comparable to the usually addition of silica fume. The increase in the strength can be attributed to the improved aggregate-matrix bond resulting from the formation of a less porous transition zone in the zeolites concrete as silica fume concrete. The zeolites with calcium ions can be used to produce high strength concrete in the range 60-80 MPa 28 days compressive strength, with high workability (slump flow more than 100 mm), using total cementious content of about 400 kg/m3. Therefore, high strength mortars and concrete with 10% calcium zeolite by weight of cement can be produced and marketed to provide technical and economical advantages in specific local uses.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Ortega E.A., Cheeseman C., Knight J., Loizidu M., Properties of alkaliactivated clinoptilolite, Cement Concrete Research 30 (10), pp. 16411646, 2000. Poon C.S., Lam L., Kou S.C., Lin Z., A study on the hydratation rate of material zeolite blended cement pastes, Construction and building materials, 13 (8), pp. 427-432, 1999. Fragoulis D., Chaniotakis E., Stamatakis M.G., Zeolitic tuffs of Kimolos Island, Aegean Sea, Greece and their industrial potential, Cement Concrete Research, 27 (6), 889-905, 1997. Shi C., Day R.L., Pozzolanic reaction in the presence of chemical activators: Part II — Reaction products and mechanism, Cement Concrete Research, 30 (4), pp.607-613, 2000. Chan S. Y.N., Ji X., Comparative study of the initial surface absorption and chloride diffusion of high performance zeolite, silica fume and PFA concretes, Cement Concrete Composites, 21 (4), pp. 293-300, (1999). Taylor H.F.W., Cement Chemistry, Academic Press. London, 1992. Naiqian F., Properties of zeolitic mineral admixtures concrete, in: S.L. Sarkar & S. N. Gosh Eds, Mineral Admixtures in cement and concrete. ABI Books. India pp. 393-447, 1993. Sersale R., Frigione G., Portland-zeolite-cement for minimizing alkaliaggregate expansion, Cement Concrete Research, 17, pp.404-410, 1987. Atkins M., Glasser F. P., Jack J.J., Zeolite P in cements: its potential for immobilizing toxic and radioactive waste species, Waste Management, 15 (2), 127-135, 1995. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

458 High Performance Structures and Materials III [10] [11]

[12] [13] [14]

Fu Y., Ding J., Beaudoin J. J., Zeolite-based additives for high alumina cement products, Advanced Cement Basic Materials, 3 (1), pp. 3742,1996. Gervais C., Ouki S. K., Performance study of cementitious systems containing zeolite and silica fume: effects of four metal nitrates on the setting time, strength and leaching characteristics, Journal of Hazardous Materials 93 (2), pp.187-200, 2002. Mazzasa F., Pozzolana and Pozzolanic Cements in P.C: Hewlett. Editors. Lea’s Chemistry of Cement and concrete. Ed. Arnold, pp.241, 1995. Su N., Fang H.-F., Chen Z –H, Liu F.–S, Reuse of waste catalysts from petrochemical industries for cement substitution, Cement Concrete Research, 30 (11), pp. 1773-1783, 2000. Ujike I., Properties of concrete containing artificial zeolite made by chemical conversion of fly ash in R. K. Dhir. M. J. Mccarthy Editors. Concrete Durability and Repair Technology. Thomas Telford pp.123-131, 1999.

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Behaviour model and experimental study for the torsion of reinforced concrete members C. E. Chalioris Department of Civil Engineering, Democritus University of Thrace, Greece

Abstract The typical behavioural curve of a reinforced concrete element with longitudinal bars and stirrups under torsion comprises two distinct regions; the elastic part until the first cracking and the part after cracking. The different character of the response in these regions reveals the different nature of the load resisting mechanism in each case. The present work addresses an approach that combines two different analytical models in order to predict the entire torsional behaviour. The prediction of the elastic part until the first cracking is achieved using a smeared crack analysis for plain concrete in torsion, whereas for the description of the post-cracking response the softened truss model is used. Further, the results of an experimental investigation on the behaviour of 15 reinforced concrete beams subjected to pure torsion are also presented in this paper. The reported results include the behavioural curves and the values of the initial torsional stiffness, the cracking torque moment and the ultimate torque capacity of the beams. Analyses for the torsional behaviour of the tested beams using the proposed approach were performed and analytical curves are produced and compared with the experimental ones. A good agreement between predicted and experimental results is observed. Keywords: beams, reinforced concrete, smeared crack model, softened truss model, torsion, tests.

1

Introduction

Several theoretical and experimental researches have emphasized the differences between the pre-cracking and post-cracking behaviour of reinforced concrete members in pure torsion [1]. The first, pre-cracking part is characterized by the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06045

460 High Performance Structures and Materials III high value of torsional stiffness. The element behaves as a homogeneous one and the influence of reinforcement is of minor importance. The second, after cracking part is characterized by a further increase of the torque moment at a lower rate, depending on the volume of the transversal reinforcement. The consistently decreasing torsional stiffness reveals the different nature of the bearing mechanism. This justifies the fact that theories for the prediction of the ultimate torsional strength of reinforced concrete elements generally fail to describe the elastic response and to predict the torque moment at cracking. Application of the two well-known space truss models to reinforced concrete elements under torsion exhibits very reliable results for the prediction of the post-cracking behaviour and the ultimate torque. The space truss model with spalling of concrete cover [2] and the space truss model with softening of concrete [3, 4] are experimentally verified and they can predict the ultimate torsional strength, the angle of twist, the steel and concrete strains throughout the post-cracking response of reinforced concrete beams, assuming that the section is cracked from the beginning. However, deficiency of the space truss models consists the fact that predictions of the elastic stiffness of analytical torque - twist curves lies considerable below the test curves in all the examined cases in the literature [3, 5–9]. The elastic torsional behaviour till the development of the first cracks of concrete of a reinforced concrete element is quite similar to a plain concrete member response. Thus, the ultimate torque moment of a plain concrete element is approximately equal to the torque moment at cracking of the same element with longitudinal bars and stirrups. The classical Saint Venant theory to the torsion problem of plain concrete members, although it properly describes the elastic behaviour, fails to predict the ultimate torsional strength even in the case of plain concrete [3, 10]. Recently, a new method for the analysis of plain concrete elements in torsion has been proposed by Karayannis [10]. This approach is using an efficient numerical scheme for the torsional analysis of concrete that although initially is based on the elastic theory, it utilizes a special numerical technique properly modified to include the smeared cracking approach. Extensive comparisons between analytical results yielded by this smeared crack analysis and experimental data derived from a broad range of parametrical studies established the validity of this analytical model [11–13]. This paper reports experimental results of tests on 15 beam specimens subjected to pure torsion and addresses the combination of two analytical models for the prediction of the entire torsional behaviour of reinforced concrete elements. The prediction of the elastic behaviour and the estimation of the torque moment at cracking are achieved using the smeared crack analysis for plain concrete in torsion [10], whereas for the description of the post-cracking response and the calculation of the ultimate torque moment the softened truss model developed by Hsu and Mo [3, 4] is used. Analyses for the prediction of the torsional behaviour of the tested beams using the proposed approach were performed. Predicted torsional behaviour curves are produced and compared with the experimental ones. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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461

Experimental program

The experimental program comprises 15 beams of rectangular cross-section sorted in five groups (R4, R6, RH4, RH6 and RH8) and tested under pure torsion action. The cross-section dimensions of the beams of the groups R4 and R6 were 10/20 cm (height to width ratio h/b = 2), whereas the beams of the groups RH4, RH6 and RH8 had cross-section dimensions 10/30 cm (h/b = 3). Each group comprises three specimens. The transversal reinforcement used in these three specimens was 8 mm diameter closed stirrups at a uniform spacing of 200 mm, 150 mm and 100 mm, respectively for each beam. The longitudinal reinforcement used in the beams of groups R4 and RH4 was four longitudinal bars of diameter 8 mm (4∅8) located at the corners of the closed stirrups. The longitudinal reinforcement of beams of groups R6 and RH6 comprised 6∅8; four of them located at the corners of the stirrups and the other two bars located at the midheight of the closed stirrups (total longitudinal reinforcement 6∅8 uniformly distributed). Finally, the beams of group RH8 had eight bars of diameter 8 mm (8∅8); four of them located at the corners of the stirrups and the other four bars were uniformly distributed at the height of stirrups sides. Steel yield strength was 518 MPa for the high bond longitudinal steel bars and 365 MPa for the mild plain steel stirrups. Reinforcement arrangement for the tested beams is also presented in Table 1. Table 1: Group R4

R6

RH4

RH6

RH8

Reinforcement and concrete strength of tested beams.

Beam code name R4-20 R4-15 R4-10 R6-20 R6-15 R6-10 RH4-20 RH4-15 RH4-10 RH6-20 RH6-15 RH6-10 RH8-20 RH8-15 RH8-10

Longitudinal bars 4∅8

6∅8

4∅8

6∅8

8∅8

Stirrups ∅8/20 cm ∅8/15 cm ∅8/10 cm ∅8/20 cm ∅8/15 cm ∅8/10 cm ∅8/20 cm ∅8/15 cm ∅8/10 cm ∅8/20 cm ∅8/15 cm ∅8/10 cm ∅8/20 cm ∅8/15 cm ∅8/10 cm

fc (MPa)

fsp (MPa)

20.96

2.89

24.59

3.33

26.56

3.04

24.90

3.42

27.39

3.09

The cement used in this experimental work was a locally manufactured general purpose ordinary Portland type cement. Sand with a high fineness WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

462 High Performance Structures and Materials III modulus and coarse aggregates with a maximum size of 9.5 mm were used. The concrete mixture was made using cement, sand and crushed aggregates in a proportion 1:2.8:1.2, respectively, and water to cement ratio equal to 0.43. Also included in Table 1 are the concrete compressive and tensile strength values as deduced from supplementary compression and splitting tests, respectively. The total length of the beams was common for all specimens equal to 1.60 m. Experimental setup and instrumentation are shown in Figure 1. All specimens were tested monotonically under pure torsion [14]. They were supported on two roller supports that allowed beam to twist and to elongate longitudinally at both ends. The load was applied on the ends of two steel arms fixed at the end parts of each tested beam, through a steel spreader. The end parts of the beams were properly over-reinforced so that they can bear without cracking the imposed loading. Cracking and, finally, failure localized at the middle part of each beam. The load was imposed consistently in low rate and was measured by a load cell. The average angle of twist per unit length of the tested beams was estimated using two linear variable differential transducers with high accuracy. These two LVDTs measured the opposite deformations of each specimen as it rotates. Initial elastic torsional stiffness, torque moment at cracking and ultimate torque moment, as measured from the tests, are presented in Table 2. Further, the entire torsional behaviour of the beams of groups R4, R6, RH4, RH6 and RH8 is also presented in Figures 2, 3, 4, 5 and 6, respectively, in terms of torque moment versus angle of twist per unit length experimental curves. P

load cell

steel spreader beam LVDT LVDT

160 0

steel arm tested beam roller support

Figure 1:

3

Test setup.

Behavioural model and comparisons with test results

In this study, for the prediction of the entire torsional behaviour of reinforced concrete elements, the combination of two different theories is adopted. The elastic till the first cracking part is described by a smeared crack analysis for WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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plain concrete in torsion [10] and the post-cracking part is described by the wellknown softened truss model [3, 4]. It is justified that for the elastic till the first cracking part the percentage of steel has a minor effect on the torsional response and reinforced concrete elements behave as plain concrete members [1]. Therefore, the analytical smeared crack model for plain concrete in torsion proposed by Karayannis [10] is applicable to reinforced concrete beams for the prediction of the first elastic part till the developing of concrete cracking. This approach is using an efficient numerical scheme for the torsional analysis of concrete that is initially based on the elastic theory and utilizes a special numerical technique properly modified to include the smeared cracking approach. The model is based on an analytical technique that employs constitutive relations expressed in terms of normal stress and crack width, for the behaviour of the crack process zones. Detailed derivation of the equations and the solution technique of this theory can be found in reference 10. Table 2: Beam code name R4-20 R4-15 R4-10 R6-20 R6-15 R6-10 RH4-20 RH4-15 RH4-10 RH6-20 RH6-15 RH6-10 RH8-20 RH8-15 RH8-10

Test results and analytical predictions of the tested beams.

Initial stiffness (×10-3 kNcm2/rad) exp . exp. calc. calc. 3771 3084 3229 3452 3557 3442 5913 6689 6186 6494 7215 6755 7744 7191 6658

3346

3624

6500

6292

6596

1.13 0.92 0.97 0.95 0.98 0.95 0.91 1.03 0.95 1.03 1.15 1.07 1.17 1.09 1.01

Cracking torque (kNcm) exp . exp. calc. calc. 217.05 201.25 200.79 240.12 259.63 264.60 322.00 364.54 418.78 378.70 355.10 372.10 345.27 330.37 368.30

204.65

235.89

337.82

379.97

343.39

1.06 0.98 0.98 1.02 1.10 1.12 0.95 1.08 1.24 1.00 0.93 0.98 1.01 0.96 1.07

Ultimate torque (kNcm) exp . exp. calc. calc. 238.49 264.87 325.40 287.28 318.38 374.22 394.76 501.27 583.42 481.14 586.91 661.63 503.69 611.95 694.97

263.26 277.52 292.30 288.75 322.81 346.65 439.99 500.89 525.01 444.21 509.88 541.61 462.20 551.14 603.72

0.91 0.95 1.11 0.99 0.99 1.08 0.90 1.00 1.11 1.08 1.15 1.22 1.09 1.11 1.15

Among the various theories available in the literature, researchers have a common consensus that the space truss models (the spalled and the softened truss model) are the most rational and powerful models for dealing with torsional problems. In the present study, the theory of the softened truss model is used [3], [4]. This method relies on solving three equilibrium and three compatibility equations along with the constitutive laws of an element taken from a member subjected to pure torsion. The equations of equilibrium and compatibility are based on the assumption that the material is continuous. Especially for the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

464 High Performance Structures and Materials III concrete in compression, it is considered that concrete struts strength is greatly reduced by the diagonal cracking caused by tension in the perpendicular direction (concrete softening). For further details refer to references 3 and 4. 500

Experimental results

Torque moment (kNcm)

2 8

Smeared crack analysis Softened truss model

8/s

400

Proposed approach 2 8

300

200

100 s = 20 cm 0

0.0

0.5

s = 15 cm

1.0 1.5 0.0

0.5

1.0

s = 10 cm 1.5 0.0

-3

0.5

1.0

1.5

Angle of twist per length (x 10 rad/cm) Figure 2:

Experimental and calculated curves of the beams of group R4.

600

Experimental results

2 8

Torque moment (kNcm)

500

Smeared crack analysis Softened truss model

8/s

2 8

Proposed approach

2 8

400 300 200 100 0

s = 20 cm 0.0

0.5

1.0 1.5 0.0

s = 15 cm 0.5

1.0

s = 10 cm 1.5 0.0

0.5

1.0

1.5

-3

Angle of twist per length (x 10 rad/cm) Figure 3:

Experimental and calculated curves of the beams of group R6.

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High Performance Structures and Materials III

900

Experimental results

2 8

800

465

Smeared crack analysis

Torque moment (kNcm)

Softened truss model

700

Proposed approach

8/s

600 2 8

500 400 300 200

s = 20 cm

100 0

0.0

0.5

s = 15 cm

1.0 1.5

0.0

0.5

1.0

s = 10 cm 1.5 0.0

0.5

-3

1.0

1.5

Angle of twist per length (x 10 rad/cm) Figure 4:

Experimental and calculated curves of the beams of group RH4.

1000

Torque moment (kNcm)

Experimental results

2 8

900

Smeared crack analysis Softened truss model

800

Proposed approach

2 8 8/s

700 600

2 8

500 400 300 200 s = 20 cm

100 0

0.0

0.5

1.0

s = 15 cm 1.5 0.0

0.5

1.0

s = 10 cm 1.5 0.0

-3

0.5

1.0

1.5

Angle of twist per length (x 10 rad/cm) Figure 5:

Experimental and calculated curves of the beams of group RH6.

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466 High Performance Structures and Materials III Analyses for the prediction of the torsional behaviour of the tested beams using the proposed combined approach were performed. Smeared crack analysis was performed in order to determine the elastic response and the torque moment at cracking, whereas the softened truss model was performed for the prediction of the post-cracking response and the ultimate torque moment. The predicted values of the initial torsional stiffness, the torque moment at cracking and the ultimate torque moment are presented and compared with the measured ones in Table 2. Furthermore, analytical torque curves for the torsional behaviour of the beams of groups R4, R6, RH4, RH6 and RH8 are presented and compared with the experimental ones in Figures 2, 3, 4, 5 and 6, respectively. 1000

Torque moment (kNcm)

Experimental results

2 8

900 800

2 8

700

2 8

600

2 8

Smeared crack analysis Softened truss model

8/s

Proposed approach

500 400 300 200 s = 20 cm

100 0

0.0

0.5 1.0 1.5 0.0

s = 15 cm 0.5

1.0

s = 10 cm 1.5 0.0

-3

0.5

1.0

1.5

Angle of twist per length (x 10 rad/cm) Figure 6:

4

Experimental and calculated curves of the beams of group RH8.

Conclusions

The following concluding remarks are drawn from the results reported herein. Volume of the transversal and the longitudinal reinforcement significantly affects the torsional behaviour of the tested beams, as it was expected. Two distinct regions can be observed in a typical experimental torque – twist curve of the tested reinforced concrete beams. The different character of the response in these regions reveals the different nature of the load resisting mechanism in each case. In order to describe the entire torsional behaviour of the tested beams and based on the above observation, the combination of two different theories is adopted. For the estimation of the elastic till the first cracking response, a smeared crack analysis for plain concrete in torsion is used, whereas the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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prediction of the post-cracking behaviour is achieved using the softened truss model. Comparisons between the predicted torsional behaviour and the measured one showed that this combined approach yields realistic torsional curves for the entire response of the element and is capable to predict with satisfactory accuracy the initial torsional stiffness, the torque moment at cracking and the ultimate torsional strength.

References [1]

Hsu, T.C., Unified Theory of Reinforced Concrete, CRC Press, Inc., Boca Raton, Fla., 1993. [2] Mitchell, D. & Collins, M.P., Diagonal Compression Field Theory – A Rational Model for Structural Concrete in Pure Torsion, ACI Proceedings, 71(8), pp. 396-408, 1974 [3] Hsu, T.C. & Mo, Y.L., Softening of Concrete in Torsional Members – Theory and Tests, ACI Proceedings, 82(3), pp. 290-303, 1985. [4] Hsu, T.C., Toward a Unified Nomenclature for Reinforced-Concrete Theory, Structural Engineering, ASCE, 122(3), pp. 275-283, 1996. [5] Mansur, M.A., Nagataki, S., Lee, S.H. & Oosumimoto, Y., Torsional Response of Reinforced Fibrous Concrete Beams, ACI Structural, 86(1), pp. 36-44, 1989. [6] Wafa, F.F., Shihata, S.A., Ashour, S.A. & Akhtaruzzaman, A.A., Prestressed High-Strength Concrete Beams under Torsion, Structural Engineering, ASCE, 121(9), pp. 1280-1286, 1995. [7] Rahal, K.N. & Collins, M.P., Simple Model for Predicting Torsional Strength of Reinforced and Prestressed Concrete Sections, ACI Structural, 93(6), pp. 658-666, 1996. [8] Ashour, S.A., Samman, T.A. & Radain, T.A., Torsional Behavior of Reinforced High-Strength Concrete Deep Beams, ACI Structural, 96(6), pp. 1049-1058, 1999. [9] Chalioris, C.E., Study of the Behaviour and the Failure Mechanisms of Plain and Reinforced Concrete Elements in Torsion, PhD dissertation, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece, 1999. [10] Karayannis, C.G., Smeared Crack Analysis for Plain Concrete in Torsion, Structural Engineering, ASCE, 126(6), pp. 638-645, 2000. [11] Karayannis, C.G., Nonlinear Analysis and Tests of Steel-fiber Concrete Beams in Torsion, Structural Engineering and Mechanics, 9(4), pp. 323-338, 2000. [12] Karayannis, C.G. & Chalioris, C.E., Experimental Validation of Smeared Analysis for Plain Concrete in Torsion, Structural Engineering, ASCE, 126(6), pp. 646-653, 2000. [13] Karayannis, C.G. & Chalioris, C.E., Strength of Prestressed Concrete Beams in Torsion, Structural Engineering and Mechanics, 10(2), 165-180, 2000.

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468 High Performance Structures and Materials III [14]

Chalioris, C.E., Cracking and Ultimate Torque Capacity of Reinforced Concrete Beams, Proc. of the Int. Symposia Celebrating Concrete: People and Practice, Vol. Role of Concrete Bridges in Sustainable Development, University of Dundee, Scotland, UK, pp. 109-118, 2003.

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A study on use of blended ferrocement: a high performance material for repair/strengthening of brick masonry columns T. Kibriya National University of Sciences and Technology, Risalpur, Pakistan

Abstract The use of ferrocement, containing blended cement, to repair and enhance the strength of brick masonry columns of old buildings and structures showing distress has been investigated for use as a rehabilitation/strengthening material. Intensive experimental study has been carried out on plain brick masonry columns and brick masonry columns with single, double and triple layers of ferrocement, with/without the use of chemical bonding agents. Use of blended cements containing agricultural waste i.e. rice husk ash have not as yet been investigated at all for use in ferrocement. Blended cement containing 75% OPC and 25% rice husk ash has now been investigated for use in ferrocement. Ferrocement containing blended cement has been observed to substantially increase the load carrying capacity of brick masonry columns along with decreasing the strains by about 50%. Strength improvements of up to 160% in the load carrying capacity of columns with single layered ferrocement have been observed whilst about 100% strength improvements have been observed in columns with 2/3 layers of ferrocement applications. The slope of the stress-strain curve resembles typical curves characteristic of high strengths. The results of this study have been highly encouraging and the use of ferrocement with blended cement is recommended for the repair and rehabilitation/strengthening of brick masonry columns. The use of rice husk ash for blending with ordinary Portland cement produces a stronger, impermeable and durable material in addition to ease of in-situ application, reducing the disposal problem of this agricultural waste and reducing the costs of repairs. Keywords: strengthening, rehabilitation, brick masonry columns, blended cement, old buildings, retrofit material.

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470 High Performance Structures and Materials III

1

Introduction

Many old buildings along with few bearing historical architecture of past civilisations were constructed by using brick masonry. Most of these old buildings/structures have shown signs of distress/damage due to aging and action of various offensive agencies. To maintain the safety requirements, most such structures now require replacement/ rehabilitation/ strengthening. Strengthening of existing columns may also be required for expansion works in the existing buildings/structures. With the trend of conserving the heritage and specific architecture/ façade along with avoiding environmental/disposal problems of demolition waste, stress is laid on rehabilitation/ strengthening of the existing elevations/structures. Resource conservation and paucity of development funds in developing/under developed countries suggests economical options of rehabilitation/ strengthening of existing structures. To maintain economy and environment, use of waste materials in concrete have been investigated widely. Lot of work has been done on use of waste brick aggregates (widely available from old dilapidated buildings in the South Asian region) in normal/high strength concrete by Kibriya [1–4] and use of agricultural waste in blended cement and normal/high strength concrete Kibriya [5–8]. Ferrocement carries certain advantages in rehabilitation/strengthening works like in-situ application, ease of construction, use of locally available materials, no formworks, no special skills/training required, easily moulded in any shape, very low permeability, easily repairable, resource saving and durability vide Kibriya [9, 10], Paul and Pama [11], ACI [12] and BOSTID [13]. The repair and strengthening of existing structures has been a more challenging problem for civil engineers than construction of new structures. Use of blended cements containing agricultural waste i.e. rice husk ash have not at all been investigated as yet for use in ferrocement. Higher content of silica found in rice husk ashes along with low calcium oxide content is advantageous as far as its cementing properties are concerned. Blended cements containing rice husk ash as a partial replacement material have been found to improve strength and durability characteristics substantially when used as cementitious material and in high strength concretes for various purposes. Blended cement containing 75% OPC and 25% rice husk ash has now been investigated for use in ferrocement. Selection of 25% replacement of OPC with rice husk ash has been on the basis of excellent performance of such blended cements and concrete in the recent studies by Kibriya [5–8].

2 Research methodology This experimental investigation was carried out on nine different groups of columns. Each group consisted of three specimen 225x225x750mm. The groups were labeled in alphabetical order and the individual specimens were thus numbered as 1, 2 and 3 and so on. 1st Group i.e Group A comprised of plain brick masonry in 1:4 cement mortar and were considered as control specimen. Group B comprised of brick masonry columns in 1:4 cement mortar with WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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12.5mm thick, 1:4 cement mortar plaster. Group C were brick masonry columns in 1:4 cement mortar, simply plastered with 1:2 cement mortar. Groups D, E and F comprised of brick masonry columns in 1:4 cement mortar with one, two and three layers of ferrocement applied respectively. Each layer of ferrocement applied was 15mm thick applied in two steps i.e. a layer of 7mm 1:2 cement mortar, application of wire mesh and another 8mm layer of 1:2 cement mortar. W/C ratio of mortar was maintained at 0.3. Blended cement containing 25% rice husk ash and 75% ordinary Portland cement was used. Rice husk ash used for blending was obtained by burning rice husk in an industrial furnace with temperatures maintained around 500 to 6000C. Ashes were then cooled to 200C and subsequently ground in a laboratory ball mill for about 120 minutes and thereafter sieved through a #325 sieve. Ashes passing 100% through the sieve were then used for blending with ordinary Portland cement. Sand with medium grading was used for mortar. Upto 1% high range water reducing admixture was used for preparing workable mortar mix. Wire mesh of expanded metal having openings of 15mm and wire thickness of 1mm was applied with yield strength of 227 N/mm2 and ultimate strength of 379 N/mm2. 1mm thick, 2.5cm long nails were used to bind wire mesh layers to the core. Groups G, H, and I comprised of specimen similar to groups D, E and F with the exception that a chemical bonding agent Sika-Latex was used in addition to nails to bind the ferrocement layer to the core. The specimen were moist cured for 28 days at a temperature around 200C before testing. The details are shown in Table 1.

3

Testing procedure

The column specimen were tested for compression using universal compression testing machine with a 255x255mm, 5mm thick plate on top and bottom. The observations were made for stress-strain behavior, crack pattern, cracking and failure loads. The measurements for strain were done with the help of two compressometers attached to the specimen. The results of the tests are tabulated in Table 2. Permeability testing was carried out by the help of capillary rise test according to RILEM CPC13 and Initial Surface Absorption (ISAT) tests according to BS 1881. In addition, 50mmx50mm cubes were caste, moist cured for 30 days and immersed in water for 30 days to observe any weight changes.

4

Analysis of test results

The ultimate failure and cracking load capacities of the specimen are given in Table 2. The stress-strain curves and trend lines based on the average readings of the specimen groups are shown in Figures 1 and 2. 4.1 Brick masonry columns (group A) The failure loads were generally identical. The average failure load for case A was 142kN. The crack initiation started at about 70% of the failure loads. Major cracks were vertically oriented, however, a few horizontal cracks were also WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

472 High Performance Structures and Materials III observed. The crack widening and propagation was quite fast after their first appearance. The stress - strain behaviour of this group is shown in Figs 1and 2. Table 1: Type of Specimen

Details of specimen.

Group Designation A B

No of Layers _ _

Bonding Agent _ _

Mortar Ratio _ 1:4

Masonry Col Masonry Col With Plaster Masonry Col with C _ _ 1:2 Plaster Masonry Col + D 1 Nails 1:2 Ferro cement Masonry Col + E 2 Nails 1:2 Ferro cement Masonry Col + F 3 Nails 1:2 Ferro cement Masonry Col + G 1 Nails & 1:2 Ferro cement S. Latex Masonry Col + H 2 Nails & 1:2 Ferro cement S .Latex Masonry Col + I 3 Nails & 1:2 Ferro cement S. Latex Note:• Brick Masonry in 1:4 Cement Mortar. • One layer of ferrocement mortar 15mm thick with 1:2 cement mortar. • Wire mesh with 1mm wire, 15mm mesh. • Cementitious material – 75% OPC + 25% Rice Husk Ash • W/C ratio 0.3 of ferrocement. • Moist curing - 28 days 4.2 Brick masonry columns with 1:5 cement mortar plaster (group B) The results of all specimen of group B clearly suggest that simple mortar application can increase the failure loads by up to 35%. However, it seems to have no effect on crack initiation and it’s growth as the crack growth was rapid as in case of group A. The cracks generally started at about 70% of failure load. The average failure load was 195kN as given in Table 2. The strains also reduced somewhat, as compared to plain brick masonry columns as shown in Figure 1. 4.3 Brick masonry columns with 1:2 cement mortar plaster (group C) The specimen of group C exhibited fairly good results in terms of failure loads. The average failure load was calculated as 208kN. This clearly shows an increase of about 46% over the failure load of the control specimen of Case A WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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and about 10% increase over columns plastered with 1:5 mortar ratio. The stressstrain curve is fairly straight with reduced strains as shown in Figure 2. Table 2: Type of Specimen

Group

Masonry Col

A

Masonry With Plaster

Col

B

Masonry Col with Plaster

C

Masonry Col + Ferro cement (1 Layer) Masonry Col + Ferro cement (2 Layers) Masonry Col + Ferro cement (3 Layers) Masonry Col + Ferrocement(1) Bonding Agent Masonry Col + Ferrocement(2) Bonding Agent Masonry Col + Ferro cement(3) Bonding Agent

D E F G H I

Summary of test results. First cracking Load (kN) 86.19 113.00 104.10 146.40 124.60 157.10 142.00 128.00 146.60 187.30 196.30 212.20 157.30 186.00 175.80 202.10 195.80 186.40 204.70 192.10 224.20 162.50 197.60 166.26 142.90 189.70 178.80

Failure Load (kN) 138.4 141.2 146.8 192.1 198.3 196.4 211.1 208.4 204.7 340.9 394.2 367.2 299.4 268.6 279.5 310.4 301.5 270.8 354.8 396.2 377.1 280.4 359.7 260.9 281.7 320.1 318.4

Mean Failure Load (kN)

% Increase

142.13

Control

195.6

37.6

208.1

46.5

367.4

158.7

282.5

98.9

294.3

107.3

376

164.8

300.3

111.5

306.7

116

4.4 Brick masonry columns with single layer of ferrocement (group D) The specimen of this group having one layer of ferrocement showed a marked difference in results by increasing the failure load to 367kN, an increase of 158% over the control. The crack propagation was not rapid and widening of cracks occurred gradually thus showing a ductile failure. The increased failure load and mode of failure clearly suggests usefulness of ferrocement in strength enhancement of brick masonry columns. Failure strains were observed to be around 60% of the strains in the control.

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474 High Performance Structures and Materials III A

B

C

G

H

I

D

E

F

1000 900 800 700

STRESS

600 500 400 300 200 100 0 0

0.0005

0.001

0.0015

0.002

0.0025

0.003

STRAIN

Figure 1:

Average stress-strain curves of all groups.

4.5 Brick masonry columns with double/triple layer of ferrocement (groups E and F) The specimen of groups E and F treated with ferrocement having 2 and 3 layers respectively of wire meshes did not give results better than group D having one layer of ferrocement. The ultimate loads were 282kN and 294kN respectively for 2 and 3 layers of ferrocement, an increase of about 100% in the load carrying capacity of columns as compared to 158% increase due to a single layer of ferrocement application. The absence of higher contribution by ferrocement with more meshes was due to budging and early bond failure between the mesh layers and smaller size of top and bottom steel plates (255mm) than the over all cross section of the samples having multi-layers of wire meshes. The stress-strain curve is shown in Figures 1 and 2.

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1200

1000

STRESS

800

600

400

200

0 0

0.0005

0.001

0.0015 STRAIN D

0.002

0.0025

0.003

A

B

C

E

F

G

H

I

Linear (A)

Linear (B)

Linear (C)

Linear (D)

Linear (E)

Linear (E)

Linear (F)

Linear (G)

Linear (H)

Linear (I)

Figure 2:

Trend line - stress-strain curves.

4.6 Brick masonry columns with single layer of ferrocement and bonding chemical applied (group G) In the course of samples preparation of this group, a bonding agent, Sika-Latex was used in addition to nailing the mesh layer. The average failure load 376kN of this group although significantly higher i.e. 164% than the control specimen, did not show much improvement over its corresponding group D of samples having one layer of mesh nailed to the core, without any bonding agent. However, the Sika-Latex bonding was effective in controlling the spalling of outer mortar layer on failure. The failure strains were almost 10% lower than the specimen with single layer of ferrocement without any bonding agent. The propagation of cracks upto failure was gradual.

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476 High Performance Structures and Materials III 4.7 Brick masonry columns with double/triple layer of ferrocement and bonding chemical applied (groups H and I) The columns with 2/3 layers of ferrocement application failed at 300kN and 306kN respectively showing 111% and 116% strength improvements over the control specimen. The specimen of these groups did not show much improved results over similar groups E and F specimen having same number of meshes but without application of bonding agent. Sika-Latex agent was not effective in maintaining bond between the mesh layers. The stress-strain diagrams are shown in Figures 1 and 2. 4.8 Permeability testing on ferrocement specimen The capillary rise test indicated that ferrocement used was watertight. The w/c ratio of ferrocement mortar used was 0.3 which theoretically should also be watertight. Same results were confirmed by ISAT. Due to impermeability of ferrocement, it is expected to be highly durable in offensive environments. Weight recordings on 50mm cubes kept in water for 30 days indicated constant weight before and after the soaking period indicating the ferrocement was watertight and impermeable.

5

Discussion of results

From the summary of these results it can be clearly observed that the load carrying capacity of ferrocement encased brick masonry columns increased significantly. It is evident that a significant increase in failure load capacity can be achieved using only one layer of ferrocement. Failure strains were observed to be reduced by about 50% on application of single layer of ferrocement. Additional layers (2 or 3) of ferrocement did not contribute much due to bond failure between them and while bulging outwards these layers pulled the inner mesh along and thus affecting the results. The contribution of Sika-Latex bonding agent was not much except for controlling the spalling of mortar at failure. The permeability of ferrocement layer is negligibly low thereby increasing the durability of the ferrocement applied columns. It has been observed that ferrocement containing blended cements has performed even better than ferrocement with ordinary Portland cement on comparing results of recent studies. Strength improvement of about 30% along with complete watertightness in ferrocement with blended cement have been observed. The results of this study have been highly encouraging and use of ferrocement, with blended cement containing 25% rice husk ash, for repair and rehabilitation/strengthening of brick masonry columns has been recommended for practical use in the field. Use of rice husk ash for blending with ordinary Portland cement produces a stronger and durable material in addition to ease of in-situ application, reducing the disposal problem of this agricultural waste and reducing the costs of repairs. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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477

Conclusions

From the above study the following conclusions are drawn:• The ferrocement (using blended cement) treatment enhances the load carrying capacity of brick masonry columns by as much as 150% to 160%. This increase can be economically achieved with one layer of wire mesh used in ferrocement nailed to core and with or without application of any chemical bonding agent. There were almost 50% reduction in strains at failure and much improved modulus was observed. • Ferro cement having 2 and 3 layers of wire meshes are uneconomical and did not appreciably improve the load carrying capacity as noted in this experimental study. • The initial resistance to cracking and thereafter crack growth mechanism of brick masonry columns improved quite significantly due to ferrocement treatment. • The permeability of ferrocement layer is negligibly low thereby increasing the durability of the ferrocement applied columns.

References [1]

[2] [3] [4] [5]

[6]

[7] [8]

Kibriya. T., Crushed burnt clay bricks as aggregates for histrength concrete, Proceedings of McMat 2005, Joint ASCE/ASME/SES Conference on Mechanics and Materials, Baton Rouge, Louisiana, USA, 2005. Kibriya, T., Properties of Concrete with crushed brick aggregates, Dissertation submitted for partial fulfillment of Ph.D requirement, City University, London, U.K. 1991. Kibriya, T. and Speare, P.R.S., The use of crushed brick coarse aggregates in concrete, International Congress on “Concrete in the Service of Mankind”, Dundee, Scotland, pp 204 – 207, 1996. Kibriya. T., Durability of concrete with crushed bricks coarse aggregates, 8th Islamic Countries Conference on Statistical Sciences, University of Bahrain, Bahrain, pp 161 – 164, 2002. Kibriya. T. and Baig. N., High performance concrete with blended cements using agrowastes, Proceedings of McMat 2005, Joint ASCE/ASME/SES Conference on Mechanics and Materials, Baton Rouge, Louisiana, USA, 2005. Kibriya. T. and Khan. S., High performance concrete pavements using agrowaste blended cement, Proceedings of McMat 2005, Joint ASCE/ASME/SES Conference on Mechanics and Materials, Baton Rouge, Louisiana, USA, 2005. Kibriya. T. and Khan. S., Performance of ecologically friendly waste in high strength concrete, IV Regional Conference on Civil Engineering Technology, Joint ASCE/ESIE Conference, Cairo, Egypt, 2005. Kibriya. T. and Baig. N., Agricultural wastes in construction, Blended cements containing rice straw ash, IV Regional Conference on Civil WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

478 High Performance Structures and Materials III

[9] [10]

[11] [12] [13]

Engineering Technology, Joint ASCE/ESIE Conference, Cairo, Egypt, 2005. Kibriya. T., Ferrocement as retrofit material for old structures, Proceedings of McMat 2005, Joint ASCE/ASME/SES Conference on Mechanics and Materials, Baton Rouge, Louisiana, USA, 2005. Kibriya. T., Ferrocement – An innovative material for rehabilitation/strengthening of brick masonry columns, IV Regional Conference on Civil Engineering Technology, Joint ASCE/ESIE Conference, Cairo, Egypt, 2005. Paul. B.K. and Pama. R.P., Ferrocement, 1989. ACI 549, Ferrocement, American Concrete Institute, 1988. National Academy of Sciences, Ferrocement - Application in Developing Countries, A report of an Adhoc Panel of the advisory Committee on technological innovation, BOSTID, Washington, D.C., 1973.

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Section 6 Damage and fracture mechanics

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Crush behaviour of open cellular lattice structures manufactured using selective laser melting M. Santorinaios, W. Brooks, C. J. Sutcliffe & R. A. W. Mines Department of Engineering, University of Liverpool, England, UK

Abstract Open cellular lattice structures, made out of stainless steel, have been manufactured using the micro fabrication procedure of selective laser melting. A single geometry has been considered, namely vertical struts with cross bracing with a cell geometry of a cubic form. Blocks of this material, with dimensions of 25 mm cube, have been tested under compression and shear loading. Three cell sizes have been considered, namely 1.25, 2.5 and 5 mm. Uniaxial compression and shear force versus displacement and failure characteristics of these materials are discussed, In this way, the micro mechanical characteristics of the material and manufacturing parameters are related to bulk properties. Keywords: cellular, micro fabrication, selective laser melting, mechanical properties.

1

Introduction

Foams have been used for a number of years as core materials in sandwich construction for transport and general engineering systems. There is a wide variety of foam ranging from low density, high end technology applications, e.g. the Polymethacrylimide foam Rohacell 51WF for aerospace use [1], to high density/low end technology applications, e.g. the aluminium foam Alporas for civil engineering use [2]. The usual design parameters for sandwich construction focus on elastic response (specific stiffness and strength) [3], but more specialised behaviours include fatigue behaviour, crashworthiness and foreign object impact performance [3, 4]. The mechanical behaviour of sandwich structures is dependent on the mechanical properties of the foam core and composite skin. As far as the foam core is concerned, the core can be subject to WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06047

482 High Performance Structures and Materials III multi axial stresses and progressive failure in applications such as foreign object impact loading of sandwich panels [4]. The properties of foam cores are dependent on the parent material and the cell geometry [5]. Hence, for a given parent material, properties are adjusted by varying the foam density. This means that foams tend to be quasi isotropic, and that they tend to have a restricted range of properties. It should also be noted that low end foams have highly variable microstructure, which means that sophisticated modelling of mechanical behaviour is negated by the variation in bulk properties [4]. There is currently large interest in creating cellular structures with tailored cellular properties. These can be closed cell, open cell and framework type geometries. A number of manufacturing procedures are possible. This includes investment casting, deformation forming, brazing etc. [6]. Limitations of such techniques include the fineness of the structure, and the actual cell geometry. From the theoretical point of view, research is currently underway that addresses the analysis and optimisation of lattice structures in the context of core materials in sandwich construction [7]. Rapid prototyping techniques have been developed over the years to create complex geometries [8]. Selective laser melting is a new technique which allows the creation of metal structures with fine detail [9]. The technique that has been developed has the potential to create open cellular lattice structures with features with a resolution of 50 micro meters [10].

2

Selective laser melting of open cellular lattice structures

A schematic of the selective laser melting process is shown in Figure 1. The metallic powder is levelled off using a plate. A pre-programmed laser then traverses the powder surface, selectively melting areas of the surface. A new layer of powder is then deposited, and the process is repeated. In this way, a structure can be created in 50 micrometer layers.

Figure 1:

The Selective Laser Melting manufacturing process [9].

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There are a large number of possible geometries possible, which gives large design freedom. The laser spot has a size of 50 micrometers, and any structural feature can be built to this resolution, as long as the newly solidified material can be built on top of the previous (solidified) layer. Figure 2 shows two cellular structures that can be created. The geometry is based on standard RP support geometries, but has features typical of general open cellular lattice structures. The structure consists of vertical members with diagonal cross members only, as currently, the process cannot build horizontal members. The powder used was Stainless steel 316L from Sandvik [11], and the powder grain size was between 10 and 38 micrometers. The finer the powder, the more expensive and dangerous (from combustion) it is.

Figure 2:

Builds for cell sizes of 2.5mm and 1.25mm.

Figure 2 shows the cellular structures with 1.25mm and 2.5mm cell sizes. A 5mm cell size was also considered, but this proved to be problematic to make. The struts tended to ‘sag’ during build, which meant that the final quality was unsatisfactory. In a single manufacture, six blocks were built on a steel plate base. The manufacture time for these six blocks was about 4 hours. The time of the process is dependent on the size and complexity of the cellular structure. There are about 500 laser passes to create the height of blocks. The blocks were cut from the steel bottom plate using a sharp knife, in an effort to reduce damage to cells. An important issue is the size and quality of the struts. This has been discussed in detail in Brocks et al. [10]. Two laser exposure times and two laser powers were used (see Table 1). Table 1:

Summary of builds (X number of specimen, 1 to 5).

Build ID

Cell size (mm) 5 2.5 2.5 1.25

1-X-5 2L-X-2.5 2H-X-2.5 3-X-1.25

Rel. Dens. (%) 0.8 4.1 5.4 12.5

Laser Exp (ms) 2.5 2.5 3.0 3.0

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Las. Pwr (A) 2.25 2.25 2.60 2.60

484 High Performance Structures and Materials III Figure 3 shows the detailed quality of the structure, in the form of an actual picture and in the form of a shadowgraph. The laser power was 2.6A and the laser exposure time was 3 milliseconds. It can be seen that the struts are by no means cylindrical, and that there is the possibility of local imperfections. A feature of selective laser melting is molten material ‘balling’, which gives rise to beads being formed [12]. This shows the importance of careful control of manufacturing parameters. The strut diameter is on average 100 micrometers [10].

Figure 3:

3

Photo and shadow graph of detail of 2.5mm cell sized specimen.

Compression tests and results

25 mm cube specimens were crushed along the cell vertical direction in a 50kN Instron testing machine at a displacement rate of 5mm/min. The specimens were crushed between two steel platens and the stress and strain data was derived from load cell and crosshead data. Engineering stress and strain measures are used. Figure 4 gives stress v strain data for the three cell sizes, with stress being plotted on a log scale, for clarity. Figure 5 gives the failure mode for the 2.5mm cube cell size case. It can be seen that the progressive crush is stable and that the major failure mode results from elastic-plastic buckling of the vertical struts. This gives rise to a ‘shear band’ forming in the complete specimen. Cells also fail near the base, which shows the need to improve the test to ensure failure in the bulk material. Zhou et al. [13] give an analysis of plastic buckling of struts, and show that the critical buckling load, Pcr is given by:

Pcr =

12C l

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(1)

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where C is a constant relating the plastic hinge couple of the strut, and l is the length of the strut. Hence the critical failure load should be inversely proportional to the strut length. Figure 6 checks out this hypothesis. Equation (1) is calibrated from the 2.5mm cell data. The effect of the change of the number of vertical struts as a function of cell size is taken into account. The general trend is correct, showing that compression failure for this block and loading configuration is governed by strut length, strut cross section and parent material property. Figure 7 gives the effect of manufacturing parameters. It compares the average crush stresses from the 2H tests with two 2L tests. For details of specimens, see Table 1. It can be seen that manufacturing parameters are very important, and it should be noted that these need to be optimised before optimising the cell geometry. It would be interesting to study the crush of the block in the lateral direction. It is proposed that, crush strength would be reduced, as there are no struts in that direction. Therefore this cellular material will be highly anisotropic.

a b c

Figure 4:

Stress v strain for three cell sizes (a. 1.25mm, b. 2.5mm, c. 5mm).

Figure 5:

Failure of 2.5mm cell size sample (H-2-2.5).

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486 High Performance Structures and Materials III 8.00E+06

Crush Stress (N)

7.00E+06 6.00E+06 5.00E+06 4.00E+06 3.00E+06

Eqn.1

2.00E+06 1.00E+06 0.00E+00 0

2

4

6

8

10

Cell size (mm)

Figure 6:

Predicted (from Equation (1)) and actual failure loads.

2H

Figure 7:

4

Comparison of the effect of manufacturing parameters on compression stress strain behaviour (2.5mm cell size).

Shear tests and results

Foam cores in sandwich structures are subject to multi axial stresses, and so these have to be studied [4]. A tensile Arcan test [14] has been developed, in which combinations of tensile direct and shear stress can be applied to a block of foam. In this paper, shear only will be described. The 25mm cube blocks are glued onto steel plates, which are then mounted in the Arcan Rig [14]. Figure 8 gives shear stress v strain data, and Figure 9 gives the failure modes for 1.25mm and 2.5mm cell sizes. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3-2

2H

Figure 8:

Figure 9:

Stress strain data from Arcan shear tests.

Comparison in failure modes for shear loading (2.5 and 1.5mm cell sizes).

In both cases, failure initiates at the boundaries, showing the need to develop the test further, to ensure the initiation of failure for the bulk material. In the 2.5mm cell size case, failure initiates by the buckling of the diagonal struts, whereas in the 1.25mm cell size case, the whole block rotates, and failure is by tensile rupture of the vertical struts. The latter failure mode gives a more abrupt reduction in stress after failure. It would be interesting to conduct both biaxial and hydrostatic tests, as these are important for localised loadings in sandwich structures [4,15]. For the 2.5mm cell size, the shear failure stress is about 6MPa which should be compared to the compression value of 10MPa. This would indicate a failure surface typical of foams [15]. However, more unusual cellular structures may well give more unconventional failure surfaces.

5

General discussion

Figure 10 compares the bulk performance of the current cellular structure with other foams [16]. It can be seen that the structure performs well, and it should be WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

488 High Performance Structures and Materials III noted that no attempt has been made to optimise the cell geometry. Therefore, it can be concluded that there is potential to markedly improve mechanical properties, over and above those of foams.

B

Figure 10:

A

Comparison of current cellular structure with envelope for foams [16] (A – 1.25mm cell size, B – 2.5mm cell size).

To summarise, there is scope to optimise the current cellular geometry. This needs to be addressed from a fundamental point of view – basically the manufacturing process can create any cellular structure with a resolution in structural features of about 50 micrometers. The topology of the open cellular lattice structure can be optimised for elastic stiffness in bulk properties, and for non-linear (collapse) behaviour. A promising approach has been discussed by Bendsoe and Sigmund [17], in which a two dimensional solid is defined, and is populated by blocks (20 by 40, for example). Boundary conditions are then applied to the geometry, and a linear finite element analysis is carried out. Blocks that are least structurally effective are then deleted, giving rise to the most efficient overall structure. Simple cases of beam bending have been analysed using MATLAB [17] and the approach can be extended to three dimensions. The optimisation approach has been applied to elastic stiffness problems [17], buckling problems [18] and crashworthiness problems [19].

6

Concluding remarks

A rapid prototyping manufacturing method is being developed to realise open cellular lattice structures that can be tailored for specific structural applications. A wide variety of structural geometries are possible. The synthesis of optimum structures (for a given application) is theoretically complex, and design tools need to be developed to maximise the potential of the selective laser melting manufacturing process. The analysis method of topology optimisation shows good potential for quantifying optimum performance. A related issue concerns the simulation of the mechanical behaviour of these types of materials in the context of the actual structure. Periodic structures with WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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small cell size give rise to a large number of structural features, which leads to unrealistic numerical models for full micro mechanical modelling. Also, in the case of foreign object impact loading, material behaviour is dynamic and highly non linear with damage occurring. There is a need to develop homogenised models, that characterise the effective behaviour of periodic structures [17].

Acknowledgements The SLM machine has been funded by the EPSRC Grant number GR/598405/01 with contributions from MCP and Sandvik Osprey.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13]

Rohacell 51 WF Foam, Roehm Ltd., Bradbourne Drive, Tilbrook, Milton Keynes, Bucks, MK7 8AU, England, 1999. Alporas Data Sheet, Shinko Wire Company, Amagasaki, Japan, 2001. Zenkert, D.A., Introduction to Sandwich Construction, EMAS Publishers, 2000. McKown, S and Mines, R.A.W., Impact behaviour of metal foam cored sandwich beams, European Conference on Fracture 16, Alexandropoulos, Greece, Paper No. 254, July 2006. Gibson, L.J., and Ashby, M.F., Cellular Solids, Cambridge University Press, 1999. Wadley, H.N.G., Fleck N.A., Evans, A.G., Fabrication and structural performance of periodic cellular metal sandwich structures, Composites Science and Technology, 2004. Liu, J.S., Lu, T.J., Multi objective and multi loading optimisation of ultra lightweight truss materials, International Journal of Solids and Structures, Vol. 41, pp619-635, 2004. Chau, C.K., Leong, K.F., and Lim, C.S., Rapid Prototyping (2nd Edition), World Scientific, 2004. MCP Realiser II, MCP Tooling Technologies Ltd., Whitebridge Way, Whitebridge Ind. Park, Stone, Staffordshire, ST15 8LQ, England, 2005. Brooks, W.K., Tsopanos, S., Stamp, R., Sutcliffe, C.J., Cantwell, W.J., Fox. P., Todd, J., The production of open cellular lattice structures using selective laser melting, 6th National Conference on Rapid Design, Prototyping, and Manufacturing, Buckinghamshire Chilterns University College, June 2005. Data sheet for microfine powders, 316L Austenitic stainless steel, Sandvik Osprey, Neath, Scotland, 2005. Kruth, J.P., Froyen L., Van Vaerenbergh, J., Mercelis, P., Rombouts, M., Lauwers, B., Selective melting of iron based powders, Journal of Materials Processing Technology, No. 149, pp616-622, 2004. Zhou, J., Shrotriya, P., Soboyejo, W.O., On the deformation of aluminium lattice block structures: from struts to structures, Mechanics of Materials, Vol. 36, pp723-737, 2004. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

490 High Performance Structures and Materials III [14] [15] [16] [17] [18] [19]

McKown, S., Mines, R.A.W., Measurement of material properties for metal foam cored polymer composite sandwich construction, Journal of Applied Mechanics and Materials, Vol. 1-2, pp211-216, 2004. Li, Q.M., Mines, R.A.W, and Birch, R.S., The crush behaviour of Rohacell 51WF structural foams, International Journal of Solids and Structures, Vol. 37, pp 6321-6341, 2000. Ashby, M.F., Materials selection in engineering design, Butterworth Heinemann, Oxford, 2005. Bendsoe, M.P. and Sigmund, O., Topology Optimization: Theory, Methods and Applications, Springer, Berlin, 2004. Neves, M.M., Sigmund, O., Bendsoe, M.P., Topology optimisation of periodic microstructures with buckling criteria, WCCM 5, Vienna, July 2002. Pedersen, C.B.W., Topology optimisation of energy absorbing frames, WCCM 5, Vienna, July 2002.

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491

Stress intensity factors for cracked cold-drawn steel wires under tensile loading B. Lin1 & G. Lu 2 1

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, People’s Republic of China 2 Department of Bridge Engineering, Tongji University, People’s Republic of China

Abstract Cold-drawn steel wires have the excellent mechanical properties of strength and toughness and are widely used in prestressed concrete structures. During cold-drawn operations residual stresses are generated in steel wires. In this paper, by the residual stress distribution from the finite element analysis (FEA), a weight function method (WFM) has been used to evaluate the effective stress intensity factors (SIFs) for cracked cold-drawn steel wires under tensile loading. The calculation results have been compared with those obtained by the two-dimensional (2D) FEA, which considers the residual stress redistribution in the presence of a crack. In the present study, the effective SIFs calculated using the WFM have shown a good agreement with those derived from the 2D FEA. Keywords: stress intensity factor, residual stress, weight function, cold-drawn steel wire, finite element analysis.

1

Introduction

In the cold-drawn procedure, the tensile strength and toughness of steel wires increase, but a residual stress field also appears in them. After the treatment of a further drawing with a very small area reduction or a combination of heating and stretching the wire, the decreased residual tensile stresses still exist in the steel wire surface [1]. It is well known that the presence of residual tensile stresses will reduce the fatigue life of steel wires. With increasing in residual tensile stresses, the fatigue crack growth rate increases for stress ranges close to the fatigue limit [2]. The calculation of the SIF is one of key factors in fatigue life WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06048

492 High Performance Structures and Materials III determination of the components. The superposition principle, which assumes that the stresses applied by external loading superimpose linearly on residual stresses, can be used to calculate the effective SIFs in a cracked body in which a residual stress field acts [3]. If the weight function, which depends only on the geometry of the body, is obtained, the contribution of residual stresses on the effective SIFs for cracks in mode I can be determined by the WFM [4]. As for cold-drawn steel wires under tensile loading, the net stresses in the steel wires under service loading remain tensile. When a crack is produced such that it lies wholly in a tensile region the faces of the crack will remain fully open and the applicability of the linear superposition principle is valid [5]. Additionally, the produced crack will bring the residual stresses ahead of a crack tip to redistribute, [5,6]. So, the availability of the WFM, which assumes that the residual stress distribution keeps invariable, must be validated when dealing with residual stresses and fatigue crack growth. In this paper, the 2D FEA, which considers the residual stress redistribution in the presence of a crack, is carried out to validate the WFM for calculations of effective SIF for cracked cold-drawn steel wires under tensile loading. Section 2 presents the 2D model for the cracked steel wire. Calculations of the effective SIF by the weight function analysis are described in Section 3 and by the 2D FEA in Section 4. Section 5 gives the corresponding analysis results and discussion. The present work is concluded in Section 6. σapp

σapp

A C x

L

y

O

D

σapp (a)

Figure 1:

2

O C

x

R

O

a a

A

2b D A£ -A

L

z

D

L

L

z

σapp

(b)

(c)

3D geometry and 2D model of a wire used in the analysis.

Two-dimensional model

The surface crack of the cold-drawn steel wires produced by mechanical fatigue can be assumed to be a semi-elliptical geometry (a/b<1, a is the crack depth or minor axis of the ellipse, b the major axis of the ellipse), perpendicular to the tensile loading direction [7], as show in fig. 1(a), 1(b), and the maximum SIF value was obtained at the crack center C, according to the results given by Astiz [8]. Following a well known purely linear elastic solution in the vicinity of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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a 3D crack with an arbitrary curved front [9], the calculation of the SIF can be based on 2D plane strain hypothesis on the longitudinal section of a cracked cold-drawn steel wire under tensile loading, fig. 1(c).

3

Weight function analysis

As for a cracked cold-drawn steel wire under tensile loading, the effective SIF at the crack front is decided by the combined effect of local residual stresses and the stresses applied by external loading. According to the linear superposition principle, the effective SIF (Keff) is the algebraic sum of the external SIF (Kapp) due to the external applied loads and the residual SIF (Kres) due to the residual stresses, and can be expressed as: Keff = Kapp + Kres (1) 3.1 Residual SIF It is well known that, once the weight function is determined for a particular cracked body, the SIFs can be obtained for any stress field by an appropriate integration. So, in this section, the WFM is used to evaluate the residual SIF (Kres). Based on the FEA [1], the distribution of the residual stresses for colddrawn steel wires with a thermomechanical treatment is shown in fig. 2, where r is the polar coordinate of the wire and R the radius of the wire, and the equation used for the calculation of the residual stresses in the plane problem is as follows:

σres( r R ) = −235.71 + 411.88 ⋅ r R − 987.84 ⋅ (r R )2 + 3133.93 ⋅ (r R )3

4 5 (2) − 3257.62 ⋅ (r R ) + 1029.93 ⋅ (r R ) . In the edge-cracked 2D model, fig. 1c, a residual SIF can be calculated by integrating the weight function m(x,a) and the residual stress distribution σres(x) acting on the crack length a [10], which is expressed as follows:

a

Kres = ∫ σres( x )m( x , a )dx

(3)

500

(MPa)

Longitudinal redidual stress

0

Figure 2:

0

-500 1.0

0.8

0.6

0.4

Relativ depth, (r/R)

0.2

0.0

Residual stress distribution for a steel wire used in the analysis.

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494 High Performance Structures and Materials III The analytical weight function for an edge crack in a finite width plate, as shown in fig. 3, was given by Kaya and Erdogan [11]:

m( x ,a ) =

x a G ,  a D

2

πa 

a 1 −   D

3/ 2

 x 1−   a

(4)

2

where 2

3

 a  x   a   x a x a x a G ,  = g1   + g 2   ⋅ + g3   ⋅   + g 4    ,  D  a  D a D a D a D 5

2

2

a a a a   a g1   = 0.46 + 3.06 ⋅ + 0.841 −  + 0.66  1 −  , D D  D  D D 2

a a g 2   = −3.52  , D D 2

3

3/ 2

a a a  a a g3   = 6.17 − 28.22 + 34.54  −14.39  − 1 −  D  D  D  D  D 5

2

2

a a a   − 5.881 −  − 2.64  1 −  , D  D  D 2

a a a a g 4   = −6.63 + 25.16 − 31.04  + 14.41  D D D  D a  + 21 −   D

3/ 2

5

2

3

2

a a a   + 5.041 −  + 1.98  1 −  . D  D  D y x P=1

a

Figure 3:

x

D

Weight function notation for an edge crack in a finite width plate.

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3.2 External SIF An empirical solution of the SIF for a single edge cracked plate applied under the axial stress, fig. 1(c), given by Tada et al. [12] is used to calculate the external SIF (Kapp), which is expressed as follows:

Kapp = σapp πa ⋅ F ( a D ) (5) where σapp is the external applied stress, D the diameter of the steel wire and F (a/D) given by: F( a / D ) =

πa  a  0.752 + 2.02 + 0.371 − sin  π a 2D D 2D   tan . πa πa 2D cos 2D ıapp

L=10 mm

y

3

x

a D/2=2.5 mm D/2=2.5 mm

Figure 4:

4

2D finite element model.

Finite element analysis

4.1 2D finite element model 2D finite element model of the cracked steel wire was developed for numerical stress analysis. A detail of a typical mesh was shown in fig. 4. A different mesh was used for each crack length to present the equivalent level of mesh refinement. The ABAQUS finite element program [13] was chosen for the present analysis. 8-node biquadratic solid element for plane strain available in the ABAQUS code element library was used in the model, and the quarter point technique, used for numerical evaluation of the SIF, was applied to improve precision of the results. Due to symmetry, as shown in fig. 1(c), only the top half-plane was modelled for the specimen with a homogeneous linear elastic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

496 High Performance Structures and Materials III material using a Young’s modulus, E, of 207 GPa, Poisson’s ratio,υ, of 0.3 and Limit strength, fptk, of 1940 MPa . The SIF for the elastic singular stress field was calculated by means of the J-integral for 10 different paths, which was obtained using the domain integral formulation in ABAQUS. Calculations coupled with the residual stress field were carried out for each crack length. Two levels of applied loading, σapp= 0.4 and 0.7fptk, were employed and applied to the top edge of the model. From the average J-integral value, the effective SIF was computed. 4.2 Verification of the finite element model In order to verify the finite element model’s accuracy to calculate SIFs for different crack lengths, comparisons between these solutions and empirical solutions given by eqn. (5) were made only under tension loading. The normalised SIF, KI σapp πa , is presented in fig. 5, showing a fairly good agreement and all differences being within 1%. Based on these results, the present finite element model was considered suitable for the 2D analysis of cracked cold-drawn steel wires under tensile loading.

Normalised SIF, Κ I/σapp(πa)

2.4

FEM

Tada et al.

2.1 1.8 1.5 1.2 0.0

Figure 5:

0.1

0.2 0.3 Normalised crack length, a/D

Comparison of KI σ calculations.

app

0.4

πa between FEM and Tada et al.’s

4.3 Residual stresses The initial stress method illustrated by Lei et al. [14] can be used to introduce a residual stress distribution into the finite element model as an initial condition, and a fine mesh has been employed to provide for an accurate description of the residual stress distribution in the model. But the introduced residual stresses are WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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not self-equilibrating; usually resulting in a stress distribution that differs significantly from desired one. This problem was overcome by using two load steps in the linear elastic analysis. In the first step, reactions required to balance the out-of-balance load vector caused by the residual stresses were derived with all degrees of freedom constrained and no other applied loadings. In the second step, the obtained reactions were converted to applied force, and an equilibrium step analysis was executed for the uncracked model with the standard boundary conditions. After this procedure, the contour of the residual stresses (unit: MPa) is presented in fig. 6. It is shown that the residual stress distribution in the uncracked model accords well with the target distribution shown in fig. 2.

Figure 6:

Contour of residual stresses in the uncracked model.

4.4 Residual stress redistribution After the residual stresses are introduced into the uncracked model, a crack can be introduced into the FE model by releasing nodes located on the crack face until the desired crack length is reached. When a node is released from its displacement constraint, the reactions previously acting on the node disappear, and moreover, the equilibrium state in the elements around the node is also changed. Subsequently, a new equilibrium state is re-established. The shift of the equilibrium state will bring the residual stresses in these elements to redistribute.

5

Analysis results and discussion

Both the WFM and the FEM described above were used to evaluate the variation of the effective SIF as a function of the crack length. The effective SIFs calculated from the WFM and those derived from the FEA are compared in fig. 7 for seven different crack lengths. From this figure, one can find that, for different crack lengths under two loading levels, the maximum difference in values between the two methods is 0.87%, which exhibits a good mutual agreement. Comparison results between the FEA and the WFM showed that, when calculating effective SIFs with a linear elastic analysis for cracked cold-drawn WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

498 High Performance Structures and Materials III steel wires under tensile loading, the phenomenon of redistribution was not obvious to residual stress problems, which keeps a good agreement with the results for cold-worked cracked holes given by Moreira et al. [15].

240

Effective SIF (MPa√m)

200

WFM, σapp=0.4fptk WFM, σapp=0.7fptk

FEA, σapp=0.4fptk FEA, σapp=0.7fptk

160 120 80 40 0.0

Figure 7:

6

0.1 0.2 0.3 Normalised crack length, a/D

0.4

Comparison of residual SIF between weight function and finite element results.

Conclusions

In this paper, we have demonstrated that, according to the obtained residual stress fields from documents, the effective SIF can be calculated for a cracked body using either the WFM or the FEA. The following conclusion can be drawn: the reliability of the WFM results is demonstrated by comparing with FE calculations, and the complexity of the analysis is remarkably reduced.

References [1] [2] [3] [4]

Elices, M., Influence of residual stresses in the performance of cold-drawn pearlitic wires. Journal of Materials Science, 39(12), pp. 3889-3899, 2004. Llorca, J. & Sanchez-Galvez, V., Fatigue limit and fatigue life prediction in high strength cold drawn eutectoid steel wires. Fatigue and Fracture of Engineering Materials & Structures, 12(1), pp. 31-45, 1989. Parker, A.P., Stress intensity factors, crack profiles and fatigue crack growth rates in residual stress fields. ASTM STP 776, pp. 13-31, 1981. Glinka, G. & Shen, G., Universal features of weight functions for cracks in mode I. Engineering Fracture Mechanics, 40(6), pp. 1135-1146, 1991. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

[5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15]

499

Wilks, M.D.B., Nowell, D. & Hills, D.A., The evaluation of stress intensity factors for plane cracks in residual stress fields. Journal of Strain Analysis for Engineering Design, 28(3), pp. 145-152, 1993. Lee, Y.B., Chung, C.S., Park, Y.K. & Kim, H.K., Effects of redistributing residual stress on the fatigue behavior of SS330 weldment. International Journal of Fatigue, 20(8), pp. 565-573, 1998. Toribio, J. & Toledano, M., A fracture criterion for prestressing steel cracked wires. Proc. Of the 2nd Int. Conf. On Advances in Steel Structures, eds. S.L. Chan & J.G. Teng, Elsevier: New York, pp. 947-954, 1999. Astiz, M.A., An incompatible singular elastic element for two- and threedimensional crack problems. International Journal of Fracture, 31(2), pp. 105-124, 1986. Hartranft, R.J. & Sih, G.C., Stress singularity for a crack with an arbitrarily curved front. Engineering fracture Mechanics, 9(3), pp. 705718, 1977. Cartwright, D.J., Stress intensity factor determination (Chapter 2). Developments in Fracture Mechanics, eds. G.G. Chell, Applied Science: London, pp. 29-66, 1980. Kaya, A.C. & Erdogan, F., Stress intensity factors and COD in an orthotropic strip. International Journal of Fracture, 16(2), pp. 171-190, 1980. Tada, H., Paris, P.C. & Irwin, G.R., (eds). The Stress Analysis of Cracks Handbook, Del Research Corporation: Hellertown, Pennsylvania, 1973. Hibbitt, B., Karlsson, B. & Sorensen, P., (eds). ABAQUS/Standard User’s Manual, version 6.1, Hibbitt, Karlsson & Sorensen Inc.: Rode Island, RI, 2001. Lei, Y., O'Dowd, N.P. & Webster, G.A., Fracture mechanics analysis of a crack in a residual stress field. International Journal of Fracture, 106(3), pp. 195-216, 2000. Moreira, P.M.G.P., De Matos, P.F.P. & Pinho, S.T., The residual stress intensity factors for cold-worked cracked holes: A technical note. Fatigue and Fracture of Engineering Materials and Structures, 27(9), pp. 879886, 2004.

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Void growth and damage ahead of a crack in pressure-sensitive dilatant polymers H. B. Chew, T. F. Guo & L. Cheng Department of Mechanical Engineering, National University of Singapore, Singapore

Abstract The deformation of amorphous polymers is highly sensitive to hydrostatic pressure and the plastic flow is non-volume preserving. Failure of these materials is often preceded by the formation of small crack-like defects known as crazes, bridged by the fibrils of the polymer. In this work, a population of discrete voids is placed ahead of a crack within a boundary layer configuration to study the effects of pressure-sensitivity α and plastic dilatancy β on void growth and damage in polymeric materials. Results show that high pressure-sensitivity significantly increases both the intensity and spatial extent of damage, which could promote crazing. These effects can be compounded as the deviation from the associated flow rule, |β – α|, increases. This study offers some evidence of pressure-sensitivity and plastic dilatancy as the major contributing parameters to the formation and growth of microporous crazing zones in polymers. Keywords: void growth, coalescence, pressure-sensitive yielding, plastic dilatancy, polymer.

1

Introduction

The ductile fracture of many engineering materials involves the micromechanical process of void growth and coalescence. The fracture surfaces of these materials contain voids of various size-scales. In polymeric materials, the larger voids can originate from cavitated rubber blends or from decohesion of filler particle/polymer matrix interfaces, while shear banding and crazing can induce the formation of the smaller microvoids. The ductile fracture process can be described by mechanism-based computational approaches which employ voidcontaining cell elements [1]. Porous material models, such as the Gurson WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06049

502 High Performance Structures and Materials III constitutive model [2], are used to describe void growth for each cell element, and the macroscopic softening during the fracture process. This approach has been widely used in the prediction of ductile crack growth in metals. More recently, Guo and Cheng [3] extended the Gurson model to incorporate vapour pressure effects. The extended Gurson model has been used to study the vapour pressure assisted void growth and rupture of hygroscopic polymeric materials, like the die attach and molding compound in electronic packages [4–6]. While the micromechanisms of ductile fracture for metals and polymers are somewhat similar, two important characteristics of polymers that differ from metals are its pressure-sensitive yielding, and its non-volume preserving plastic flow. To this end, Chew et al. [7] examined the effects of microvoid interaction in pressure-sensitive dilatant polymeric materials. They showed that increasing pressure-sensitivity severely reduces the material’s stress carrying capacity, while multiple void interactions was responsible for the sharp post-peak stress drop and could act as a catalyst for rapid failure. In addition, plastic dilatancy was found to have some effect in raising the post-peak stress levels, resulting in a larger work of separation. An accurate simulation of the void growth behaviour in pressure-sensitive dilatant polymeric materials can be achieved by explicitly modelling the voids using refined finite elements. This discrete void approach was previously adopted by Tvergaard and Hutchinson [8] and Kim et al. [9] for the study of the ductile fracture of metals, and by Chew et al. [10] for the modelling of moisture induced failure of hygroscopic polymeric materials. In this work, a single row of discrete voids is placed ahead of a crack within a boundary layer configuration to study void growth and damage in pressuresensitive dilatant polymeric materials (Fig. 1). The pressure-sensitive yielding of the material is described by a linear combination of the mean stress and effective stress under a non-associated flow rule. The motivation behind this study is to understand the contributions of pressure-sensitivity α and plastic dilatancy β to the formation and growth of long craze zones widely observed in polymeric materials [11]. X2

R

ș

D

D

2R0

X1

Figure 1:

2

Schematic of cylindrical discrete voids ahead of a crack in a boundary layer configuration.

Problem formulation

2.1 Material model The plastic behaviour of polymeric materials differs considerably from the von Mises material. Such behaviour can be explained by assuming a yield criterion WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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based on a linear combination of the mean stress and effective stress. Here, the pressure-dependent yielding of the material is described by

3J 2 + σ m tan α − σˆ = 0

(1)

where J2 is the second invariant of the deviatoric part of the Cauchy stress tensor σ, σm the mean stress, σˆ the flow stress of the subsequent yield surface, and α the index for pressure-sensitivity. We assume the flow potential to take the form

Φ = 3J 2 + σ m tan β

(2)

where β is the index for plastic dilatancy. The plastic deformation rate is given by the non-associated flow rule ∂Φ d P = ε P (3) ∂σ where ε P is the equivalent strain rate. The flow stress σˆ is a function of the accumulated plastic strain P ε = ∫ ε P dt. For a power-law plastic hardening solid, one has 1/ N

σ 0  σˆ    E  σ 0 

−

σˆ E

=εP

(4)

where N is the hardening exponent ranging from 0 to 1, and σ0 the initial yield stress under pure shear which is related to the initial tensile and compressive yield stresses σ0t and σ0c, by   1  t  1 + 3 tan α σ 0 for tension    σ0 =  1    1 − tan α σ t for compression. 0   3  The pressure-sensitivity index α can be determined from σ0t and σ0c as

 σ 0c − σ 0t  . c t   σ0 +σ0 

α = tan −1  3

(5)

(6)

In polymeric materials, typical pressure-sensitivity levels range between 0º to 23º. 2.2 Small-scale yielding

Xia and Shih [1] simplified the ductile fracture process under small-scale yielding conditions by placing a single row of void-containing cells ahead of the crack. In the present study, we model void growth and damage via an array of unit-cells ahead of the crack (see Fig. 1). Each unit-cell is of dimensions D by D and contains a discrete cylindrical void of initial radius R0. The initial void volume fraction is given by f 0 = πR02 / D 2 . A total of 23 voids are used in the computation. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

504 High Performance Structures and Materials III (a)

(b)

Figure 2:

Finite element mesh for small-scale yielding analysis. (a) Entire mesh showing remote boundary. (b) Close-up view of the crack tip.

Along the remote circular boundary (see Fig. 2a), the elastic asymptotic (inplane) displacement field u1 ( R, θ ) = K I

1 +ν E

R (3 − 4ν − cos θ )cos θ 2π 2

(7) θ 1 +ν R (3 − 4ν − cos θ )sin u 2 ( R, θ ) = K I E 2π 2 is applied under plane strain conditions, where R2 = X12 + X22 and θ = tan-1(X2/X1) for points on the remote boundary. KI is the mode I stress intensity factor related to the J-integral by 1 −ν 2 2 J= KI . (8) E At various stages of loading, the value of the J-integral is calculated on a number of contours around the crack using the domain integral method. The domain integral value was found to be in good agreement with the value given by eqn. (8) for the prescribed amplitude KI. This consistency check assures that smallscale yielding conditions are satisfied. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 2 displays an example finite element mesh for the discrete void model. By taking advantage of symmetry, only one half of the geometry needs to be modelled. The mesh contains 4-node bilinear quadrilateral hybrid plane strain elements. The computations are performed within the finite strain setting using the general-purpose finite element program ABAQUS Version 6.5.1 [12].

3

Crack tip plastic zone

Before proceeding to the modeling of void growth, we first examine the effects of pressure-sensitivity α on the extent of plastic deformation ahead of the crack. Some insights can be obtained from the Dugdale strip yield model, which assumes a long slender plastic zone at the crack tip in a nonhardening pressureinsensitive material (α = 0º). The plastic zone size L under a uniaxial tensile stress σT is given as L=

π  KI  8  σ T

2

  . 

(9)

For the pressure-sensitive material in eqn. (1), the tensile stress σT at yielding is σT = σ0/(1+1/3 tan α). Denoting L0 as the plastic zone size corresponding to α = 0º, i.e. σT = σ0, we obtain 2

  1 L / L0 = 1 + tan α  . (10)   3 This relation suggests an increase in the plastic zone size L with pressuresensitivity α. Observe that an increase in α from 0º to 20º increases L by around 25%. A numerical study is next performed to shed some light on the extent of plastic dissipation in a void-free pressure-sensitive polymer. The void-free material is subjected to mode I, K-field loading in eqn. (7). An associated flow is assumed with α = β. The elastic modulus of the polymer is taken to be E = 500 MPa, with a yield strength of σ0 = 5 MPa, and a Poisson’s ratio of ν = 0.4. In accordance with the Dugdale assumption, the strain hardening exponent N is set to zero (for ideally-plastic material). Figure 3 displays the plastic zones in the deformed mesh configuration for three pressure-sensitive levels: α = 0º, 10º and 20º. The plastic zones are operationally defined by ε p ≥ 0.001 , where ε p is the accumulated plastic strain. Observe that at the same external load level, the plastic zone spreads out as the pressure-sensitivity level increases. For example, at J = 68 kNm-1, an increase in α from 0º to 20º raises the extent of plastic dissipation ahead of the crack by nearly two-fold (L/L0 ≈ 2). This general trend of increasing plastic dissipation with pressure-sensitivity corroborates with the findings from eqn. (10).

4

Damage evolution in porous polymers

In this section, an array of discrete voids is placed ahead of the crack. From dimensional considerations, the spatial distribution of the current void volume WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

506 High Performance Structures and Materials III fraction f depends on the following dimensionless geometric-material parameters: J σ0 ; , N , v, α , β ; f 0 . σ0D E

(11)

The properties of the polymeric material are specified by the parameters σ0/E = 0.01, ν = 0.4, and N = 0.1. We focus on the effects of pressure-sensitivity and plastic dilatancy on void growth and damage ahead of the crack. (a)

Į = 0º

L0

(b)

Į = 10º L

(c)

L

Figure 3:

Į = 20º

Plastic zones in the deformed mesh configuration for several pressure-sensitivity levels at J = 68 kNm-1.

4.1 Pressure-sensitivity effects

The effects of pressure-sensitivity under an associated flow rule (α = β) are shown in Fig. 4 for f0 = 0.005 and f0 = 0.05. Computations focusing on the ductile fracture of metals by Tverggard and Hutchinson [8] have identified two distinct mechanisms of fracture: (i) single void growth mechanism, and (ii) multiple void interaction mechanism. Both these failure mechanisms are also exhibited in our study for amorphous polymers. At equivalent load levels, we observe that an increase in α significantly increases both the intensity and extent of damage ahead of the crack. Herein, we operationally define the spatial extent of damage as the distance from the crack tip to the region where the void porosity has increased by f – f0 > 0.01. For f0 = 0.005 at J/(σ0D) = 0.36, an increase in α from 0º to 15º raises the peak porosity level by 160% and increases the extent of damage by nearly three-fold. Similar observations are made for f0 = 0.05. The plastic zone distribution in the deformed mesh configuration for f0 = 0.005 is displayed in Fig. 5. Observe that the plastic zone spreads over a much larger region as α increases. For α = 0º, the plastic zone is confined to the first WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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two voids ahead of the crack, resulting in near-tip void growth. For α = 15º, the plastic zone spreads across the first four voids, leading to the formation of an extended damage zone. This extended damage zone closely resembles the long craze zones in polymers. One can therefore infer that pressure-sensitivity is a major contributing parameter to the crazing phenomenon. Interestingly, we also note that there is an increasing tendency for the voids to become oblate as they grow larger. This suggests the effects of void shape to play an important role in the coalescence stages of fracture.

(a)

(b)

Figure 4:

Distribution of porosity f ahead of crack (X2 = 0) for several pressure-sensitivity levels with α = β, (a) f0 = 0.005; (b) f0 = 0.05.

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508 High Performance Structures and Materials III (a)

α =0º

(b)

α =15º

Figure 5:

Plastic zones in the deformed mesh configuration for several pressure-sensitivity levels at J/(σ0D) = 0.36. f0 = 0.005, (a) α = 0º; (b) α = 15º.

4.2 Plastic-dilatancy effects

The analysis in the previous section employs an associated flow rule, i.e. α = β. Experimental studies, however, have shown that the plastic volume change in polymers does not commensurate with the predictions of the associated flow rule, i.e. β < α. Figure 6 displays the effects of β on the damage distribution ahead of the crack for α = 10º and 20º. One can see that an increase in the deviation from the associated flow, |β – α|, significantly increases both the intensity and the spatial extent of damage. For f0 = 0.005, α = 20º, at J/(σ0D) = 0.60, an increase in |β – α| from 0º to 5º raises the damage intensity level by nearly 60%, and shifts the damage process zone from X1 = 5D to X1 = 7D. The effects of increasing |β – α| are found to be less severe for f0 = 0.05. The non-conservative nature of the associated flow in predicting damage has serious implications in the design of engineering materials and structures.

5

Conclusion

The effects of pressure-sensitivity, α, and plastic dilatancy, β, on void growth and damage in amorphous polymeric materials have been studied. Our numerical results for a void-free polymeric material show that pressure-sensitivity significantly increases the extent of plastic dissipation ahead of the crack. Void growth and damage in the pressure-sensitive dilatant polymeric material was subsequently modelled via a single row of discrete voids placed ahead of the crack. Both the voiding intensity and the spatial extent of damage were observed to increase with pressure-sensitivity, which could promote crazing. These effects were further compounded as the deviation from the associated flow rule, |β – α|, increases. These findings suggest pressure-sensitivity and plastic dilatancy to be closely linked with the formation of long craze zones in polymers.

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(a)

(b)

Figure 6:

Distribution of porosity f ahead of crack (X2 = 0) for several plastic dilatancy levels with α = 10º and 20º, (a) f0 = 0.005; (b) f0 = 0.05.

References [1] [2]

Xia, L. & Shih, C.F., Ductile crack growth - I. A numerical study using computational cells with microstructurally-based length scales. Journal of the Mechanics and Physics of Solids, 43, pp. 233-259, 1995. Gurson, A.L., Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology, 99, pp. 2-15, 1977. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

510 High Performance Structures and Materials III [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Guo, T.F. & Cheng, L., Modeling vapor pressure effects on void rupture and crack growth resistance. Acta Materialia, 50, pp. 3487-3500, 2002. Chew, H.B., Guo, T.F. & Cheng, L., Vapor pressure and residual stress effects on the toughness of polymeric adhesive joints. Engineering Fracture Mechanics, 71, pp. 2435-2448, 2004. Chew, H.B., Guo, T.F. & Cheng, L., Vapor pressure and residual stress effects on failure of an adhesive film. International Journal of Solids and Structures, 42, pp. 4795-4810, 2005. Chew, H.B., Guo, T.F. & Cheng, L., Vapor pressure and residual stress effects on mixed mode toughness of an adhesive film. International Journal of Fracture, 134, pp. 349-368, 2005. Chew, H.B., Guo, T.F. & Cheng, L., Effects of pressure-sensitivity and plastic dilatancy on void growth and interaction. International Journal of Solids and Structures, in Press, 2006. Tvergaard, V. & Hutchinson, J.W., Two mechanisms of ductile fracture: void by void growth versus multiple void interaction. International Journal of Solids and Structures, 39, pp. 3581-3597, 2002. Kim, J., Gao, X. & Srivatsan, T.S., Modeling of crack growth in ductile solids: a three-dimensional analysis. International Journal of Solids and Structures, 40, pp. 7357-7374, 2003. Chew, H.B., Guo, T.F. & Cheng, L., Vapor pressure and voiding effects on thin film damage. Thin Solid Films, in Press, 2006. Kambour, R.P., A review of crazing and fracture in thermoplastics. Journal of Polymer Science, 7, pp. 1-154, 1973. Hibbit, Karlsson & Sorensen Inc., Abaqus/Standard User’s Manual, Version 6.5.1, Vol. 1, 2005.

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An orthotropic damage model for crash simulation of composites W. Wang1, F. H. M. Swartjes1 & M. D. Gan2 1

BU Automotive Centre of Lightweight Structures TUD-TNO, TNO Science and Industry, Delft, The Netherlands

2

Abstract In this paper, a practical orthotropic damage model is developed and tested for composite materials during crash. The model uses the Hashin’s failure criteria in which the fibre and matrix failures are described explicitly, both in tension and compression. A linear softening degradation is proposed and a close-form solution of the corresponding damage parameter is provided. To reduce the mesh dependency, an embedded discontinuous element is proposed. It is a virtual embedded element in the sense that an actual element is divided into two zones, one elastic (undamaged) zone and one localization (damaged) zone. Their equivalence is preserved by constraining the kinematics and equilibrium equations. Since the damage zone is introduced into the element, the corresponding dissipated energy due to damage is fixed, independent of the element size. The numerical simulations using the developed damage model show a mesh objective result and correlate well with the energy dissipation in dynamic 3-point bending experiments. It is concluded that an accurate material calibration is crucial for the success of failure simulations. The traditional strain mapping cannot be applied to failure problems in which the strain distribution is highly non-uniform. Material parameter calibration should take into account the ratio of the failure zone and the gauge length. A practical approximation formula is provided. Keywords: composite, damage model, crash simulation, embedded element.

1

Introduction

The crash simulation of composites is particularly difficult due to the complexity of its physics. First, the failure process involves multiscales and is difficult to WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06050

512 High Performance Structures and Materials III observe experimentally. Second, the softening effect leads to the well known numerical simulation difficulties, cf. mesh dependency, instability and bifurcation. In this paper, a practical orthotropic damage model is developed and tested for composite materials during crash. This new model is implemented in LSDYNA as a user-defined material model. The related issues such as the mesh dependency and material calibrations of the corresponding damage parameters are addressed.

2

Continuum damage constitutive relationship

For plane stress situations in the continuum damage approach, the effective ˆ is related to the total strain ε by the elastic stiffness tensor C0 stress σ

σˆ = C 0 ε

(1)

The effective stress is mapped to the true stress by the damage operator M

σ = Mσˆ ,

0  1 − D1 0  M= 0 1 − D2 0   0 0 1 − D12 

(2)

where D1, D2 and D12 are the damage parameters. Inserting Equation (1) into Equation (2) yields

σ = M C0 ε = C ε

(3)

in which the damaged stiffness tensor C reads

 (1 − D1 )E11 1  C = (1 − D2 )ν 12 E11 φ  0 

(1 − D1 )ν 21 E22 (1 − D2 )E22 0

  0  (1 − D12 )φ G 

φ = 1 − ν 12ν 21

0

(4)

(5)

with E11, E22, G, ν 12 and ν 21 are the elasticity parameters of the undamaged lamina. Note that when damage occurs (Di>0), the stiffness tensor is not symmetric. To preserve a symmetric stiffness tensor, the Poisson’s ratio can be related to the damage parameters [3].

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513

Hashin’s failure criteria

Hashin’s criteria [2] refers to an unidirectional fibre reinforced composite with five failure modes: 1) Fibre tensile and compression modes 2

≥ 0 failed σ  e =  11  − 1   X  < 0 elastic 2 f

(6)

2) Matrix tensile and compression modes 2

≥ 0 failed σ  em2 =  22  − 1   Y  < 0 elastic 3)

(7)

Shear mode 2

≥ 0 failed τ  e =   −1  S  < 0 elastic 2 s

(8)

where X and Y are fiber and matrix strengths and S is the shear strength. Note that the compression and tensile modes are treated in the same way. For notation simplicity, the sub indexes for the tensile and compression strength parameters are omitted. Note that Hashin’s criteria have a complete decoupling of the fibre and matrix failure modes from shear failure. Therefore, the failure criteria may overestimate the strength of a lamina when shear is present. However, it should be mentioned that measuring the softening part of the stress-strain curve is very difficult even in a uniaxial test condition. A multi-axial loading surface is almost impossible to be validated. Furthermore, the damage evolution or degradation law is, in most cases, in an uncoupled fashion. Therefore, it is convenient to treat all failure criteria to be independent. It must be emphasised that although the failure modes and the degradation law (which will be introduced in the next section) are decoupled, the complete stress-strain relationship still contains a coupling due to the stiffness matrix C (see Equation 3).

4 Degradation rule Damage occurs when one of the Hashin’s criteria is satisfied. For simplicity, we assume a linear degradation law (see Figure 1) after the initial strength σ0 is exceeded. Consequently, the damage parameter D can be determined by the following relationship

σ = E ( D) ε = E0 (1 − D ) ε = σ 0 − h (ε − ε 0 ) WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(9)

514 High Performance Structures and Materials III where E0 is the elastic modulus, ε0 = σ0/E0 the initial failure strain and h is the softening parameter, respectively.

Figure 1:

A linear stiffness degradation.

It must be emphasised that Equation (9) holds for materials in the localisation zone. Two issues related to this localisation problem should be addressed: 1) The material calibration of localisation related parameters such as the softening parameter h and the final failure strain εf. Note that the strain field is not uniform as soon as localisation occurs. Therefore, the standard strain mapping in uniaxial tensile examples: ε = u/L where L is the gauge length and u the deformation, is not valid. The results of the experiments should be interpreted carefully. 2) The mesh dependent problem. For a tensile bar with a uniform cross section area A, only one element will follow the failure path. Consequently, the total dissipation energy of the tensile bar reads

(

)

1 Wdiss = σ 0 ε f − ε 0 l A 2

(10)

where l is the length of the failure element. Clearly, the energy dissipation is dependent on the element size and, therefore, the simulation result is mesh dependent. These two issues will be discussed in the next section. First, the mesh dependence problem is solved by introducing a localisation zone into a standard finite element. Next, the corresponding localisation related parameters are investigated and their relationships are outlined. Finally, formula will be provided for practical approximation of the final failure strain and the softening parameter. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Figure 2:

5

515

Embedded localisation element (left) and its equivalent element (right).

Embedded localization element

To reduce the mesh dependency, an embedded discontinuous element is proposed. It is a virtual embedded element in the sense that an actual element is divided into two zones, one elastic (undamaged) zone and one localization (damaged) zone. Their equivalence is preserved by constraining the kinematics and equilibrium equations. Since the damage zone is introduced into the element, the corresponding dissipated energy due to damage is fixed, independent of the element size. Therefore, mesh dependency with respect to dissipated energy can be removed. Assume that the length of the damage zone is ξ and the length of the element is L, see Figure 2. When damage occurs, the strain localised in the damage zone is εh and the strain at the elastic zone is εe. If we smear the localisation zone and make an equivalent element, the following kinematics and equilibrium equations must be satisfied

ξ ε h + (L − ξ )ε e = Lε = u

(11)

E0 ε e = Eh ε h = E L ε = σ

(12)

where E0 is the elastic modulus, Eh the stiffness at the damage zone and EL the equivalent (element) elastic modulus, respectively

E h = (1 − d ) E

(13)

E L = (1 − D ) E

(14)

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516 High Performance Structures and Materials III Note that d and D are damage parameters of the embedded localisation element and the equivalent element, respectively. From Equations (11-14) the localisation strain εh can be related to the equivalent (element) strain ε

εh =

ε

1 + d (r − 1)

,

r=

ξ L

(15)

and the local damage parameter d can be evaluated and then transferred to the global (element) damage parameter

D=

rd 1 + d (r − 1)

(16)

Note that the localisation strain εh is a function of damage parameter d. This local damage parameter can be determined by inserting Equations (13)–(15) into the linear softening law (9)

d=

(E + h ) (ε − ε 0 ) Eε + (E + h ) ε 0 (r − 1)

(17)

With a known element strain ε, the local damage parameter can be directly determined. This is the advantage of a linear softening assumption, which makes the stress update simple and straightforward. Note that the softening parameters should be carefully calibrated. The traditional strain mapping cannot be applied to failure problems in which the strain distribution is highly non-uniform. Material parameter calibration should take into account the ratio of the failure zone and the gauge length. For the final failure, d=D=1, from Equation (15), the local failure strain εf and the element failure strain χ have relationship

εf =

χ

r=

r

ξ L

(18)

Suppose one uses one element to simulate a tensile bar with a gauge length L. The material failure strain εf reads

εf =

χ r

=

Lu u = ξ L ξ

(19)

where ξ is the length of the damage zone. Note that the material failure strain is obtained by dividing the deformation by the length of the damage zone and not the gauge length. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

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517

Numerical simulations

6.1 Mesh sensitivity analysis A mesh sensitivity analysis is performed by simulating the fracture behaviour of the juvenile part of a dogbone specimen with a multi-element FE model. Three different meshes are used with element size of 2 mm, 1 mm and 0.5 mm for mesh 1, 2 and 3, respectively (see Figure 3). The same material data are used for the LS-DYNA material model 58 and the new user-defined material model (with the assumption that the failure zone is equal to 0.1 mm). In Figure 4 the results of the mesh sensitivity analysis are plotted. Clearly, the global force is independent on the mesh size.

Figure 3:

Mesh sensitivity analysis: three meshes with element size 2 mm (left), 1 mm (middle) and 0.5 mm (right).

Figure 4:

Mesh sensitivity analysis for the LS-DYNA material model 58 (left) and the user-defined model (right).

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518 High Performance Structures and Materials III 6.2 Three-point bending tests Next, numerical simulations on three-point bending experiments were performed using the new developed composite model. These US bumper standard experiments (NHTSA 49 CFR Part 581: Bumper Standard) were performed on U-shaped GMT bumper beams with an aluminum impactor having a total mass 778 kg and initial impact velocity of 2.30 m/s.

Figure 5:

FE model for the three-point bending tests on composite U-beams. Force-deflection curves (filtered with CFC60 filter)

6

test data mat58 new model

5

Force [KN]

4

3

2

1

0

Figure 6:

0

10

20

30

40 50 deflection [mm]

60

70

80

90

Force-deflection curves from test and simulation for the dynamic three-point bending experiments on composite U-beams.

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The FE model of the experimental set-up is plotted in Figure 5. The same material parameters are used for the LS-DYNA material model 58 and the new user-defined material model. In Figure 6 the contact forces for the experiment and numerical simulations are plotted (all data are filtered with a CFC60 filter). Both models predict the shape of the force-deflection curve accurate until the onset of failure. It is observed that the new user-defined material model captures the softening range better than the original LS-DYNA material model 58.

7 Conclusions An orthotropic damage model is developed and tested for composite materials. The model uses the Hashin’s failure criteria in which the fibre and matrix failures are described explicitly, both in tension and compression. A linear softening degradation is proposed and a close-form solution of the corresponding damage parameter is provided. This new model is implemented in LS-DYNA as a user-defined material model. To reduce the mesh dependency, an embedded discontinuous element is proposed. It is a virtual embedded element in the sense that an actual element is divided into two zones, one elastic (undamaged) zone and one localisation (damaged) zone. Their equivalence is preserved by constraining the kinematics and equilibrium equations. Since the damage zone is introduced into the element, the corresponding dissipated energy due to damage is fixed, independent on the element size. Therefore, mesh dependency with respect to the dissipated energy is removed. It is concluded that an accurate material calibration is crucial for successful failure simulations. The traditional strain mapping cannot be applied to failure problems in which the strain distribution is highly non-uniform. Material parameter calibration should take into account the ratio of the failure zone and the gauge length. A practical approximation formula is provided.

Reference [1]

[2]

[3]

Haan, P.A.J. de, Swartjes, F.H.M and Wang, W. “Failure of composite material under impact loading- WP4: Numerical simulation of three-point bending experiments on bumper beams”, TNO Science and Industry, Internal report 04.OR.SA.032.1/FSW, 2004. Hashin Z. “Failure criteria for unidirectional fiber composites”, Composite struct. 32, 255-264 (1980). Schweizerhof, K., Weimar, K., Münz, Th. and Rottner, Th. “Crashworthiness analysis with enhanced composite material models in LS-DYNA -Merits and limits”, 5th Ls-Dyna User’s Conference, Southfield, Michigan, USA (1998).

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High Performance Structures and Materials III

521

Ring-shaped crack propagation in a cylinder under nonsteady cooling V. A. Zhornik, Yu. A. Prokopenko, A. A. Rybinskaya & P. A. Savochka Department of Theoretical Physics, Taganrog Pedagogical Institute, Russia

Abstract A quasistatic problem of thermoelasticity for a solid infinite cylinder with a ringshaped crack is considered. The cylinder is enclosed in a rigid thin heat absorbing shell with a slip joint which reduces thermal impact. The external surface of the shell is subjected to a linear heat transfer by radiation to the surroundings. We assumed that the cylinder had initially a temperature changing along the radius and the surrounding temperature was a function of time. The main result is an expression obtained for the stress intensity factor depending on time. Various particular cases are considered obtained both by the authors of this paper and by other scientists. An analysis of the dependences of the stress intensity factor (SIF) on time shows that with an assigned critical SIF, there are minimum and maximum crack sizes below and above which the crack will not grow under the given cooling conditions. For these dimensions, the maximum of SIF does not reach the critical value. For intermediate crack sizes, growth of the crack begins at the moment SIF reaches the critical value. Here, the crack first grows irregularly to the size for which at the given moment of time SIF is equal to the critical value. Extension of so called “hot ”cracks in a solid cylinder is also considered. These cracks arise when a cold cylinder is placed in a fusion of the same material at the melted temperature. Further on this cylinder is rapidly taken out of the fusion and on its surface the melting layer linked with the cylinder surface arise. Under cooling of the cylinder, residual tensile stresses arise in the layer which may cause growth of ring-shaped surface cracks which are dangerous for further exploitation of this system. Keywords: thermoelasticity, crack propagation, SIF, powdery covering.

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522 High Performance Structures and Materials III

1

Introduction

A solid infinite cylinder of radius rc with an unloaded co-axial ring-shaped crack (rd < r < rc) coming to the surface of the cylinder was chosen for crack propagation analyses in deformed solids under nonstationary thermal loading. We assume that the cylinder is surrounded by the shell with a slip joint, i.e. the surfaces of the cylinder and the shell can slide freely relatively to each other to the axial direction. It is assumed that the covered cylinder, with a constant initial temperature, is cooled over is entire surface through heat exchange with its constant– temperature environment. In this case, the heat flows are radial and the crack has no effect on heat propagation within the cylinder.

2

Mathematical modeling

In light of this, the solution of the thermoelastic problem is represented as the sum of two solutions,

σ ij = σ ijT + σ ijP , u i = u iT + u iP .

(1)

The first solution, obtained with the assumption that no cracks are present, satisfies the equations of thermoelectricity for an infinite cylinder in a generalized plain-strain state. This solution satisfies all boundary conditions except for those for the edges of the crack, which are loaded by a normal load σ T (1 − ν ) σ zz* = zz αT E (T0 − θ ) ∞

= 2 Bi ∑

yn J 0 ( yn ) ( Bi − ωky  a root of equation n =1

yn –

exp ( − yn2 Fo )

)

2 2 n

  r   2 J 1 ( yn ) − yn J 0  yn     rc   + y n2 (1 + 2ωk )  

(

(2)

)

yJ1 ( y) = Bi − ωky 2 J 0 ( y) . Here E is the elastic modulus of the cylinder material; ν is the Poisson ratio;

αr c is the Biot criterion; λ ρ c 1 + α 0 d 2λ T 1 α = α0 ; ω= Π Π ; 1 + α 0d λ Π ρc 1 + α 0 d λ T k = d/rc; d – coating width.

αT is the coefficient of lineral expansion; Bi =

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High Performance Structures and Materials III

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523

Calculations

For numeral analyses a solid cylinder of the radius rc = 5⋅10–3 m made of steel 45 (it’s chemical composition: Fe, 0,42-0,49% C, 0,17-0,37% Si, 0,5-0,8% Mn; ρ = 7800 kg/m3, c = 470 J/kg⋅K, λ = 50 W/m⋅K) is chosen, coating is made of bronze Бр ОФ–10–1, (it’s chemical composition: Cu, 9-11% Sn, 0,8-1,2% P; ρ0 = 8700 kg/m3, c0 = 380 J/kg⋅K, λ0 = 100 W/m⋅K); it’s width is d = 0,25⋅10–3m. The coefficient of heat transfer α0 = 1,5⋅105 W/m2⋅K. The porosity θ = 0,54. Using the formula for the porous medium λΠ = λ0(1 – 1,5θ) for θ < 0,6 (3) and the formula cΠ = c0(1 – θ) (4) (5) ρΠ = ρ0(1 – θ) we obtain: ρΠ = 4000 kg/m3, cΠ = 175 J/kg⋅K, λΠ = 19 W/m⋅K. As a result k = d/rc = 0,05 << 1, ωk = 0,0065, Bi = 5. In the case of covering absence (d = 0) we obtain ωk = 0, Bi = α0rc/λ = 15. Figures 1 and 2 show the relations between time Fo = at/ rc2 (a – heat diffusing) and axial strains σ zz* depending on the radius r/rc for the cases of the cylinder with the covering (Figure 1) and without it (Figure 2). As we can see from the graph on the figures powdery coverings essentially reduce thermoelastic stress in cylinder.

*

Figure 1: Diagram for σ zz depending on Fo for different r/rc (without covering). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

524 High Performance Structures and Materials III

*

Figure 2: Diagram for σ zz depending on Fo for different r/rc (with covering). In the case of plane-strain stress axial displacement is zero in the plane of the crack. When added to the first solution, the second solution – from the isothermal theory of elasticity – satisfies all boundary conditions including those for the crack edges. Because of symmetry conserning the plane of the crack, the second solution is for semi – infinite cylinder (z > 0) with mixed boundary conditions on the end – wall. Through boundary condition for tangent stress is given on the end – wall. The condition of slipping joint leads to zero radial displacement and zero tangent stress component. Mathematical setting of the problem is:

σpzz = −σTzz , u pz = 0 ,

rd < r < rc,

z = 0,

(6)

0 < r < rd,

z = 0,

(7)

0 < r < rc,

z = 0,

(8)

r = rc,

0 < z < ∞,

(9)

p

σrz = 0 p rz

σ = 0, p r

u =0

r = rc, 0
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525

High Performance Structures and Materials III

As is known, the solution of the problem from the isothermal theory of

(

P

P

elasticity σij , u i

) reduces to finding the function χ(ρ, ζ ) , having the form rc2 Bn (λ n )J 0 (λ n ρ )e− λ n ζ n =1λ n ∞

χ(ρ, ζ ) = ∑

(11)

Here, λn are the positive roots of the equation of Jl (λ) = 0, ζ = r / rc. The boundary conditions for the rigid shell (8),(9),(10) are satisfied identically through selection of the function χ(ρ, ζ ) . By substitution

Bn (λ n ) = b n (λ n ) +

rd  2 t g(t ) cos λ n dt ∫ 2 rc J 0 (λ n ) 0  rc 

(12)

the unknown Bn is determined from a pair of the ordinary equations constructed from the mixed boundary conditions assigned in the plane of the crack

σ pzz = −2µF(r ) ,

0 < r < rd,

z = 0,

(13)

u pz

rd < r < rc,

z = 0,

(14)

= −δ ,

Here, µ is the shear modulus, bn are the coefficients of expansion of the



function F(r ) = σ zz 2µ in terms of J 0  λ n T



r  , δ is the constant normal rc 

displacement along the crack which is determined from the balance condition of the cylinder in the plane of the crack. rd

rc ∫ g (t )dt = − 0

rc P − ∫ F(x )xdx 4πµ rd

(15)

P is a preset force, applied to the cross-section of the cylinder at infinity. The final solution of this problem reduces to finding the solution of a Fredholm equation of the second kind relative to the unknown function g(t) independent on δ (in particular δ can be equal to zero).

g (τ ) = −

1  1  P 211 a ( ) − + yF(yrc ) − dy yF yr arcsin ∫ ∫ c 2  παα α  4πµrc α y 

2 1 yF(yrc )dy 4 ∞ K1 (ξ )  sh ξα  1 ch + − ξ ξτ ∫ ∫  dξ ∫ yF(yrc )I0 (ξy )dy + π α y 2 − τ 2 π2 0 ξI1 (ξ )  α α +

∞ 4 α K1 (ξ )  sh ξα  ( ) g d ν ν ∫ ξI (ξ ) ch ξν  α − ξch ξτ dξ , 2 ∫ π 0   0 1 0 < τ < α, α = rd rc . (16)

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526 High Performance Structures and Materials III Integral equation (16) with respect to g( τ ) was solved by a successive approximation technique where function F(r) was approximated by polynomial with even degrees from 0 to 8.In this case the maximum approximation error which occurs at small Fo is no more than 2 %.Stress intensity factor SIF K I is determined according to [3]

K I = 2µ

π

rd

r c g (r d )

(17)

In Figure 3 the dependence of stress intensity factor SIF

K *I =

(1 − ν ) (T0 − θ)

2K I 1/ 2 rc α T E

π

on time for Fo for different ring-shaped crack sizes α = rd/rc in the case of “rigid” slip joint where d = 0 is shown. In Figure4 – the same for d = 2,5⋅10–4m. In case of “flexible” joint SIF is 12 % larger for the same sizes of the cylinder and the shell.

Figure 3: Dependence SIF on time Fo fir different crack sizes (without covering).

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Figure 4: Dependence SIF on time Fo fir different crack sizes (with covering).

4 Analyses and conclusions Let us analyze ring-shaped crack propagation in a covered cylinder starting from relation shown in Figure 4. * Let us assume critical SIF K IC = 0.136 (horizontal dotted line). Then, for example, a very small ring-shaped crack of the size α = 0.95 (the depth is 0.2 mm) at the moment of time Fo ≈ 0.02 begins to grow unstable, jump-like (point a) and passes the sizes from α = 0.95 to α = 0.8 (vertical arrows). After that the * crack propagates stable (horizontal arrow) as K I* reaches it’s critical value K IC = 0.136. Finally the crack stops at the moment of time Fo = 0.12 (point b) as the crack size is α = 0,5 (the depth is 2,5 mm). This fact should be taken into account when covering the cylindrical details by powder materials.

References [1] Zhornik A.I. Nonsteady problem of the theory of elasticity for a solid infinite cylinder with a penny-shaped crack, enclosed in a thin shell, International Applied Mechanics, USA – 1993, July – P. 47 – 52. [2] Zhornik A.I., Zhornik V.A., Kashitsyn L.P. Nonsteady problem of thermoelectricity for a solid infinite cylinder will a ring-shaped crack, enclosed in a thin shell, Advanced Computational Methods in Heat Transfer WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

528 High Performance Structures and Materials III IV: Book of Proceedings. Fourth International Conference on Advanced Computational Methods in Heat Transfer “Heat Transfer 96” – Southampton, UK, Boston, USA, 1996.– P. 505 – 516. [3] Das B.R. Thermal stress in a long cylinder containing a penny-shaped crack on the distribution of stress in a long circular cylinder, Int.J. Engng.Sci.1963,v.1 - P.391-406

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529

Investigation of the hygrothermal performance of wooden beam ends embedded in inside insulated outside walls H. Stopp & P. Strangfeld Department of Building Physics, University of Applied Sciences FH Lausitz, Germany

Abstract The inside insulation has been investigated in the past by means of scientific institutions as a necessary method in connection with the thermal improvement of worth-preserving facades of architectonic heritage. In the case of fair faced brickwork it is established on homogenous areas of the wall, for instance as a variant of capillary active systems or as a classical version with a humid-dependent vapor barrier. With regard to the insulation of special details, e.g. embedded wooden beam ends within areas of outside walls, the opinion of experts differs considerably. Both the total ventilation of the beam ends and complete air tightness are proposed. In line with the new European directive on the energy efficiency of buildings, innovative ideas should also be developed in the field of building physics in order to realize the restrictions without damage to building envelope parts. The main point of this paper is the investigation of the hygrothermal situation of embedded beam ends in inside insulated outside walls, dependent on time and climate boundary conditions. The topics are treated by means of in situ measurements in two test houses in the eastern part of Germany under condition of use and by experiments in connection with an erected test stand. A comprehensive numerical simulation of the coupled heat- and mass transfer supports the measurements to reduce the experimental work. In principle there are several options to avoid the effects of the hygrothermal deficiencies of the structure caused or intensified by inside insulation. Hygrothermal effects are quantified and solutions are found for workmanship without damage. Keywords: wooden beam ends, worth-preserving facades, heating energy saving, hygrothermal performance, damages to structures.

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530 High Performance Structures and Materials III

1

Introduction

In case of worth preserving facades there are some conflicts: for an example on the one hand the changing climate conditions challenge the mankind to develop political strategies and to implement measures that reduce greenhouse gas emissions. Plans of European Member States call for 20% ...40% decrease by 2020. Another key objective, which is linked to the environmental protection, is to make European states less dependent on oil and gas. Current forecasts indicate that without measures this dependency will jump to approximately 70% by 2030. A directive of the European Parliament and of the Council on the energy performance of buildings was published and discussed in the past. The updated regulation came into force in Dec.16th 2002 [1]. The directive should be realized in the member states of the European Union up to 2006. About 41% of the total final energy demand of Europe is used in the residential and tertiary sectors. Space heating is by far the largest energy end-use of households in EU Member States, namely 57%. Therefore the European directive covers four main elements: -Establishment of a general framework of a common methodology for calculation the integrated energy performance of buildings. -Application of minimum standards on the energy performance to new buildings and certain existing buildings when they are renovated. -Certification schemes for new and existing buildings on the basis of the above standards and public display of energy performance certificates and recommended indoor temperatures and other relevant climatic factors in public buildings and buildings frequented by the public. -Specific inspection and assessment of boilers and heating and cooling installations. About 32% of the current stock of the 150 million residential dwellings in the 15 old EU Member States was built prior to 1945, about 40 % between 1945 and 1974 and 28% since 1974. An upgrade of thermal insulation regulations and improved efficiency for installed equipment for existing dwellings, bringing them close to current buildings codes, would help to realize an important saving potential, making it a very desirable and in most cases a cost-effective option. So the directive is explicit valid for existing buildings too. In the face of the low turnover rate of buildings with a lifetime from 50 to more than 100 years it is clear that the existing stock of buildings contains the largest potential for improving energy performance in short and medium periods of time. On the other hand due to an architectonic heritage the use of a thermal insulation composite system or another outside insulation is not possible. An inside insulation may be an alternative if the hygrothermal and acoustic behaviour is considered. But in regard with the inside insulation of wooden beam ends embedded within areas of outside walls, the opinion of experts differ considerably. For this reason the investigation is to be done both by means of numerical simulation of the coupled heat- and mass transfer included the effects of ventilation and a lot of in situ measurements in test houses under condition of use and by experiments in connection with an installed test stand in a laboratory. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

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531

Calculation of the hygrothermal performance

Owing to the requirements of the new European directive and to the demand for a comfortable room-climate the thermal insulation of outside walls of existing buildings should be increased. A good solution for homogenous areas in case of worth- preserving facades is the so-called capillary-active inside insulation or a classical version with a humid-dependent vapor retarder [2]. The results of the numerical simulation of the coupled heat and moisture transfer, represented in this paper are carried out by means of the program “Delphin” [3], validated also in international projects. Fig. 1 represents the moderate moisture distribution by means of a capillary active calciumsilicate inside insulation without use of a classic vapor barrier. This method guarantees the drying out of moisture caused for instance by driving rain at the facades. Owing to the capillary diffusivity function of the material calciumsilicate with a maximum of 10-5m2/s the condensation at the cold site of the inside insulation is limited in wintertime and due to the deleted vapor barrier a sufficient drying out process takes place during summer period.

Figure 1:

Moisture field of an inside insulated brick masonry wall by 50mm calciumsilicate. Indoor climate: 20°C; 50%. Outdoor climate: TRY Middle Europe, hills, Westside

Embedded wooden beam ends in connection with an inside insulated worth preserving facade can be a hygrothermal problem. A non-reflected application of energy saving directives particularly with regard to the interior insulation can cause defects of restoration. The formulated requirements of the energy directives are raising many questions to the architect, contractor and client of a restoration project. Therefore the purpose of a special recommendation of the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

532 High Performance Structures and Materials III WTA [4] is to provide support in interpreting and applying the regulations (e.g. in case of half-timbered framework the additional thermal resistance of an inside insulation system should be limited to 0.8 m2·K/W). 20+ 17.5 to 20 15 to 17.5 12.5 to 15 10 to 12.5 7.5 to 10 5 to 7.5 2.5 to 5 0 to 2.5

100

300

6

4

8

12

10

15

y in mm

500

20 20

700

5

15

10

900

1,100

1,300

0

100

Figure 2:

200

300

x in mm

400

500

700

Temperature field Jan. 29th, variant “heating channel”, heating mobilized, (air temp within channel: 35°C, partial pressure of water vapor: 1170 Pa), TRY Middle Europe.

20+ 18 to 20 16 to 18 14 to 16 12 to 14 10 to 12 8 to 10 6 to 8 4 to 6 2 to 4 0 to 2

100 3

5

300 8

500

y in mm

600

5

700

10

15

900 5

10

15

1,100

1,300

0

100

200

300

400

500

600

700

x in mm

Figure 3:

Temperature field Jan. 29th, variant: “complete inside insulation system”, TRY Middle Europe.

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Figure 4:

Effect of thermal bridge caused by a mobilized heating cannel, additional heat losses of the wall, related high of wall 0.52 m and 2.40 m, width of wall: 1 m.

Figure 5:

Temperature field of the area of the wooden beam end influenced through a so called passive heating bar (thermal short-circuit by means of the inserted metallic conductor aluminium) Climate boundary conditions: indoor 20°C/50%; outdoor -10°C/80%.

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534 High Performance Structures and Materials III The selected figures 2 and 3 demonstrate the deformation of isotherm-lines by the operating flow pipe and return, installed within a „channel“ of the insulation near the floor (the insulating calciumsilicate panel with a thickness of 50 mm is replaced at the bottom for a height of 80mm) in comparison with an inside insulation of an outside wall completely (fig. 3). Fig. 2 shows the positive effect of an operating heating pipe in a channel. The increased temperature of 3 K at the front side of the beam should not conceal the fact that the calculated decreasing of the wooden moisture at the same place amounts to about 1mass% only. The fig. 4 represents the additional heating energy losses caused by the effect of thermal bridge with different reference areas of the wall. In the fig. 5 the shifted temperature field is shown caused by a so-called passive heating bar.

Figure 6:

3

Course of the wooden moisture measured in front of the four wooden beam ends H1...H4 after retrofitting is finished.

Experimental investigation

3.1 In situ measurements The in situ measurements are carried out under conditions of use in a climate of Middle Europe in test houses near Berlin [5]. The courses of the wooden moisture content of four investigated wooden beam ends prove the considerable quantities of moisture built in during the retrofitting activities. They show the long periods of drying out in laps of several years also in case of indoor climate similar to an office, fig 6. The start of the supply of heating at the wooden beam end H4 by means of a lengthened flow pipe with return as a bypass is to recognize clearly. The wooden moisture decreases by 2M% in comparison to a no operating flow pipe (Comment: beam ends H4, H3 and H2 are influenced through the same pipeline of flow and return for the heaters. Only H4 is to WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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separate from the heating energy supply completely – no heater is installed parallel to the pipeline). The reduction of the relative humidity of the air gape surrounding the beam head amounts up to about 10%, fig.7.

Figure 7:

Course of the relative humidity of the air gape in the front side of the beam end H4.

Figure 8:

Course of the temperature in front of the four wooden beams ends H1...H4 dependent on the heating supply through the flow pipe and return.

The effect of the additional heating supply through the heating channel to the temperature in front of the beam ends demonstrates fig. 8. After the heating flow to H4 is mobilized (Jan.4th.2002) the differences between the temperatures at the beam ends H4 and H3 are compensated. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

536 High Performance Structures and Materials III Experiments depending on the users activities deal with the influence of the indoor air on the humidity of the air gaps surrounding the inside insulated beam ends. The influence of a natural ventilation by opening of windows or/and doors is an indifferent result in practice and can bring in warm-humid air into the area of the air gaps, fig. 9.

Figure 9:

Penetration of indoor air to the front side of wooden beam ends owing to the activities of inhabitants

3.2 Test stand The effect of an additional energy supply is small in case of a thermal shortcircuit of the natural temperature gradients between the indoor and the air gape by means of a good conductor without thermal resistance, e.g. through a metallic bar inserted into the wooden beam end. In opposite to that passive bar the fig. 10 shows the abrupt increasing of the temperature from March 8th in 2003 clearly in spite of the many measuring points at the prepared wooden beam head. The period of heating energy supply of an active heating bar is realized in combination with an underfloor heating.

4

Conclusions

With regard to the hygrothermal performance of wooden beam ends embedded in inside insulated worth-preserving outside walls, some results for practice are represented and evaluated in the following briefly. They are worked out by numerical simulation and measurements under conditions of use. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 10:

537

Effect of a so called active heating bar within a wooden beam head mounted in an outside wall of the Building Physics testing floor of the FH Lausitz.

- A supply of additional heat to the wooden beam ends by an aimed conductivity of heating energy including available building components represents the best solution. - For instance without additional expenses and few effort the heating system’s flow pipe and return fitted in a heating channel should be used. The effect of thermal bridges is to neglect. - The idea of a so-called passive or active heating bar, inserted into the wooden beam end connected with an underfloor heating, is an interesting variant. It should be tested detailed in future. - Concerning the influence of the natural ventilation on the hygrothermal situation of the air gaps surrounding the beam ends the investigations show that the wooden beam ends are individualists.

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538 High Performance Structures and Materials III - In connection with a bigger stock of existing buildings of an increasing European Union the topic provides an option for the preservation of a lot of the building heritage in the new eastern countries. The thermal insulation of outside walls with worth-preserving facades and embedded wooden beam ends can be improved without damage.

References [1] [2]

[3] [4]

[5]

Richtlinie 2002/91/EG des Europäischen Parlaments und des Rates vom 16. Dezember 2002 über die Gesamtenergieeffizienz von Gebäuden, Amtsblatt der Europäischen Gemeinschaften Nr. L1 pp. 65. Stopp, H.; Strangfeld, P.: The hygrothermal performance of external walls with inside insulation. In proceedings of the conference, Performance of exterior envelopes of whole buildings VIII: Integration of building envelopes. Clearwater Beach FL, U.S.A., Dec.2001. Grunewald, J.: Diffusiver und konvektiver Stoff- und Energietransport in kapillar-porösen Baustoffen. PhD thesis, TU Dresden, Germany, 1996. Restoration of historical buildings. EnEV: possibilities and limits. WTA:Wissenschaftlich-Technische Arbeitsgemeinschaft für Bauwerkserhaltung und Denkmalpflege e.V., D-80686 München, Germany, 2002. Stopp,. H.; Strangfeld, P. et al.: Heizungstechnisch gestützte kapillaraktive Innendämmung bei Holzbalkendecken. Bau- und Wohnforschung–Bericht F 2431, Fraunhofer IRB Verlag 2004.

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Bond repair of cracked beams H. Cruz1, J. Custódio1 & D. Smedley2 1

Structures Department, Wood Structures Division, Laboratório Nacional de Engenharia Civil, Portugal 2 Rotafix, Rotafix Ltd., Rotafix House, Abercraf, Swansea, UK

Abstract The repair of cracked timber beams on site frequently involves crack injection with adhesives followed by the application of the chosen strengthening system, involving bonded or bolted metal plates or rods or bonded pultruded GFRP or CFRP profiles. Although strengthening design will not generally take into account the contribution of injecting adhesives in the crack volume region it is felt that this contribution to overall strengthening of the component or structure may be of some significance. A test programme was therefore developed at LNEC to assess the strength and stiffness of timber beams which were treated with epoxy adhesives after they had developed failure in a 4 point bending test. It is recognised that partially broken beams may have very narrow, almost invisible, fissures, which are difficult to impregnate. As well as the forgoing partially separated broken beams may present extremely irregular split ends, which are difficult to bring back into perfect alignment and contact. These two factors may reduce the effectiveness of the strength restoration and were considered in the experimental study. However, the specific beam failure appearance is not likely to give a clear indication of how efficient the consolidation turns out to be. Test results show that epoxy adhesive injection of cracked timber beams may lead to a not insignificant, but somewhat variable, strength recovery (0.45 to 1.11) and to a significant stiffness recovery (0.74 to 0.99) of broken beams. Having these results in mind, although on site repair of broken timber beams should not rely on epoxy adhesives injection only, its contribution to the stiffness of strengthened beams is significant and one should make use of this advantage. Keywords: cracked timber beams, structural adhesives, consolidation, strength and stiffness recovery.

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540 High Performance Structures and Materials III

1

Introduction

Old timber structural members may present failure due to biological damage, which is responsible for timber strength reduction or cross section loss, or due to unacceptable stresses resulting from inadequate design or from structural accidental or intentional modifications. Furthermore older structures intended for a particular purpose are currently being subjected to types of use for which the original design parameters were not intended. Although the replacement of damaged members by new timber or other materials may seem the easiest solution, on site repair may be a better cost option in cases where only a small timber length is damaged, or when that member is part of a complex structural system and its removal may jeopardise decorative ceilings and/or limit the occupancy of the building, Even the replacement of one carrier beam may with traditional and well-proven carpentry techniques involve the whole or partial removal of a roof structure with its corresponding time and cost delay. In fact removal of sound surrounding components often puts hitherto undamaged and adequate parts of the structure at risk. The repair of cracked timber beams on site frequently involves crack injection with adhesives followed by the application of the chosen strengthening system, namely involving bonded or bolted metal plates or rods or bonded glass fibre profiles. It is interesting to note that in the early days of injection technology with timber (borrowed from Concrete repair practice in the late 1950’s early 1960’s) attempts were made to restore overstressed classical truss rafter systems with inappropriate low viscosity 2 part epoxy so-called injection systems (Ref Aberdeen Music Hall) - the lack of success resulted in a secondary repair method using bolted ‘U’ beam sections. Current strengthening design will not generally take into account the contribution of injecting adhesives in the cracked area, so-called consolidation. It is felt that this contribution might be significant. There was also the question as to whether it would be possible to identify in advance some typical failure configurations likely to benefit in practice from the so-called consolidation technique. A test programme was therefore developed at LNEC with the intention of assessing the strength and stiffness of timber beams which were consolidated with epoxy adhesives after they had developed failure in 4 point bending tests. References to studies involving timber crack injection could not be found in the literature, although they have some similarity with glued timber delamination repair. For the specific case of crack consolidation, with complex irregular surfaces of the adherend it is necessary to find adhesives with suitable rheology. Thick gap-filling adhesive which usually contain fillers are not deemed to be suitable, whereas very fluid adhesives will penetrate more easily but will not be capable of filling small voids left inside the timber member, reducing bonding efficiency at those points. This was observed in previous trials by the authors [1], where a poor crack consolidation (and corresponding low strength recovery) was obtained by the combined use of a thick gap-filling adhesive followed by

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injection with a very fluid one. To avoid this problem, a thixotropic adhesive was used in the current experimental programme. Another peculiarity of this kind of repair lies in the fact that preparation of clean bonding surfaces is not possible. Wood surface oxidation and any possible air borne or liquid contamination will have taken place since the time between fracture and repair intervention could be days or even years depending upon discovery of the fracture and application of the chosen remedial solution.

2

Materials and method

Test specimens consisted of sixteen 1500x120x70 mm Maritime pine (Pinus pinaster, Ait.) timber beams. Timber defects like knots were within the limits of the EE strength grade defined in NP4305 [2], although no restrictions concerning the presence of pith and the associated juvenile wood (which is not allowed in grade EE) were imposed. Two series of beams were studied: 1 to 10 and A to F. Every beam was first tested up to failure, in 4-point bending, to evaluate its maximum strength and modulus of elasticity. Moisture content of timber varied between 12% and 14% (mean value = 13%). After a period of time (making approximately nine months for beams 1-10 and fifteen months for beams A-F), each beam was consolidated with the thixotropic epoxy adhesive standard Slow Set CB10T, from ROTAFIX. No regularization of the split ends was done and the beams were brought back to the initial (straight) shape by using clamps and steel bars which were removed after adhesive cure. Two months later, the consolidated beams were subjected to a 4-point bending test again. The same tension edge was chosen in both the bending tests, before and after consolidation. Moisture content of timber at the time of the second bending test was checked with the electrical moisture meter and it was seen to be similar to their initial value. Bending tests were carried out in general accordance with the specifications of EN408 [3]. The 1.50 m long specimens, simply supported over a span of 1.40 m, were symmetrically loaded in bending at two points 0.50 m apart. Small steel plates, 5 cm large, were inserted between the test piece and the loading heads, and between the test piece and the supports, to minimise local indentation (figure 1). The total load F (equal to the sum of the vertical loads at the two load points) was applied at a constant displacement rate of 1 cm/min. Total deformation (wtotal) was measured at mid span, at mid depth of the cross section. Bending strength and elasticity modulus of each beam were to be assessed in a single test; therefore, the referred displacements were measured till failure. The global modulus of elasticity in bending was determined by applying the relevant equations in EN 408 [3].

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542 High Performance Structures and Materials III

Figure 1:

3

Test set up and specimen being tested in bending.

Test results and discussion

Table 1 presents individual test results in the first (initial sound state) and in the second (consolidated) round of tests, concerning the global modulus of elasticity and bending strength. In annex A are presented some examples of the beams after the second round of bending tests. The fibre separation lines on both faces and bottom edge are enhanced on the photos (fine lines for old cracks – before consolidation, thick lines for cracks developed on the consolidated beams). Load-deformation diagrams obtained for each beam, in both rounds of bending tests (initial state and consolidated beam), are presented in the annex B. The average recovery of strength due to consolidation with the epoxy adhesive was 0.72 (ranging from 0.45 to 1.11). In most cases, the initial part of the load-displacement test diagrams obtained before and after consolidation were very similar. Therefore, similar global modulus of elasticity was obtained – 0.90 efficiency was obtained with individual beams (ranging from 0.74 to 0.99). This seems to indicate that consolidation was effective till a certain point – under moderate loads – although it was not enough to recover the performance of the solid timber beam for stress levels closer to the initial bending strength. This is confirmed by the fact that bending failure in the consolidated beams generally affected the same material, that is to say that it occurred in the same areas where the initial beam had previously failed. There is not a clear correspondence between crack pattern and efficiency of the repair: apparently similar cracks have conducted to distinct results within the range of strengthening values obtained. Despite the small number of beams in each group, there is no clear distinction between groups 1-10 and A-F. In both cases, the application of the epoxy

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adhesive took place several months after failure, thus simulating the application of adhesive to old timber surfaces. Table 1:

Global modulus of elasticity and bending strength for initial beams and beams after consolidation (ini/cons).

Beam

Modulus of Elasticity (MPa) Econs

Econs/Eini

Fini

Fcons

Fcons/Fini

1

11921

9684

0.81

29.88

14.00

0.47

2

16193

15504

0.96

56.99

63.13

1.11

3

17371

14680

0.85

45.74

20.77

0.45

4

12479

12033

0.96

43.93

33.18

0.75

5

12504

10660

0.85

31.49

20.93

0.66

6

13378

12843

0.96

41.10

42.48

1.03

7

11706

(1102)

-

32.68

6.11

-

8

15261

15105

0.99

40.42

33.84

0.84

9

14431

12162

0.84

44.39

24.26

0.55

10

15683

14989

0.96

50.40

27.49

0.55

A

14169

10519

0.74

59.05

43.54

0.74

B

13364

11668

0.87

44.54

36.68

0.82

C

14183

13913

0.98

51.66

27.49

0.53

Mean (1-10)

4

Maximum Load (KN)

Eini

0.91

0.71

D

12636

11400

0.90

48.39

23.42

0.48

E

13565

12345

0.91

58.67

55.30

0.94

Conclusions

It is recognised that partially broken beams may have very narrow, almost invisible, fissures, which are difficult to impregnate. Together with the fact that partially separate broken beams may present extremely irregular split ends, which are difficult to bring back into perfect contact. These two factors may reduce the effectiveness of the consolidation and were considered in the experimental study. However, the specific beam failure appearance is not likely to give a clear indication of how efficient the consolidation turns out to be. Nevertheless, test results show that epoxy adhesive injection of cracked timber beams may lead to a not negligible, but somehow variable, strength recovery (0.45 to 1.11) and to a significant stiffness recovery (0.74 to 0.99) of broken beams. Having these results in mind, although on site repair of broken timber beams should not rely on epoxy adhesives injection only, its contribution to the stiffness of strengthened beams is significant and should not be ignored. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

544 High Performance Structures and Materials III

Annexes Annex A

Beam 1 WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Beam 3

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Beam 6

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Beam E

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546 High Performance Structures and Materials III

Annex B

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Acknowledgement This work was carried out in the scope of the European Research Project CRAFT-1999-71216 LICONS – “Low Intrusion Conservation Systems for Timber Structures” (www.licons.org), which had a much wider scope. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

548 High Performance Structures and Materials III The authors wish to thank STAP (www.stap.pt) for carrying out crack consolidation and ROTAFIX (www.rotafix.co.uk) for providing the thixotropic adhesive used in this study.

References [1] COLORETIM CT 98-9548, COmposites LOcal REinforcement for TIMber structures, Final Report – “Consolidating tests: bending tests on timber beams strengthened with FRP. [2] Norma Portuguesa NP 4305: “Madeira serrada de Pinheiro bravo para estruturas. Classificação visual” – “Visual strength grading of maritime pine”. [3] European Standard EN 408: “Timber structures – Structural timber and glued laminated timber – Determination of some physical and mechanical properties”. Draft revision, November 1999.

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Microtexture and nanoindentation study of delamination cracking in Al-Cu-Li-X alloys R. Crooks1, M. S. Domack2 & J. A. Wagner2 1 2

National Institute of Aerospace, Hampton, Virginia, USA NASA Langley Research Center, Hampton, Virginia, USA

Abstract Commercial Al-Li alloys have strength and weight advantages over non-Al-Li alloys. The fracture behavior of these alloys is unusual and has limited their use. The fracture mode, described as delamination, is intergranular, along the broad grain boundaries parallel to the rolling plane of the plate. Microtexture analyses have shown that delaminations occur along boundaries with greater than 30º misorientation. However, it was observed that relatively few of the high angle boundaries exhibited this behavior. Some grains of the retained deformation texture show high internal misorientation, which is a measure of stored strain energy. Delamination tends to occur between these grains and adjacent, recrystallized grains. Nanoindentation studies indicate a higher hardness for the high internal misorientation grains. These results suggest that the delamination could be reduced by processing the alloys to minimize grain-to-grain property disparities. Keywords: microtexture, nanoindentation, Al-Li, delamination.

1

Introduction

Fracture in aluminum aerospace alloys typically occurs by ductile rupture and/or transgranular shear. Al-Li alloys saw renewed interest in the 1980’s as a low density, high strength alternative to conventional aerospace Al alloys [1]. In addition to exhibiting ductile rupture and transgranular shear, many Al-Li plate alloys in the T8 temper (stretched and age hardened), exhibit delamination fracture. The intergranular fracture in these alloys is referred to as delamination because fracture along the characteristically thin, elongated grains resembles the separation of a laminated material. When the primary crack is perpendicular to WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06054

550 High Performance Structures and Materials III the rolling plane in the LT or TL orientation, the triaxial stress state near the crack tip can be sufficient to separate grain boundaries susceptible to delamination. The propensity of Al-Li alloys to delaminate has limited their use in aerospace applications and is due in part, to the absence of aerospace design tools to account for delaminations. Development of mechanistic models that describe delamination fracture behavior could help guide the modification of ingot processing to reduce the tendency for this type of fracture, thereby allowing designers and fabricators to realize the full benefits of Al-Li alloys. There are several theories to explain this tendency for intergranular fracture. Based on recent journal publications, there are at least three current and competing explanations for this behavior. These can be described as planar slip, grain boundary precipitation, and grain boundary segregation. The planar slip explanation has been strongly advocated by Sanders and Starke [2] and Csontos and Starke [3]. The reasoning is that intense planar slip bands arise due to the presence of shearable precipitates, e.g. the δ’, Al3Li phase. Shearing of the precipitates reduces the resistance to dislocation glide within the slip system and promotes planar slip. Planar slip bands cause intense stress concentrations at grain boundaries and promote intergranular fracture. Planar slip and precipitate shearing also occur in Al-Li-X alloys where the δ’, Al3Li precipitates are not the dominant strengthening phase. This is evident in the shearing of the higher strength ternary T1, Al2CuLi precipitates [3-5] and may be related to disruption of order within the Li-containing matrix. An unusual feature of these alloys is the presence of large, grain boundary precipitates which cover large area fractions of high angle grain boundaries. Vasudevan and colleagues [6-9] have published several studies which conclude that the grain boundary particles are responsible for the relative weakness of the boundaries. Lynch et al. [10, 11], on the other hand, maintain that intergranular fracture in Al-Li-X alloys is due to grain boundary segregation of Li. Lithium segregation is postulated to cause a modification of grain boundary structure and a lowering of grain boundary cohesive strength. The publication of several conflicting explanations for low energy intergranular fracture of these alloys shows a lack of conclusive evidence for a delamination fracture mechanism. In this study some relatively new analytical techniques, namely microtexture and nanohardness, have been used to obtain more precise information on the grains adjacent to the fracture path of delamination cracks in the Al-Li alloy 2090. The grains in unrecrystallized AlLi-X alloy plate tend to be less than 20 µm thick, so a plate of 1 cm thickness would have 500 or more grains in the thickness direction. Delamination is, however, limited to a small fraction of boundaries (Figure 1). The properties and microstructures of the grains adjacent to the delaminations may help provide a more accurate description of the fracture mechanism. Microtexture, as a commercially available technology, is a development of the last decade [12, 13]. Microtexture studies rely on the collection of grain specific orientation information from polished sections of materials. The WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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information is collected in a scanning electron microscope equipped with an electron backscattered pattern detector (EBSD) system, and uses a computer interface to position the electron beam at designated points on a tilted, polished sample; collect diffraction patterns with a phosphor screen coupled to a low-light camera and a series of computer algorithms to determine the crystallographic orientation. Data can be collected with spatial resolutions better than 0.1 µm and at rates better than 50,000 points per hour. Most data is collected in an automatic mode over a grid of selected size and point separation, and the data can be displayed as maps showing various aspects of the crystallite orientation, the boundary angle between grains, and even the extent of deformation of unrecrystallized grains. Thickness, 0.95 cm Delamination

Side Groove

Figure 1:

Side Groove

Cross section of a fractured specimen of Al-Li, showing delamination cracks. Growth direction of primary crack is into the plane of the page. Microtexture results show that delaminations occur along relatively few high angle boundaries.

In earlier work [14, 15], microtexture methods were used to examine the misorientation angles of delaminating boundaries, the possible role of special boundaries, and similarities between boundary planes and precipitate planes. These studies showed that delamination occurs at high angle boundaries in excess of 30º misorientation. No conclusive correlation was found with grain boundary types, crack path and precipitate distributions [15]. However, Kalu and Wagner [14] showed that delaminations in alloy 2090 tended to occur adjacent to – grains of the {110}<112>, Brass texture component orientation. The Brass component is an extremely frequent orientation of the rolling texture in this alloy. Nanoindentation is another recently developed technique, with a history including the last twenty or thirty years [16]. With currently available instruments it is possible to determine hardness and Young’s modulus from extremely small regions. Loads as small as 1 nN and displacements as small as 0.1 nm can be measured. Measurements of other properties, e.g. local yield stress, strain-hardening and creep behavior are also possible. Recent nanoindentation studies of aluminum alloys have examined the micromechanical WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

552 High Performance Structures and Materials III properties of particle reinforced aluminum matrix composites [17, 18], and the properties of precipitate free zones in Al-Zn-Mg alloys [19]. In this study, nanoindentation was used to explore the grain-to-grain property variations of the Al-Cu-Li-Mg-Zr alloy 2090. Results were correlated with microtexture studies which characterized both the location of delaminations and the variation in recovery and recrystallization in the alloy plate.

2

Experimental

Fractured 2090 compact tension samples were sectioned normal to the fracture surface and crack direction in a region containing several delaminations. Sectioning was performed with a slow speed diamond saw. Fracture surfaces were protected by painting with an acetone soluble stop-off lacquer (Microstop), mounted in 1.25” diameter epoxy mounts and metallographically polished with a specimen load of ~ 4lbs./sample. The polishing procedure involved the following abrasives and times: 600 grit SiC until flat, 1200 grit SiC for 15 minutes, 3 µm Diamond paste with Struers Blue lubricant for 30 minutes on Buehler Texmet 2000 cloth, 1 µm Diamond paste with Struers Blue lubricant for 30 minutes on Buehler Texmet 2000 cloth. Final polish used a 1:1:2 mixture of Stuers colloidal silica (OP-S), soap (Liquinox) and water. Final polishing was for 1 hour on Buehler Mastertex cloth. Samples were then freed from the epoxy and cleaned in acetone and alcohol with ultrasonic agitation. This method provided a flat, fairly strain-free surface, which was well suited for collection of microtexture data along the delamination cracks. The surface quality was comparable to that obtainable from electropolishing methods, but with a much flatter surface. A flat surface is important since even the minor undulations common to electropolished samples interfere with the microtexture data collection at the required 70° of tilt, especially in vicinity of cracks. Grains on either side of a delamination were studied via microtexture. A JEOL 6400 scanning electron microscope (SEM) equipped with a microtexture system was used to study microtexture and the distribution of nanoindents. Microtexture data were collected in large sets of ~ 40,000 solved diffraction patterns per set, with a spatial separation of 0.5 µm. Analyses of the data included grain boundary misorientation; internal misorientation; discrete texture from points of the data set; identification of grains by proximity to specific, characteristic orientations (or texture components); and internal misorientation. Internal misorientation is a measure of the spread of orientations bound by a high angle boundary; i.e. it is essentially a quantification of the amount of substructure. Since the recrystallization process removes substructure, a low IM is an indication of a recrystallized condition. The data were plotted as colorcoded maps, where appropriate. Data were collected from around delamination cracks and from undeformed material. A simple method of indicating the grain orientation will be used in this paper, namely showing the distribution of indexed points on a discrete pole figure. Discrete pole figures associated with microtexture maps can be used to depict the grain orientations. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Regions studied by microtexture were also evaluated by nanoindentation. Nanoindentation was performed with an MTS Nano Indenter XP system. Low, constant maximum load indents (1mN) were applied with a Berkovitch diamond indenter. Nanoindentation studies required use of an SEM for examination of the individual nanoindentations. The location of nanoindent rows in the SEM was facilitated by placement of two indents with a much larger load. Single rows of ten nanoindents were applied across a delamination, with a spacing of 5 µm, and the hardness values were calculated by the method of Oliver and Pharr [16]. Results from microtexture studies of delamination cracks and undeformed material were compared with nanoindentation measurements straddling delamination cracks.

High IM

Figure 2:

3

Internal misorientation (IM) map from a wide area microtexture scan of a region below a delamination crack in a L-T, J-R, 2090, compact tension sample. The range of grey levels corresponds to an IM range of 0º to 5º. The indicated grain shows a high IM and is within the retained deformation texture brass orientation.

Results and discussion

Microtexture study of undeformed material (1 cm below a delamination crack) revealed a wide variation in internal misorientation from grain to grain. The contrast was particularly evident between grains of the retained deformation texture and neighboring, recrystallized grains with dramatically different levels of IM. The indicated grain in Figure 2, for instance, shows an internal misorientation near 5º, whereas some grains show IM values near 0º. Examination of several delaminations indicated common features which were, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

554 High Performance Structures and Materials III presumably, related to the fracture process. There was a tendency for fracture to occur between grains with large differences in IM. Low internal misorientation grains were expected to be “soft” due to recrystallization. Such grains were found adjacent to delaminations either as single grains or small “necklace” grains. The other side of the delamination was found to be a large, unrecrystallized grain with a greater amount of work hardening. This unrecrystallized grain was typically larger, with a much larger internal misorientation than the neighboring grains. These grains were presumed to be harder prior to aging. a

b

c

d

GOSS Figure 3:

Cu

Microtexture plots showing internal misorientation of grains adjacent to a delamination crack (arrows) in 2090-T8. From a T-L, J-R, compact tension sample. a) Image Quality (IQ) map, b) Internal Misorientation (IM) map, c) schematic of (111) pole figure showing ideal Goss and Cu orientations and d) discrete (111) pole figure from the data of b). Darker grain is near a Cu deformation component orientation. Region below the crack is in the Goss, recrystallized orientation.

An example of this effect is shown in Figure 3, which shows microtexture maps of delamination fracture between a Goss, recrystallized grain, and a grain near the Cu deformation component orientation. The unrecrystallized (Cu) grain WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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shows an IM of near 5º, which is about twice as high as the Goss grain across the crack. Large internal misorientation was taken as an indication of high stored strain energy, such as would be expected for a grain of the deformation texture. Low internal misorientation was an indication of recrystallization subsequent to rolling, which is likely to have developed during solution heat treatment of the sheet prior to artificial aging. For sheet or plate to be given a T8 aging treatment, a stretch of about 6 – 8% is applied subsequent to solution heat treatment and prior to aging. Recrystallized grains (without substructure) are then given a significant dislocation density which aids in precipitate nucleation. The increased density of precipitates increases the strength considerably. However, the grains with well developed substructure should exhibit precipitation on subgrain boundaries, and should develop an even higher dislocation density due to dislocation multiplication effects during deformation. These grains may develop greater hardness during aging. A nanoindentation study was used to investigate this possibility. Figure 4 shows a series of nanoindents traversing a delamination crack. The grain on the right side of the crack, which was near the Brass orientation, showed a hardness increase of about 20% greater than the grain across the crack. This was despite the indent being at the very edge of the crack, which might be expected to result in a low hardness value.

Figure 4:

Scanning electron micrograph of a delamination crack and nanoindents. Longitudinal (LS) plane from a 2090 L-T, J-R test. Nanohardness indents are shown along with larger marker indents. A nanohardness increase, of about 20%, was found for one point within a grain adjacent to the delamination (arrow). The higher nanohardness grain shown has a locally high IM and is close to the Brass orientation.

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556 High Performance Structures and Materials III Delamination frequently occurs along grain boundaries which are interfaces between hard and soft grains. Measurements of IM, texture components and direct measurements of nanohardness support this observation. This may be due to the tendency of the softer grains to preferentially deform and suffer from poor accommodation of plastic distortion at the grain boundary, since the minimum resolved shear stress would be greater in the harder grain. An additional factor may be the tendency of recrystallized grains to end their growth at obstacles, such as incoherent intermetallic particles, which establish weak interfaces at the new grain boundaries.

4

Conclusions

The following conclusions can be reached from this study: Delamination occurs preferentially along interfaces between “hard” and “soft” grains, representing, in the first case, grains of the deformation texture with high stored strain energy and in the second case, recrystallized grains. This is supported by 1) texture information, which can be compared to well-known sequences in the processing history, 2) measurements of the intragranular orientation variation and 3) direct measurements of the nanohardness adjacent to delamination fractures. Higher hardness is attributed to stored strain energy of some components of the unrecrystallized material vs. the softer condition of the recrystallized material. This observation of fracture along the hard/soft interface does not eliminate any one of the mechanisms proposed for delamination fracture, but adds yet another factor. The correlation of nanoindentation and microtexture data provide significant and new insights into the mechanical properties of Al-Cu-Li-X alloys.

References [1] [2] [3]

[4]

Sanders, Jr., T.H., & Starke, Jr., E.A., (eds). Al-Li Alloys I, Proceedings of the First International Aluminum-Lithium Conference, Stone Mountain, Georgia, 1980. Sanders, Jr., T.H. & Starke, Jr., E.A., The effect of slip distribution on the monotonic and cyclic ductility of Al-Li binary alloy. Acta Metall., 30, p. 927, 1982. Csontos, A.A. & Starke, Jr., E.A., The effect of inhomogeneous plastic deformation on the ductility and fracture behavior of age hardenable aluminum alloys. International Journal of Plasticity, 21, pp. 1097-1118, 2005. Howe, J.M. & Vasudevan, A.K., Structure and deformation behavior of T sub 1 precipitate plates in an Al-2Li-1Cu alloy. Metall. Trans. A, 19A, p. 2911, 1988. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15]

[16] [17]

[18] [19]

557

Crooks, R., Wang, Z., Levit, V.I. & Shenoy, R.N., Microtexture, microstructure and plastic anisotropy of AA2195. Materials Science and Engineering A, A257(1), pp. 145-152, 1998. Jata, K.V. & Vasudevan, A.K., Effect of fabrication and microstructure on the fracture initiation and growth toughness of Al-Li-Cu alloys. Materials Science and Engineering A, A241, pp. 104-113, 1998. Suresh, S., Vasudevan, A.K., Tosten, M. & Howell, P.R., Acta Metall., 35, p. 25, 1987. Vasudevan, A.K. & Doherty, R.D., Acta Metall., 35, p. 1193, 1987. Vasudevan, A.K., Ludwiczak, E.A., Baumann, S.F., Howell, P.R., Dougherty, R.D. & Kersker, M.M., Mater. Sci. Technol., 2, p. 1205, 1986. Lynch, S.P., Materials Science and Engineering A, A136, p. 25, 1991. Lynch, S.P., Muddle, B.C. & Pasang, T., Ductile-Brittle Fracture Transitions in 8090 All-Li Alloys. Acta mater, 49, pp. 2863-2874, 2001. Adams, B.L., Orientation Imaging Microscopy: Emerging and Future Applications. Ultramicroscopy, 67, pp. 11-17, 1997. Schwartz, A.J., Kumar, M. & Adams, B.L., Electron Backscatter Diffraction in Materials Science, Kluwer Academic/Plenum, New York, 2000. Kalu, P.N. & Wagner, J.A., A Microtexture Investigation of the Fracture Behavior of Al-Li Alloy, 2090-T81. The 3rd Light Weight Alloys for Aerospace Applications, ed. E.W. Lee, TMS, Warrendale, PA, pp. 157167, 1995. Hales, S.J. & Crooks, R., Recrystallized and unrecrystallized Al-Li alloys for elevated temperature service. Proc. 3rd International Conf. on Recrystallization and Related Phenomena, ed. T.R. McNelley, MIAS, pp. 511-518, 1997. Oliver, W.C. & Pharr, G.M., An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments. J. Mater. Res., 7(6), pp. 1564-1583, 1992. Torralba, J.M., Velasco, F., Costa, C.E., Vergara, I. & Caceres, D., Mechanical behaviour of the interphase between matrix and reinforcement of Al 2014 matrix composites reinforced with (Ni3Al)p. Composites: Part A, 33, pp. 491-496, 2002. Liu, C., Qin, S., Zhang, G. & Naka, M., Micromechanical properties of high fracture performance SiCp-6061/6061 Al composite. Materials Science and Engineering A, A332, pp. 203-209, 2002. Ogura, T., Hirosawa, S. & Sato, T., Quantitative characterization of precipitate free zones in Al-Zn-Mg(-Ag) alloys by microchemical analysis and nanoindentation measurement. Science and Technology of Advanced Materials, 5, pp. 491-496, 2004.

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559

A contribution to the rehabilitation of reinforced concrete structures by non-destructive electrochemical methods J. L. Rovira Santa Olaya & P. Pardo Tràfach Department of Construction Engineering, Materials Division, Universitat Politècnica de Catalunya, Barcelona, Spain

Abstract The aim of this work has been the study of an electrochemical treatment that allows the prevention and halting of chloride corrosion processes in passive reinforced concrete. This treatment is based on a desalination system that removes free chloride in concrete simultaneously increasing the pH around reinforcement steel until its repassivation. With this purpose a concrete has been manufactured with mixing proportions close to that used in civil engineering. The chloride attack has been carried out at room temperature, and the chloride removal process has been started at several concrete ages. Chloride ions have been determined before and after the attack, as well as the corrosion potential, in order to establish comparison between the grade of corrosion of the reinforcement steel before and after treatment, according to exposition time and initial corrosion situation. Relationships between electrochemical parameters have been determined. The studied procedure is a non-destructive and non-intrusive method that does not change either the strength of the concrete or its external appearance. The procedure uses non-pollutant materials, so no environmental damage is caused. Keywords: electrochemical chloride extraction, steel-reinforced concrete, repassivation.

1

Introduction

Corrosion in steel reinforcement caused by exposition to chlorides is the main cause of deterioration in steel-reinforced concrete structures. Electrochemichal WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06055

560 High Performance Structures and Materials III chloride extraction (ECE, also named desalination) is a method to repair steel reinforced structures that have suffered chloride attack [1, 2], although it can be also applied as a prevention treatment [3]. The treatment consists of the application of an electric current between the steel, acting as a cathode, and a temporary external anode placed on the concrete surface. As reported by Arya et al. [4], chloride removal increases with increasing applied potential, number of reinforcing bars at a particular depth and initial chloride content, although chloride remaining seems to be independent of the initial chloride content. This method requires the chloride ions to be in the pore water to be transported out of the concrete [2]. Removal of 40 to 70% of the initial chloride has been observed using this technique [1, 5, 6]. On the other hand, repassivation of steel due to the increase in the potential of corrosion after the chloride removal has been studied by several authors [5, 6, 7]. In this work, electrochemical chloride removal has been studied on reinforced concrete that has been subjected to different degrees of corrosion. The measurement of the potential of corrosion has allowed the probability of corrosion to be determined before and after the treatment according to these values for the potential [8]: - Higher than -200 mV: low probability (0-5%) - Between -200 and -350 mV: medium probability (5-50%) - Lower than -350 mV: high probability (50-100%) Finally, the time period during which the treatment should be maintained to ensure the rehabilitation of concrete has been studied.

26 cm

11 cm

Figure 1:

2

Disposition of the reinforcement steel in the test specimens.

Materials and methods

2.1 Production of tests specimens 40 cylindrical standard test specimens were cast and molded according to UNE EN 83301:91 (150 mm in diameter and 300 mm in length). 33 of the specimens were reinforced with eight steel bars (AEH-400) connected by two circular fences as it is shown in fig. 1. The steel surface was mechanically cleaned before WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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use. In order to accelerate the corrosion process, 2 cm of covering layer was selected to cover the steel bars. This must not affect the results, since the goal of this work is a comparative study between several corrosion and regeneration conditions. The concrete composition, shown in table 1, corresponds to a typical mixing for building purposes. Steel characteristics are also shown in table 1. 40 test specimens 7 control specimens (without reinforcement) 33 reinforced specimens

11 YY-specimens

11 XX-specimens 1 control specimen chloride attack (7d)

11 ZZ-specimens 1 control specimen

chloride attack (56d)

1 control specimen

chloride attack (90d)

1 control specimen

1 control specimen

1 control specimen

ECE (7d; 3 specimens)

ECE (7d; 3 specimens)

ECE (7d; 3 specimens)

ECE (21d; 3 specimens)

ECE (21d; 3 specimens)

ECE (21d; 3 specimens)

ECE (42d; 3 specimens)

ECE (42d; 3 specimens)

ECE (42d; 3 specimens)

Figure 2:

Schedule followed for the experiment.

Test specimens were compacted in a shaking table and demolded after 24 hours. Then, they were maintained in a wet room for six days. After this period, they were subjected to chloride attack and electrochemical treatment following the schedule shown in fig. 2. 2.2 Chloride attack Chloride attack was carried out in two pools of 350 l, containing a solution with 150 g NaCl·l-1 (52 g Cl-·l-1). Water evaporation was controlled, maintaining chloride concentration within a ±2% variation. 2.3 Electrochemical chloride extraction treatment In order to save time, the treatment for chloride removal was applied to batches of nine specimens at a time. A current intensity of 1 A·m-2 with a potential of 20 V was supplied and maintained constant during the required period of time. Fig. 3 shows the electrochemical chloride extraction system for a specimen. A first 2 cm layer of wet cellulose paste covered the specimen. This layer acts as an exchange electrolyte between reinforcement bars and a titanium anode mesh. Finally, the whole system was coated with a second 1.5 cm wet cellulose layer in order to maintain the wet conditions. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

562 High Performance Structures and Materials III 2.4 Potential measurements Corrosion potentials were measured using an in situ measurement system “Colebrand” with a Cu/CuSO4 reference electrode. Table 1:

Concrete composition and steel characteristics.

Concrete Cement CEM I 42,5R Calcareous Sand 0/5 Calcareous gravel 5/12 Calcareous gravel 12/18 Water w/c Compressive strength 28d Steel bars AEH-400N Diameter Modulus of elasticity Breaking stress

346 kg/m3 1289 kg/m3 521 kg/m3 739 kg/m3 225 L/m3 0.65 45 MPa 3 mm 4100 kp/cm2 4500 kp/cm2

Steel reinforcement (cathode) Concrete First wet cellulose layer (2 cm) Titanium anode mesh Second wet cellulose layer (1.5 cm)

Figure 3:

3

Electrochemical extraction system.

Results and discussion

For a better monitoring of the corrosion and electrochemical removal processes, the corrosion potential was measured in two points of each of the two bars which constituted the reinforcement structure, at 1/3 and 2/3 height from the specimen basis. The measurements were taken before and after electrochemical treatment. Table 2 shows the increase in the potential of corrosion in each set of specimens produced by the chloride attack. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3.1 X-specimen set (7 day chloride attack) The attack on this set was a low intensity attack, simulating an initial stage of corrosion or a concrete with an excess of chloride. Fig. 4 shows the evolution with time of the potential of corrosion for this set. As it can be seen in the table 3, the corrosion potential decreases a 28.3% during the first week of the chloride removal treatment. From the second week to the end of treatment, the reduction in the corrosion potential was about 6-7% per week. The total decrease with respect to the initial corrosion potential was 60.2% (from -283 mV to -112.7 mV). For this set, the first week of treatment caused a reduction in the potential of corrosion enough to reach the low corrosion probability zone, as it is shown in fig. 4. This is due to the fact that the corrosion process was hardly initiated after 7 days to chloride exposition, and chloride ions had not penetrated deeply into concrete. Consequently, chloride removal was carried out quickly. From these results, it can be concluded that a four week removal treatment would be enough for structures with a low corrosion degree. Table 2:

Increase in the potential of corrosion for each set after attack.

Potential before attack (mV) Potential after attack (mV) Variation (%) Table 3:

X-specimen set (7 day attack) -110

Y-specimen set (56 day attack) -110

Z-specimen set (90 day attack) -110

-283

-524

-621

157

376

464

Decrease of the potential of corrosion for each set after treatment.

Decrease 1st week of ECE Decrease (per week) after 3 weeks of ECE Decrease (per week) after 6 weeks of ECE Cumulative decrease 1st week of ECE Cumulative decrease after 3 weeks of ECE Cumulative decrease after 6 weeks of ECE

X-specimen set (7 day attack) 28.3

Y-specimen set (56 day attack) 13.3

Z-specimen set (90 day attack) 17.1

5.84

12.6

10.9

6.73

9.38

8.90

28.3

13.3

17.1

40.0

38.6

34.8

60.2

66.7

61.6

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

564 High Performance Structures and Materials III 3.2 Y-specimen set (56 day chloride attack) The attack on this set was a medium-high intensity attack, simulating a concrete with a corrosion degree such that the structure is damaged, but not in a way to be at risk for the moment. Fig. 4 shows the evolution with time of the potential of corrosion for this set. As table 3 shows, the reduction of the potential of corrosion for this set is maintained between 9 and 13% per week, decreasing with a regular rate along the period of time in which the experiment was carried out. The total decrease with respect to the initial corrosion potential was 66.7% (from -523.8 mV to -174.3 mV). For this set, the low corrosion probability zone was reached after six weeks of treatment, although a few more weeks could be necessary to ensure the concrete to stay in this low risk zone. From these results, it can be concluded that a ten or eleven week removal treatment would be needed to ensure a correct rehabilitation of structures with a medium or high degree of corrosion.

Potential of corrosion (mV * -10)

70 X-specimen set

62,1

60

Y-specimen set 52,38

50

Z-specimen set

51,5 45,4 40,46

40

32,18

30

28,3 23,87

20,26

20

17,43

16,99

11,27

10 0 0

1

2

3

4

5

6

7

Week

Figure 4:

Evolution of the potential of corrosion during the electrochemical removal of chloride. Horizontal line: 50% probability of corrosion. Horizontal plotted line: 5% probability of corrosion.

3.3 Z-specimen set (90 day chloride attack) The attack on this set was a high intensity attack, simulating a very damaged concrete, which would imply its demolition if it was a structural concrete. Fig. 4 shows the evolution with time of the potential of corrosion for this set. As it is shown in table 3, the reduction per week of the potential of corrosion for this set decreases with time from 17.1% to 8.90%. The total decrease with respect to the initial potential of corrosion was 61.6% (from -621 mV to -239 mV). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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0,35 0,32

Cl- concentration (%)

0,30

0,28

0,25

0,22

0,20 0,15

0,12

0,10 0,05 0,00

0,1

0,08

Exterior

0,02 0,02

0

0,05

0,04

Interior

1

2

3

4

5

6

7

Week

Cl- concentration (%)

0,60 0,51

0,50 0,44

0,40 0,33 0,29

0,30 0,20 0,10 0,00

0,19

0,17

Exterior

0,12 0,02 0,02

0

0,07

Interior

1

2

3

4

5

6

7

Week

Cl- concentration (%)

0,60

0,57

0,50

0,6

0,48

0,40

0,35

0,34

0,30 0,21

0,20

0,17

Exterior

0,10 0,00

0,09

0,02 0,02

0

Interior

1

2

3

4

5

6

7

Week

Figure 5:

Evolution of chloride concentration inside and near surface.

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566 High Performance Structures and Materials III For this set, after six weeks of electrochemical removal, the zone of medium risk of corrosion (5-50% probability) is reached. An extrapolation of the data obtained for this set, showed that 12 weeks of treatment would be needed to reach the low risk of corrosion zone for the rehabilitation of highly damaged concrete elements. 3.4 Evolution of chloride concentration Chloride concentration was monitored both inside the test specimens (near reinforcement) and near the surface of the specimen. Results for the three sets are pictured in fig. 5. Chloride concentration decreased between 65 and 71%, and the results were in concordance with that obtained from the study of the potential of corrosion, supporting the conclusions previously drawn.

4

Conclusions

After electrochemical removal treatment, chloride concentration decreased between 65% and 71% in the three sets of specimens studied. The potential of corrosion decreased about 60% in all cases, lowering the probability of corrosion to less than 5% for specimens subjected to low and medium intensity chloride attack. Depending on the degree of the aggression by chloride ions, between 4 to 12 weeks of electrochemical treatment can be needed to ensure a correct rehabilitation of concrete. The results obtained have shown that electrochemical chloride removal is a useful technique for rehabilitation of reinforced concrete structures subjected to corrosion of steel reinforcement by chloride attack.

References [1] [2] [3] [4] [5]

Elsener, B., Molina, M. & Bôhni, H., Electrochemical removal of chlorides from reinforced concrete structures. Corrosion Science, 35(5-8), pp. 1563-1570, 1993. Tritthart, J., Petterson, K. & Sørensen, B., Electrochemical removal of chloride from hardened cement paste. Cement and Concrete Research, 23(5), pp. 1095-1104, 1993. Glass, G. K. & Buenfueld, N. R., The inhibitive effects of electrochemical treatment applied to steel in concrete. Corrosion Science, 42(6), pp. 923927, 2000. Arya, C., Sa’id-Shawqi, Q. & Vassie, P. R. W., Factors influencing electrochemical removal of chloride from concrete. Cement and Concrete Research, 26(6), pp. 851-860, 1996. Polder, R. B. Electrochemical chloride removal from concrete prisms containing chloride penetrated from seawater. Construction and Building Materials, 10(1), pp. 83-88, 1996. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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[6]

[7] [8]

567

Fajardo, G., Escadeillas, G. & Arliguie, G. Electrochemical chloride extraction from steel-reinforced concrete specimens contaminated by “artificial” seawater. Corrosion Science, in press, available on www.elsevier.com/locate/corsci. Elsener, B. Half-cell potential mapping to assess repair work on RC structures. Construction and Building Materials, 15(2-3), pp. 133-139, 2001. Andrade, C., Monitoring Techniques (Chapter 6). Corrosion of Steel in Concrete, ed. P. Schiessl, Chapman & Hal: London and New York, pp. 95-79, 1988.

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Section 7 Adhesion and adhesives

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571

Timber specimens parametrized design for numerical analysis E. Martín Gutiérrez, J. Estévez Cimadevila, D. Otero Chans & S. Muñiz Gómez Department of Technology of Construction, A Coruña University, Spain

Abstract The characterization of materials or the study of resistant response associated with new constructive solutions often convey progressive adjustment of models of analysis starting from the results obtained by means of test campaigns. The reliability of the conclusions will obviously be conditioned by the amplitude of the experimental development which, as a result, should contain a sufficiently representative sampling of cases. In the situations where significantly multiple parameters intervene, specimens that involve possible values and combinations should be examined, which in practice remarkably increases the number of specimens to be treated. The case being presented deals with the evaluation of the effectiveness of certain fixed joints carried out by gluing metallic bars into timber pieces by means of adhesives of different formulations. The set of calculus patterns, including the extensive variability of the associated magnitudes, has been treated in a parametrized form. The geometry of the studied pieces, as well as the sustenance conditions and the load, are automatically generated by means of software specially designed for those purposes. The resulting information is structured in an orderly way to facilitate its reading and also its manipulation if needed. The files of the process obtained in this way can optionally incorporate the meshing sequences, as well as the commands of the calculus, analysis and postprocess. The adopted strategy considerably reduces the time assigned to the phases of the geometrical definition and resolution, and enormously simplifies the interpretation of the results. Keywords: numerical analysis, parametrized design, glued joints.

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572 High Performance Structures and Materials III

1

Introduction

This work is placed in an investigation project directed to the study of the joints behaviour in timber pieces by means of steel bars glued-in with an adhesive. The approach causes the development of two parallel complementary lines of advance: the first one is of experimental nature and the second one is based on the use of the numerical analysis technique. On the one hand it is pretended to use the results obtained in the experiments for calibrating gradually the theoretical models in such a way that they, in their turn, could be applied for extrapolating not specifically experimented situations or facilitating the adoption of design decisions in certain practical cases. On the other hand the numerical analysis can be useful for the selection of those mechanisms of connection that prove to be more operative in advance, reducing the temporary and economic costs associated with the creation of the specimens.

Figure 1:

Reconstruction of different specimens.

Specimens are timber prisms of square transversal section made of timber, at both ends of which two threaded bars are placed, properly fixed by means of adhesives of epoxi formulation (fig. 1). The resulting specimen is subjected to a failure test of growing centered tensile load, registering the limit number and the form of the dominant failure. Throughout the process the equipment registers the total relative displacement between the ends of the specimen, associated to each load value, which later permits to configure the corresponding cause-effect diagrams. In principle a first experimental campaign consisting of a total of 45 pieces for each of the used adhesives is planned. The given number is distributed in 3 series of 15 specimens in which the joint with 3 possible diameters (8, 10 and 12 mm) and 5 anchorage lengths are analyzed (table 1). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Table 1: d

8

D

e

10

1

10 12

1

12 14

1

573

The structure of the first experimental campaign. L

60 90 120 150 180 60 90 120 150 180 60 90 120 150 180

Figure 2:

Li

Lm

Li=L

3.L ≤ 500

60 90 120 150 140 60 90 120 150 140 60 90 120 150 140

180 270 360 450 500 180 270 360 450 500 180 270 360 450 500

Le

150

150

150

Lb 210 240 270 300 330 210 240 270 300 330 210 240 270 300 330

a

Reference

a=6.d

Units: mm

48

60

72

≡1a∅ ≡1b∅ ≡1c∅ ≡1d∅ ≡1e∅ ≡2a∅ ≡2b∅ ≡2c∅ ≡2d∅ ≡2e∅ ≡3a∅ ≡3b∅ ≡3c∅ ≡3d∅ ≡3e∅

Initial configuration of the specimens.

The attached diagram shows the proposed combinations for the 9 geometrical parameters to deal with, as well as the type of the timber and the type of the applied adhesives. This set appears to be sufficiently full for establishing relevant conclusions in relation with the incidence of each magnitude and even facilitating formulations that will permit to define the necessary anchorage lengths.

2

The structure of the process file

The numerical models are calculated with the help of the application of Ansys Multiphysics, Swanson Analysis Systems Inc., initially assuming an elastic and lineal behaviour, noticeably correct in accordance with the experimentally obtained stress-deformation diagrams for load values not surpassing the 75-80 per cent of the limit number. The volume of the pieces, the possible parametrization of its geometrical and mechanical values lead us to the development of a specific software that WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

574 High Performance Structures and Materials III automates the definition of the model, taking advantage of the common commands of the preprocessor. It deals with the creation of a process file for each specimen that Ansys could manage in an autonomous way but in which the information would be structured in an orderly way with the purpose of simplifying its reading and the possible manipulation. In general the mentioned files respond to the following configuration: numerical definition of parametrized data; meshing specifications; assignment of the load value; access to the preprocessor of the system; description of the conditions of visualization of the model; definition of elements and materials; an orderly localization of the keypoints; positions that designate lines and arcs; creation of areas; construction of volumes by means of extrusion operations; material assignment; meshing operations; access to the calculus module; definition of coercion and application of the load; access to the postprocessor of the application; energy estimation of errors structured by materials; and presentation of results. It should be paid attention to the fact that the file in question is defined in ASCII format, therefore it turns out to be operable with any conventional line editor. This circumstance allows to assess, with remarkable simplicity, diverse alternatives of modelling, for example: typology of the involved finite elements; adequacy of the parameters that define the mechanical behaviour of the materials; variations in the conditions of the meshing and adjustment of dicretization in concrete areas. The model is configured by means of tridimensional elements of 8 nodes (Solid45), that present 3 degrees of liberty in each position (translations x, y, z). After diverse experiments the use of elements of 20 nodes has been discarded, taking into consideration the fact that it greatly increases the storage requirements and the time of the calculus without significant differences neither in the results nor in the reductions in the energy estimations of the possible committed errors. The associated materials are: steel 8.8 (fy=640 N/mm², fu=800 N/mm²) for the threaded bars of the ends, sawn timber (basically chestnut) and 3 types of adhesives of epoxi formulation (Sika AnchorFix-3, Hilti Hit Re-500, Loctite Hysol 9464A&B). All of them are integrated in the model by their parameters of elastic behaviour; basically the modules of transversal and longitudinal elasticity and the coefficient of Poisson in relation with the axes that define the Cartesian system of reference (is designated as Z the relative to the direction of the load application). The mentioned magnitudes are initially decided on the basis of the recommended values of different references [1,2,6], and considering the experiments previously made upon the isolated materials. Later a successive adjustment is realized with a purpose of approximating the obtained results by means of experimental and numerical technique.

3

Geometrical organization of the model

Attending to the considerations of symmetry it is decided to define a fraction corresponding to the eighth part of the specimen. At first, the configuration of the generic transversal section is established that will give place to the complete volume of the modelling by repetition of its elements, totally or partially. In WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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principle, ample divisions are managed to tackle the first assessment of the problem. Later comes the restructuring of the configuration on the basis of a radial scheme that serves to a better estimation of the stress distribution in different materials, especially in the areas close to the contact surfaces (fig. 3).

Figure 3:

Figure 4:

Sections and axonometry of the model.

Basic numeration of nodes, lines and arcs.

At the beginning of the process the coordinates (x, y) of the 105 points, that define the base plane, are stored in matrices. In the area nearest to the centre of the section the mentioned values are deduced by means of a polar system of reference, while in the exterior panels a Cartesian type of analysis is proposed. It should be paid attention to the fact that the parametrization of the process of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

576 High Performance Structures and Materials III discretization permits the successive application of the adjustments that will improve the behaviour of the model. The generation program supposes that the specimen is formed by 3 well differentiated areas (fig. 2): the central zone (zl), corresponding integrally to the timber piece; the relative to the glueline (z2); and the external fraction of the steel bar (z3). Each of these sets can be subdivided independently in the desired number of bands. With these premises the definition of the model keypoints is realized by means of the command: K,NPT (reference number for keypoint),X,Y,Z (keypoint location) The generation sequences of the nodes are structured by means of nested buckles for each of the described areas: an external one that runs across all the fringes of the zone that has been fractioned; and another internal one that analyzes the 105 constituting points of each transversal section (18 in the third sector). In each case the number of the node is resolved by means of a counter, the coordinates (x,y) are deduced from the previously formed auxiliary matrices, and the position of (z) is estimated in the external buckle in relation with the section that is being configured at every moment. The following phase supposes the creation of lines and arcs with an arrangement corresponding respectively to the orders: L,P1,P2 (keypoints at the end of the line or circular arc line) LARC,P1,P2,PC (center),RAD (radius of curvature of the arc) In this case, Ansys does not require the numeration of such entities in the command line, that is why it is considered as appropriate to specify it as a comment to its term. This possibility simplifies, in its case, the localization of the elements in the list. The process is configured equally by means of buckles, updating in each transversal plane the number of its lines and arcs with a total arrangement of the previously treated sections. Once the bases of the wire structure of the model are formed, the definition of the areas based on the lines and the arcs that configure its perimeter is realized. The process is very similar to the described one in accordance with the previous phase and is used in any case of the command: A,P1,P2,P3,P4,P5,P6,P7,P8,P9 (list of keypoints defining area) For configuring the volumes an extrusion of the definite areas in each section is realized in such a way that the numeration of both types of entities turns out to be coincident (fig. 5). To achieve this operation it is necessary to create in advance auxiliary lines over the axis of the specimen, linking the central nodes (1) of each of the consecutive two sections. The order that formalizes the extrusion responds to the following syntax: VDRAG,N1,N2,N3,…,N6 (areas in the pattern),NLP (line defining the path) WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

Figure 5:

577

Obtaining volumen by extrusion.

At the end of this phase the volumes that are situated between the zones z1 and z2 (additional band of 2 mm) relative to the bottom of the bar and the adhesive are eliminated. This elimination is decided after the real specimens are analyzed with a very reduced efficiency of the adhesive in this area, as a consequence of the filling process itself. In figure 4 the resulting slot caused by the elimination commands is estimated (VDELE, NV1, NV2, NINC: delete volumes from NV1 to NV2 in steps of NINC). To finish the preprocess of the model, the obtained volumes are related to the involved materials, a task in which controlling the numeration of the first items proves to be fundamental. The selection is realized by means of ranks, a reason for which in each section at first the central steel elements are coded, then the relatives to the band of the adhesive and finally the constituent elements of the timber. VSEL,Type,Item,,VMIN,VMAX (values of the item range),VINC (increment) VATT,MAT,REAL,TYPE (parameters to be assigned) The meshing only affects the final volumes (VMESH,ALL). With a previous discretization, a subdivision of each edge in two halves (ESIZE,2) is considered correct. The application permits to refine the mesh in determined areas, but advanced studies indicate that the resultant model is sufficiently approximate without further modifications of the set.

4

Analysis and interpretation of the results

In the module of the calculus the transversal displacements coerce the symmetry planes (used for reducing the real specimen to its eighth part). So the external load is applied to the extreme surface of the steel bar, and the phase of analysis WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

578 High Performance Structures and Materials III begins. After the calculus is finished, the postprocessor is opened studying the magnitudes that define the solution: basically the energy estimation of the errors, displacements (fundamental for its comparison with the experimental results), and stress conditions (fig.6). An important question is the distribution of the tangential stresses over the surface of contact between the adhesive and the timber, a parameter that could be determinant in numerous extreme situations (fig.7). For that reason the nodes of the mentioned area included in the plane (xz) are isolated and a list of resisting normal and tangential stresses is requested. A third computer program extracts the desired values (τxz in this case) and starts to represent them in an environment of computer aided design (fig. 8).

Figure 6:

Figure 7:

Displacements (z) and tangencial stresses (τxz).

Failures associated with the excessive tangential stresses.

The diagrams show curved lines with maximal numbers focused on the ends of the anchorage area which are much greater than the average value traditionally used in the references. Both peaks turn to be similar for reduced longitudes, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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579

increasing noticeably the exterior one when the union is realized with more profundity.

Figure 8:

5

Distributions of the tangential stress.

Conclusions

The means in parallel to experimental technique and numerical analysis, for the study of the resistant behaviour of timber-steel joints as the described ones, supposes a sufficiently complete and illustrative strategy. The comparison of the results allows to calibrate gradually the virtual models in such away that they prove to be useful for verifying practical cases that are not specifically subjected to tests, or for predicting possible situations of risk. Likewise, the mentioned idealizations are useful for assessing constructive solutions at the very beginning, thus reducing the relative costs of production and of the load of possible prototypes. In this generic frame the computerized configuration of the models remarkably simplifies the generation and postprocess tasks appealing to the parametrization of its geometry.

Acknowledgements This research is sponsored by the Ministry of Science and Technology through research project title “uniones metálicas encoladas con adhesivos en barras de WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

580 High Performance Structures and Materials III madera” (glued anchoraged timer joints). The financial support is gratefully acknowledged.

References [1] [2] [3]

[4]

[5]

[6] [7]

Broughton, J.G. & Hutchinson, A.R. Effect on timber moisture content on bonded-in rods. Construction and Building Materials, volume 15, issue 1, pp. 17-25, 2001. Deng, J.X., Moss, P.J. & Buchanan, A.H. Glued bolts in glulam. An analysis of stress distribution. 5º World Conference on Timber Engineering, Montreux, Switzerland, 1998. Estévez, J., Pablos, J., Muñiz, S., Freire, M., Vázquez, R. & Álvarez, J. Double-layer space structures of laminated timber tubular members. Proc. of the 4º International Conference on Space Structures, University of Surrey, Guildford, UK, Ed. Thomas Telford Ltd.: London, pp. 563-572, 1993. Estévez, J. & Vázquez, J.A. Spatial truss of hollow bars made of laminated timber supported by walls of reinforced masonry. Journal of the International Association for Shell and Spatial Structures, volume 45, n.1, April n.144. Estévez, J., Vázquez, J.A. & Otero, M.D. Diseño y dimensionado del nudo extremo de una barra hueca de madera laminada. 1º Congreso Ibérico A Madeira na Construçao, Universidade do Minho, Guimaraes, Portugal, pp. 689-698, 2004. Senno, M., Plazza, M. & Tomasi, R. Axial glued-in steel timber jointsexperimental and numerical analysis. Holz als Roh – und Werkstoff, volume 62, number 2, 2004. Swanson Analysis Systems, Inc. Ansys User’s Manual. Houston, 2005. (See also http://www.ansys.com).

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581

Theory of the elasticity of the materials of the second order V. Shorkin1 & V. Gordon2 1 2

Department of Physics, Oryol technical states university, Russia Department of Mathematics, Oryol technical states university, Russia

Abstract It is admitted that a solid body is obtained by means of extraction from an infinite medium, and at the same time its state changes. To describe this phenomenon there a linear theory of elasticity of isotropic materials of the second order was used, generalized for the case of availability of the initial stressed state described by tensors of various ranks. They depend upon the physical nature of the body, the form of its boundary and position of points for their determination. The boundary effect – existence of surface tension and surface energy is described. The analysis of known test data regarding values of these quantities, wave dispersion in resilient media, calculus results on the basis of the model offered and one-dimensional theory of the crystalline lattice of elastic energy density for a number materials allowed values of elastic modulus and initial stresses to be determined. The created model of medium is used for the description of the stressed state of bodies adhered together, when their adhesion is considered to be one of the kinds of contact interaction. The calculus results of adhesion energy are confronted with the known data. Keywords: theory of the elasticity, materials of the second order, the initial stressed state, and surface energy, adhesion energy.

1

Introduction

It is known that particles of solid body - atoms, ions, interact with the aid of the potential forces [1]. Therefore a change in the position at least of one particle of a body affects the position and the mutual influence of all remaining particles. The formation of the new section of free surface, for example, with the division, is accompanied by the separation of the sets of particles. On the one side, the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06057

582 High Performance Structures and Materials III adhesion of two bodies is connected with the association of large quantities of particles. In these processes the inter-atomic bonds break or are formed only along the interface or association. But the result affects all particles of the divided or united bodies. The possibility of the calculation of the aforesaid exists with the discrete description of the behavior of material [2]. This can be made, also, with the continuous non-local description [3, 4]. But the first method is difficult to use for describing the materials with the complex structure. The second method requires information about the potential of non-local interaction. For describing the surface phenomena it is possible to use gradient theories (for example [5, 6]). They use a higher order differential equation of motions. Surface effects can be considered as the effects of boundary layer. However, in the existing models is not traced clearly the connection of the predicted effects with the special features of potential interaction of the particles of solid body. In the model proposed below the underlining of this connection is the basis of theory.

2

Model of continuous elastic medium

2.1 Basic assumptions 1.

Interaction of the atoms of real body is described with the aid of the potential of pair wise interaction. It is assumed that for the continuous simulator interaction of infinitely small particles is also potential, paired. The potential of interaction is proportional to their volumes dV and dV1 ( ρ = const - material density), that have at the moment of the time t of G G G G position R and R 1 . Constant of proportionality Φ = Φ R 1, R (further

(

2.

potential) - the continuous and differentiated function of its arguments. The studied processes are isothermal, reversed. The mechanical properties of medium are completely determined by the form of the function G G Φ = Φ R 1, R . The uniformity of properties indicates the dependence G G G G G G G G G G G Φ R 1 , R = Φ R 1 − R , R1 − R = a + ∆ u , where a = r1 − r – the state vector of particle dV1 relatively dV in the initial state of the considered body, G G G ∆ u = u1 − u relative particle displacement upon transfer of body into the G G G current state. Values u and u 1 are considered small. Therefore ∆ u –

(

3.

)

(

) ) (

)

also low value. For the exception of the influence of boundary it is assumed that any body B is obtained by isolation from infinitely extensive homogeneous G G continuous medium Ω . Isolation occurs instantly. Potential Φ = Φ R 1, R does not change. Initial for the body B is the state, in which it was located after instantaneous isolation. The motion, caused by disruption of the equilibrium of internal forces, begins at this moment. It is considered that it rapidly goes out. Therefore, in further reasoning it is not examined.

(

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)

High Performance Structures and Materials III

583

2.2 Stress-strained state and mechanical properties The characteristics of the strained and stressed state of medium are introduced as follows. The potential d ϕ of particle dV inside the homogeneous medium is calculated through the formula   G G d ϕ =  Φ R 1 − R dV1  dV .   Ω 

∫ (

)

(1)

Its variation relative to the current state of medium Ω is determined by the equality 

G ( ( ∫

G

G

) (G

G

))



δ (dϕ ) = δ w dV =  Φ R1 − R + δ (∆ u ) − Φ R1 − R dV1  dV = Ω

  G G T G =  ∇Φ R1 − R ⋅ δ (∆ u )dV1  dV .   Ω 

∫( (

(

)

(

)

 

(2)

))

G G G Potential gradient F = ∇Φ R 1− R is written in the form

( )

G G G G G GT F R1 − R = F (a ) + ∇F

G ∆ u =0

G ⋅∆u .

(3)

G In this case for the relative displacement ∆ u is considered valid the idea in the G form Taylor series according to the external degrees of vector a [7] G ∆u =

∑ k ! [∇rG(k )u ]G ∞

k =1

1

G

G ⋅ ... ⋅ a k .

(4)

a =0

The consequence of these operations is the idea of a variation in the density of a change in the potential of internal forces δ w , developing in the environment G dV of point r , which is equal to the work of internal forces in this region, in the form

δ w=

∑ P (m )T ⋅ ...⋅ δ (∇ rG (m )u ) . ∞

G

m =1

where: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(5)

584 High Performance Structures and Materials III P (m ) = P 0(m ) + P

0(m )

∑ C (m , n)T ⋅ ... ⋅(∇ (n )rG u ) , ∞

G

n =1

=

G G Gm ∫ [F (a )a ]dV1 ,

Ω

C (m , n ) =

(6)

∫ (

(7)

)

G G G 1 a n ∇ aG F a m dV1 . n! m! Ω

{

(8)

}

G It follows from idea (5) that the infinite sequence ∇ rG (m )u of the gradients of displacements is the characteristic of the strained state in the infinitely small G environment dV of point r . Its elements - these are the generalized displacements, determined in dV . By the generalized forces, which work on them, appear the stress tensors, determined by equalities (6) - (8). They compose

{ }

the infinite sequence P (m ) of the characteristics of the stressed state of region dV . It is characteristic that the calculation of interaction of the particles of the G G medium by the introduction of potential Φ = Φ R1 , R led to the need for the calculation of the initial stressed state in the medium, caused exclusively by its G G intrinsic properties. These properties are described by function Φ = Φ R1 , R . It determines the values of the mechanical characteristics (in the classical understanding) of medium Ω , computed with the aid of (8). For the homogeneous infinite medium Ω tensors (7) and (8) do not depend on the position of the point of their determination. The same is correct for the body B , when isolated it in Ω mentally only with surface S with the external G unit normal n . Picture changes with the real isolation, described in the third assumption of introduction. The action of the appearance of body B on its particles disappears at this moment, integration in (7) and (8) must be produced not on Ω , but on B . In this case the position of boundary S relative to each G G point r , from which is counted off the vector a , proves to be different. In spite G G of the safety of function Φ = Φ R1 , R with the real isolation of body B from Ω it ceases to be uniform in the classical understanding.

(

)

(

(

)

)

2.3 External actions and equation of motion The system of the differential equation of motions and boundary conditions takes the form G G ∂ 2u ∇ ⋅ P (1) − ∇ ⋅ P (2 ) − ∇ ⋅ P (3) − ... + f = ρ ∂t2 G G n ⋅ P (1) − ∇ ⋅ P (2 ) − ... − ∇ S ⋅ n ⋅ P (2 ) − ... = Π (0 )

(

(

(

( ( )) G n ⋅ (P ( ) − ∇ ⋅ (P ( ) − ...)) − ∇ 2

3

S

))) ( ( )) G ⋅ (n ⋅ (P ( ) − ...)) = Π ( ) 3

0

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G r ∈V ,

(9)

G r ∈S ,

(10)

G r ∈S .

(11)

High Performance Structures and Materials III

585

∇ ≡ ∇ rG – the symbol of volumetric gradient; ∇ S – the symbol of surface gradient. It was assumed during the construction of system (9) - (11) that, as in the classical case, external actions on any body and any part of it are divided into those volumetrically distributed and those superficially distributed. The first G make classical sense. Their bulk density is f . The second, just as in the classical case, characterize contact interaction of different bodies. However, this interaction is characterized not by one form of forces, but by infinite sequence,

that possess surface densities Π (k ) , k = 0 ,1, ... . With k = 0 this is the density of the classical surface forces, which accomplish work on the true displacements of material points. With k > 0 - this is of density over forces [8]. System (9) - (11) must be augmented by the differential form of the law of conservation of momentum of pulse and by initial conditions for pour on displacements and their speeds.

(

)

G G 2.4 Version of the potential Φ = Φ R1 , R determination

It can be realized for the isotropic medium. In this case it suffices to examine one-dimensional situation. Equation (9) taking into account (6) - (8) which G f = 0 takes the form 1 ∂ 2 u1 c2 ∂t 2

=

∂ 2 u1 ∂ x12

+ b12

∂ 4 u1 ∂ x14

+ b2 4

∂ 6 u1 ∂ x16

+ ...

(12)

Coefficients are determined by the formulas C11(1,1)

, D j = D1b j −12( j −1) , j = 1, 2 , ... , ρ D1 = C11(1,1) , D 2 = C111(1,3) − C111(2 ,2 ) + C111(3 ,1) ,

c=

(13)

................................. ................

The search for the solution in the form u1 = e i (ω t − kx1 ) , i = − 1 , gives the possibility to obtain dispersion law – dependence ω (k ) [1], in the form of the power series

ω2 c

2

= k 2 − b12 k 4 + b2 4 k 6 − ... .

(14)

The same law can be built theoretically, with the use of ideas of solid state physics [1], or on the basis of data of experiments [9]. With the comparison it is G possible to determine values b j . Function Φ = Φ (a ), a = a can be WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

586 High Performance Structures and Materials III represented, for example, in the form of a number of the diminishing exponential curves Φ (a ) =

∞

∑α p e

−η p

, ηp =

p =1

a bp

(15)

Substitution (15) in (7), (8), and then in (13) reduces to the infinite system of linear algebraic equations for the unknowns α p . Its solution determines potential (15) of material, for which are obtained the values b j .

3

The theory application

3.1 Additional assumptions 1.

Instead of the infinite sequences of the generalized displacements and stresses they are used only on their two first elements. This gives the possibility to use for its solution classical methods, in particular, the method of separation of variables - Fourier's method. The analytical method of solving the ordinary differential equations of final order with the arbitrary differentiated coefficients, which are appeared in the one-dimensional case, is represented into [10]. 2. Maximum simplification in the task is possible to achieve, if we assume that in the range of values in question under the condition of coefficients are constant. 2. It is assumed that the material is isotropic. In its near-boundary layer the values of the tensors of initial stresses and characteristics of elastic state are constant and equal to those values, which are characteristic for the boundary plane of semi-infinite body. In this case the density of a change in the internal forces potential is determined by the equality w = µ g ij g ij +

(

λ

2

g kk g ll +

)

+ (2µ + λ ) b1α Z ijk Z ijk + b1β Z ijk Z kij + π 0δ ij Ε k Z ijk

(

2

2

(16)

)

1 ui , j + u j ,i and Z ijk = ui , jk – the component of the classical strain 2 tensor and second gradient of displacements; µ , λ − the Lame coefficients; g ij =

b1 , b2 , π 0 – additional constants; Ε k – the component of the unit vector, G G normal to the surface, parallel to boundary, where Ε = n ; P (1)ij and P (2 )ijk they are determined by differentiation (16) on g ij and Z ijk respectively.

Relationship (16) corresponds to the model of the linearly elastic medium of the second order with the initial stressed state. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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3.2 Quantity checking of theory Within the framework of the theory proposed is examined the task about the stress-strained state of semi-infinite body, on which act no external forces. On the basis of its solution and data of experiments regarding the surface energy W p of solid bodies [11] in accordance with the formulas

π0 =

Wp k

,b =

3W p 2

4k A1

, k=λ

2µ + λ

, 2 A1 = 2 µ + λ , b 2 = b1α 2 + b2 β 2 . (17)

the estimations of additional elastic constants are made for the number of materials b and π 0 . The results of calculations are represented in table 1. Table 1:

Estimations of additional elastic constants.

(

№

Element

W p Dg / m 2

1 2 3 4 5 6 7

Mg Al Si Ni Cu Ag Au

0,73 1,04 1,24 1,44 1,12 0,95 1,23

)

(

b ⋅1011 (m )

π 0 Dg / m 2

2,67 4,65 7,54 2,49 1,89 1,98 1,29

1,28 1,92 3,21 2,50 1,57 1,28 1,46

)

The use of a dispersion law [1] for determining the parameter b leads to the results, commensurate with those given. 3.3 Adhesion of two solid bodies By adhesion of two solid deformed bodies B (1) and B (2 ) in the model of medium is understood this form of their contact, when the rectilinear material fiber, normal to the surface of contact S a and which intersects it, in the process

of the deformation of the united of body B = B (1) ∪ B (2) preserves not only its integrity (as in the case of classical, solid contact), but also smoothness of the distribution of its deformations. This idea gives the possibility to solve the problem about the stress-strained state of the body, which are in a state of adhesion as the combined task about the state of piecewise-uniform body B = B (1) ∪ B (2 ) with the condition of the absence of external actions on it.

Adhesion energy F of bodies B (1) and B (2 ) is determined by loss per the unit surface area, along which occurred the adhesion, their free energy [12] F = W p(1) + W p(2 ) – W p(1,2 ) .

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(18)

588 High Performance Structures and Materials III Here: W p(1,2 ) – the surface energy of the system of bodies, concentrated bi S a ;

W p( j ) – the surface energies of the disconnected bodies. The theory, examined in

this work, makes it possible to consider the volumetric distribution of these is specific internal energy by means of calculated value (16). For two plates with

the thickness, for much the great significance of the parameters b ( j ) , is obtained the formula F =

W p( 1 )W p( 2 ) ( k ( 1 ) + k ( 2 ) )2 W p( 1 ) k ( 2 )2 + W p( 2 ) k ( 1 )2

, k( j) =

ν ( j) , 1 −ν ( j )

(19)

ν ( j ) – Poisson ratio. The results of the comparison of the results of theory with the data of experiment are represented in table 2. Table 2:

The comparison of the results of theory with the data of experiment.

Metal

adhesion energy on silicon

(Dg

m

2

)

adhesion energy on the

(

glass Dg m 2

aluminum (Al)

0,214

0,150

silver (Ag)

0,181

0,120

copper (Cu)

0,210

0,110

4

)

Conclusion

At present is discussed a question about the correspondence of the discrete and continuous description of real material [2]. In the represented work it is shown that it is possible to attain an increase in the degree of the adequacy of description by the allotment of continuous medium with the properties, characteristic for the discrete medium. In this work this property is the ability of particles to interact at the significant distance with the aid of the potential forces. The gradient model of the second order for the medium with the initial stresses became the result this of calculation. It proved to be applicable for the use in the calculations of surface energy and adhesion energy.

References [1] [2]

Kittel, Ch., Introduction to Solid State Physics, Science: Moscow, pp 111–148, 1978. Krivtsov, A.M. & Myasnikov, V.P., Simulation by the method of the dynamics of particles of the change in the internal structure and stressed WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

[3] [4] [5] [6]

[7] [8] [9] [10] [11] [12]

589

state in the material under the strong thermal influence. By notify of Russia academy of science. Mechanic of solid body, 1, pp. 88 - 103, 2005. Gordon, V.A. & Shorkin, V.S., The non-local theory of the surface layer of solid body. Proceedings of Ttua State University. Mathematic. Mechanic. Informatics. 4(2), pp. 55 – 57, 1998. Eringen, A. C., Nonlocal continuum field theories, Springer-Verlag: Berlin, 2002. Mikaelyan, K.N., Gutkin, M. Yu & Ayfantis, Ye. S., Edge dislocations at the phase boundaries in the gradient theory of elasticity. Solid state physics, 2(9), pp. 1613 – 1620, 2000. Vitcovsky, I.V, Konev, A.N., Shorkin, V.S., Kzaev, N.D., Rusanov, A.E., Khoroshikh, V.M., & Leonov. S.L. Adhesion energy estimation of some composite materials. Plasma Devices and Operations, 11(2), pp 81-87. 2003. Korn, G. & Korn, T., Mathematical handbook for scientists and engineers, eMGRAW-HILL BOOK COMPANY, INC: New York, Toronto, London, 1961. Tupin, R.A., The theory of elasticity, which considers moment stresses. Voltages the mechanics coll. of transfers, 3, pp. 113 – 140, 1965. Woods, A.D.B., Cochran, W. & Brockhouse, B.N. Lattice Dynamics of Alkali Halide Crystals. Phys. Rev., 119, pp.980-999, 1960. Gordon, V.A., Shorkin V.S. & Borsenkov, M.I., The method of solution of problems of mechanics of heterogeneous bodies, Oryol State University: Oryol, pp. 35 – 46, 2005. Samsonov, G.V., (ed). Properties of elements. Physical Property, Metallurgy: Moskow, 1976. Vvedensky, B.A., (ed). Physical encyclopedias dictionary, Soviet encyclopedia: Moskow, 1960.

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Section 8 Optimal design

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593

Optimal design of fibre reinforced tubular structures H. Martikka & E. Taitokari Department of Mechanical Engineering, Lappeenranta University of Technology, Lappeenranta, Finland

Abstract The main purpose of this study is to present results of the design and damage analysis of industrial vessel shell microstructures made of fibre reinforced plastic laminates and subject to mechanical and aggressive chemical loads due to sulphuric and hydrochloric acids and high temperatures. Methods to obtain optimal performance microstructure are use of balanced and symmetric laminate layering with optimal fibre strengthening directions and volume fractions. Optimal selection of chemically resistant fibres and matrix is essential for endurance. Use of thick enough walls is cost-effective for vessel bottoms. Two vessel winding manufacturing technologies are compared for optimal selection. Keywords: composite materials and structures, optimal design, stress corrosion cracking.

1

Introduction

Composites used for structural engineering are commonly formed from five commonest material groups, metals, ceramics, glasses, elastomers and polymers. Design rules and codes are based continuum models. Another trend is to use more microscopic modelling. Useful information of composite design is presented by Agarwal and Broutman [1] and Barbero [2]. Chemical resistances are studied by Aveston and Sillwood [3] and, Kawada and Srivastava [4]. In aggressive enough loadings all material alternatives and combinations suffer accumulation of damage. Typical damage events are stress corrosion wearing of outer surface, fracture of fibres and degradation of matrix. The main purpose of this study is to present results of design and damage study analysis of industrial vessel walls. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06058

594 High Performance Structures and Materials III

2

Goals, materials and methods

2.1 General goals of designing thin shelled structures for process industry In metallurgical chemical industry concentrated sulphuric and hydrochloric acids are processed at about boiling temperature in thin shelled large vessels. The focus is now on shells made inorganic composites, Figure 1. Biopolymers Natural fibres

Polymers, elastomers

Ceramics, glasses, inorganics

Old

Metals T=10y A B New T=0.1y

FRP,inorganic FRP biobased

MMC

Rm,new

Composites Metal -metal-interphases

a)

Rm,old

b)

Figure 1:

a) Composite structure relationships. b) Environmental attack on a two layered composite causing wear and decrease of strength Rm.

2.2 Materials and loads The principle of designing composite walls is to use functional property gradients. Chemical resistance is needed close to inner wall. Load bearing capacity is obtained by making the walls strong and thick enough. The fibre materials are typically glass fibres. Now elastic modulus is Ef = 70000MPa, fracture strain εfu = 0.012. Options are normal low cost E-glass and special chemically resistant glass. Matrix modulus is typically Em =3400, tensile strength 83MPa and shear strength of same magnitude. Load is 20% sulphuric and hydrochloric acid at 104oC.

3

Background theory

3.1 Micromechanical properties of plane strain orthotropic lamina layers In principal LT material directions the relation between principal strains and principal stresses is given by the principal material direction stiffness matrix Q. It is the Hooke’s law

 σ L  Q11 σ  = Q  T   12 τ LT   0

Q12 Q22 0

0  ε L  0   ε T  Q66  γ LT 

(1)

3.1.1 Stresses in global and principal strength directions Each lamina gets its input loads from the global strain distribution. Use of the direction transformation matrix gives the components in principal directions. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

 c 2 s2 2sc   σ x  σ L   σ  =  s 2 c 2 −2sc   σ    y   T τ LT  k − sc sc c 2 − s 2  τ xy  k  k

{σ L } k = [T]{σ x } k

,

595

(2)

Transformation of stiffness matrix components from principal to global xy coordinates gives Q11 → Q11, k = Q11c 4 + Q22 s 4 + 2(Q12 + 2Q66 ) s 2 c 2 Q22 → Q22, k = Q11s 4 + Q22 c 4 + 2(Q12 + 2Q66 )s 2 c 2

(

Q12 → Q12, k = (Q11 + Q22 − 4Q66 )s 2 c 2 + Q12 c 4 + s 4 Q66 → Q66, k = (Q11 + Q22 − 2Q12 − 2Q66 )s 2 c 2 + Q66

) (c

(3) 4

+ s4

)

Q16 → Q16, k = (Q11 − Q12 − 2Q66 )sc 3 − (Q22 − Q12 − 2Q66 )s3c Q26 → Q21, k = (Q11 − Q12 − 2Q66 )s3c − (Q22 − Q12 − 2Q66 )sc 3

The global midplane strain induces global stresses at layer k with angle θ to xaxis

σ x   Q11     σ y  = Q12 τ xy     k Q16

Q12 Q22 Q26

Q16   ε x 0     Q26   ε y 0  , Q66  γ xy 0   k

{σ x } k = [Q ]{ε x 0 }

(4)

3.1.2 Force and moment resultants of all layers Force resultants are needed. They are obtained by integrating over the thickness

 Nx    Ny  =  N xy   

σ x    ∑ ∫hk -1 σ y  dz = k =1 τ xy   k

k =n h k

k=n h k

∑ ∫h

k =1

k -1

σ k dz

(5)

σ = Q ε = Q (ε 0 + zκ ) = Q ε 0 + zQ κ In component vector form the force resultant vector is  Nx     N y  = Aij  N xy   

[ ]

 ε x0  κ x   0   ⋅  ε y  + Bij  κ y  γ 0  κ xy     xy 

[ ]

where the A and B matrices are WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(6)

596 High Performance Structures and Materials III k =n

[ Aij ] = ∑ Qij k =1

k

k=n

[Bij ] = 12 ∑ Qij k •(hk 2 − hk-12 )

• ( hk − hk-1 ) ,

(7)

k =1

Use of these is illustrated in the case study model, shown in Figure 2. k =1

t1

θ1=+45 h0 = -6

k=2 x k=3

t2

θ2=-45

h1=--3 h2 = 0

t3

θ3=-45

h3=3

k = 4 = n t4

θ4=+45

x

θ z T y Figure 2:

θ

h4 = 6

L Principle of a symmetric and balanced composite +45/-45/-45/+45.

The ABD matrix relates the midplane strains and plate curvatures to resultant force vector N and moment vector M. The ABD matrices are extensional stiffness matrix A, coupling stiffness matrix B and bending stiffness matrix D.

 N  A B  ε 0  A B  ε 0  = ⇒ M   B D     B D   →      κ     0 

Aε 0   0  Bε 

(8)

In formulating the design goals and constraints the symmetry condition is the stiffest one. These rules can be utilised to obtain simple and well functioning microstructures. 3.1.3 Symmetric laminate structure as one optimisation goal The useful result of using symmetry goal is that the B matrix is made zero. Using the case study data

[Q ]k=1,+45deg = [Q ]k=4,+45deg

,

[Q ]k=2,-45deg = [Q ]k=3,-45deg (9)

Gives for the B matrix

[Bij ] = 12 ([Q ]1 (t 2 − 4t 2 ) + [Q ] 2 (0 2 − t 2 ) + [Q ] 2 (t 2 − 0 2 ) + [Q ]1 (4t 2 − t 2 )) = 0

(10)

The simplified result form is

 N   A 0  ε 0   M  =  0 D   ⇒      κ 

 A 0  ε 0   0 D   →    0 

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

 Aε 0     0 

(11)

High Performance Structures and Materials III

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3.1.4 Balanced laminate structure as one optimisation goal The next optimisation goal is to simplify the A matrix as much as possible. This goal is reached by adjusting the transformed stiffness components. If for every layer there is another almost identical layer but with opposite orientation somewhere in the laminate, the sign of sine function is changed. This means that Q16 and Q26 are zero meaning that A16, A26 are zero. For instance Q16 → +θ → sin θ = + s = (Q11 − Q12 − 2Q66 )sc 3 − (Q22 − Q12 − 2Q66 )s3c

(12)

Q16 ' → −θ → sin( −θ ) = − s = −(Q11 − Q12 − 2Q66 )sc 3 + (Q22 − Q12 − 2Q66 )s3c = − Q16

Using this result in A matrix and setting laminate layers constant ∆h = t

[ A16 ] =... Q16 • t + Q16 '•t = t • (Q16 − Q16 ) = 0

(13)

Using this one obtains for the A matrix

[ ] ∑ Qij k = t((Qij )1 + (Qij ) 2 + (Qij )3 + (Qij ) 4 ) k =1 Aij = t

4

 N x   A11     N y  =  A12  N xy   A16  

A12 A22 A26

A16   ε x 0   A11   A26   ε y0  ⇒  A12 A66  γ xy0   0  

A12 A22 0

, T = Σt

0   ε x0    0   ε y0  A66  γ xy0   

(14)

(15)

Force resultant depends on strains caused by global load stresses  N x   A11 N  =   y   A12

A12  ε x 0  N x  σ x  1 pr  1  = pr    0  ,   = Σt   = Σt    σ N A22  ε y  Σt 0.5 0.5  y  y

(16)

N xy = A66γ xy 0

The globally directed stresses acting on or within lamina k are σ x,k  Q11 σ  =   y,k  Q12

Q12  ε x 0     Q22  k ε y 0 

(17)

From this equation the midplane strains can be solved ε x 0   a11  0 =  ε y  a12 a66 =

1 A66

a12   N x  1  A22  = a 22   N y  det A − A12 ,

det A = A11 A22 − A12

− A12   N x    A11   N y 

2

whence the globally directed stresses acting on lamina depend on pressure WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(18)

598 High Performance Structures and Materials III σ x,k  σ  =  y,k 

 Q11  Q12

Q12   a11   Q22  k a12

a12   1  pr a 22   0.5

(19)

3.1.5 Stresses in laminate layers Using the xy global stress in layer k the stresses in principal directions LT of a layer k can be calculated. These are needed in stress based failure criteria  σ L,k   c 2 s 2    2 c2  σ T,k  =  s τ LT,k  − sc sc   

2sc   σ x,k     −2sc   σ y,k  c 2 − s 2  τ xy = 0

(20)

k

3.2 Environmental interaction effects The degradation of composite materials may result from several factors [1] 1) Loss of strength of reinforcing fibres by stress-corrosion 2) Loss of adhesion and interfacial bond strength degradation 3) Chemical degradation of the matrix material 4) Dependence of the matrix modulus and strength on time and temperature. One present observation is that rise of temperature from 20C ⇒ 100C lowers the modulus to half 3400 ⇒1700MPa 5) Accelerated degradation by temperature and chemical environment.

3.2.1 Failure mechanisms in fibre reinforced structures Some failure mechanisms in fibre reinforced structures are shown in Figure 3.

d 5

b) 2a

1

2

3

hackles 4

c)

a) Figure 3:

d)

a) Crack growth in fibre reinforced material by various mechanisms, 1. Fibre pull-out. 2. Fibre bridging, 3. Fibre/matrix debonding, 4. Fibre failure, 5. Matrix cracking. b) In E/VE composite the solution gets into contact with E-glass to corrode fibres. In E-glass there are Al2O3 and Fe2O3 compounds which corrode forming hydroxides. c) In C/VE glass water diffuses into matrix swelling it. Acid corrodes interfaces but less fibres. The result is that fibres are loosened and pull-out takes place. d) Fibre cross-section fracture models.

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High Performance Structures and Materials III

599

3.2.2 Condition of chemical endurance The industrial goal is to ensure that the wall does not leak and burst. Both the strength and the load bearing effective wall thickness T are decreased by the acid gas and liquid. Margin of safety goal is that stress should be less than strength. Dominant stress is the hoop stress. It increases when the wall corrodes. The strength decreases also and the factor safety N should be large enough σx = p

r ≤ T

1 N

Rm =

. Rm = 10 a t b ≈ 100 • t −0163

1 N

Rm0

1

(21)

tp , Rm [ MPa ], t[ years]

(22)

300 Bending Rm= 250MPa, new samples tested in air.

Bending Rm=150MPa, after 2 years in liquid acid

200

Bending Rm= 100MP, after 2 years in gas

Ultimate strength Tensile (MPa) Rm= 100

Tensile test Log(Rm)-Log(T) prediction model for samples held in gas 5 years Rm= 69 MPa k

190MPa new

Tensile Rm=100MPa 2 years in liquid Tensile Rm= 80MPa 2 years in gas

0 0

1

2

3

T, time in environmental test conditions t1= 0.1years

Figure 4:

Tensile test Rm-T linear model

4

5

Tensile test Lin(Rm)-Log(T) linear -log prediction model

Experimental testing of E-glass vinylester 25mm thick laminates at Sulphuric -hydrochloric 20% acid at 104 oC and in gas above it. Average stress was zero.

3.3 Model of stress corrosion aided crack growth of fibres Aveston and Sillwood [3] have studied conditions for crack growth in E-glass unidirectional polyester in 1 N H2SO4.They made several tests a1) static load, a2) static load with pre-soak for 1.8⋅106 sec, a3) Fatigue 0.1 Hz square wave. All points were close each other allowing a single model for crack growth rate to be used nearly independent of the time dependence of load stress as WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

600 High Performance Structures and Materials III V=

da = AK I n = 9.55 ⋅10 −11 ⋅ K I 3 dt

units: V

[ ], m s

t[sec], σ [ MPa]

(23)

1 Y ' = Yπ 2

KI = σ aY' a½ , n ≈ 3 , σa is applied stress, static or dynamic, a is crack length at time t, Y is a geometrical factor for crack. For a crack of length 2a in a plate, Y = 1 , Y’ = π½ Since the life of fibres is decisive for the whole composite, the fibre life can be regarded as the life time of the composite under full loading. Initial crack size is ai and final aIC. Stress corrosion life is from initial to final crack size dt =

da AK I n

=

(

da

A σ aY' a½

a

tC

)

1

→ ∫ dt = t C = AY ' n σ a n 0

n

C 1 1  2 ( 2 − n )  (24)   • a 1 − ½n     ai

For short time testing, the applied stress is raised up to tensile strength while the crack starts at initial size. Stress intensities at initial and final crack sizes are K Ii = σ a Y ' a i½ , K IC = σ a Y ' a C½ ⇒ K IC = σ max Y ' a i½ , σ max = Rm (25)

4

Analytical optimisation

This is based on the previous theory.

4.1 Logic of finding optimal structure 1. For Itt = 1 to 3, T(iItt) : T(1) =.01, T(2)=.022, T(3)= .04,wall thickness 2. For Ivf = 1 to 2, Vf (IVf): Vf (1) = 0.14, Vf (2) = 0.46, volume fraction 3. For Iss = 1 to 2, s(Iss) : s = 5000,20,1, aspect ratio choices Table 1:

Material and model data.

Ef = 70000

ν f = 0.22

Em = 3400

ν m = 0.38

s=

2L ,e d

ηL =

=

e −1 e + 2s

Ef Em

ε fu ( Im) = 0.012

ηT =

e −1 e+2

Ef 2(1 + ν f ) Em Gm = 2(1 + ν m ) Gf =

v=

Gf Gm

ηG =

, z =1

v −1 v+z

Goal is to calculate elements for the principal LT matrix Q 1 + 2 sη LVf 1 + 2η TVf E L = uE m , u = , E T = wE m , w = 1 − η LVf 1 − η TVf E 1 + η GV f G LT = gG m , g = , ν LT = ν f Vf + ν mVm , ν TL = ν LT T EL 1 − η GV f WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(26) (27)

High Performance Structures and Materials III

601

These models are input to principal LT stiffness matrix of orthotropic lamina EL ν LT E L   0  1 − ν ν − 1 ν ν LT TL LT TL σ 11     ε 11  (28) ET σ  =  ν LT E L  ε 22  , ν 21 = ν 12 0 22  1 − ν ν    E2 E1 LT TL 1 − ν LTν TL  τ 12   G12  γ 12  0 0     The Q matrix is in Hooke’s law 0  ε L   σ L  Q11 Q12  σ  = Q (29) 0   ε T   T   12 Q22 τ LT   0 0 Q66  γ LT 

4. for Ith = 1 to 3, Angle θ(Ith), is varied θ(1) = 20, θ(2) = 40, θ(3) = 60…

Next the transformed Q k for layer k are calculated. The two optimising conditions of symmetry and balancing have been applied For k’= 1 to 4, calculation of global stresses in lamina k’ (30) s = sin(θ k ) , c = cos(θ k )

{σ x } k = [Q ]{ε x 0 }

(31)

Next k’ 5. For k = 1 to 4 loop for calculating lamina k stresses and strengths Calculate A matrix 4

[ Aij ] = t ∑

(( ) ( ) ( ) + ( Q ) )

(32)

Q12   a11   Q22  k a12

(33)

Qij = t Qij + Qij + Qij k 1 2 k =1 -1 Calculate inverse A = A = a Calculate globally directed stresses in lamina layer k

σ x,k  σ  =  y,k 

 Q11  Q12

3

ij 4

a12   1  pr a22  0.5

Calculate lamina stresses in LT directions σ L,k   c 2 s 2  σ x,k  σ x,k   , τ LT,k = [ − sc sc]σ  σ  =  2 2  σ c  k  y,k   T,k  − s  x,k 

(34)

Strength in layer k Strength in L direction is related to volume fractions of fibres

EL V = p ≈ fL → VfT = 1p VfL ET VfT

, VfL + VfT = Vf 0

(35)

Total average volume fraction of fibres if conserved and divided to L and T directions WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

602 High Performance Structures and Materials III Vf 0 + Vm 0 = 1 ,VfL + VmL = 1, VfT + VmT = 1

(36)

Volume fraction portions for L and T directions are p 1 VfL = Vf 0 , VfT = VfL = VmL 1+ p p Strength formulas Strength depends on length of fibres and mechanism of strengthening If s > sc then the overcritical strength formula applies

(37)

σ LU = σ fuVfL + σ mε *VmL , σ TU = σ fuVfT + σ mε *VmT , τ LTU = 83 (38) f

f

Else if s < sc, then undercritical then strength formula applies It is assumed m = 1 meaning full slipping

σ LU = 2msτ i ⋅ (1 − ½ m) • VfL + σ mε *VmL

(39)

σ TU = 2msτ i ⋅ (1 − ½ m) • VfT + σ mε *VmT → s < sc

(40)

f

f

End if condition s<>sc Failure criterion Fk for each layer k Now failure criteria for each layer k, fibre material Im, aspect ratio s and environmental loadings are obtained For k = 1 to 4, get F (k) 2

2

2

 σ   σ   σ   σ   τ  Fk (θ k ) =  L  −  L   T  +  T  +  LT  ≤ 1  σ LU  k  σ LU  k  σ TU  k  σ TU  k  τ LTU  k

(41)

Next k 5. Next k, for each lamina stresses and strengths and failure criteria 4. Next Ith, Angle θ(Ith) is varied 3. Next Iss, s = 5000, 20, 1, aspect ratio is decreased if environmental attack by stress corrosion cuts fibres 2. Next IVf, Vf = 0.14 to 0.46 is lower at inner and higher for support layer 1. Next Itt, wall thickness

Ef Em

Figure 5:

τ

σf

σ1

σm

2L σ

ε1

2r=d

τ mL

Basic models of laminate and fibre strength behaviour.

4.2 Goals and results The main goal is minimisation of cost subject to desired values of failure criteria of layers. The cost is related to thickness times cost of a unit mass. The second WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

603

goal is to constrain product of failure criterions. The third goal is to obtain stress corrosion life of fibres sufficiently large by fibre and matrix material selections. A. At new materials, interfacial matrix strength is τi=83MPa, LT strengths are σLUk=320, σTUk=125. When long fibres s=5000, failure F=0.073. Strains εx=0.0015, εy=0.0028 are close to allowed strain 0.0015. B. Next matrix degradation is assumed from τi =83 to 40 causing F rise to 0.1 C. If cracking decreases aspect ratio to s=20, with lowing of matrix strength to τi=40, then F increases further but not much to 0.11. D. If cracking of fibres is maximal, then aspect ratio is minimal, s =1. If also matrix is degraded to τi=40, then σLUk=40 and σTUk=40. Failure criteria are large predicting full failure F >4. Also strains are large εx=0.004, εy=0.01. E. Stress corrosion cracking life at pressure 1 MPa gave hoop stress 45. At 1N H2SO4 acid load reasonable threshold stress intensity is KIi=5 for typical E-glass giving 0.02 years. Somewhat more chemically resistant E-glass with KIi=1 may give 0.1 years. Most optimal is to use such fibres which endure these acids well.

5

Microscopy characterisation

Scanning electron microscopy was applied to study the microstructure of a sample of thickness 9 mm which had been 2 years in saturated acid vapour gas, Figure 6. Then it had been tested to fracture in three point bending test. During the 2 years of acid attack the stress was low. At the final test it was applied up to fracture load. Evidently fibres had been weakened substantially as shown by its lowered strength value. Also little sign of matrix was found. 1

σ

2 3

Figure 6:

SEM characterisation. Location (1) is at neutral surface and fracture mode is mainly shearing and fibre tensile fractures, at left up (230x) and left down (500x). Location (2), (at middle top 650x and top down 230x) shows fibres at transverse direction. c) Right down (at 180 x) at location (3) close to the tensile surface shows axial and transverse fibre fractures but little matrix left.

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604 High Performance Structures and Materials III

6 Optimisation of pipe manufacturing and structure An industrial pipe made of continuous fibre and mat as sketched in Figure 2. Diameter is 2m and load is internal pressure of 1 MPa. Volume fraction is 0.46, moduli and Poisson’s ratios: Ef =72000MPa, νf =0.22, Em =3450, νm =0.38, densities ρf =2590kg/m3, ρm =1200. Individual ply lamina LT properties were EL=35300, ET=6150, GLT=2244, νLT=0.3, νTL=0.053. Allowable strains of 0.0015 were achieved. Two manufacturing methods are competing options: A. Manufacturing with 90/0deg winding can be done with a simple machine and control but requires costly axial mats. This gives pipes with about 30% fibres in hoop and 70% in axial direction. B. Manufacturing with 60 deg gives equal strains to the hoop and axial directions but requires a programmable costlier machine. Advantages are that low cost continuous roving can be used.

7

Conclusions

The following conclusions can be drawn. • In this study the results are presented of designing optimally industrial large vessel shell microstructures made of fibre reinforced plastic laminates and subject to mechanical and aggressive acids at high temperatures. • Optimal performance microstructure is obtained by use of balanced and symmetric laminate layering with optimal fibre directions and chemically resistant fibres and matrix. Thicker wall is cost-effective for vessel bottoms. • Industrial end-users require pipes to have same allowed strains in main directions. Manufacturing choices determine also microstructures: the 90/0deg winding requires simple machine but costly mats, the 60 deg winding requires complex machine but with inexpensive continuous roving.

Acknowledgements The assistance in SEM analyses by Mr. Markku Levomäki, MSc, is gratefully acknowledged. This research is supported by EU Asia Link Project (Contract Reference no.: ASI/B7-301/98/679-023)

References [1] [2] [3] [4]

Agarwal, B.D.and Broutman L.J.Analysis and performance of fiber composites, John Wiley & Sons, Inc.1990. Barbero, E.J. Introduction to composite materials design, Taylor & Francis, 1999. Aveston, J. and Sillwood, J.M. Long term strength of glass-reinforced plastics in dilute sulphuric acid. Journal of materials science 17 (1982) 3491-3498. Kawada, H. and Srivastava, V.K., The effect of an acidic stress environment on the stress-intensity factor for GRP laminates. Composites science and technology 61 (2001) 1109-1114. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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MINLP optimization of steel frames S. Kravanja & U. Klanšek University of Maribor, Faculty of Civil Engineering, Smetanova 17, 2000 Maribor, Slovenia

Abstract In this paper we deal with the topology and standard optimization of unbraced steel frames with rigid beam-to-column connections. The optimization has been performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP performs a discrete topology and standard dimension optimization, while continuous parameters are simultaneously calculated inside the continuous space. As the discrete/continuous optimization problem of steel frames is non-convex and highly non-linear, the Modified Outer-Approximation/EqualityRelaxation (OA/ER) algorithm has been used for the optimization. Two practical examples with the results of the optimization are shown at the end of the paper.

1

Introduction

The paper presents the topology and standard dimension optimization of unbraced steel frames with rigid beam-to-column connections. The optimization of frames is performed by the Mixed-Integer Nonlinear Programming, MINLP. The MINLP is a combined discrete-continuous optimization technique. In this way, the MINLP performs the discrete topology (i.e. the number of columns and beams) and standard dimension optimization (i.e. standard cross-section sizes) simultaneously with the continuous optimization of parameters (e.g. internal forces, deflections, mass, costs, etc.). The MINLP discrete/continuous optimization problems of frames are in most cases comprehensive, non-convex and highly non-linear. This optimization approach is proposed to be performed through three steps. The first one includes the generation of a mechanical superstructure of different topology and standard dimension alternatives, the second one involves the development of an MINLP model formulation and the last one consists of a solution for the defined MINLP optimization problem. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06059

606 High Performance Structures and Materials III The Modified Outer-Approximation/Equality-Relaxation algorithm [1], [2] and [3] is used to perform the optimization. The objective of the optimization is to minimize the mass of the structure. The finite element equations are defined as the equality constraints for the calculation of internal forces and deflections of the structure. Constraints for the dimensioning of steel members are determined in accordance with Eurocode 3 [4]. Beside the theoretical basics, an example of the optimization of a steel frame is presented at the end of the paper.

2

Steel frames

Steel frames, see Figure 1, are proposed to be analyzed under the combined effects of the self-weight of frame members, uniformly distributed variable load, concentrated variable load on each storey and an initial frame imperfection. Second-order elastic structural analysis is performed by considering a geometric nonlinearity due to P-δ and P-∆ effects. While the P-δ effect is associated with the influence of the axial force on the beam-column member flexure, the P-∆ effect denotes the influence of axial force acting through the relative sideways displacements of the element ends. In this paper, both effects are included in the nonlinear stiffness matrix of the individual frame member by usage of stability function approach. Chen and Lui [5] have comprehensively described this approach and have presented stability functions sii and sij. These stability functions are different for compressive and tensile axial forces. Since they give the indeterminate numerical solution when axial force is zero, simplified expressions for stability functions S1 and S2, introduced by Kim et al. [6], are rather used. The shear deformation effect is neglected considering the fact that only slender structural members are subjected to buckling for which shear deformation is insignificant. Design/dimensioning of steel frames is performed in accordance with Eurocodes 3 for the conditions of both the ultimate limit and serviceability limit states. When the ultimate limit state of beam-column members is considered, the elements were checked for bending moment, vertical shear, shear buckling, interaction between bending, shear and axial force and interaction between bending and axial compression/buckling. The ultimate moment capacity is calculated by the elastic method. Since the second-order elastic global analysis is used, the in-plane buckling lengths of compression members are calculated considering the non-sway mode. Considering the serviceability limit state, the vertical deflections of beams in the individual storey were calculated by the elastic method. The total deflections subjected to overall load δmax and the deflections subjected to the variable imposed load δ2 are calculated to be smaller than the limited maximum values: span/250 and span/300, respectively. The horizontal deflections are also checked for the individual storey and for the structure as a whole. Both types of horizontal deflections are checked for the recommended limits: the relative horizontal deflection of each storey should be smaller than each storey height/300 and the horizontal deflection of the top of the frame must be smaller than an overall frame height/500. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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F

Lc

q F

H Lc

Lb

Lb

Lb

L

Figure 1:

3

Unbraced steel frame.

MINLP model formulation for mechanical superstructures

The MINLP optimization approach requires the generation of an MINLP frame superstructure composed of various topology and standard dimension design alternatives that are all candidates for a feasible and optimal solution. It is assumed that a general nonconvex and nonlinear discrete/continuous optimization problem can be formulated as an MINLP problem (MINLP-G) in the form: min z = c T y + f ( x ) h( x ) = 0 g( x ) ≤ 0 By + Cx ≤ b

s.t.

(MINLP-G)

n

x ∈ X = {x ∈ R : xLO ≤ x ≤ xUP} m y ∈ Y ={0,1} where x is a vector of continuous variables specified in the compact set X and y is a vector of discrete, mostly binary 0-1 variables. Functions f(x), h(x) and g(x) are nonlinear functions involved in the objective function z, equality and inequality constraints, respectively. Finally, By+Cx ≤ b represents a subset of mixed linear equality/inequality constraints. The above general MINLP model formulation has been adapted for the optimization of mechanical superstructures (MINLP-SMS). The resulted formulation that is more specific, particularly in variables and constraints. It can also be used for the modelling of steel frames. It is given in the following form: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

608 High Performance Structures and Materials III min z = c T y + f ( x ) s.t.

h( x ) = 0

g(x ) ≤ 0

A( x ) ≤ a Ey ≤ e

(MINLP-SMS)

Dy + R( x ) ≤ r e

( ) Py + S (d ) ≤ s

Ky e + L d cn ≤ k st

n

x ∈ X = {x ∈ R : xLO ≤ x ≤ xUP} m y ∈ Y ={0,1} In the model formulation, included are continuous variables x={d, p} and discrete binary variables y={ye, yst}. Continuous variables are partitioned into design variables d={dcn, dst} and into performance (nondesign) variables p, where subvectors dcn and dst stand for continuous and standard dimensions, respectively. Subvectors of binary variables ye and yst denote the potential existence of structural elements inside the superstructure (the topology determination) and the potential selection of standard dimension alternatives, respectively. The mass or economical objective function z involves fixed mass or cost charges in the linear term cTy and dimension dependent mass or costs in the term f(x). Parameter nonlinear and linear constraints h(x)=0, g(x) ≤ 0 and A(x) ≤ a represent the rigorous system of the design, loading, stress, deflection, stability, etc. constraints known from the structural analysis. Integer linear constraints Ey ≤ e are proposed to describe relations between binary variables. Mixed linear constraints Dye+R(x) ≤ r restore interconnection relations between currently selected or existing structural elements (corresponding ye = 1) and cancel relations for currently disappearing or nonexistent elements (corresponding ye = 0). Mixed linear constraints Kye+L(dcn) ≤ k are proposed to define the continuous design variables for each existing structural element. The space is defined only when the corresponding structure element exists (ye =1), otherwise it is empty. Mixed linear constraints Py+S(dst) ≤ s define standard design variables dst. Each standard dimension dst is determined as a scalar product between its vector of standard dimension constants q and its vector of binary variables yst. Only one discrete value can be selected for each standard dimension since: d st = ∑ qi yist i∈I

st ∑ yi = 1

i∈I

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609

Optimization

The Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm is used to perform the optimization. The OA/ER algorithm consists of solving an alternative sequence of Non-linear Programming optimization subproblems (NLP) and Mixed-Integer Linear Programming master problems (MILP). The former corresponds to the optimization of parameters for a frame structure with fixed topology and standard dimensions and yields an upper bound to the objective to be minimized. The latter involves a global approximation to the superstructure of alternatives in which a new topology and new standard sizes are identified so that its lower bound does not exceed the current best upper bound. The search is terminated when the predicted lower bound exceeds the upper bound. The optimization is performed sequentially in two different phases in order to accelerate the convergence of the OA/ER algorithm. The optimization starts with the topology optimization of the frame, while standard dimensions are relaxed temporary into continuous parameters. In the case of the standard dimension optimization only, the optimization begins with the continuous NLP optimization of the frame. When the optimal topology (continuous parameters) is found, standard sizes of cross-sections are re-established and the standard dimension optimization of cross-sections is then continued until the optimal solution is found.

5

Examples

5.1 Example 1: three-storey frame The first example shows the standard dimension optimization of the three-storey, three-bay plane steel frame (see Figure 2). The frame is subjected to the selfweight, to the uniformly distributed imposed load of 50 kN/m and to the concentrated imposed load of 10 kN on each storey. The frame is considered as a laterally supported plane frame. The frame superstructure has been generated in which all possible structures are embedded by different standard sizes variation. The superstructure comprises 24 different standard hot rolled European wide flange HEA sections (from HEA 100 to HEA 1000) for each beam and column separately. The material used is steel S 355. The optimization was performed by the MINLP optimization approach. The MINLP optimization model for steel frames was used. The task of the optimization was to find the optimal structure mass. The optimization was carried out by a user-friendly version of the MINLP computer package MIPSYN, the successor of PROSYN [1] and TOP [7]. As an interface for mathematical modelling and data inputs/outputs GAMS (General Algebraic Modelling System), a high level language, was used [8]. The Modified OA/ER algorithm and the two-phased optimization were applied (a single MINLP), where GAMS/CONOPT2 (Generalized reduced-gradient method) [9] WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

610 High Performance Structures and Materials III was used to solve NLP subproblems and GAMS/Cplex 7.0 (Branch and Bound) [10] was used to solve MILP master problems.

5.0 m

F = 10.0 kN

q = 50.0 kN/m

q = 50.0 kN/m

5.0 m

F = 10.0 kN

15.0 m

5.0 m

F = 10.0 kN

5.0 m

5.0 m

5.0 m

15.0 m

Figure 2:

Three-storey, three-bay steel frame.

The optimization model of the frame contained 1572 (in)equality constraints, 1552 continuous and 357 binary variables (61 after the prescreening). The final optimal solution of 6214 kg was obtained in the 61st main MINLP iteration (the subsequent feasible result was not so good). The optimal standard sizes were also obtained. Only 152 seconds of CPU time were spent on a 2 GHz PC. The optimal structure of the frame is shown in Figure 3. 5.2 Example 2: single-storey industrial building The second example presents the topology and standard dimension optimization of a single-storey industrial building. The building is 20 meters wide, 40 meters long and 6.5 meters high (see Figure 4). The structure is consisted from equal non-sway steel portal frames, which are mutually connected with the purlins. The optimization was performed by the MINLP optimization approach. The task of the optimization was to find the optimal structure mass, the optimal topology (the optimal number of portal frames and purlins) and optimal standard cross-sectional sizes.

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Variable imposed loads s = 1.60 kN/m2 (snow) and w = 0.137 kN/m2 (wind) are defined as the uniformly distributed surface load in the model input data. Both, the horizontal concentrated load at the top of the columns (wind) and the vertical uniformly distributed line load on the beams (snow and wind) are calculated considering the intermediate distance between the portal frames. Internal forces are calculated by the elastic first-order analysis for non-sway frame mode. The dimensioning of steel members is performed in accordance with Eurocode 3 for the conditions of both ultimate limit state (ULS) and serviceability limit state (SLS).

5.0 m

5.0 m

HEA 180 HEA 180

HEA 340

HEA 180 5.0 m

15.0 m

5.0 m

HEA 260

HEA 340

HEA 260

5.0 m

HEA 260

HEA 180

HEA 260

HEA 260 HEA 180

HEA 260 HEA 260

HEA 180

HEA 260

HEA 260

HEA 180

HEA 260

HEA 180

HEA 260

5.0 m

15.0 m

Figure 3:

Optimum design of the frame.

The portal frame superstructure was generated in which all possible structures were embedded by 30 portal frame alternatives and 20 purlin alternatives. The superstructure also comprised 24 different standard hot rolled European wide flange I beams, i.e. HEA sections (from HEA 100 to HEA 1000) for each column, beam and purlin separately. The material used was steel S 355. The optimization model contained 124 (in)equality constraints, 111 continuous and 122 binary variables (55 after the prescreening). The final optimal solution of 62029 kg was obtained in the 3rd main MINLP iteration. The optimal result includes the obtained optimal topology of 15 portal frames and 12 purlins (see Figure 4) as well as the calculated optimal standard sizes of columns, beams and purlins (see Figure 5). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

6.5 m

612 High Performance Structures and Materials III

× 14

6m .0 m 40

2.8

20.0 m

0.50 m

Optimal design of the single-storey industrial building.

HEA 120

HEA 260

HEA 260

6.5 m

HEA 500

HEA 500

6.0 m

Figure 4:

10 × 2.0 m 20.0 m

Figure 5:

6

Optimal design of the portal frame.

Conclusions

This paper presents the topology and standard dimension optimization of steel frames with rigid beam-to-column connections. The optimization has been performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP was found to be very successful optimization technique for solving the frame structures. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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References [1] [2]

[3]

[4] [5] [6] [7]

[8] [9] [10]

Kravanja, Z. and Grossmann, I.E., New Developments and Capabilities in PROSYN - An Automated Topology and Parameter Process Synthesizer, Computers chem. Eng., 18(11/12), pp. 1097-1114, 1994. Kravanja, S., Kravanja, Z. and Bedenik, B.S., The MINLP optimization approach to structural synthesis. Part I: A general view on simultaneous topology and parameter optimization, Int. J. Numer. Methods Eng., 43, pp. 263-292, 1998. Kravanja, S., Kravanja, Z. and Bedenik, B.S., The MINLP optimization approach to structural synthesis. Part II: Simultaneous topology, parameter and standard dimension optimization by the use of the Linked two-phase MINLP strategy, Int. J. Numer. Methods Eng., 43, pp. 293-328, 1998. Eurocode 3, Design of steel structures, European Committee for Standardization, 1992. Chen, W.F. and Lui, E.M., Structural Stability: Theory and Implementation, New York: Elsevier Science Publishing Co., Inc., 1987. Kim, S.E., Lee, J.S., Choi S.H. and Kim, C.S., Practical second-order inelastic analysis for steel frames subjected to distributed load, Eng. Struct., 26, pp. 51-61, 2004. Kravanja, S., Kravanja, Z., Bedenik, B.S. and Faith, S., Simultaneous Topology and Parameter Optimization of Mechanical Structures, Numerical Methods in Engineering '92, Proceedings of the First European Conference on Numerical Methods in Engineering, ed. Ch. Hirsch et al., pp. 487-495, Elsevier, Amsterdam, 1992. Brooke, A., Kendrick, D. and Meeraus, A., GAMS - A User's Guide, Scientific Press, Redwood City, CA, 1988. Drudd, A.S., CONOPT – A Large-Scale GRG Code, ORSA J. Comput., 6 (2), pp. 207-216, 1994. CPLEX User Notes, ILOG inc.

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Optimization of timber trusses considering joint flexibility S. Šilih, M. Premrov & S. Kravanja University of Maribor, Faculty of Civil Engineering, Slovenia

Abstract This paper presents the optimization of metal-plate-connected plane timber trusses with special emphasis on joint flexibility. The optimization is performed by the non-linear programming approach. Since various truss design parameters such as type of truss configuration, span/depth ratio, number and type of diagonal and vertical members and type of joint connections simultaneously affect each other, all of these parameters are proposed to be considered simultaneously in a single mathematical model. The optimization model for cost optimization of timber trusses was thus developed. The economic objective function was defined to minimize the structure’s self-manufacturing costs, subjected to the design, stress and deflection (in)equality constraints. The finite element equations were as the equality constraints defined for the calculation of the internal forces and the deflections of the structure. The stiffness matrix of the structure was composed by considering the fictiously decreased cross-section areas of all the flexible connected elements. Constraints for the dimensioning of the timber members were determined in accordance with Eurocode 5 in order to satisfy the requirements of both the ultimate and the serviceability limit states. The cross-section dimensions and the number of fasteners were defined as independent optimization variables. A numerical example demonstrates the applicability of the presented approach.

1

Introduction

Timber construction is an important part of the infrastructure in a number of areas around the world. Wood has proved to be quite a resilient material, showing relatively high ductility and low density. In addition, the flexibility of mechanical fasteners provides a high damping capacity between the connected WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06060

616 High Performance Structures and Materials III timber elements. Well-built timber structures maintain a good performance particularly under the influence of wind and especially earthquake forces. In the last decades, the application of timber trusses has frequently been noticed in all aspects of building construction. Timber trusses have become known for their pleasing architectural appearance, lightweight design and easy fabrication. The use of timber trusses to bridge over large open areas with a few or no intermediate supports is still on the increase. These trusses are essentially lighter than the analogous beam solutions. Many magnificent space and plane timber trusses have been constructed all over the world, covering public halls, stadiums, exhibition centres and many other buildings. In this field, metal-plateconnected timber trusses have been found to be favourable structures for roof framings for spans greater than 20 meters. In order to design a satisfactory and optimal timber truss structure with the given load, span and boundary conditions, some main design parameters need to be considered, on which timber truss behaviour basically depends: • type of truss configuration, • span/depth ratio, • number and type of intermediate members (diagonals and verticals), • type of joint connections. When a high number of truss design parameters, designer decisions and factors are involved in the analysis, the designing of timber trusses can become a difficult and expensive process. This has forced designers to find simpler and cheaper alternative design methods, adequate at least for the preliminary design state. Several approximate methods have been developed in the recent past with different accuracies of suitability and simplification according to real truss conditions, see [1]. Approximate designing methods which additionally consider the flexibility of the joints in timber trusses with respect to different diagonal members can for example be found in [2-5]. The idea of the present study was to together simultaneously consider all the mentioned design parameters and factors in a single mathematical truss model, where structural optimization is performed rather than classical analysis. For more than four decades, trusses have not only been successfully optimized but also very frequently used to present, test and improve various optimization techniques. Numerous research papers on this topic have been published since the early 1960s, e.g. [6]. While many papers discuss the topology, shape and discrete sizing optimization particularly of steel trusses, e.g. [7-10]; also the optimization of composite trusses, [11]; timber trusses have been quite neglected. The paper presents the sizing optimization of metal-plate-connected timber trusses considering the flexibility of the embedded fasteners. The optimization was performed by the non-linear programming (NLP) approach, where all the mentioned design parameters were simultaneously considered as (in)equality constraints. The optimization model for the cost optimization of the timber trusses was developed. An economic objective function was proposed to minimize the structure’s self-manufacturing costs, subjected to the design, stress WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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and deflection (in)equality constraints. The finite element equations were as the equality constraints defined for the calculation of the internal forces and the deflections of the structure.

2

Timber truss design criteria

The design constraints for the timber trusses were determined in accordance with Eurocode 5 [12] in order to satisfy the requirements of both the ultimate (ULS) and the serviceability limit state (SLS). Considering the ULS, the truss members were checked for the tensional as well as the compressive/buckling resistance. The required number of fasteners was also calculated for each joint. At the SLS the vertical deflections of the truss girders were checked. Since the bracing members (diagonals and verticals) are flexibly connected, their stiffness decreases. In finite element analysis we consider the joint flexibility in such a way that cross-sectional areas Am of all bracing members are replaced by a fictiously decreased cross-section area Am* [4]: A*m =

Am E m ,mean ⋅ Am  1 1 1+ ⋅ +  K ⋅k Lm K ser ⋅ k m ,2  ser m ,1

   

(1)

where km,1 and km,2 represent the numbers of fasteners at both ends of the considered m-th bracing element and Kser denotes the fasteners’ slip modulus, taken for different types of fasteners from Table 7.1 of Eurocode 5. Em,mean stands for the mean value of the modulus of elasticity.

3

Optimization of timber trusses

As the optimization problem of timber trusses is non-linear, e.g. the objective function and (in)equality constraints are non-linear, the non-linear programming optimization (NLP) approach is used. The general NLP optimization problem can be formulated as follows: Min z = f(x) subjected to: h(x) = 0 g(x) ≤ 0 x ∈ X = { x x ∈ Rn, xLO ≤ x ≤ xUP }

(NLP)

where x is a vector of continuous variables, defined within the compact set X. The variables x are calculated between their lower and upper bounds xLO and xUP. Functions f(x), h(x) and g(x) are non-linear functions involved in the objective function z, equality and inequality constraints, respectively. All functions f(x), h(x) and g(x) must be continuous and differentiable. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

618 High Performance Structures and Materials III In the context of structural optimization, variables include dimensions, cross-section characteristics, strains, materials characteristics, stresses, economic parameters, etc. Equality and inequality constraints and the bounds of the variables represent a rigorous system of the design, loading, stress, deflections and stability functions taken from the structural analysis. The optimization of the structures may include various objectives worthy of consideration. The most popular criterion used today is the minimization of mass. In this paper, an economic objective function is proposed to minimize the structure's selfmanufacturing costs. Hence, the trade-offs between different materials can be appropriately accounted for. The optimization model TTO (Timber Truss Optimization) for the optimization of timber trusses was developed according to the above NLP model formulation. GAMS (General Algebraic Modeling System), [13], was used as the interface for mathematical modeling and data inputs/outputs. A general timber truss layout with its characteristic elements (upper and lower chord, verticals, diagonals) is presented in Figure 1, where xi represents the local longitudinal axis of element i, while yi and zi represent the principal axes of the cross-section of the element i; the axes X and Y form the global coordinate system of the structure. Ai and li stand for the cross-section area and the length of member i, respectively. The cross-sections are considered to be rectangular, where bi and hi represent the width and the height of the cross-section of the truss member i. xi

Y

upper chord i,

li

vertical

A i,

dia

l na go

hi X

lower chord

Figure 1:

bi A i = bi×hi

Plain timber truss.

An economic objective function is defined in the model to minimize the structure’s self-manufacturing costs, subjected to design, stress and stability constraints, known from structural analysis. Internal forces are proposed to be determined by the finite element equations, while the dimensioning is performed in accordance with Eurocode 5. The objective function is thus defined: min

I

(

)

M

(

)

J

cost = ct ⋅ ∑ bi hi li + c fm + c fl ⋅ 2 ⋅ ∑ k m + c fm + c fl ⋅ ∑ k j i =1

m =1

j =1

(2)

where cost represents the self-manufacturing (material and labour) costs of the structure; ct denotes the price of the manufactured and embedded timber material per m3; the sum of the products between widths bi, heights hi and lengths li of i, i∈I, timber members represents the volume of the truss in m3 (see Figure 1); cfm is the material cost of one fastener together with the adjoining steel plates, while WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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cfl denotes the manual labour costs for handling, assembling, drilling and bolting, defined per one fastener. Considering that the required number of fasteners is equal for both ends of each intermediate member, the total number of fasteners in the m-th member is 2km, where km = km,1 = km,2. The last term of the objective function represents the sum of all fasteners required in joints of the chord members. Variable kj stands for the number of fasteners of the j-th joint, j∈J. It is evaluated considering the resultant force on account of the axial forces of all intermediate elements connected to joint j. Since the dimensions of steel plates depend directly on the number of calculated fasteners, the costs of steel plates are included in the values cfm and cfl. The input data of the optimization model is the truss geometry (coordinates of joints), the supporting and loading conditions, the diameter of the considered fasteners, the thickness of the metal plates, as well as the material characteristics of all the used components (timber, fasteners, plates). The cross-section dimensions bi and hi of i, i∈I, truss timber members and the number of fasteners km and kj are defined as independent optimization variables. The finite element equations for the calculation of internal forces and deflections of the structure are defined as equality constraints. The stiffness matrix of the structure is composed by considering the fictiously decreased cross-section areas of all the intermediate timber elements (diagonals and verticals) in accordance with Eq. (1). The ULS and SLS design conditions, described in Section 2 are defined as inequality constraints.

4

Numerical example

As a numerical example, a timber truss girder of 30 m span is presented. The layout of the structure is shown in Figure 2. The truss elements are composed from a glued laminated timber GL32h according to the EN 1194 [14] classification. M14 dowels made of steel S 235 are used as fasteners. Steel plates of 8 mm thickness made of steel S 235 are additionally placed as the central members of a double shear connection. The truss is subjected to a permanent load of 2 kN/m and a variable load of 5 kN/m (snow load). The self-weight of the truss members was automatically determined through the optimization process with respect to the actual calculated cross-sectional dimensions. The uniform loads are, in the calculation, approximated as nodal forces (see Figure 2).

Fi α

i ΣF = q⋅L

h0 L = 30 m

Figure 2:

Layout of the timber truss.

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h

620 High Performance Structures and Materials III The economic parameters considered for the objective function (2) were costs ct = 900 EUR/m3 for the GL32h timber, while cfl = 1.5 EUR and cfm = 1.0 EUR for one M14 S 235 dowel (including the corresponding parts of the steel plates). The lower bound on the cross-section dimensions bilo/ hilo were taken as 12/12 cm. Three different optimization cases of the considered truss were performed, namely A, B and C. At truss A the maximal height h amounted to 400 cm (span/depth ratio = 7.5), while at truss B a lower value of h = 250 cm (span/depth ratio = 12) was considered. In both examples A and B the flexibility of joints was taken into account. In addition, the lower truss (h = 250 cm) was optimized again by ignoring the flexibility of joints (truss C). In all cases the inclination of the upper chord α amounted to 7.5%. The vertical displacements of joints at SLS were limited to L/300 = 10 cm. Since the NLP optimization handles continuous variables, the obtained variables of the final/optimal solution take some real values between their defined lower and upper bounds. At this stage, the structure is fully exploited considering either the ultimate or the serviceability limit state design conditions. As the aim of this research is to obtain a structure of practical applicability, the final continuous optimal solution was reanalysed with the variables rounded to their nearest upper integer values (i.e. 1 cm for cross-sectional dimensions and 1 fastener). The NLP optimization was performed by the computer program GAMS/CONOPT2 [15]. Table 1:

The obtained optimal results. Truss A

Truss B

Truss C

[EUR]

2984.33

3575.14

3322.59

[kg]

1291.29

1486.19

1356.99

16/13 18/15

17/21 21/17

17/20 21/17

181

248

244

Max. deflection without considering joint flexibility

44.82

79.85

82.62

Max. deflection by considering joint flexibility

59.88

95.79a

-

Total costs Timber mass

Chord dimensions b/h [cm] lower chord upper chord Total number of dowels

a

It should be noted, that the max. displacements of truss B is not equal to L/300 = 100 mm due to the rounding of cross sections before the reanalysis. After the first optimization phase (continuous optimization), the SLS conditions were active and the displacement was equal to 100 mm.

The obtained results of the three performed optimizations are presented in Table 1. In order to arrive to appropriate conclusions, it is convenient to present the final results by comparing the optimal costs and the obtained masses between WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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different truss alternatives, see Table 2. The influence of the height of the truss on the final results is presented by the ratio B/A, which represents the ratio between the obtained costs (masses) of the truss B (h = 250 cm) and A (h=400 cm). The possible error, caused by neglecting the flexibility of the joints in the timber trusses, is exposed through the ratio B/C which represents the ratio of the obtained costs (masses) between truss B (considering joint flexibility) and C (neglecting joint flexibility). Table 2:

Comparison of timber mass and total costs.

costs ratio

timber mass ratio

B/A

B/C

B/A

B/C

1.198

1.076

1.151

1.095

From the results presented in Table 1 it is evident, that truss A represents the best solution. At the final result the deflection constraints are not active i.e. the ULS design conditions are decisive. A lower span/depth ratio of the truss is thus more favourable. Despite lower timber mass and total costs, the vertical deflections of Truss A are considerably smaller when compared to the deflections of truss B. However, both the results of trusses A and B show a significant influence of joint flexibility on the final displacements. When joint flexibility is considered, the deflections increase by over 30 percent (truss A). Comparing trusses B and C, it is evident that the influence of fastener flexibility should not be neglected at all. The total costs increase by 7.6%, and the timber mass by almost 10%. It should be noted, that a more detailed study, presented in [16], has shown, that the influence of joint flexibility increases with a higher number of flexibly connected bracing members (diagonals and verticals).

5

Conclusions

The paper presents the optimization of metal-plate-connected plane timber trusses with respect to joint flexibility. The optimization was performed by the non-linear programming approach. Since various truss design parameters like the type of truss configuration, the span/depth ratio, the number and type of diagonal and vertical members as well as the type of joint connections simultaneously affect each other, all these parameters were proposed to be simultaneously considered in a single mathematical model. The optimization model TTO (Timber Truss Optimization) for cost optimization of timber trusses was thus developed. The economic objective function was defined in order to minimize the structure’s self-manufacturing costs, subjected to the design, stress and deflection (in)equality constraints. The finite element equations were as the equality constraints defined for the calculation of the internal forces and the deflections of the structure. The undesirable slips in the connections of timber trusses additionally resulted in the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

622 High Performance Structures and Materials III reduction of truss stiffness. A stiffness matrix of the structure was therefore composed by considering the fictiously decreased cross-section areas of all the flexibly connected elements. The constraints for dimensioning the timber members were determined in accordance with Eurocode 5 in order to satisfy the requirements of both the ultimate and the serviceability limit states. The crosssection dimensions and the number of fasteners are defined as independent optimization variables. The obtained results are not only optimal, but the optimization also enables that the conditions of either the ultimate or the serviceability limit states are fully exploited and there is no reserve in the resistance of the structures. At this point the comparison between results of different truss design alternatives was performed. The presented numerical example shows the applicability of the presented optimization approach as well as the influence of considering the fasteners’ flexibility on the optimal self-manufacturing costs. Based on numerical results, it is recommended to design higher timber trusses with a lower span/depth ratio, with a smaller number of diagonal and vertical elements and, consequently, by using chord elements with smaller cross-sections.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11]

Hoyle RJ, Woeste FE. Wood Technology in the Design of Structures, Ames, Iowa: Iowa State University Press, 1989. El-Sheikh A. Approximate Analysis of Space Trusses. International Journal of Space Trusses 1996; 11: 321-30. Stalnaker J, Harris E. Structural Design in Wood, NY: Van Nostrand Reinchold, 1989. Steck G. Fachwerbinde aus Brettschictholz un Vollholz, Holzbauwerke: Bauteile Step 2, Konstruktionen, Details nach Eurocode 5, Düsseldorf: Fachverlag Holz, 1995. Brüninghoff H et al. Eine Ausführliche Erläuterung zu DIN 1052, Teil 1 bis Teil 3, Beuth –Kommentare, Berlin: Beuth Bauverlag, 1988. Schmidt LA Structural Design by Systematic Synthesis. Proceedings of 2nd Conference on Electronic computations; NY: ASCE, 1960, 105-22. Lipson SL, Agrawai KM. Weight Optimization of Plane Trusses. ASCE Journal of the Structural Division 1974; 100: 865-79. Prager W. Optimization of Structural Design. Journal of Optimization Theory and Applications 1970; 6: 1-21. Šilih S, Kravanja S, Bedenik BS. Shape optimization of plane trusses. In: Hendriks, MAN, Rots JG editors, Finite Elements in Civil Engineering Applications, Proceedings of the Third DIANA World Conference, Tokyo, 2002, 369-73. Kaveh A, Kalatjari V. Topology optimization of trusses using genetic algorithm, force method and graph theory. International Journal for Numerical Methods in Engineering 2003; 58: 771-91. Kravanja S, Šilih S. Optimization based comparison between composite I beams and composite trusses. Journal of Constructional Steel Research 2003; 59: 609-25. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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[12] [13] [14] [15] [16]

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CEN/TC 250/SC5 N173, Eurocode 5: Design of Timber Structures, Part 1-1 General rules and rules for buildings, Final draft prEN 1995-1-1, Brussels, European Committee for Standardization, 2002. Brooke A, Kendrick D, Meeraus A. GAMS - A User's Guide, Redwood City, CA: Scientific Press, 1988. EN 1194, Timber structures – Glued laminated timber – Strength classes and determination of characteristic values, Brussels, European Committee for Standardization, 1999. Drud AS. CONOPT – A Large-Scale GRG Code. ORSA Journal on Computing 1994; 6:207-16, 1994. Šilih S, Premrov M, Kravanja S. Optimum design of plane timber trusses considering joint flexibility. Engineering structures 2005; 27: 145-154.

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The Hendrickx–Vanwalleghem design strategy W. Debacker1, C. Henrotay2, W. P. De Wilde1 & H. Hendrickx2 1

Department of Mechanics of Materials and Constructions (MeMC), Vrije Universiteit Brussels, Faculty of Engineering, Belgium 2 Department of Architecture (ARCH), Vrije Universiteit Brussels, Faculty of Engineering, Belgium

Abstract The optimisation of structures and materials is a justifiably popular engineering topic. Contemporary research is concentrated, among others, into cost minimisation, structural efficiency and intelligence, in compliance with environmental and social preservation. As a counterpart this paper puts the accent on the time dependent aspect of constructions, such as the life cycle cost, the possibility to make (non-) structural changes and recycling or reuse of building material. In search of an optimisation of this ‘dynamic’ efficiency of constructions, a design strategy has been developed at the Vrije Universiteit Brussels (Departments of MeMC and ARCH). This strategy is presented here. It considers the temporal character of constructions from the first sketches onwards. Keywords: adaptability, reuse, design strategy, construction kits, generating system, temporal/temporary.

1

Introduction

In society buildings have been - and still are - designed in terms of end states. The moment the first sketches are drawn, the construction’s finality is planned or denied. Because of their static nature, which they acquired ab initio, most buildings are not suited to meet the demands of a quickly changing society. As a result many building components end up as waste or are brought back in circulation by means of expensive and consuming industrial processes. Changing functions, quickly evolving living and building trends, the amount of new materials and improved techniques… are some mutations the built environment has to go through and has to provide appropriate answers to. Although humans WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06061

626 High Performance Structures and Materials III have to cope with an unpredictable future, full of uncertainties, there is one universal ‘constant’: the environment always changes! Hence, a sustainable built environment requires a dynamic concept; a step-by-step redesign process of gradual changes in which no end states or final goals can be defined [2]. The strategy “Hendrickx–Vanwalleghem” [3] includes this dynamic view on the built environment. By designing adaptable construction systems, which are compatible with each other, a dynamic – and by this a sustainable – answer can be given to an unexpected and unpredictable future. These construction systems are made of a minimum number of basic elements and a set of combination rules. They allow the conversion of each artefact to a different configuration, by means of adding, removing or transforming the basic elements which it is made of. It offers a high potential of recycling and (direct) reuse. The outcome can be compared with the ‘Meccano’ building set, which, in this view, encloses all materials and techniques, and is applicable to all scales. Hendrickx and Vanwalleghem proposed a set of standardisation rules, which they called a “generating form and dimensioning system”, The generating system is a central concept in the design strategy, in the sense that it ensures full compatibility of form and dimensions between all basic elements. The rules are translated into a fractal model, based on basic forms, such as the square, the inner circle and its diagonal, and a dimensional range using the operator “multiply or divide by 2” (Fig. 2).

2

The Hendrickx–Vanwalleghem strategy

Hendrickx and Vanwalleghem developed a “dynamic design strategy” (further called “Hendrickx–Vanwalleghem strategy”). It allows for the design of flexible construction systems, based on a minimal number of elements and combination rules. The design strategy is explained in Fig 1. It consists of 4 layers. The lower layer is characterised by a material solution, using 2 design tools, shown in the upper conceptual layer: the generating form and dimensioning system and the theoretical design catalogues. The text hereafter explains the set-up of the design strategy, starting with the material solutions (layer 4). 2.1 Layer 4: Adaptable material solutions According to European standards [4], a durable house is designed for a life cycle of 50 years. It is easy to understand that such constructions will undergo radical transformations and repairs during this extensive period. The “Hendrickx– Vanwalleghem” design strategy has consequently been developed for constructions subjected to these transformation processes. In Fig.1 the reader can see how a minimal habitation unit can be transformed into alternative configurations. This minimal habitation unit provides its user(s) with basic functionalities. It can be expanded from its core, according to evolving possibilities of user(s) and/or the building site. This principle has been WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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successfully applied in (developing) countries, such as India, as it takes into account a gradual development of the users and the possibility to perform reversible changes. Of course, such processes should make use of simple and cost effective building methods. 2.2 Layer 3: Adaptable construction elements As a consequence of previous argumentation, construction elements must be developed to be easily adaptable and reusable. This should be done not only for the non structural elements but also for the load bearing ones, such as walls, floors, foundations, columns, beams (Fig.1) and, even more important, the connections. Nevertheless, all these elements can be deconstructed into very simple “basic elements”. 2.3 Layer 2: Compatible construction kits The basic elements can be compared to the letters of an alphabet: they do not carry a semantic meaning. The basic elements can be combined in different ways and form a variety of construction elements, which in the actual comparison can be considered as words. Three types of basic elements can be discerned: -

line elements (one-dimensional) plane elements (two-dimensional) volume elements (three-dimensional)

Figure 1:

The Hendrickx–Vanwalleghem design strategy.

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628 High Performance Structures and Materials III Point elements (zero-dimensional) can be categorised in the previous classification: e.g. bolts can be considered as scaled volume elements. A “construction kit” is defined as a grouping of a FEW simple basic elements aiming at the assembly of one or more adaptive construction elements. One must be aware that the building of a complete construction will very often require several construction kits. It is thus absolutely necessary that all basic elements found in the same construction kit or even another one, be compatible with each other. The establishment of explicit standardisation rules is here for stringent. 2.4 Layer 1: Development of adaptable construction systems 2.4.1 Design tool 1: a generating form and dimensioning system Using their own developed design tool, called a “generating form and dimensioning system”, Hendrickx and Vanwalleghem proposed a set of standardisation rules. It is a central concept in the design strategy, in the sense that it ensures full compatibility of form and dimensions between all the simple basic elements. Hendrickx and Vanwalleghem presume that any tangible element, in any construction phase, can be approximated with a minimal diversity of basic forms. They have chosen the square, its diagonals and the inscribed circle, due to an important property of the former, i.e. its orthonormality. This makes sense since right angles are found in many material solutions and certainly in the area of construction.

Figure 2:

Model of the generating system.

To make effective use of the proposed system, the set of basic forms should be provided with basic dimensions. In order to achieve optimal flexibility and combination, the basic elements should have the same dimensions. If differences are unavoidable “Hendrickx–Vanwalleghem” proposed to solve the problem using the rules of either halving or doubling. Both are the result of an easy mathematical manipulation and create a geometrical series. Halving is easy if one uses flexible elements: they can be folded. Starting with a square with side ‘x’ one finds: x, 2x, 4x, 8x, 16x… The fractal model in Fig. 2 can be projected on all materials and all scales and thus can define the basic elements for each material type. Grouping ALL possible WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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variations within a chosen set of basic elements is named as a “construction system”. The types of basic element (first order elements) are defined by both their form and their constitutive material, as these define how basic elements should be joined and combined into basic elements of a higher level. Here the attention should be drawn to the fact that a construction system is not an object, but rather a set of entities, i.e. smallest elements of the system, between which predefined relations exist. Those are the dimensional and formal rules imposed by the generating system. The concept of “construction kit” can thus be redefined as a rational selection of SOME basic elements out of one ore more construction systems. The objective is to generate one or more flexible constructions and their constitutive parts. 2.4.2 Design tool 2: theoretical design catalogues An aid to the development of construction systems is achieved developing theoretical design catalogues. This development is carried out in the following way. In a first step each material solution, or more precisely each of its construction elements, in whichever phase, is objectively and verbally described based on characteristics, strengths and weaknesses. Each characteristic has one or more parameters as a counterpart, all bracketed between predefined limits. This delimitation, for each parameter, is done at the level of the entities. Considering that all artefacts are measurable and can be depicted, most of the parameters can be visualised with simple symbols or pictograms and be categorized in different series. If a graphic representation is not wanted or impossible, a short verbal description will be sufficient. Through interpolation and/or combination of the outer elements in the series all variants can be achieved.

Figure 3:

Theoretical design catalogue of a load bearing corrugated plate.

This can be illustrated with a simple construction element. The bearing capacity of a steel corrugated plate, subject to transverse loads – e.g. used as a roof element – can be described with three parameters: its thickness, the number WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

630 High Performance Structures and Materials III of waves per unit length and the height of the waves (or also the predefined form of the wave). These parameters can be entered into series by defining the extreme values. A thin plate is the opposite of a thick one: this means that there is a limit for the ratio thickness/span. The number of waves per unit length and the height of the waves are limited by plate thickness and fabrication process. All variants should lie between the thus defined limits. With arithmetic and geometric calculation rules changes in each series can be described; every value of the parameter thus gets its place within the series. Whether the succession of elements is continuous or discontinuous has no importance, at least on the theoretical level. But practically seen stepwise variations are preferred: they reduce the number of values and more easily achieve the goal of using a minimal number of (standardised) basic elements. The adopted geometrical “rule of the game” stems from the fractal model in the generating system, which can easily be seen in following sketch:

Figure 4:

Compatible basic elements.

If all series of individual parameters are grouped, any historical, actual or future artefact can be theoretically described by a series of lines or vectors which intersect in a common point. Variations or new solutions are found by translating one or more lines through the intersection point. A theoretical design catalogue is thus established, combining and juxtaposing elements. The emphasis has been put on “theoretical”, as in practice not all combinations are possible or technically sound. This means that they are erased in the practical catalogue. This approach can be applied to a whole construction. As the bracketing of an entire construction is difficult, the preference is given to deconstructing it into physically and non-physically observable elements, without forgetting to keep in mind the global context. For example, the maximum height of a building is often defined through ridge height or roof slope of adjacent constructions.

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The concluding result is a fan of design catalogues, each one based on combinations of selected parametric rules. They allow to describe any artefact, existing or not, through translation of one or more series. The above presented strategy matches perfectly with the idea of open industrialisation, wherein a minimum of construction elements, belonging to several construction systems and distributors, based on the same design rules, can be combined together to form multiple (adaptable) projects [5].

3

Temporary character of constructions

The awareness of the limited life time of our heritage is ever increasing; slums are demolished, old train stations are replaced by prestigious ones, offices are refurbished and monuments are carefully renovated or the object of restoration… Still professional developers and real estate owners pay little attention to the temporary character of a construction: even during the study and the drawing phase this aspect is often forgotten or even simply ignored. The point is that if you want to face changing uses during the life time of the construction, static solutions will make transformations extremely difficult if not impossible. It could happen that some structural elements still perform in a satisfactory way, but the owner will often prefer to demolish and start over. This causes a lot of debris. Consequently, the actual society is missing a dynamic design strategy, allowing transformations and adaptations during the life cycle of a construction. The “Hendrickx – Vanwalleghem” design strategy takes these characteristics of temporality and adaptability of constructions into account, from the first sketches on. It allows every construction part to transform into another configuration by adding, deleting and transforming basic elements of the same system and combining it with elements coming from other. One of the consequences is that so-called “dry connections” are used: bolts and nuts, screws, click-systems… [6, 7].

Figure 5:

Actual model of the life cycle of a construction.

Figure 6:

Proposed model of the life cycle of a construction.

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632 High Performance Structures and Materials III Damaged basic elements can be reused allowing small transformations, even in fields where the structural, i.e. load bearing properties, are less important (furniture, window frames…). Hence the “total life cycle cost” of the elements will decrease.

4

Modular versus generating

The main asset of a modular construction system is an economical one. Thanks to (modular) standardization, simplified and cheaper prefabrication processes are made possible, which consequently speeds up the construction phase. Modular construction systems are also known as flexible. However, this is not without any shortcomings!

Figure 7:

The 4 design levels [8].

Changing a module or unit is excluded, because it is has been technically and structurally denied. Adaptability – and by this the designer’s freedom – is therefore limited to the addition and reduction of fixed modules. A common employed module is ‘the foot’ (in the horizontal plane). This module is approx. 30cm and is rightfully successful as a functional, ergonomic and spatial unit. But it cannot be used at all levels of the design: for technical dimensions it is often too large, for structural purposes too small (Fig.7). A multi-modular grid provides an improvement; i.e. a superposition of modular design grids with a different module – related to the respective design level (structural, spatial, functional or technical). Design at different levels is thus possible, but not without possible conflicts. Using an arbitrary or no mathematical relation between the module sizes dimensional problems occur where different grid lines intersect (Fig. 8). [8] The standardisation rules of the generating system are based on a fractal model (Fig.2). Thanks to a single operator (divide or multiply by 2), switching to different design levels is always possible, and this without jeopardising compatibility between each basic element. A generating system thus allows the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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development of (multi-) modular systems, but with the additional property that they can be used with different design scales. Furthermore, it is not “the module” which is standardised but the (dimensional) modifiable basic elements from which it is composed. The latter is the key difference with modular construction systems.

Figure 8:

5

5M

8M

3M

4M

2M

2M

M

M

Multi-modular grid.

Figure 9:

Generated compatible design grids.

Conclusion

The Hendrickx-Vanwalleghem design strategy offers a sustainable answer to the optimisation of ‘dynamic’ characteristics of the construction world. Thanks to its explicit standardisation rules, it maximises adaptability and reuse possibilities during the construction’s entire life. Full compatibility between basic elements of the same construction kit and elements belonging to others, makes an unequivocal generating system even more efficient than modular systems.

References [1] De Wilde W.P., Conceptual design of lightweight structures: the role of morphological indicators and the structural index. Proc. of 3rd International Conf. on High Performance Structures and Materials 2006, ed. Wessex Institute of Technology, United Kingdom, 2006. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

634 High Performance Structures and Materials III [2] Hendrickx, H., Vanwalleghem, H., Sustainable Urban Development. Local Agenda 21 in Development Perspective, 48p. http://home.tiscali.be/ momentumcongressen/Paper%20Van%20Walleghem.pdf [3] Hendrickx H., Solutions derived from natural processes harmonising nature and material culture. Proc. of the 1st Conf. on Design and Nature 2002. Comparing Design in Nature with science and engineering, ed. C.A. Brebbia & L.J. Sucharov, Wessex Institute of Technology, United Kingdom and P. Pascolo, Universita degli di Udini, Italy, 10p. 2002. [4] European Committee for Standardization (CEN), Eurocode – Basis of structural design: NBN EN 1990, pp. 23-26, 2002. [5] Henrotay, C., Research report. A generally applicable building system for shelter in emergency situations. , Vrije Universiteit Brussels, Brussels, Belgium, 2005 [6] W. Debacker, Sustainable development of the material culture approached by system thinking, Master’s dissertation, Vrije Universiteit Brussels, Brussels, 2003. [7] C. Henrotay, Sustainable Development. A general description and research of its application towards the material support in emergency situations. Master’s Thesis, Vrije Universiteit Brussels, Brussels, Belgium, 2003 [8] De Troyer, F., Naert, N., Grids: help or obstacle?. Building with prefab concrete: design guide, ed. FEBELCEM / FEBE, Brussels, Belgium, 12p.

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The optimization of a truss facade B. Verbeeck1 , W. P. De Wilde1 & Ph. Samyn2 1 MeMC,

2

Vrije Universiteit Brussel, Belgium Samyn and Partners, Belgium

Abstract This paper presents the study of a facade. One of the aims of the designing team is the minimisation of the weight of the facade. Due to architectural constraints, the geometry of the facade is fixed to a highly statically indeterminate truss. Therefore, the minimisation of the weight of the facade is limited to a section optimisation. The objective function is the indicator of volume. Since the number of members is of the order of 1000, exhaustive search methods are impractical. We used a simple iterative process to find optimal sections. In the first iteration all sections are equal. This allows the forces to flow through the facade as if the facade has a constant stiffness. In the next iterations sections are adapted to the forces that arise from the previous iteration. This method quickly converges to an optimal section layout. The results of this method are corroborated by a genetic algorithm. We find that the truss facade with optimal sections consumes less material than an arch with push rods, that transfer the forces to the arch. Furthermore, the influence of buckling can be ignored. Keywords: morphological indicators, optimization, genetic algorithm, statically indeterminate, truss.

1 Introduction This paper describes a methodology to optimize sections of a (highly) indeterminate truss. It is especially applicable for fixed geometries, where only sections need to be optimized. The method is iterative and consists of updating the stiffness of the members according to forces in that member in the previous iteration. The main advantage of this process is that it consumes considerable less time than an ordinary optimization algorithm (genetic algorithm). Typically it will take the process on the order of 10 iterations before convergence occurs. This means that WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06062

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Figure 1: Geometry of the facade, which spans 48.6m. only on the order of 10 finite element analyses are performed (compared to 104 or more for genetic algorithms). The application of the method is a facade of a large building. A basic geometry is calculated to illustrate the principle. The results for an alternative model is shown and compared to an arch. The “update” method we use was first proposed in [2]. 1.1 Facade The facade is composed of 9 by 12 cruciformed modules, which measure respectively 5.4m and 3.5m. Due to the constraint that a train has to pass under the facade, supports are only possible at both extremes (Figure 1). Forces are applied on all nodes. On the top nodes the force is 306kN (roof). The force on all other nodes is 89kN . The other model (Figure 2) has a larger span, but a middle support. 1.2 Morphological indicators In order to compare the different solutions of the facade, an objective function is necessary. The aim of the designers is the minimization of weight (or volume). We therefore use the Morphological Indicators (MI). These dimensionless numbers enable the designer to compare different structural systems on the basis of weight. The most important MI, the Indicator of Volume W = FσVL allows the comparison of the volume of material for different structural systems. It is the volWIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 2: Alternative model (forces are shown).

ume of an isomorphic structure with unit span, with at least one section of each element dimensioned on its unit allowable stress, subjected to a load system with Eδ compares the displacement unit resultant. The Displacement Indicator ∆ = σL of different structural systems. It is the maximum displacement of an isomorphic structure with unit span in a material with unit Young’s Modulus, with at least one section of each element dimensioned on its unit allowable stress, subjected to a load system with unit resultant. The analytical expressions of both W and ∆ have been established in [3] and [2] for trusses, beams, arches, cables, cable stayed structures, masts and frames subjected to a limited number of (simple) load cases and support conditions. For statically determinate structures these Morphological L if instabilities, self Indicator are only function of the geometrical slenderness H weight and second order effect are neglected. Efficiency curves with respect to minimum volume of used material can be established (Figure 3).

2 Method The first iteration consists of making all sections equal, thus making all modules equally stiff. The system is then solved, using a simple finite element program. The solution can be interpreted as follows: it is a set of sections that is needed to transfer the loads to the supports, if the stiffness is homogeneously distributed over the framework (Figure 4). In reality the stiffness will not be homogeneous, WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 3: Efficiency curves.

since member sizes vary. Therefore an update of the framework stiffness will yield a more accurate flow of forces. This is done in the second and every successive iteration, until convergence of the objective function W .

3 Results The process converges rapidly (Figure 5(a)). We observe that after 1 iteration the value of W is 1.38. After 10 iterations this value becomes 1.27. The optimum is reached after 31 iterations (W = 1.26) (Figure 5(b)). The objective function decrease between the first and final (best) iteration is 10%. This decrease becomes WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Figure 4: Forces in framework after 1 iteration.

1% between iteration 10 and the final one. This means that early on, the process gives a very good estimation of the optimum. Of course this method has to be compared to other optimization techniques. Using the genetic algorithm toolbox of MatLab, a slightly better optimal solution of W = 1.20 is reached. However, it takes on the order of 104 more time to reach this optimum. This has consequences on the design process: because one finite element analysis takes about 2 seconds, the answer of the iterative process is known after 1 minute, whereas the GA yields its result after 5 hours. The advantage of having a slightly better solution does not weigh up against the enormous loss in calculation time, especially during the conceptual design stage. The results are obtained by incorporating buckling. Optimal solutions vary slightly when buckling is ignored. There are two reasons for this: member lengths 2 are small and sections are buckling efficient, with form factor ( ΩH = 8 with Ω I section, H height and I moment of inertia) low (circular tubes). The alternative geometry (Figure 6) yields a W -value of 1.07. The previous solution obtained by the “update” iteration is compared to an arch with push rods. According to [3] the value of W of this structural system is 1.13. This means that the (alternative) truss facade (will transfer the vertical loads more efficiently than an arch with push rods. The inclusion of horizontal (wind) loads will only widen the gap between the two structural systems, because the arch is inefficient at transferring this type of loads. It is not surprising that the truss solutions exhibit an arch shape of its most compressed members and a suspension cable shape of its most tensioned members. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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(a) evolution of the objective function

(b) final solution

Figure 5: The result of the iterative process.

Figure 6: Result for alternative model.

4 Conclusion and future work The method proposed in this paper gives the designer of structures a very fast optimization tool for (highly) statically indeterminate trusses. The process consists of starting with the truss that consists of members with identical sections. In every consecutively step, the sections are adapted to the forces in the members of the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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truss with the previous sections. Within the limits of the conceptual design phase, the results are accurate enough to guide the designer in her/his choices. With regard to the application, this method yields member sections that make the facade very light. The truss solution is even better than the arch with push rods, wind loads excluded. A setback of this method is that some boundary conditions (e.g. maximum displacement) are difficult to incorporate. This is due to the method itself. The process can be used for two situations: 1. fixed geometry: only the member sections are to be optimized; 2. truss members used as grid: the truss works as base structure and members that are not useful tend to have sections that converge to zero (Figure 7). This type of optimization echoes the topology optimization method developed by Bendsoe [1]. The problem will be to find the optimal “grid”.

Figure 7: Optimization using “update” method.

References [1] M.P. Bendsoe. Optimization of Structural Topology, Shape and Material. Springer, New York, 1995. [2] P. Latteur. Optimisation et pr´edimensionnement des treillis, arcs, poutres et cˆables sur base d’indicateurs morphologiques. VUB, Brussels, 2004. ´ [3] Ph. Samyn. Etude de la morphologie des structures a` l’aide des indicateurs de volume et de d´eplacement. Acad´emie Royale de Belgique, Brussels, 2004.

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MINLP optimization of the single-storey industrial building steel structure T. Žula, U. Klanšek & S. Kravanja University of Maribor, Faculty of Civil Engineering, Maribor, Slovenia

Abstract In this paper the optimization of the single-storey industrial building steel structure is presented. The structure consists of the main portal frames, which are mutually connected with the purlins. All structural elements are proposed to be built up of standard hot rolled I sections. The structural optimization is performed by the Mixed-Integer Non-linear programming approach, MINLP. The MINLP performs a discrete topology and standard dimension optimization, while continuous parameters are simultaneously calculated inside the continuous space. Since the discrete/continuous optimization problem of steel frames is non-convex and highly non-linear, the Modified Outer-Approximation/EqualityRelaxation (OA/ER) algorithm has been used for the optimization. Alongside the optimal structure mass, the optimal topology (an optimal number of main portal frames and purlins) as well as all standard cross-section sizes have been obtained. The paper includes the theoretical basis and a practical example with the results of the optimization.

1

Introduction

Single-storey frame structures are extensively used for industrial, leisure and commercial buildings. In order to obtain efficient frame designs, researchers have introduced various optimization techniques, appropriate either for the continuous or the discrete optimization. O’Brien and Dixon [1] have proposed a linear programming approach for the optimal design of pitched roof frames. Guerlement et al. [2] have introduced a practical method for single-storey steel structures, based on a discrete minimum weight design and Eurocode 3 [3] design constraints. Recently, Saka [4] has considered an optimum design of pitched roof steel frames with haunched rafters by using a genetic algorithm. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06063

644 High Performance Structures and Materials III One of the latest researches reported in this field is the work of Hernández et al. [5], where authors have considered minimum weight design of steel portal frames with software developed for structural optimization. This paper deals with topology and standard dimension optimization of unbraced single-storey industrial building steel structures. The optimization of portal frames is performed by the Mixed-Integer Nonlinear Programming, MINLP. The MINLP is a combined discrete and continuous optimization technique. In this way, the MINLP performs the discrete topology (i.e. the number of frames and purlins) and standard dimension (i.e. the standard crosssection sizes of columns, beams and purlins) optimization simultaneously with the continuous optimization of parameters (e.g. a structure mass, internal forces, deflections, etc.). The MINLP discrete/continuous optimization problems of frames are in most cases comprehensive, non-convex and highly non-linear. The optimization is proposed to be performed through three steps. The first one includes the generation of a mechanical superstructure of different topology and standard dimension alternatives, the second one involves the development of an MINLP model formulation and the last one consists of a solution for the defined MINLP optimization problem. The objective of the optimization is to minimize the mass of the single-storey industrial building. The mass objective function is subjected to the set of the equality and inequality constraints known from the structural analysis and dimensioning. The dimensioning of steel members is performed in accordance with Eurocode 3. The Modified Outer-Approximation/Equality-Relaxation algorithm is used to perform the optimization, see Kravanja and Grossmann [6], Kravanja et al. [7-8]. The two-phase MINLP optimization is proposed. It starts with the topology optimization, while standard dimensions are relaxed temporary into continuous parameters. When the optimal topology is found, standard dimensions of crosssections are re-established and standard dimension optimization of beams, columns and purlins is then continued until the optimal solution is found.

2

Single-storey industrial building

The paper presents the topology and standard dimension optimization of unbraced rigid single-storey industrial building steel structures, see Figure 1. Columns, beams and purlins are proposed to be built up of standard hot rolled steel I sections. The considered portal frame structures are analyzed under the combined effects of the self-weight of frame members, uniformly distributed surface variable load (snow and wind), concentrated horizontal variable load (wind) and an initial frame imperfection. The purlins are designed to transfer permanent load (self-weight of purlins and weight of roof) and variable load (snow and wind) to frame structure. Internal forces are calculated by the elastic first-order analysis. The dimensioning of steel members is performed in accordance with Eurocode 3

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for the conditions of both ultimate limit state (ULS) and serviceability limit state (SLS).

H

q

Lf

P

Lf

P

Lf

P

Lf Lf Lf

L

Figure 1:

LL

Single-storey industrial building.

When the ultimate limit state of structural members is considered, the elements are checked for axial resistance, shear resistance, bending moment resistance, interaction between bending moment and axial force, interaction between axial compression/buckling and buckling resistance moment. Considering the serviceability limit state, the vertical deflections of beams and purlins are calculated by the force method. The total deflection δmax subjected to the overall load and the deflections δ2 subjected to the variable imposed load are calculated to be smaller than limited maximum values: span/200 and span/250, respectively. The horizontal deflections ∆ are also checked for the recommended limits: the relative horizontal deflection should be smaller then the height/150 of the portal frame.

3

MINLP model formulation for mechanical superstructures

It is assumed that a general nonconvex and nonlinear discrete/continuous optimization problem can be formulated as an MINLP problem (MINLP-G) in the form: min z = c T y + f ( x ) h( x ) = 0 g( x ) ≤ 0 By + Cx ≤ b

s.t.

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(MINLP-G)

646 High Performance Structures and Materials III n

x ∈ X = {x ∈ R : xLO ≤ x ≤ xUP} m y ∈ Y ={0,1} where x is a vector of continuous variables specified in the compact set X and y is a vector of discrete, mostly binary 0-1 variables. Functions f(x), h(x) and g(x) are nonlinear functions involved in the objective function z, equality and inequality constraints, respectively. Finally, By+Cx ≤ b represents a subset of mixed linear equality/inequality constraints. The above general MINLP model formulation has been adapted for the synthesis of mechanical superstructures (MINLP-SMS). The resulted formulation that is more specific, particularly in variables and constraints. It can be used also for the modelling the steel frames. It is given in the following form: min z = c T y + f ( x ) s.t.

h( x ) = 0

g(x ) ≤ 0

A( x ) ≤ a Ey ≤ e

Dy + R( x ) ≤ r

(MINLP-SMS)

e

( ) Py + S (d ) ≤ s

Ky e + L d cn ≤ k st

n

x ∈ X = {x ∈ R : xLO ≤ x ≤ xUP} m y ∈ Y ={0,1} The MINLP model formulation for mechanical superstructures is proposed to be described as follows: • Included are continuous variables x={d, p} and discrete binary variables y={ye, yst}. Continuous variables are partitioned into design variables d={dcn, dst} and into performance (nondesign) variables p, where subvectors dcn and dst stand for continuous and standard dimensions, respectively. Subvectors of binary variables ye and yst denote the potential existence of structural elements inside the superstructure (the topology determination) and the potential selection of standard dimension alternatives, respectively. • The mass or economical objective function z involves fixed mass or cost charges in the linear term cT y and dimension dependant mass or costs in the term f(x). • Parameter nonlinear and linear constraints h(x)=0, g(x) ≤ 0 and A(x) ≤ a represent the rigorous system of the design, loading, stress, deflection, stability, etc. constraints known from the structural analysis. • Integer linear constraints Ey ≤ e are proposed to describe relations between binary variables. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

•

•

•

Mixed linear constraints Dye+R(x) ≤ r restore interconnection relations between currently selected or existing structural elements (corresponding ye = 1) and cancel relations for currently disappearing or nonexisting elements (corresponding ye = 0). Mixed linear constraints Kye+L(dcn) ≤ k are proposed to define the continuous design variables for each existing structural element. The space is defined only when the corresponding structure element exists (ye = 1), otherwise it is empty. Mixed linear constraints Py+S(dst) ≤ s define standard design variables dst. Each standard dimension dst is determined as a scalar product between its vector of standard dimension constants q and its vector of binary variables yst. Only one discrete value can be selected for each standard dimension since: st ∑ yi = 1

d st = ∑ qi yist

i∈I

i∈I

4

647

(1)

The optimization model

The optimization model for a single-storey industrial building steel structure contains the mass objective function, (in)equality constraints and the variables (dimensions, internal forces, deflections, mass, etc.). Equality and inequality constraints represent a rigorous system of the design, loading, resistance, stress, deflections and stability functions. The dimensioning constraints are determined according to Eurocodes 3 (ultimate and serviceability limit states). Resistance to bending moment of beams, columns and purlins: M Sd ≤ M el ,Rd

M el , Rd =

Wel ⋅ f y

γM0

(2) (3)

where MSd is the design bending moment, Mel,Rd is the elastic resistance moment, fy is the yield strength of structural steel, Wel is the elastic section modulus and γM0 is the partial safety coefficient. Resistance to axial force of the beams and columns: N Sd ≤ N pl ,Rd

N pl , Rd =

A ⋅ fy

γM0

(4) (5)

where NSd is the design axial force, Npl,Rd is the compression resistance, A is the cross-section area, γM0 is the partial safety coefficient. Compression/buckling resistance of columns: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

648 High Performance Structures and Materials III (6)

N Sd ≤ N b, Rd N b, Rd = χ

A ⋅ fy

(7)

γ M1

where Nb,Rd is the compression/buckling resistances, χ is the reduction factors for the relevant buckling mode and γM1 is the partial safety coefficient. Shear resistance of beams, columns and purlins: (8)

VSd ≤ V pl , Rd V pl ,Rd = Av ⋅

fy

1 ⋅ 3 γM0

(9)

where VSd is the design shear force, Vpl,Rd is the design shear resistance and Av is shear area. Interaction between axial force and bending moment: N sd M sd + ≤ 1 .0 N pl , Rd M el , Rd

(10)

Interaction between axial compression/buckling and bending moment lateraltorsional buckling: N sd

χ ⋅ A ⋅ f y / γ M1

+

k LT ⋅ M sd

χ LT ⋅ Wel ⋅ f y / γ M 1

≤ 1.0

(11)

where kLT is the coefficient at lateral-torsional buckling, and χLT is the reduction factors for the relevant buckling mode at lateral-torsional buckling. Inequality constraints of serviceability limit states for the vertical deflection of beams and purlins: δ max,L ≤

L 200

δ max,Lf ≤

δ 2, L ≤

L 250

δ 2,Lf ≤

Lf 200 Lf 250

(12)

(13)

where δmax.L is the vertical deflection of beams and δmax.Lf is the vertical deflection of purlins for the terminated state, δ2,L is the vertical deflection of beams and δ2,Lf is the vertical deflection of purlins for the variable load.

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Inequality constraints of serviceability limit states for the horizontal deflections of portal frame: ∆≤

5

H 150

(14)

The optimization

The Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm (Kravanja and Grossmann [6]) was used to perform the optimization. The OA/ER algorithm consists of solving an alternative sequence of Non-linear Programming optimization subproblems (NLP) and Mixed-Integer Linear Programming master problems (MILP). The former corresponds to the optimization of parameters for a frame structure with fixed topology and standard dimensions and yields an upper bound to the objective to be minimized. The latter involves a global approximation to the superstructure of alternatives in which a new topology and standard sizes are identified so that its lower bound does not exceed the current best upper bound. The search is terminated when the predicted lower bound exceeds the upper bound. The optimal solution of complex non-convex and non-linear MINLP problem with a high number of discrete decisions is in general very difficult to be obtained. The optimization is thus performed sequentially in two different phases to accelerate the convergence of the OA/ER algorithm. The optimization is proposed to start with the topology optimization of the frame, while standard dimensions are relaxed temporary into continuous parameters. When the optimal topology is found, standard sizes of cross-sections are re-established and the standard dimension optimization of beams, columns and purlins is then continued until the optimal solution is found.

6

The example

The paper presents an example of the topology and standard dimension optimization of a single-storey industrial building. The building is 30 meters wide, 50 meters long and 8 meters high, see Figure 2. The structure is consisted from equal non-sway steel portal frames which are mutually connected with the purlins. The portal frame is subjected to self-weight g, uniformly distributed surface variable load q (snow s and wind w), concentrated variable load P (wind F and initial frame imperfection Fφ). Variable imposed load (s = 1.60 kN/m2 and w = 0.137 kN/m2) is defined as the uniformly distributed surface load in the model input data. Both, the horizontal concentrated load at the top of the columns and the vertical uniformly distributed line load on the beams are calculated considering the intermediate distance between the portal frames. Design/dimensioning was performed in accordance with Eurocode 3. The design loads were calculated for the conditions of both ultimate limit states and serviceability limit states: (a) for ultimate limit states: 1.35·(g + s + w + F + Fφ), WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

650 High Performance Structures and Materials III

8.0 m

(b): for serviceability limit states: 1.00·g + 0.90·(s + w + F + Fφ). While internal forces were calculated by the elastic first-order analysis the deformation of frame members were calculated by the force method. The portal frame superstructure was generated in which all possible structures were embended by 30 portal alternatives, 20 purlin alternatives and different standard size variation. The superstructure also comprised 24 different standard hot rolled European wide flange I beams, i.e. HEA sections (from HEA 100 to HEA 1000) for each column, beam and purlin separately. The material used was steel S 355. The optimization was performed by the MINLP optimization approach. The task of the optimization was to find the optimal structure mass, the optimal topology (the optimal number of portal frames and purlins) and optimal standard sizes. The optimization was carried out by a user-friendly version of the MINLP computer package MIPSYN, the successor of PROSYN [6] and TOP [9]. As an interface for mathematical modelling and data inputs/outputs GAMS (General Algebraic Modelling System), a high level language, was used [10]. The Modified OA/ER algorithm and the two-phased optimization were applied, where GAMS/CONOPT2 (Generalized reduced-gradient method) [11] was used to solve NLP subproblems and GAMS/Cplex 7.0 (Branch and Bound) [12] was used to solve MILP master problems. The optimization model contained 130 (in)equality constraints, 183 continuous and 122 binary variables. The final optimal solution of 150,87 tons was obtained in the 12th main MINLP iteration. The optimal result represents the mentioned optimal structure mass of 150,87 tons, the obtained optimal topology of 12 portal frames an 20 purlins (see Figure 2) and the calculated optimal standard sizes of columns, beams and purlins (see Figure 3).

m .54 u4 m 1 0 . 1 50

30.0 m

Figure 2:

Optimal design of the single-storey industrial building.

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651

0.50 m

High Performance Structures and Materials III

HEA 140

8.0 m

HEA 900

HEA 900

7.5 m

HEA 800

HEA 800

18 × 1.67 m 30.0 m

Figure 3:

7

Optimal design of the portal frame.

Conclusions

The paper presents the topology and standard dimension optimization of the single-storey industrial building steel structure. The optimization is proposed to be performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP was found to be a successful optimization technique for solving this type of structures.

References [1] [2] [3] [4] [5] [6] [7]

E.J. O'Brien, A.S. Dixon, Optimal plastic design of pitched roof frames for multiple loading, Comput. Struct. 64, 737-740, 1997. G. Gurlement, R. Targowski, W. Gutkowski, J. Zawidzka and J. Zawidzki, Discrete minimum weight design of steel structures using EC3 code, Struct. Multidisc. Optim. 22, 322-327, 2001. Eurocode 3, Design of steel structures, European Committee for Standardization, 1992. M.P. Saka, Optimum design of pitched roof steel frames with haunched rafters by genetic algorithm, Comput. Struct. 81, 1967-1978, 2003. S. Hernández, A.N. Fontán, J.C. Perezzán, P. Loscos, Design optimization of steel portal frames, Adv. Eng. Software. 36, 626-633, 2005. Kravanja, Z. and Grossmann, I.E., New Developments and Capabilities in PROSYN - An Automated Topology and Parameter Process Synthesizer, Computers chem. Eng., 18, 1097-1114, 1994. Kravanja, S., Kravanja, Z. and Bedenik, B.S., The MINLP optimization approach to structural synthesis. Part I: A general view on simultaneous topology and parameter optimization, Int. J. Numer. Methods Eng. 43, 263-292, 1998.

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652 High Performance Structures and Materials III [8]

[9]

[10] [11] [12]

Kravanja, S., Kravanja, Z. and Bedenik, B.S., The MINLP optimization approach to structural synthesis. Part II: Simultaneous topology, parameter and standard dimension optimization by the use of the Linked two-phase MINLP strategy, Int. J. Numer. Methods Eng. 43, 293-328, 1998. Kravanja, S., Kravanja, Z., Bedenik, B.S. and Faith, S., Simultaneous Topology and Parameter Optimization of Mechanical Structures, Numerical Methods in Engineering '92, Proceedings of the First European Conference on Numerical Methods in Engineering, ed. Ch. Hirsch et al., pp. 487-495, Elsevier, Amsterdam, 1992. Brooke, A., Kendrick, D. and Meeraus, A., GAMS - A User's Guide, Scientific Press, Redwood City, CA, 1988. Drudd, A.S., CONOPT – A Large-Scale GRG Code, ORSA J. Comput. 6, 207-216, 1994. CPLEX User Notes, ILOG inc.

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Genetically optimised placement of piezoelectric sensor arrays: linear and nonlinear transient analysis J. N. Rao, S. Lentzen & R. Schmidt Institute of General Mechanics, RWTH Aachen University, Germany

Abstract In modern structural control the application of discrete modal sensor arrays is a commonly used technique to obtain the modal state vectors. In this paper, a genetic algorithm is used to find the optimal positions to place modal sensor arrays on simple structures such as beams, plates and shells. The performance criterion is taken as the steady state observability Grammian of the system and includes spillover prevention as well. The performance of optimally placed modal sensors in the linear range is discussed. The variation in the performance of these modal sensors in the nonlinear range is investigated. Keywords: geometrical nonlinearity, modal sensors, Lyapunov equation, genetic algorithm.

1 Introduction There has been consistent research on smart materials and structures for the last two decades. Geometrically linear theories and numerical methods have been developed by many authors, e.g. Crawley and de Luis [1], Tzou and Tseng [2]. Considerably less work can be found in the area of geometrically nonlinear modelling of smart structures. Structural nonlinearity has been taken into account for interlaminar stress analysis by Icardi and Di Sciuva [3], for large deflection shape control in Yi et al. [4], Mukherjee and Chaudari [5] and Lentzen and Schmidt [6]. Active buckling control and post-buckling analysis has been done by Krishna and Mei [7] and Chandrashekhara and Bhatia [8]. Piezothermoelastic analysis including nonlinearity is discussed in Tzou et al. [9] and Pai et al. [10]. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06064

654 High Performance Structures and Materials III In order to implement structural vibration control in a modern fashion, one should be able to sense the modal amplitudes. Several possibilities of modal sensing can be found in the literature. Clark and Burke [11], Lee and Moon [12], Gawronski [13] are some of the earlier researchers, who worked on modal sensors. The optimal locations of sensors in intelligent structures is an important issue in the research of present day, since an arbitrary decision can degrade their performance. In order to find the optimal positions of these modal sensors, genetic algorithms (GAs) can be used as a searching method. Some of the earlier successful attempts can be found in the works of Sadri et al. [14], Han and Lee [15], where GA is used to find optimal places for both sensors as well as actuators. In the present paper discrete array sensors are used to filter the modal amplitudes. To find the optimal locations of modal sensors the observability of the system should be investigated to obtain the best performance with the least number of sensors. The performance criterion for the selection of the sensor positions is taken as the steady state observability Grammian of the system. The steady state observability Grammian is obtained by solving the Lyapunov equation. The performance of genetically optimised sensor arrays in the linear range is discussed, and the influence of structural nonlinearity is investigated.

2 Structural model The FE analysis of the structural response is performed using the theory of composite laminated shells given by Schmidt and Reddy [16]. The geometrically nonlinear strain displacement relations are based on the Reissner-Mindlin hypothesis and are valid for small strains and moderate rotations of the midsurface normals [16]
 2 1 0 εαβ = εαβ +Θ3 εαβ + Θ3 2 εαβ 1

0

εα3 = εα3 +Θ3 εα3 ε33 = 0 with

0

0

εαβ =θαβ +

1

εαβ =

1 2

1 0 0 ϕα ϕβ , 2

  0 0 0 0 1 1 1 1 v α|β + v β|α −bλα ϕλβ −bλβ ϕλα + ϕα bλβ v λ + ϕβ bλα vλ ,

 1 λ κ 1 1 1 1 2 εαβ = bα bβ v λ v κ −bλα vλ|β −bλβ v λ|α , 2   1 0 1 1 1 0 1 1 0 1 ϕα + v α + vλ ϕλα εα3 = and εα3 = v λ v λ|α . 2 2 The following abbreviations were used:  0 1 0 0 0 v α|β + v β|α − bαβ v 3 , θ αβ = 2 WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(1)

High Performance Structures and Materials III 0

0

0

ϕαβ =v α|β −bαβ v 3

0

0

655

0

ϕα =v 3,α +bλα v λ .

and

The internal and external virtual work are evaluated in a total Lagrangian fashion. The second Piola-Kirchhoff stress and Green-Lagrange strain are chosen to express the mechanical part of the internal virtual work. Consequently, the electric variables are defined as referring to the initial undeformed configuration [17].

3 Performance criterion The performance criterion to optimise the modal sensor positions requires modal voltages. The linear equations of motion of a laminated composite structure with embedded piezoelectric layers can be written as 

   q Fe M 0 q¨ Kqq Kqφ (2) = + φ Kφq Kφφ Qe 0 0 0 where the above matrices are found in Lentzen et al. [17]. There, M is the mass matrix, Kqq is the linear elastic stiffness matrix, Kqφ = Kφq are the linear electromechanical coupling matrices and Kφφ is the piezoelectric capacitance matrix. The electromechanical response of the structure is denoted by {q, φ}, where q and φ are the system’s generalised displacements and sensor voltages, respectively. The externally applied forces (Fe ) and charges (Qe ) are denoted by the right subscript e. After static condensation of the electric variables, equation (2) can be written in modal form as R 
BjL VL (3) p¨j + ωj2 pj = L=1

where p¨j , p˙ j and p represent the j modal acceleration, velocity and displacement, respectively. The natural frequency of the j th mode is denoted by ωj . The piezo sensor voltages can be represented as th

ΦL =

∞ 




CjL pj .

(4)

j=1

Here the induced voltage of the Lth sensor and the control voltage at the Lth actuator are denoted by ΦL and VL , respectively. The sensing constant of the Lth sensor
due to unit modal displacement of the j th mode is denoted by CjL and the j th modal
actuation constant due to unit applied voltage of the Lth actuator is denoted by BjL . In order to address the observability problem one should convert the standard FE equations into state space equations as follows [15]: x˙ = [A]{x} + [B]{u}

(5)

y = [C]{x}

(6)

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656 High Performance Structures and Materials III where x=



p˙1

ω1 p1

. . . p˙n  Ai =

A = diag(Ai ),

 






B11

  0   . B =  .. 
  Bn1 0  0  −1
 ω1 C11   .. and C =  .   0  −1
ωn Cn1

V1  .   u =  ..  , VR





Φ1  .   y =  ..   Φs

ωn pn

T 

−ωi 0

0 ωi

(7)

···

B1R






··· .. . ···

0 .. .
BnR

       

···

0

··· ··· .. .

0 −1
ω1 C1S .. .

··· ···


ωn−1 CnS

0

T      .   

A linear time invariant system (A, B, C), with s outputs is completely observable, if any of the following conditions are satisfied [19]. 1. The (sn × n) observability matrix [O] has rank n, where

T [O] = [C] [C][A] [C][A]2 · · · [C][A]n−1 (8) 2. The observability Grammian [Wo ] is full-rank  ∞ T e[A]t [C][C]T e[A] t dt [Wo ] =

(9)

0

In fact [Wo ] is the solution of the Lyapunov equation: [A][Wo ] + [Wo ][A]T + [C][C]T = 0

(10)

The observability test based on the rank is binary in nature (i.e. it tells us whether the system is observable or not). Additionally the extent of observability is required. Therefore, in the present work the objective function proposed by Hac and Liu [18] is used, given by:       2(no +nr )  2no 2(no +nr ) 2no      2n r 2n o     J= λj (λj ) − γ λj (λj ) , (11) j=1

j=1

j=2no +1

j=2no +1

where no and nr are the number of observed modes and residual modes, respectively. For sensor optimal places, λj is the eigenvalue of the steady state observability Grammian (i.e. [Wo ]). Note that in the objective function (i.e. Eq. 11) a WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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product term is included to account for least controlled modes. If any of the modes is least controlled then the total objective function value goes to zero.

4 Genetic algorithm Genetic algorithms (GAs) are random search techniques based on the mechanics of natural selection and genetics. Genetic algorithms are used to explore the global extremum of the given linear or nonlinear function. Although randomised, genetic algorithms can efficiently explore the new generation with better fitness. The GA is used to maximise J (i.e. Eq. (11)) for a given number of sensors. 4.1 Algorithm 1. Create a random initial population of sensors. 2. Evaluate each member of the current population by computing its fitness value (i.e. J as given in Eq. (11) ), and select parents based on their fitness value. 3. Children are produced by mating a randomly selected pair of parents at a randomly selected site, known as crossover and by making random changes to a single parent, known as mutation. 4. Replace the current population with the children from the new generation. 5. Repeat the algorithm for a prescribed number of generations. 4.2 Modal sensors The principle of discrete modal sensor arrays is depicted in Figure 1. By choosing the gains αi in a particular way, the modal sensor will respond only to the mode j. The gains are obtained by solving the orthogonal system of equations [G]{αj } = {ej },

(12)

where [G] = gki , which is the modal voltage of the k th sensor due to the unit modal displacement of mode i, {αj } are the linear gains of all sensors to sense mode j and {ej } is the unit vector in the direction j. Modal sensor signal

α 1 ϕ 1

ϕ2

αn

α2

Linear combiner Sensors

ϕn

Structure

Figure 1: Principle of the discrete modal sensor array. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

658 High Performance Structures and Materials III

5 Numerical results 5.1 Clamped plate A plate, with the dimensions [110×110×1] mm as shown in Figure 2, is taken as a numerical example. The plate consists of aluminium and is meshed with a [11x11] grid. The material properties of the aluminium and PVDF layers are displayed in Table 1. The position of the four patches has to be optimised in order to observe the first four modes. The GA parameters used in the analysis are population size (30), crossover probability (50%), mutation probability (10%) and number of generations (40). After application of GA, the optimal sensor positions for the plate are obtained as depicted in Figure 2. The first four natural frequencies are calculated as 728 Hz, 1486 Hz, 1486 Hz and 2193 Hz. Table 1: Aluminium and piezo material properties. Aluminium

PVDF

E [Gpa] ν [−]

70 0.3

2 0.3

ρ [kg/m3 ] d31 [m/V ]

2700 −

2800 2.2 · 10−10

d33 [m/V ]

−

1.062 · 10−11

111

111

121

85

37

41

82

86

38

42

1

11 110 mm

Figure 2: Optimised configuration.

110 mm

110 mm

81

121

1

11 110 mm

Figure 3: Arbitrary configuration.

In order to examine the effectiveness of the modal sensors, the transient modal sensor signals have to be investigated. One of the popular explicit time integration techniques (i.e. central difference method) is used to integrate the equations (2) WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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in the time domain. As an initial displacement field, the superposition of the first four modal displacements is prescribed, with the respective modal amplitudes of 0.01, 0.007, 0.007 and 0.003 resulting in a mid-point deflection of 0.3616 mm. Figure 3 shows an arbitrary configuration of sensors that is considered in the present work for comparison purpose. Figures 4-7 compare the results obtained with the optimised and the arbitrary configuration. It can be observed in Figures 4-7 that the modal sensor signals for the optimal sensor configuration are correct and are barely aliased with those of higher modes. In case of the arbitrary sensor configuration the modal sensor signals are strongly aliased with those of higher modes. Figures 8-9 show the modal signals for both optimal sensor configuration and arbitrary sensor configuration in the nonlinear case. It can be concluded that the induced membrane stresses which are not considered in the linear and modal analysis are the main cause for the failure of modal sensor arrays in the nonlinear range of deformations.

2

2 x 10

modal sensor signal, mode2

modal sensor signal, mode1

x 10

−

optimal configuration arbitrary configuration −

−

optimal configuration −

arbitrary configuration −

−

Figure 4: First mode signal.

Figure 5: Second mode signal.

2 x 10

−

modal sensor signal, mode4

modal sensor signal, mode3

2 x 10

−

optimal configuration −

−

time[sec]

time[sec]

Figure 6: Third mode signal.

−

optimal configuration arbitrary configuration −

arbitrary configuration

time[sec]

−

−

time[sec]

Figure 7: Fourth mode signal.

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

−

660 High Performance Structures and Materials III

modal sensor signal, mode2

modal sensor signal, mode1

−

−

−

optimal configuration arbitrary configuration

−

−

−

optimal configuration arbitrary configuration −

time[sec]

−

Figure 8: Nonlinear first mode signal.

time[sec]

Figure 9: Nonlinear signal.

second

−

mode

6 Conclusions In the present work, GA is used to find optimal placement for modal sensors. Transient analysis is performed with linear and nonlinear FE, based on first-order shear deformation moderate rotation theory. By numerical example it is shown that the principle of modal sensor arrays yields good results for the genetically optimally placed sensor patches in the range of small displacements. In the geometrically nonlinear case, it is found that the induced membrane stresses are the prime cause for the failure of modal sensor arrays.

References [1] E. F. Crawley and J. de Luis, Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal, 25, 1373-1385, 1987. [2] H. S. Tzou and C. I. Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. J. Sound Vib., 138, 17-34, 1990. [3] U. Icardi and M. Di Sciuva, Large-deflection and stress analysis of multilayered plates with induced-strain actuators. Smart Mater. Struct., 5, 140-164, 1996. [4] S. Yi, S. F. Ling and M. Ying, Large deformation finite element analyses of composite structures integrated with piezoelectric sensors and actuators. Finite Elements in Analysis and Design, 35, 1-15, 2000. [5] A. Mukherjee and A. S. Chaudari, Piezolaminated beams with large deformations. Int. J. of Solids and Structures , 39, 4567-4582, 2002. [6] S. Lentzen and R. Schmidt, Nonlinear finite element modelling of composite structures with integrated piezoelectric layers. High Performance Structures and Materials II, WIT Press, Southampton-Boston, 67-76, 2004. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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[7] M. R. M. Krishna, and C. Mei, Finite element buckling and post-buckling analyses of a plate with piezoelectric actuator. In Proc. of the Conference on Recent Advances in Adaptive and Sensory Materials and Their Applications, Blacksburg, USA. 301–313, 1992. [8] K. Chandrashekhara and K. Bhatia, Active buckling control of smart composite plates – finite element analysis. Smart Mater. Struct., 2, 31-39, 1993. [9] H. S. Tzou, Y. Bao and R. Ye, In Smart Structures and Materials 1994: Smart Structures and Intelligent Systems, Proc. of SPIE, Vol. 2190. 206-214, 1994. [10] P. F. Pai, A. H. Nayfeh, K. Oh and D. T. Mook, A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors. Int. J. of Solids and Struct., 30, 1603-1630, 1993. [11] R. L. Clark and S. E. Burke, Practical limitation in achieving shaped modal sensors with induced strain materials. J. Vib. Acoust., 118, 668-675, 1996. [12] C. K. Lee and F. C. Moon, Modal sensors/actuators. J. Appl. Mech, 57, 434441, 1996. [13] W. Gawronski, Modal actuators and sensors. J. Sound Vib., 229, 1013-1022, 2000. [14] A. M. Sadri, J. R. Wright and R. J. Wynne, Modelling and placement of piezoelectric actuators in isotropic plates using genetic algorithms. Smart Mater. Struct., 8, 490-498, 1999. [15] J. H. Han and I. Lee, Optimal placement of piezoelectric sensors and actuators for vibration control of a composite plate using genetic algorithms. Smart Mater. Struct., 8, 257-267, 1999. [16] R. Schmidt and J. N. Reddy, A refined small strain and moderate rotation theory of elastic anisotropic shells. J. Appl Mech, 55, 611-617, 1988. [17] S. Lentzen and R. Schmidt, Nonlinear FE-simulation of piezolaminated plates and shells, Proc. International Congress on Computational Mechanics & Simulation, ICCMS-04, IIT Kanpur (India), vol. I, 77-85, 2004. [18] A. Hac and L. Liu, Sensor actuator location in motion control of flexible structures. J. Sound Vib., 167, 239-261, 1993. [19] W. Gawronski, 1998, Dynamics and Control of Structures: A Modal Approach, Springer. [20] I. Bruant, G. Coffignal and F. Lene, A methodology for determination of piezoelectric actuator and sensor location on beam structures. J. Sound Vib., 243, 861-882, 2001. [21] S. L. Padula and R. K. Kinkaid, Optimisation strategies for sensor and actuator placement. National Aeronautics and Space Administration (NASA), Langley Research Center, Langley, Virginia, 1999. [22] D. Halim and S. O. Reza Moheimani, An optimisation approach to optimal placement of collocated piezoelectric actuators and sensors on a thin plate. Mechatronics, 13, 27-47, 2001.

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Numerical analysis of the process of trapezoidal thread rolling L. Kukielka & K. Kukielka Department of Mechanical Engineering, The Technical University of Koszalin, Poland

Abstract In this paper the physical and mathematical models of deformations (displacements and strains) and stress in the cold process of trapezoidal thread rolling is presented. The process is considered as a geometrical and physical non-linear, initial as well as boundary value problem. The phenomena on a typical incremental step were described using a step-by-step incremental procedure, with an updated Lagrangian formulation. The state of strains was described by Green-Lagrange’s tensor, while the state of stress by the second symmetrical Pioli-Kirchhoff’s tensor. The object was treated as an elastic (in the reversible zone) and visco-plastic body (in non-reversible zone) with mixed hardening. The variational equation of motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. The application developed for the method of finite elements ANSYS 8.1 provides a complex time analysis of displacement, strains and stresses occurring in the object. The boundary conditions for a displacement increment determined in model investigations were used. Examples of calculations of influence of a friction coefficient on the state’s deformation and stress were presented.

1

Introduction

One of the most widespread machine elements is the thread. Above 60% parts of modern machines, devices and mechanism has threaded holes, whose performance by screw-tap in high-plastic steels, some non-ferrous metals and their alloy, pose difficult problems in technological aspects. The difficulty at threading hard materials characterized by a large ductility and a high elastic WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06065

664 High Performance Structures and Materials III limit, come especially from the tendency to seizure the screw - tap in a threading hole. However, in spite of peck constructional improvements and many recommendations in the field of selection conditions for treatment quality executed, threads do not comply with standard technical requirements, the required proprieties do not apply to the surface layer, and apart from this, this tool wears away too fast, while the capacity of the process is low [1]. At the Technical University of Koszalin in Poland, in the Chair of Working Machines a scientific research is being conducted into methods of realization of male threads through plastic shaping (e.g. embossing screw threads) with a high velocity. Thread rolling is a non-cutting cold-forming process, and this gives the following further advantages: the actual time for thread rolling lasts only seconds and, therefore, it is unimportant in respect to the cycle time; the time required for inspection is very short and the maintenance costs are extremely low; the rolling attachment operation is of a very high accuracy; changing to another thread size is possible by using another set of thread rolls and a gauge, provided that the thread diameter is in the range of the respective rolling attachment size; the short thread run-out (about ½ pitch) of the thread rolls allows to produce a thread reaching up to the collar of the component; high resistance to corrosion due to smooth press polished surfaces; significant in the surface strength; an uninterrupted grain flow (Fig. 1). The advantageous exploitation proprieties of screw threads are the result of a plastic strain in surface layer of elements.

Figure 1:

The surface layer after trapezoidal thread rolling.

Company FETTE successfully applies the technology of the rolled screw thread for embossing on various pieces (Fig. 2). Currently, we are examining three varieties of threading: tangential, radial and axial. A stand for threads rolling of the axial method, made by FETTE, is in our Laboratory (Fig. 3).

Figure 2:

Example of work pieces, in which tread rolling was applied.

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High Performance Structures and Materials III

Figure 3:

665

Machine for axial thread rolling, produced by the FETTE.

The trapezoidal thread rolling is a geometrical and physical non-linear process, with initial and boundary conditions as a function of time and space. The boundary conditions in the contact zone are not known. The paper has focused mostly on the preparation of a physical model of the process and then on a mathematical model and the solution algorithms obtained for the discrete systems of equations along with the respective initial and boundary conditions. The basic problem in this technology is the knowledge of physical phenomena, especially influence of friction coefficient on state’s deformation and stress in the surface layer.

2

Mathematical model of process

A mathematical model of the process is formulated in increments and contains the following: a material model, an equation of motion, with initial and boundary conditions. 2.1 Material model 2.1.1 Incremental model of yield stress Yield stress σy is the most important parameter characterizing the resistance of a visco-plastic deformation. The incremental model of the yield stress for a typical step time t→τ=t+∆t is defined as [2]: τ t t τ t ∆σ y = t F2 ( t y) t ∆y +

∂ tt F1[•] τ ( VP ) ∂ tt F1[•] t t ( VP ) τ ( VP )  eq + t t ∆ε t F3 ( t ε eq ) t ∆ε eq , ∂ t σ st ∂ tt ε (eqVP )

= 2 τt ∆e ij(VP) τt ∆e ij(VP) /3 , where τt ∆ε (VP) eq

τ  (VP) t ∆ε eq

=

2 τt ∆ e ij(VP) τt ∆ e ij(VP) /3

(1)

are

the incremental of effective visco-plastic strain and strain rate; σst is the state stress depending on the accumulated effective visco-plastic strain and time, t t τ t F2 ( t y) t ∆y is the component of change in the initial yield stress Re with a WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

666 High Performance Structures and Materials III change of chemical composition; [ ∂ tt F1 [ • ]/ ∂ tt σ st ] tt F3 ( tt ε (VP) ) τt ∆ε (VP) is the eq eq component of change in the temporary yield stress tt σ y with change of the viscoplastic strain, [ ∂ tt F1 [ • ]/ ∂ tt ε (VP) ] τt ∆ ε (VP) is the component of change in the eq eq temporary yield stress with change of the visco-plastic strain rate. 2.1.2 Elastic/visco-plastic material model A new model of mixed hardening for isotropic material which includes the combined effects of elasticity (a reversible domain), visco-plasticity (a non reversible domain) (E/VP) is used. The model takes into account the history of the material. The constitutive equation of increment components of a total strain tensor takes form [3]: τ t ∆ε ij

=

1 [ t D (E) τ ∆σ kl − tt A] t ~ ** t ijkl t 1− t S

(2)

and of increment components of the total stress tensor: τ t (E) τ t ∆σ ij = t C ijkl t ∆ε kl

~ ~ (E) τ t −ψ tt Sij*[ tt Sij tt Cijkl t ∆ε kl − t A],

(3)

where: t ~** t ~* t (TE) t ~ t S = t Sij t Cijmn t Smn

(4)

is a positive scalar variable, t ~* t Sij

=

t~ t Sij t ~ t (E) t ~ t Sij t C ijkl t Sij

+

3

(5)

,

2t 2 t~ t

t σ y ( t C+ t E T )

is a component of a stress tensor, t tA

is a positive scalar variable, Piola-Kirchhoff t (E) tD

stress

=

2t 3

t σy

τ t ∆σ ij

tensor,

∂σ y τ (VP) ∆ε eq t (VP) t ∂ t ε eq

(6)

is the increment component of the second t (E) t D ijkl

are

the

components

of

tensor

=[ tt C (E) ]−1 in time t, τt ∆ε ij is the increment component of Green-Lagrange

(E) strain tensor, t Cijkl are the components of elastic constitutive tensor.

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2.2 Incremental model of motion

In this section we develop the equation of a deformation of the object in the updated Lagrangian formulation. Assuming that numerical solutions are obtained at discrete time t, the solution for t+∆t is to be obtained. In such a case, a functional increment tτ∆J (⋅) is formulated for the increment displacement and reads as follows: τ τ τ  i , τt ∆u i )= τt ∆J (⋅) t ∆J ( t ∆u i , t ∆u

,

(7)

where tτ∆u i , tτ∆u i , tτ∆u i are the ith increment components of the displacement, velocity and acceleration vectors, respectively. While using the conditions of stationary of functional tτ ∆J and a finite element method, we can write an equation of motion in the form:

[ tt M]{tτ∆r} +[ tt CT ]{tτ∆r} + ([tt K T ] +[tτ∆K T ]){tτ∆r} ={tτ∆R} +{tτ∆F} +{tt F} +{tt R} (8) where mass matrix [ tt M ] , damping matrix [ tt CT ] , stiffness matrix [ tt K T ] and force vector {tt FT } are known at time t. However, increment stiffness matrix [ tτ∆K T ] , external incremental load vector {tτ ∆R} , internal incremental forces

vector { tτ ∆F} , incremental vectors of displacement {tτ ∆r} , velocity {tτ ∆r} and acceleration {tτ ∆r} of finite element assembly at a typical step time are not known. In order to solve this problem we apply the integration methods.

3 DEM solution Assuming that an increase of temporary step ∆t is very small, it is possible to execute a linearization of equation (8) and using the incremental decomposition we obtain: [M ]{t − ∆tt r} + [CT ]{t − ∆tt r} + [K T ]{t − ∆tt r} = {tt −− ∆∆tt FT } +{t − ∆tt Q}.

(9)

Then using the central difference method (DEM), in which it is assumed that: {t r} =

1 1 ({t + ∆t r} − {t − ∆t r}) , {t r} = 2 ({t + ∆t r} − 2{t r} + {t − ∆t r}) (10) 2 ∆t ∆t

and substituting the relations in (10) into (9) we obtain: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

668 High Performance Structures and Materials III ~ ~ [M]{τt r} = {t − ∆tt Q T } ,

(11)

where: ~ 1 1 [CT ], [M ] = 2 [ M ] + 2∆t ∆t ~ {t −∆tt Q T } = {tt −−∆∆tt FT } +{t −∆tt Q} − [K T ]{t −∆tt r} + +

t

t −∆t

2{ r} − { ∆t 2

r}

[M ] +

{

(12)

t −∆t

r} [CT ]. 2∆t

The integration method requires that the time step ∆t is smaller than critical value ∆tkr, which can be calculated from the mass and stiffness properties of the complete element assemblage: ∆t ≤ ∆t kr = TN / π, where TN is the smallest period of the finite element assemblage with n degrees of freedom.

4

Model investigation

The model investigation was conducted in order to settle the course deformation layer top sample executed from the plastic material, as well as with the aim to qualify boundary conditions for displacements indispensable to numeric analysis of the trapezoidal tread rolling process. To model investigations were applied to the samples in a rectangle shape with the following dimensions: a)

b)

Figure 4:

The stamp (a) and the meshed sample before deformation (b).

Two samples were joined by sides with a plot mesh, and were closed in a metal form. Then the samples were subjected to the deformation by a perpendicular shift of rectilinear motion in the model stamp of an outline of trapezoidal thread rolling. The view of the deformed mesh of complete elements is presented in Fig. 5.

a) Figure 5:

b) The mesh after deformation for µ=0,2 (a) and µ=0,39 (b).

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669

Numerical analysis of state of displacement, strain and stress of material during trapezoidal thread rolling

The application developed with regard to the method of finite elements in ANSYS 8.1 programme provides a complex time analysis of states of displacement, strains and stresses. Digital computing for the process was carried out with the use of two methods. The first method requires introducing the boundary conditions for displacements in the contact zone determined by the model investigation, whereas the second one requires the adequate determination of the contact zone without an introduction of boundary conditions. The main aim of the simulation was to define the influence of friction coefficient on the state of deformation and stress in the surface layer of the object. The numerical analysis for 2-D states of deformation and 3-D states of stress was applied on the example of steel C55 (DIN). The example results of deformation, strain and stress obtained by the numerical analysis for friction coefficient µ=0,2 and µ=0,39 are presented in Fig. 6 and Fig.7.

Figure 6:

Composition of numerical results for friction coefficient µ=0,2: 1 – initial finite element mesh, 2 - finite elements after deformation, 3 – intensity of strain, 4 – intensity of stress.

The distribution of strain and stress intensity in defined nodes (Fig. 8) is shown in Figs. 9–12.

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670 High Performance Structures and Materials III 1

2 ELEMENTS

DISPLACEMENT APR 13 2005 21:59:14

Y

Y Z

APR 13 2005 21:59:14

STEP=1 SUB =100 TIME=1 DMX =15.6

Z

X

3

X

4 NODAL SOLUTION

NODAL SOLUTION APR 13 2005 21:59:14

STEP=1 SUB =100 TIME=1 EPTOINT (AVG) DMX =15.6 SMN =.00995 MX =3.616 SMX

Y

Y

MN X Z

Z .00995

.911405

1.813

2.714

3.616

APR 13 2005 21:59:14

STEP=1 SUB =100 TIME=1 SINT (AVG) DMX =15.6 SMN =.517E+08 MX SMX =.178E+10 MN

X .517E+08

.485E+09

.918E+09

.135E+10

.178E+10

Composition of numerical results for friction coefficient µ=0,39: 1 – initial finite element mesh, 2 - finite elements after deformation, 3 – intensity of strain, 4 – intensity of stress.

Figure 7:

1

DISPLACEMENT APR 19 2005 22:49:06

STEP=1 SUB =100 TIME=1 DMX =16.85

Y

X

Z

2 DISPLACEMENT STEP=1 SUB =100 TIME=1 DMX =16.85

node 158

APR 19 2005 22:49:06

node 160

node 127

Figure 8:

Defined nodes in the discretized model.

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High Performance Structures and Materials III

Figure 9:

The distribution of strain intensity for µ=0,2.

Figure 10:

The distribution of strain intensity for µ=0,39.

Figure 11:

The distribution of stress intensity for µ=0,2.

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672 High Performance Structures and Materials III

Figure 12:

6

The distribution of stress intensity for µ=0,39.

Conclusions

The paper presents a numerical analysis of the process for thread rolling. The process is considered as a geometrical, physical non-linear initial and boundary value problem. The phenomena on a typical incremental step were described using a step-by-step incremental procedure, with an updated Lagrangian formulation. The variational nonlinear equations of the object’s motion for a typical incremental step were derived from the stationary condition for these functionals. These equations were unravelled using the spatial digitization by a finite elements’ method. The adequate algorithm to solve the equations of the motion assuming Rayleigh’s proportional dissipation was developed for this method. The application developed for the method of finite elements ANSYS 8.1 provides a complex time analysis of displacement conditions, strains and stresses occurring in the object. The mathematical models of the process, the algorithms for the solution of discrete equations of motion and the application in the ANSYS system developed in the present work could be used to improve the design process for the thread rolling process.

References [1] [2]

[3]

Łyczko K.: Technologia narzędzi i wygniatania gwintów wewnętrznych. Politechnika Częstochowska, 1999. Kukiełka L., Cienkowski W., Dudek P.: Incremental model of yield stress of metals in the conditions of burnishing rolling operation with electrical current. Third Internationat Meeting on Computer Methods and Experimental Measurements for Surface Treatment Effects, eds. M. H. Aliabadi, C.A. Brebbia, WITPRESS, Boston, pp. 93-102, 1997. Kukiełka L., Krzyżyński T.: New thermo-elastic thermo-visco-plastic material model and its application. Conference in GAMM, Metz, 12-16 April 1999: Ed. WILEY-VCH, pp. 595-596, 2000. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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Three-dimensional limit analysis of ancient masonry buildings with rigid block models A. Ordu˜na Faculty of Civil Engineering, University of Colima, Mexico

Abstract Limit analysis with rigid block models is a tool successfully used in recent years for the assessment of ancient masonry structural elements and small buildings. In this paper, the interface yield functions for three-dimensional models are defined at interpolation points, instead of using the generalised stresses approach. This approach leads to very simple expressions for the yield functions and flow rules and therefore, renders the mathematical programming problem easier to solve than the generalised stresses formulation. The solution for the limit analysis problem is obtained using the load-path following approach. The validation of the present proposal shows good agreement compared with non-linear finite element results. Keywords: limit analysis, rigid block assemblages, non-associated flow, numerical integration.

1 Introduction A valuable model for the structural assessment of ancient masonry structures is the limit analysis of rigid block assemblages interacting through no-tension and frictional interfaces. The reason for this affirmation lies in the fact that masonry has low tensile strength and quasi-brittle failure; therefore, at collapse, the cracks render the structure as a set of rigid blocks rocking and sliding between them. In the rigid blocks modelling strategy, the degrees of freedom are related to the blocks, and the stress and strain variables are related to the interfaces. Therefore, blocks can be regarded as extended nodes and the interfaces as structural elements. Previous works in this subject have used mostly a generalised stress approach for the interfaces [1–4]. This means that the generalised stresses are, for instance, the normal and shear forces and the bending and torsion moments. Besides, Livesley[5, 6], using a different approach, verified the contact at points located at the interface corners. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06066

674 High Performance Structures and Materials III In this paper, the use of numerical integration techniques is investigated, as an alternative to the generalised stresses approach for the yield function description. Firstly, the formulation for non-associated limit analysis of rigid block assemblages is outlined. Then, the formulation of the yield function and flow rule using numerical integration is presented. A discussion of the torsion failure on frictional interfaces follows, where it is emphasised the importance of a correct normal stress distribution calculation over the interface. A comparison of results for a model obtained by the proposed approach against non-linear FE method results serves to validate the proposal. Finally, relevant conclusions are stated.

2 Limit analysis of rigid block assemblages Eqns. (1-6) are the conditions that a limit analysis solution with non-associated flow rule must fulfil, see e.g. [7]. Eqn. (1) combines the compatibility and flow rule conditions. Here, the columns of the matrix
N0 contain the flow directions for each one of the yield functions in the structure; the flow multipliers for each one
is the compatibility matrix and δ
u of such yield functions form the δ
λ vector; C is the vector of block displacement rates. Eqn. (2) is a scaling condition for the displacement rates that ensures the existence of non-zero but finite values. Here
Fv is the vector of variable loads. Eqn. (3) ensures equilibrium. Here
Fc is the vector of constant loads; α is the load factor that measures the amount of the variable loads
is the vector of generalised stresses at the interfaces. applied to the structure and Q
are the actual stress values at every integration In this paper the elements of Q point of all the interfaces in the model. Eqn. (4) guaranties that the yield functions, vector
ϕ, are not violated and eqn. (5) ensures that plastic flow implies energy dissipation. Finally, eqn. (6) guaranties that plastic flow cannot occur unless the stresses have reached the yield surface. Figure 1 shows a graphical representation of the static variables at two adjacent blocks, named i and j, and the common quadrilateral interface, k. Details about these vectors, matrices and functions for rigid block, three-dimensional models have been given elsewhere [4]. In this paper, modifications are made with the aim to change the generalised stresses approach to the integration points approach.
u =
0
N0 δ
λ − Cδ

FvT

(1)

· δ
u − 1 = 0

(2)


=
0
TQ
Fv − C Fc + α

ϕ ≤
0

(3)

δ
λ ≥
0
ϕT · δ
λ = 0

WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

(4) (5) (6)

High Performance Structures and Materials III r

Fi

t

Fi

675

r

Fj

t

s 1k nk

s 2k o

Z

X

Y

Fj

Block j

Block i

Figure 1: Static variables at an interface and adjacent blocks.

3 Numerical integration approach Consider a quadrilateral shaped interface, k, illustrated in Figure 1. This interface lies between two infinitely strong blocks, i and j. The cohesion-less Coulomb’s law governs the interface failure. The interface is supposed to represent a dry masonry joint, therefore, it has no tensile strength. The origin of the local coordinate system is located at the centroid of the interface, o. This coordinate system has axes x1 and x2 on the interface plane, and xn in the normal direction, forming a right-handed system. The unitary vectors along the coordinate axes x1 , x2 and xn are
s1 ,
s2 and
n, respectively, Figure 1. The stress vector
σ at a point has components τ1 , τ2 and σ along the axes x1 , x2 and xn , respectively. The normal component, σ, is positive in tension. The resultant stresses over the interface are the shear forces, V1 and V2 , along the x1 and x2 axes, respectively; the normal force, N; the bending moments, M1 and M2 , along the x1 and x2 axes, respectively and the torsion moment, T . Under a particular stress distribution, eqns. (7–12) give the values for the stress resultants. Here, dA is the differential of area. The second right hand sides of eqns. (7–12) are the numerical integration approximations for the integrals in the first right hand sides. Here, ξ represents the weights of the quadrature, n p is the number of integration points, assumed equal in both interface in-plane directions, J is the determinant of the Jacobian of the transformation between the interface coordinate system and the quadrature coordinate system [8]. Since the interfaces are supposed to be quadrilaterals, this quantity is a constant and can be placed out of the summation. In the second right hand sides, the variables are evaluated only at the integration points, according to the indexes l and m.


V1 =

S




V2 =

S

np np

τ1 dA

. = J ∑

∑ τ1lm ξl ξm

(7)

l=1 m=1 np np

τ2 dA

. = J ∑

∑ τ2lm ξl ξm

l=1 m=1

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676 High Performance Structures and Materials III


N=

S




M1 =


S

∑ σlm ξl ξm

(9)

l=1 m=1

σx2 dA

. = J ∑

∑ σlm x2m ξl ξm

(10)

l=1 m=1 np np

S




. = J ∑

np np

S

M2 = T=

np np

σdA

−σx1 dA

. = J ∑

∑ −σlm x1l ξl ξm

(11)

l=1 m=1 np np

. (−τ1 x2 + τ2 x1 )dA= J ∑

∑ (−τ1lm x2m + τ2lmx1l )ξl ξm

(12)

l=1 m=1

In the approach proposed in this paper, it is not necessary to calculate the stress resultants over the interfaces, but it can be done by means of eqns. (7–12). The yield function characterisation is performed by limiting the normal and shear stresses at each integration point. The normal stress limits are: from bellow, the effective compressive stress and from top the zero value due to the no-tension hypothesis. Therefore, eqns. (13) and (14), represent the yield functions for the normal stress. Here, fce f is the effective compressive stress, which takes into account the effect of transverse cracking and the fact that masonry presents quasi-brittle failure, while the model features perfect plastic behaviour. The shear stresses are limited by the cohesionless Coulomb criterion, as already mentioned. This criterion conducts to a quadratic function as illustrated in Figure 2 by the circle of radius −µσ, where µ is the friction coefficient. Nevertheless, in order to simplify the solution to the mathematical programming problem, a piecewise linear approximation is proposed and illustrated also in Figure 2 by an inscribed octagon. The resulting yield functions are expressed by eqns. (15) and (16). It is observed that, due to the absolute value operands, these two expressions represent the eight linear functions illustrated in Figure 2. ϕc ≡

− σ − fce f

≤0

(13)

σ ≤0 √ ϕs1−s4 ≡|τ1 | + ( 2 − 1)|τ2 | + µσ≤ 0 √ ϕs5−s8 ≡( 2 − 1)|τ1 | + |τ2 | + µσ≤ 0

(14)

ϕt ≡

(15) (16)

There are, therefore, ten linear yield functions for each integration point. The yield functions for all the integration points at every interface in a model can be cast in matrix form, in such a way that eqn. (4) takes the form of eqn. (17). Here,
N is the matrix resulting from the assemblage of all the yield functions in the model.
≤
0
ϕ ≡
N Q

(17)

There are three generalised strain or relative displacement rate components at each integration point: δs1 , δs2 and δn, parallel to the local coordinate directions WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III τ2

677

Coulomb criterion Approximation

−µσ τ1

Figure 2: Shear yield function.

Table 1: Flow directions for an integration point failure. Flow multiplier

δs1

δs2

δn

δλc

0

0

−1

δλt δλs1−4

0 τ1 /|τ1 |

0 √ ( 2 − 1)τ2 /|τ2 |

1 0

τ2 /|τ2 |

0

δλs5−8

√ ( 2 − 1)τ1 /|τ1 |

x1 , x2 and xn , respectively. The flow directions for the yield modes are in Table 1. For the compression and tension yield modes the flow consists on a normal relative displacement rate in the negative and positive directions of the xn axis, respectively. The flow directions for the sliding failure modes are obtained from the normality rule but neglecting the relative displacement rate along the xn axis, due to the hypothesis of zero dilatancy. The rows of Table 1 are used to assemble the columns of matrix
N0 , in eqn. (18), which gives the generalised strains resulting from the flow rule condition. Here the vector δ
q gathers the generalised strains for all the integration points in a model. δ
q =
N0 δ
λ

(18)

The displacement rates for a generic block i are the translation displacement rates at the block centroid δ
uti and the angular displacement rates δ
uri . These vectors are referred to the global coordinates system. Eqn. (19) gives the generalised strains at the integration point p of the interface k in terms of the displacement rates at the blocks i and j; see Figure 1. Here,
Tkg is the matrix that transforms a vector from the global coordinates system to the local interface k system;
c pk ,
ci and
c j are the position vectors of the point p, and the centroids of blocks i and j, respectively. Eqn. (19) can be used to assemble the compatibility set of equations WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

678 High Performance Structures and Materials III of a model in the form of eqn. (20). It is evident that eqn. (1) follows directly from eqns. (18) and (20).  

g
δ
q pk =
Tk δ
utj − δ
uti + δ
urj ∧
c pk −
c j − δ
uri ∧
c pk −
ci
u δ
q = Cδ


(19) (20)

4 Torsion failure on frictional interfaces Ordu˜na and Lourenc¸o [4] studied the torsion failure on rectangular frictional interfaces. In that work, the assumptions were made that dilatancy coefficient was zero and that the normal stresses over the interface have uniform distributions either over all or part of the interface area. With these assumptions, yield functions and flow rules were presented in terms of the stress resultants over the interface. Nevertheless, with the proposal presented in this work, where the normal stresses at each integration point of an interface are independent each other, there are no guarantee that uniform or even normal stresses distributions are obtained over an interface. This section presents a discussion about the influence of the normal stresses distribution over the interface on its torsion moment strength. Consider a rectangular interface with dimensions 2l1 and 2l2 parallel to the local axes x1 and x2 , respectively, as shown in Figure 3. This interface, placed between two infinitely rigid bodies, is subjected to a constant normal force in compression, N (negative), at the interface centroid, and fails under the action of a torsion moment, T . The friction coefficient is µ and an infinite compressive strength is assumed by now. At incipient failure, the sliding displacement rates are zero at the centre of twisting and vary linearly with the distance from this point. If associated plasticity is assumed (dilatancy coefficient equals friction coefficient), the normal displacement rates would be proportional to the sliding displacement rates. The limit x2 c

l2 x1 l2

l1

l1

Figure 3: Rectangular joint. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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679

analysis solution indicates that the centre of twisting is located at the interface centroid, therefore, the maximum sliding and normal displacement rates are at the interface corners. As the blocks are rigid, separation will occur at the inner points, and contact will be possible only at the interface corners. Under this conditions, it is straightforward to show that eqn. (21) proportionates the value of the failure torsion moment, where c is the distance from the interface centroid to any corner; see eqn. (22) and Figure 3. Michalowski and Gawecki [9] arrived to an analogous conclusion for a circular interface where c is replaced by the circle radius in eqn. (21). It is possible to arrive to the same result always that the dilatancy coefficient is positive, even if it does not equal the friction coefficient. T = cµ(−N)  c = l12 + l22

(21) (22)

If the dilatancy coefficient is negative, the normal displacement rates are again proportional to the sliding displacement rates. Nevertheless, no separation but penetration exists between blocks, contact is only possible at the very interface centroid and, therefore, the torsion moment strength is zero (contact on a single point). If the dilatancy coefficient is zero, contact points or areas are not determined by the failure mechanism and neither, the normal stress distribution. This means that for every possible normal stress distribution, there exists a torsion moment strength. It is evident that the values of these strengths lie between zero and that given by eqn. (21). The most fiscally meaningful case consists on a uniform normal stress distribution over the interface. For this case, eqn. (23) gives the torsion moment strength, where the torsion constant cT is given by eqn. (24) [4]. It is observed that eqns. (21) and (23) are the same except for the torsion constant definition. T = cT µ(−N)      l2 l2 1 l2 + c l1 + c + 2 ln c + 1 ln cT = 3 2l2 l1 2l1 l2

(23) (24)

The cT /c ratio can be calculated for the whole range of interface aspect ratios, and the observation is made that it has a small variation between 0.50 and 0.54. Therefore, a first but marginal conclusion is that the torsion constant can be approximated as half the distance from the centroid to the corner. A more important observation is that, for zero dilatancy, the torsion strength of the interface can take values between zero and that given by eqn. (21), but the strength for a uniform normal stress distribution is about half the way between the former values. Therefore, it is of fundamental importance, in limit analysis problems with zero dilatancy, to obtain solutions with even normal stress distributions over the interfaces. Ordu˜na and Lourenc¸o [4, 10] showed that for this type of problem it is also important to take into account the loading history. Here, a solution procedure, akin WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

680 High Performance Structures and Materials III Table 2: Calculated ultimate load factors for infinite compressive strength. Procedure

Ultimate load factor

Theoretically minimum Theoretically maximum

0.427 0.553

FEM Load-path following

0.479 0.465

to the load-path following one is used, which agrees with the integration points approach. The main idea behind this procedure is to keep the normal stresses distribution as uniform as possible, while the variable loads are applied by small increments.

5 Validation The validation of this type of three-dimensional models is not an easy task due to the lack of experimental or analytical results to compare with. Therefore, the example presented here is a comparisons against a finite element (FEM) non-linear model analysed with the package DIANA [11]. Figure 4(a) shows the model of a masonry hollow pile. The pile is built of dry masonry blocks with dimensions 0.2×0.2×0.4 m. The pile dimensions are 0.6×0.8×1.2 m. The material volumetric weight is 20 kN/m3 and the friction coefficient is 0.7. The permanent loads are the self-weight of the blocks. The variable loads are proportional to the blocks weight, but horizontally applied in the direction of the larger base side (X direction). The compressive effective stress, according to the load-path following procedure, is steadily increasing. Figure 4(b) shows the failure mechanism obtained by the FEM analysis. The same failure mechanism is obtained by limit analysis and Figures 4(c), (d) show it from two different viewpoints for a better understanding. A range of ultimate load factors is possible for this mechanism and for infinite compressive strength. It is noted that the ultimate load factor is defined as the ratio between the variable loads causing failure on the structure, and their nominal values, in this case numerically equal to the self-weight of the blocks. Table 2 presents the ultimate load factor obtained from different approaches. If the reaction on the overturning blocks set is concentrated on the interface A only, Figure 4(d), with zero stresses at interface B, the ultimate load factor would be 0.427, the minimum possible for this mechanism. If there are non-zero normal and shear contact forces on interface B, the last one opposing to the upper block overturning, the maximum possible ultimate load factor for the mechanism shown equals 0.553. The ultimate load factor calculated with the load-path following procedure is 3% lower than the FEM value. It is possible to verify that the load-path following procedure agrees very well with the FEM results. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

681

1.2

X

X

0.8

0.6

(a)

(b)

X

joint A

X

joint B

(c)

(d)

Figure 4: Masonry pile; (a) model; failure mechanisms for infinite compressive strength: (b) FEM failure mechanism; (c) and (d) different views of the limit analysis failure mechanism.

It is interesting to note that the condition of zero stresses at interface B is only possible for an infinite compressive strength. Under limited compressive stresses, the hinge on interface A forms slightly inwards and interface B must be in contact and must transmit normal and shear forces.

6 Conclusions A limit analysis formulation using numerical integration techniques at the interfaces has been proposed for rigid block assemblages. The importance of even normal stress distributions over the interfaces was demonstrated. Comparisons, not presented here due to space limitations, show that good agreement is obtained, at interface level, between the integration points approach and the yield functions obtained by constant normal stress distributions over regular shaped parts of the WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

682 High Performance Structures and Materials III interface. The presented example shows also very good agreement compared with the non-linear FEM results. It must be recognised that the computing time for the limit analysis approach and for this large model is not attractive compared with the FEM process. Nevertheless, it is expected that, optimising the numerical tasks for the limit analysis approach, advantage can be taken from the simplified nature of this formulation and the computing times can be significantly reduced.

Acknowledgment This work was supported by project PROMEP/103.5/04/1322 funded by the Secretary of Public Education (SEP) of Mexico.

References [1] Begg, D. & Fishwick, R., Numerical analysis of rigid block structures including sliding. Computer Methods in Structural Masonry 3, eds. J. Middleton & G. Pande, Portugal, pp. 177–183, 1995. [2] Baggio, C. & Trovalusci, P., Limit analysis for no-tension and frictional three-dimensional discrete systems. Mech Struct Mach, 26(3), pp. 287–304, 1998. [3] Ferris, M. & Tin-Loi, F., Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints. Int J Mech Sci, 43, pp. 209–224, 2001. [4] Ordu˜na, A. & Lourenc¸o, P., Three-dimensional limit analysis of rigid blocks assemblages. Part I: Torsion failure on frictional interfaces and limit analysis formulation. Int J Solids Structures, 2005. Accepted for publication. [5] Livesley, R., Limit analysis of structures formed from rigid blocks. Int J Num Meth Engrg, 12, pp. 1853–1871, 1978. [6] Livesley, R.K., A computational model for the limit analysis of threedimensional masonry structures. Meccanica, 27, pp. 161–172, 1992. [7] Ordu˜na, A. & Lourenc¸o, P., Cap model for limit analysis and strengthening of masonry structures. J Struct Eng, 129(10), pp. 1367–1375, 2003. [8] Bathe, K.J., Finite Element Procedures in Engineering Analysis. PrenticeHall, 1982. [9] Michalowski, R.L. & Gawecki, A., Limit torque for a frictional joint. Mech Struct Mach, 24(4), pp. 499–512, 1996. [10] Ordu˜na, A. & Lourenc¸o, P.B., Three-dimensional limit analysis of rigid blocks assemblages. Part II: Load-path following solution procedure and validation. Int J Solids Structures, 2005. Accepted for publication. [11] TNO Building and Construction Research, Delft, The Netherlands, DIANA User’s Manual Release 7.2, 1999.

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Section 9 Reliability of structures

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High Performance Structures and Materials III

685

Analysis of diffusional stress relaxation in submicron Cu interconnect structures using the model with enhanced vacancy diffusivity in grain boundary region I. Tsukrov, W. M. Grich & T. S. Gross Department of Mechanical Engineering, University of New Hampshire, USA

Abstract We propose the finite element simulation technique to model the process of diffusional creep and stress relaxation that occurs in Cu-damascene interconnects of integrated circuit devices in the processing stage. On the length scale of the interconnect lines (microns), the stress-induced mass flow constitutes the major mechanism of inelastic deformation. The mass flow problem is coupled to the stress analysis through vacancy flux and equilibrium vacancy concentration, allowing independently for the concentration profile and evolution of the stresses and strains in an iterative process to be solved. We decompose the total displacement field into the elastic part and the inelastic mass flow contribution. Performing the stress analysis in the configuration with accumulated inelastic displacements, we ensure that the shape of the interconnect line is compatible with external geometrical constraints throughout the simulation. This approach has been implemented in the software package that seamlessly integrates the problem-oriented code with the commercially available finite element program MSC.Marc. We apply the technique to model the Coble creep phenomenon by introducing the nanoscale grain boundary region having the thickness of the order of several layers of atoms. As an illustration, the problem of stress relaxation in a single grain subjected to prescribed displacements and tractions is examined. Keywords: copper interconnects, diffusional creep, grain boundary, nanoscale deformation, finite elements.

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686 High Performance Structures and Materials III

1

Introduction

The fabrication of Cu-damascene interconnect lines has become an important field of research since early 1990’s, when the first commercial samples were introduced. Much technological effort has been applied to reduce the feature size, which currently is on the order of a 100 nm (about 400 atomic radii). Operating on this length scale requires understanding of the impact of nanoscale mechanical behavior on reliability of conductors and interfaces. The major concerns arise due to the fabrication process, which includes reactive ion etching of a circuit pattern in a dielectric blanket, physical vapor deposition or chemical vapor deposition of a Ta-based diffusion barrier and deposition of copper. Thermal cycle from room temperature to 350-400 °C is used to anneal out the device damage from reactive ion etch. It is followed by chemomechanical polishing to flatten the layer and thus produce a planar layered structure. Figure 1 presents the atomic force microscope image of a real interconnect line manufactured by IBM.

Typical Cu grain Grain boundary Ta liner Low-k dielectric

µm Figure 1:

Atomic force microscope topographical map of Cu-damascene.

During thermal processing, the conductor is subjected to considerable stresses from the thermal mismatch between the substrate, the dielectric and the diffusion barrier. The reliability of the conductors and interfaces depends on the state of stress. The deformation and stress relaxation in these nanometer scale structures is highly complex. We may assume that dislocation activity is not relevant at interconnect size scales and processing temperature ranges, as suggested, for WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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687

example, by Kobrinsky et al. [1]. The deformation may be attributed to the CuCu grain boundary and Cu-Ta interfacial sliding, as well as to the material build up due to diffusional creep. The experimental background for this assumption was provided by Gross et al. [2], who developed an AFM method to measure out-of-plane deformation resulting from thermal cycling and applied this technique to observe the Cu-polyimide interconnect structure. In this paper we develop the numerical method to predict the diffusion-based deformation that occurs in the process of interconnects fabrication. To describe this phenomenon adequately, the mathematical formulation is needed that accounts for the coupled nature of mechanical variables and concentration of species such as Cu atoms, vacancies and impurities. The decoupling of creep problem is achieved by its decomposition into the linear elasticity and mass flow subproblems, which are solved in an iterational process. Rzepka et al. [3] applied similar technique to a 3-D model of interconnect. We propose a different approach by utilizing the thermodynamical coupling equations and the concept of grain boundary region of finite thickness. A general discussion of latticebased thermodynamics with respect to diffusional creep in interconnects was presented by Garikipati et al. [4]. Although our treatment is considerably simplified, it nevertheless gives adequate description of the creep/stress relaxation behaviour, and at the same time can be easily adapted to use with existing commercial finite element packages. We also devote special attention to the modelling of grain boundary regions, which are expected to be the major path of vacancy diffusion at the temperature ranges used in interconnects processing.

2

Decomposition of diffusional creep problem into elasticity and mass flow subproblems

The Arrhenius-type expression for equilibrium concentration of vacancies in the bulk at a given temperature is obtained from the minimum of Gibbs free energy ∆Gv written for dilute vacancy concentration (Porter and Easterling [5]) Cveq = exp ( −∆Gv / kT ) .

(1)

In this equation, k is Boltzmann constant and T is the absolute temperature. The Gibbs free energy can be expressed as ∆Gv = ∆H v − T ∆S v , where ∆H v and ∆S v are the enthalpy and entropy of vacancy formation. The estimate for

entropy of vacancy formation for Cu is typically given as exp ( ∆S v k ) ≅ 3 . The enthalpy is given by ∆H v = Q f + σ h Ω , where Q f is the activation energy of vacancy formation, σ h is the hydrostatic stress, and Ω is the atomic volume. The energy of vacancy formation at grain boundaries and free surfaces will be somewhat less due to reduced constraint. Since there are no hydrostatic stress gradients in a uniaxially loaded solid, the vacancy concentration gradients that drive Nabarro-Herring and Coble creep WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

688 High Performance Structures and Materials III should not exist. Nabarro, Herring, and Coble all assumed that the equilibrium concentration in the region adjacent to the grain boundary is given by σ n , the stress normal to the boundary. This can be rationalized by noting that the elastic constants of the grain boundary are different from that of the bulk as reported by Zhou and Huang [6] and other authors. Therefore, we assume that ∆H v ≅ Q f + σ n Ω which is consistent with the implicit assumptions of Nabarro, Herring, and Coble. We propose that vacancy creation or annihilation occurs instantaneously at grain boundaries and free surfaces, and that these surfaces are infinite sinks and sources of vacancies. Defining the temperature dependent stress-free vacancy concentration as C0 = exp ( ( T ∆Sv − Q f ) / kT ) , the equilibrium vacancy concentration in the GB region is given by Cveq = C0 exp (σ n Ω / kT ) .

(2)

We ignore the contribution of dislocations as vacancy sources and sinks since they are rarely observed in the interior of nanoscale grains (see Kong et al. [7] and references therein). The vacancy diffusivity is given by D = D exp ( −Q / kT ) where Qm is the v

v0

m

activation energy for vacancy motion. We assume that Qm is less in a region adjacent to the grain boundary due to disorder near the interface and reduced elastic constraint. We choose the thickness of the enhanced diffusivity region as 3-4 monolayers, assuming the absence of impurities. We define the vacancy

(

diffusivity at interfaces as Dv , gb = D0 gb exp − Qm , gb kT

)

where Qm , gb = α Qm

(α < 1) . We are not aware of a precise method to measure the activation energy for vacancy motion through the grain boundary and will treat it as an adjustable parameter that may be affected by impurities, grain boundary roughness, crystallographic orientation and whether the opposing interface has similar diffusivity. Neglecting the local vacancy relaxation strain (Hirth and Lothe [8]) the mass or atomic flux jA is opposite to the vacancy flux jv :

jA = − jv . (3) The atomic flux field can be treated as the diffusive flow velocity field in a body that is statically fixed. The gradient of the mass flow velocity −∇jv defines the

rate of creep deformation

(

εcr = − 1 2 ∇jv + ∇jv

T

).

(4)

The creep strains defined above are caused by diffusion mass flow due to stress field gradients. Assuming that total strain ε consists of elastic and creep components, the elastic part of the strain can be related to the total stress σ by Hooke’s law WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

High Performance Structures and Materials III

σ = C : ( ε − εcr )

689 (5)

where C is the elastic stiffness matrix. To analyze deformation of a grain assembly, we start with the uniform stressfree vacancy concentration C0 . Application of external load or prescribed displacements may result in stress gradients throughout the structure. This produces vacancy concentration gradients according to eqn (2). Vacancy diffusion leads to the accumulation of diffusion creep strains of eqn (4) and the evolution of stress field. The coupled vacancy diffusion – elasticity problem governed by eqns (2), (4)-(5) is solved by the finite element technique presented in the next section. We implement the iterative approach to analyze the transient processes of diffusion creep or stress relaxation. It is assumed that with time step appropriately selected, we may adopt the staggered procedure on each iteration, i.e. solve the elasticity and mass flow subproblems independently, while holding the variables of another subproblem fixed. The transient vacancy concentration subproblem is governed by the conservation law ∂Cv + ∇ ⋅ jv = 0 ∂t (6) where vacancy fluxes jv obey the Fick’s constitutive equation jv = − Dv ∇Cv . (7) On each time step we integrate eqn (6) with fixed boundary vacancy concentrations given by eqn (2). The diffusive fluxes that occur during the time step are used to compute the creep strain increment ∆εcr by integration of kinematic relation (4). The increment ∆εcr is a component of total strain increment ∆ε . We choose the displacement as the independent variable, hence ∆ε needs to be expressed in terms of displacement increments ∆u . Assuming

(

)

small strain increments, we employ the relation ∆ε ij = 1 2 ∆ui , j + ∆u j ,i , where comma denotes partial differentiation with respect to the corresponding coordinate. The principle of virtual work may be written in incremental form as

∫δε

T

: C : ( ∆ε − ∆εcr )dV = ∆P

(8)

V

where ∆P is the increment of the externally applied load. This equation is the basis for the finite element formulation of mechanical subproblem.

3

Stress relaxation in 1D periodic row of square grains

We consider stress relaxation in a 2D (plane strain) grain array shown in Figure 2 subjected to applied strain ε x = 0.002 . The numerical values used in the computation are listed in Table 1. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

690 High Performance Structures and Materials III Free surface

symmetry

y

x

Grain boundary δ

symmetry

Figure 2:

Model of periodic square grain array used for validation of finite element procedure.

In the analysis of calculated results, we assume that the time dependence of average stress σ 11 = 1 V

∫V σ 11dV

can be approximated by Maxwell model

relaxation function

(9) σ = σ 0 exp ( − t τ ) . In this equation, τ is the Maxwell model relaxation time defined through spring stiffness µ and dashpot viscosity η as τ = η µ . We use the concept of relaxation time to characterize the time period of relaxation of average stress in the model, and to scale the time t in the graphs throughout the rest of this section. Table 1:

Values of parameters used in the stress relaxation modeling of periodic grain array to compare numerical and theoretical predictions.

Parameter Grain width d Burgers vector b Grain boundary region thickness δ Atomic volume Ω Melting temperature Tm

Value 100 nm 0.25 nm 1 nm 1.18 ⋅ 10-29 m3

Young’s modulus E Poisson’s ratio ν

128 Gpa 0.33

Initial compressive stress on internal grain boundary σ 0

286.8 Mpa

Lattice vacancy diffusion pre-exponential D0 L

2 ⋅ 10-5 m2/s

Lattice activation energy for vacancy motion Qm , L

113.3 kJ/mole

Grain boundary vacancy diffusion pre-exponential δ e D0 gb

5e-15 m3/s

Grain boundary activation energy for vacancy motion Qm , gb

67.98 kJ/mole

Grain boundary activation energy for vacancy formation Q f

83.7 kJ/mole

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1356 K

High Performance Structures and Materials III 10.000

unloaded

loaded

y

Displacement y, nm

x

691

T = 0.3Tm T = 0.5Tm T = 0.6Tm

t=0

1.000

T = 0.7Tm T = 0.8Tm T = 0.9Tm

t=τ

loaded at t = 0

0.100

0.010

0.001 -50

-40

-30

-20

-10

0

10

20

30

40

50

x, nm

Figure 3:

Free surface profile at time t = τ , with stress relaxation modeled at different temperatures. Tm is the melting temperature of copper.

Application of fixed strain ε x results in initial uniform stress σ 0 . Stress relaxation occurs as material flows towards the interfaces with the lowest magnitude of normal stress. For the structure under consideration, the smallest normal stress is at the free surface. The profiles resulting from the accumulation of material along the free surface at various temperatures are shown in Figure 3. The lump at the grain junction region results from material migration along the grain boundary (Coble mechanism). The height of the lump is approximately the same for all considered temperatures. It can be explained by the fact that the increased diffusivity at elevated temperatures is compensated by higher rate of vacancy flow (eqn 7). We also observe the increased accumulation of material along the free surface at high temperatures ( T > 0.5Tm ). This effect can be explained by contribution of vacancy diffusion through the grain interior (Nabarro-Herring mechanism). Figure 4 shows the stress σ 11 distribution along the internal grain boundary at different time instances. It can be seen that the stress relaxation occurs almost instantaneously at the junction between grain boundary and free surface, and gradually advances along the internal grain boundary. The stress distribution is shown at low and high temperatures, T = 0.3Tm and T = 0.7Tm . The stress gradients are lower at high temperatures. This effect may be attributed to the contribution of Nabarro-Herring mechanism to stress relaxation. As seen in Figure 4, the difference in stress evolution at different temperatures is not substantial for a one-dimensional array of grains. Hence the relaxation of average WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

692 High Performance Structures and Materials III stress as a function of non-dimensional time t τ can be plotted as a single curve, see Figure 5. The insets illustrate the evolution of stress field σ 11 in the

grain. Note that relaxation time τ is different for different temperatures, for example τ ∼ 1 s for T = 0.5Tm and τ ∼ 0.001 s for T = 0.7Tm . 1.4

6⋅10-4τ

T=0.3Tm

1.2

T=0.7Tm

6⋅10-3τ

σ11/σ0

1

0.03τ 0.09τ

0.8

0.3τ

0.6

0.6τ

0.4

y x

0.2 0 0

− 0.05

− 0.1

− 0.15

− 0.2

− 0.25

− 0.3

− 0.35

−0.4

−0.45

− 0.5

y/d

Figure 4:

Evolution of stress distribution along the internal grain boundary (x = 0) at temperatures T = 0.7Tm (solid lines) and T = 0.3Tm (dashed lines).

1.0

t=τ

0.9

0.10

0.8

<σ11>/σ0

0.7

0.20 0.30 0.40

0.6 0.5 0.4

0.80 0.90

1.10

0.3

0.35

1.00

0.95

0.2 0.1

t = 0.01τ

0.0 0.0

Figure 5:

0.1

0.2

0.50 0.65 0.80

t = 0.1τ 0.3

0.4

0.5

t /τ

0.6

0.7

0.8

0.9

1.0

Relaxation of average stress σ 11 . Insets show the distribution of σ 11 at selected time instances.

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The numerical predictions are compared to the theoretical estimate based on the formula given by Frost and Ashby [9] δ e Dgb  42σΩ  DL 1 + π γ = (10)  2 dDL  kTd  where γ is the shear strain rate, d is the grain width, δ e is the effective thickness of grain boundary (on the order of magnitude of Burgers vector b ) and σ is the applied stress. Note that the diffusivities DL and Dgb in eqn (10) are based on the energy of both formation and motion of vacancies. They are different from the “vacancy motion only” diffusivities Dv , L and Dv , gb used in our finite element formulation which explicitly includes the vacancy formation energy in eqn (2) for stress dependence of vacancy concentration. We obtain the theoretical estimate of relaxation time by computing the dashpot viscosity from eqn (10) as η = σ γ . The shear strain rate in eqn (10) accounts for the contribution of both Nabarro-Herring and Coble mechanisms to creep rate, which is consistent with the numerical modeling approach of sections 2 and 3. Figure 6 shows good agreement of theoretical and numerical predictions for relaxation time. The largest difference is in the temperature range 600-900 K. 1.0E+09 Maxwell model

Relaxation time τ, s

1.0E+07

FEA

1.0E+05 1.0E+03 1.0E+01 1.0E-01 1.0E-03 1.0E-05 400

500

600

700

800

900

1000

1100

1200

1300

Temperature T, K

Figure 6:

4

Comparison of diffusion creep theoretical estimate of relaxation time τ to the finite element predictions.

Conclusions

The stress-induced mass flow constitutes the major mechanism of creep in Cudamascene interconnects during thermal processing of integrated circuit devices. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

694 High Performance Structures and Materials III This mechanism can be modelled by the iterative procedure which decouples the original creep problem into the static stress analysis and transient mass flow subproblems. Introduction of the grain boundary region with enhanced diffusivity into the finite element scheme allows direct modelling of effects associated with Coble creep behaviour.

Acknowledgements This work was supported by the National Science Foundation, Division of Manufacturing and Industrial Innovation, under Grant No. DMI-0300216. The donation of the Cu-damascene sample by IBM is gratefully acknowledged.

References [1] [2]

[3] [4]

[5] [6] [7] [8] [9]

Kobrinsky J., Thompson C. & Gross M., Diffusional creep in damascene Cu lines. Journal of Applied Physics, 89(1), pp. 91-98, 2001. Gross T.S., Kamsah N. & Tsukrov I.I., Scanning probe microscopy generated out-of-plane deformation maps exhibiting heterogeneous nanoscale deformation resulting from thermal cycling of Cu-polyimide damascene interconnects. Journal of Materials Research, 16(12), pp. 3560-3566, 2001. Rzepka S., Meusel E., Korhonen M. & Li C.-Y., 3-D finite element simulator for migration effects due to various driving forces in interconnect lines. AIP Conference proceedings, 491, pp.150-161, 1999. Garikipati K., Bassman L. & Deal M., A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids. Journal of the Mechanics and Physics of Solids, 49(6), pp. 12091237, 2001. Porter D.A. & Easterling K.E., Phase Transformations in Metals and Alloys, Chapman & Hall: London, 1992. Zhou L.G. & Huang H., Are surfaces elastically softer or stiffer? Applied Physics Letters, 84(11), pp. 1940-1942, 2004. Kong Q.P., Cai B., Lu L. & Lu K., The creep of nanocrystalline metals and its connection with grain boundary diffusion. Defect and Diffusion Forum, 188-190, pp. 45-58, 2001. Hirth J.P. & Lothe J., Theory of dislocations, Wiley-Interscience: New York, 1982. Frost H.J. & Ashby M.F., Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, Pergamon Press: 1982.

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Application of fuzzy sets to structural reliability of existing structures I. Mura Department of Structural Engineering, University of Cagliari, Cagliari, Italy

Abstract Assessing the safety of a structure through inspection has always been part of the practice of the structural engineer. Information obtained through these inspections can be both objective and subjective. One can update the reliability of an existing structure based on objective and subjective information through the fuzzified Bayes theorem. This paper cites the algorithm for computing the posterior probability of failure based on visual inspection of existing structures by incorporating fuzzy set theory into the Bayes theorem. The failure design probability of a structural steel frame is updated. Results indicate that the preciseness of the membership function associated with the input data has little or no influence on posterior failure probability. Keywords: reliability, Bayes theorem, fuzzy sets, existing structures, steel frame.

1

Introduction

In engineering practice one often has to deal with qualitative and vague evaluations, commonly known as “fuzzy”. Such subjective fuzzy information is to be found in the results of inspections of structures (such as bridges, frames and so on) and are accompanied by those of a deterministic and probabilistic kind. In particular, when we consider the question of evaluation of the safety level of an existing structure, it appears evident that we cannot avoid taking into account the quality of workmanship, the state of conservation of the elements making up the structure and so on. Estimation of these qualities can be expressed subjectively through variations of a linguistic nature. The values of these variables, as for example “the quality of the workmanship is good”, or “the state WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06068

696 High Performance Structures and Materials III of conservation of the bolts is poor”, being vague and imprecise (which is to say fuzzy) cannot be defined with any certainty. For this reason, it is impossible to use them efficiently in the ambit of conventional statistical theories. Modeling and the taking into consideration of vague and imprecise information can be performed through use of the fuzzy set theory, which was formulated by Zadeh in the 1960s and later developed by many other researchers (see Zadeh [1], Kaufmann and Gupta [2]). The first application of fuzzy set theory in civil engineering goes back to Blockley [3], who developed a method for evaluating the influence of the many parameters that may reduce the a priori safety factor n which determines the probability (pf = 10-n ) of the occurrence of a given structural failure by means of linguistic variables and a “fuzzifier”. Subsequently, the procedure of taking into account subjective information was developed to allow the reaching of different goals. Blockley [4] extended the method illustrated in [3] to the study of twenty-three different structural failures. Brown and Yao [5] estimated the effective strength of cast concrete in a given structure starting from the results of compression tests on test pieces through the taking into account of a fuzzy parameter appropriate for defining the quality of the cast. Itoh and Itagaki [6] and Chou and Yuan [7] considered the problem of evaluating the reliability of existing structures. The algorithms they described allow the calculation of posterior probability based on the results of the visual inspection of structural components by incorporating the fuzzy set theory into the Bayes theorem. Wu [8] applied the Bayes theorem to evaluation of the reliability of systems whose identifying parameters are assumed as fuzzy random variables with an a priori distribution of the fuzzy kind. Thus by combining the fuzzy set theory (which allows the expression of linguistic evaluations through specific functions called “membership functions”) with the Bayes theorem it is possible to solve the problem of evaluating the reliability of an existing structure and determine the posterior reliability of uncertain parameters on the basis of all the results (deterministic, probabilistic and fuzzy) of inspections. But the use of membership functions requires the solution of the problem of their modeling. The analysis and evaluation of damage to a structural element is in fact a difficult process in which human judgement plays a most important role. In the literature we can find many different methods proposed for the modeling of membership functions based on expert judgements. These operations are generally quite costly and laborious, as well as requiring long periods of time. Thus there has been a tendency to restrict membership functions to well-known forms. Triangular, left-shoulder, right-shoulder and trapezoidal or, more generally, piecewise linear, functions are common. Also used are standard Gaussian or Sigmoid type curves. In this work we first illustrate the essential functions of Bayes’ fuzzified theorem (Asai and Negoita [10], Kandel [11]). In the illustration, the theorem will be applied to the updating of design failure probability for the steel frame structure previously studied in [7], based on inspection results considered as fuzzy parameters. To evaluate the influence of the form of the membership functions necessary in defining inspection judgements on the final result, we use WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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those proposed in [3] and [7] (triangular and piecewise linear in form) in our calculations. The two solutions obtained are then compared.

2

The fuzzified Bayes theorem

The Bayes theorem provides a method that allows inclusion of new information in a priori probabilistic evaluations, thus producing a new probability value for the occurrence of an event (Benjamin and Cornell [9]). When there is no fuzzification and we take into account the theorem of total probability, the Bayes theorem is defined by the relation: P (B J | A) =

P ( A | B J ) ⋅ P (B J ) m

(1)

∑ P (A|Bk ) ⋅ P (Bk )

k =1

In Eqn (1) BJ is the Jth unknown random parameter having a known a priori distribution P(BJ ), whose probability is to be updated; A is the random sample representing the parameter to be inspected; m is the number of mutually exclusive and totally exhaustive events. P(BJ | A ) is commonly known as the posterior probability of BJ after statistical event A has taken place. Equation (1) is thus valid if event A can be expressed objectively, which is not the case of a fuzzy event. Therefore the above formulation of the Bayes theorem cannot be used with information of the type: “the crack is small” or “the state of bolt conservation is poor”, and so on, which are vague and imprecise and cannot by defined with any certainty. To express such fuzzy information, specific linguistic variables are introduced. In particular, function µÃ (x) is introduced to express the probability of fuzzy event Ã. Taking this into account, we find that the conditional probability of fuzzy event à can be expressed as:

(

)

~ P A | BJ = ∫ µ x

~ A

(x ) ⋅ f x| B J (x ) ⋅ dx

(2)

where fx|BJ (x) is the conditional probability function that predicts event BJ for support x. Equation (2) is valid if the support is continuous while if the support is discrete, the equation is written:

(

)

~ P A | BJ = ∑ µ A~ (x ) ⋅ P x | B J (x ) ∀x

(3)

Briefly stated, the posterior probability of event BJ, which takes into account the observation of fuzzy event Ã, is obtained by modifying Eqn (1) in accordance with Eqns (2) and (3). The fuzzy Bayes theorem ([10], [11]) can therefore be expressed by the following relations. Where the support is continuous, taking into account Eqn (2), Eqn (1) is written: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

698 High Performance Structures and Materials III   ∫ µ A~ ( x ) ⋅ f x | B J ( x ) ⋅ dx  ⋅ P (B J )  ~  P BJ | A =  x m   ∑  ∫ µ A~ ( x ) ⋅ f x | Bk (x ) ⋅ dx  ⋅ P (B k ) k=1  x 

(

)

(4)

where the support is discrete, taking into account Eqn (3), Eqn (1) is written:   ~ ∀∑x µ A ( x ) ⋅ P x| B J  ⋅ P (B J ) ~ P BJ | A = m   ∑  ∑ µ A~ (x ) ⋅ P x| Bk  ⋅ P (B k ) k=1 ∀x 

(

)

(5)

3 Examples of application The example under consideration (see [7]) concerns the updating of the design failure probability of a steel frame following an inspection. It is assumed that such a probability is equal to Pf = 10-5, and that this represents a mean reliability value that could be calculated by considering the construction of a large number of similar structures. Numerous components (both structural and non-structural) are inspected. Since the conditions of these components influence the overall failure probability of the structure, they assume the role of probability parameters. It is supposed that during the inspection the following parameters are considered: connections, foundations, alignment, columns, beams, braces, bolts and paint. 3.1 Statistical parameters We consider event BJ which may correspond to event B1 (= the structural component is safe) or to event B2 (= the structural component is near failure). The conditional probability function of random event BJ, owing to the support of quality x, supposedly assumes the following quadratic forms:  N −6  g ( X | B1 ) =  i  (0.1) 2 (x − 10 ) 2 + 1.0  6 

(6a)

 N −6  g( X | B2 ) =  i  (0.1) 2⋅ x 2 + 1.0  6 

(6b)

since Ni is the index of the importance of the parameter or structural component i and x is the support quality. Since the above functions (6) represent probability functions, they must satisfy the basic axiom of probability. They are therefore normalized in accordance with the following relation: g ( x | BJ ) (7) Pi ( X | BJ ) = ∑ g ( x | BJ ) ∀x

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0.14

P(X|Safe)

1 0.12

2

0.10

3 4 5 6

0.08 0.06 0.04 0.02 0

Figure 1:

2

4 6 Support of Quality, x

8

10

Conditional probability function for Quality at various importance indices given as Safe.

0.14 0.12 0.10 P(X|Failure)

6 5

0.08

4 0.06

3

0.04

2

0.02

1 0

Figure 2:

2

4 6 Support of Quality, x

8

10

Conditional probability function for Quality at various importance indices given as Failure.

Functions (6) must be defined so as to respect the “weight” assumed by each parameter in the evaluation of the overall safety of the structure. The meaningfulness of each parameter in overall safety is given by the indices of WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

700 High Performance Structures and Materials III importance. Parameters are ordered on the basis of their indices of importance. Indices of unit value correspond to the maximum importance. Numerically increasing values of the index imply a reduction in the importance of parameters. In the part that follows it is supposed possible to assume for each index, on the basis of expert judgement, the values given in Table I. It is to be pointed out that functions (6) may assume different forms for each parameter. In particular, the index of importance may in such forms be defined implicitly. The results of normalization calculations are shown in Figures 1 and 2. Finally, in order to simplify calculations, we allow the parameters to be mutually independent, as are their effects on structural reliability. 3.2 Probability of inspection results It is supposed that the quality of each of the parameters considered above can be verbally described by fuzzy evaluations of the kind: “the state of conservation of the bolts is poor”. Below, in order to evaluate the influence of the modeling on the results supplied by the fuzzified theorem, two different circumstances will be considered. The first kind of modeling, which will be referred to as type 1 and which is taken from [3], is indicated in Figure 3 and corresponds to functions (8). − 0.10 ⋅ x + 1.0 − 0.40 ⋅ x + 1.3  µ P ( x) = − 0.30 ⋅ x + 1.1 − 0.10 ⋅ x + 0.5   0

if x ≤1 if 1 ≤ x ≤ 2 if 2 ≤ x ≤ 3 if 3 ≤ x ≤ 5 if x≥5

 0  0.20 ⋅ x − 0.4   0.40 ⋅ x − 1.0 µ N ( x) =  − 0.40 ⋅ x + 3.0 − 0.20 ⋅ x + 1.6   0

if if if if if if

0≤ x≤2 2≤ x≤3 3≤ x ≤5 5≤ x≤7 7≤ x≤8 8 ≤ x ≤ 10

(8b)

if if if if if

x≤5 5≤ x≤7 7≤ x≤8 8≤ x≤9 9 ≤ x ≤ 10

(8c)

   µ G ( x) =    

0 0.10 ⋅ x − 0.5 0.30 ⋅ x − 1.9 0.40 ⋅ x − 1.7 0.10 ⋅ x

(8a)

The second kind of function modeling, which will be referred to as type 2, is taken from [7]; it is expressed by means of relations (9) and is indicated in Figure 4. WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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1.00

0.80

Normal

Good

Grade of Membership

Poor

0.60

0.40

0.20

0.00

0

2

Figure 3:

4 6 Support of Quality, x

8

10

Membership function for Quality of type 1.

1.00

Grade of Membership

0.80

Normal

Poor

Good

0.60

0.40

0.20

0.00

0

2

Figure 4:

4 6 Support of Quality, x

8

Membership function for Quality of type 2.

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10

702 High Performance Structures and Materials III −2.25 ⋅ x + 1.0  0

µP( x ) = 

 0.20 ⋅ x − 0.20 ⋅ x + 2.0

µN ( x ) = 

0.25 ⋅ x − 1.5 0

µG ( x ) = 

if

x≤4

if

x>4

if

x≤5

if

x>5

if

x≤6

if

x>6

(9a)

(9b)

(9c)

Considering that three quality judgements can be applied to the eight parameters, 6561 possible combinations of inspection results can be considered. The three possible cases of inspection results that are considered herein (taken from [7]) are illustrated in Table I below. 3.3 Updating of failure probability Having assigned the index of importance to each of the structural parameters considered, we can calculate the posterior failure probability of the frame both on the basis of functions (7) and the membership functions reflecting inspection results (8) and (9). The posterior probability of failure of the frame, taking into account the occurrence of fuzzy events Ã1, Ã2 … Ã8, is obtained by extending Eqn (1) to the case of more than one fuzzy variable. We have: ~ ~ ~ P( A1 , A 2 ... A 8 | BJ ) ⋅ P( BJ ) ~ ~ ~ P( BJ | A 1, A 2 .... A 8 ) = 2 ~ ~ ~ ∑ P( A 1, A 2 ... A 8 | Bk ) ⋅ P ( Bk )

(10)

k =1

Having supposed that events Ã1, Ã2 … Ã8 are independent, we indicate by: ~

P ( A1 | P( B1J ) = 2 ~

BJ ) ⋅ P ( BJ )

(11)

∑ P ( A 1| Bk ) ⋅ P ( Bk )

k =1

the posterior probability of event BJ on the basis of observation of the first parameter. Obviously we also have the following: P( BJ2 ) =

~ P ( A 2 | BJ ) ⋅ P ( B1J ) 2 ~ ∑ P( A 2 | Bk ) ⋅ P( Bk )

(12)

k =1

i

Having indicated by P(B J) the posterior probability of event BJ following observation of parameter i, in general we can write: WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

703

High Performance Structures and Materials III

P( BJi ) =

~ P( A i | BJ ) ⋅ P( BJi −1 ) 2 ~ i −1 ∑ P( A i | Bk ) ⋅ P( Bk )

(13)

k =1

Equation (13) above is immediately explicable by using Eqn (2) when the support is continuous and Eqn (3) when it is discrete. Table 1: CASE N°

Parameter

(1)

(2)

1

2

3

4

Posterior failure probability for the type 1 (col. 5) and type 2 (col. 6) membership functions adopted.

Connection Foundation Alignment Columns Beams Bracing Bolts Painting Connection Foundation Alignment Columns Beams Bracing Bolts Painting Connection Foundation Alignment Columns Beams Bracing Bolts Painting

Index Ni

Quality Ai

Pif ( 1 )

Pif ( 2 )

1 1 1 2 4 4 4 6 1 1 1 2 4 4 4 6 1 1 1 2 4 4 4 6

Normal Poor Normal Poor Normal Poor Poor Normal Good Normal Poor Normal Good Normal Poor Good Good Good Good Normal Good Normal Good Normal

1.00E-5 3.16E-5 3.16E-5 6.94E-5 6.94E-5 9.52E-5 1.31E-4 1.31E-4 3.17E-6 3.17E-6 1.00E-5 1.00E-5 7.29E-6 7.29E-6 1.00E-5 1.00E-5 3.17E-6 1.00E-6 3.17E-7 3.17E-7 2.32E-7 2.32E-7 1.69E-7 1.69E-7

1.00E-5 3.11E-5 3.11E-5 6.76E-5 6.76E-5 9.24E-5 1.26E-5 1.26E-5 3.22E-6 3.22E-6 1.00E-5 1.00E-5 7.32E-6 7.32E-6 1.00E-5 1.00E-5 3.22E-6 1.04E-6 3.34E-7 3.34E-7 2.44E-7 2.44E-7 1.79E-6 1.79E-6

(3)

(4)

(5)

(6)

Results of numerical processing and conclusions

Results of calculations performed are shown in Table 1 and correspond to the three circumstances of the inspection results considered. In column 5 of Table 1 we find the updated values of failure probability deduced by adopting functions (8); in column (6) we see those deduced by adopting functions (9). WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

704 High Performance Structures and Materials III Calculations were carried out by adopting the hypothesis of a discrete support by means of the use of a simple program created specifically for the purpose. The values of failure probabilities, later updated by means of Eqn (13), on the basis of inspection results, are found to be practically insensitive to the different definitions of the membership functions adopted and, to all practical purposes, substantially coincident. As can easily be seen, an “average” failure probability, like that of the design, is reduced if following inspection it is found that the parameters that globally define it turn out to be more than average. On the contrary, it increases when it is found that the parameters that globally define it are less than average. Generally speaking, the results indicate that the preciseness of the membership function associated with the input data has little or no influence on the posterior failure probability. Thus, the fuzzified Bayes theorem has practical applications in assessing the posterior failure probability of existing structures, with minimal expert input to establish the necessary membership functions and is a very powerful tool of analysis.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

Zadeh L.A., Fuzzy sets. Information and Control, Vol. 8, pp. 338-353, 1965. Kaufmann A., Gupta M.M., Fuzzy mathematical models in engineering and management science. North-Holland, Amsterdam, 1988. Blockley D.I., Predicting the likelihood of structural accidents. Proc. Inst Civ. Eng., 59(2), pp. 659-668, 1975. Blockley D.I., Analysis of structural failures. Proc. Inst Civ. Eng., 62(1), pp. 51-74, 1977. Brown C.B., Yao T.P., Fuzzy sets and structural engineering. Journal of Structural Engineering, ASCE, 109(5), pp. 1211-1225, 1983. Itoh S., Itagaki H., Application of Fuzzy-Bayesian Analysis to Structural Reliability. Proc. of ICOSSAR '89, ASCE, N.Y., pp. 1171-1174, 1989. Chou K.C., Yuan J., Fuzzy-Bayesian approach to reliability of existing structures. Journal of Structural Engineering, ASCE, 119(11), pp. 32763290, 1993. Wu H.C., Bayesian system reliability assessment under fuzzy environments. Reliability Engineering & System Safety, 83, pp. 227-286, 2003. Benjamin J.R., Cornell C.A., Probability, statistics and decision for civil engineers. Mc Graw Hill, New York, N.Y, 1970. Asai K., Negoita C.V., Introduction to fuzzy set theory. Ohm Press, Tokio,1978. Kandel A., Fuzzy mathematical techniques with applications. Addison Wesley, Reading, Massachusset, 1986.

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Non-linear response of combined system, 3D wall panels and bending steel frame subjected to seismic loading M. Z. Kabir, A. R. Rahai & Y. Nassira, Department of Civil Engineering, AmirKabir University of Technology, Tehran, Iran

Abstract 3D wall panels are used in the construction of exterior and interior bearing and non-load bearing walls and floors of building of all types of construction. This system consists of a welded wire space frame integrated with a polystyrene insulation core and two layers of concrete on both sides. In this paper, attention is focused on the experimental measurements of the seismic response of 3D wall panels surrounded by a steel bending frame. The approach of quasi-static cyclic loading is employed using horizontal actuators to the combined system. The vertical, lateral and horizontal displacements are measured by LVDT equipment. The failure mechanism of 3D wall panels is described in detail. The evaluation of strength and stiffness degradation of the whole system is presented based on the envelope force-displacement curve of actual specimens under cyclic loads. The results of the current study are shown in the form of ductility factors, hysteresis loops and load-displacement envelope curves. The comparison between the ductility of sole steel frames, 3D shear walls and the combined system as the main theme of the current research is presented. Finally, this work clarifies the benefits of using 3D wall panels as a strengthening method for existing steel frame buildings and confirms the feasibility resistance of such combined systems. Keywords: 3D-panels, combined system, cyclic loading.

1

Introduction

3D wall panels are used in construction of exterior and interior bearing and nonload bearing walls and floors of building of all types of construction. This system WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) doi:10.2495/HPSM06069

706 High Performance Structures and Materials III consists of a welded wire space frame integrated with a polystyrene insulation core. The wall panel is placed in position and wythes of concrete are applied to both sides. Wall panel receives its strength and rigidity by the diagonal cross wires welded to the welded-wire fabric on each side. This combination produces a truss behavior, which provides rigidity and shear terms for full composite behavior. Figure 1 shows schematically the 3D panel.

Figure 1:

3D Sandwich panel.

Salmon and Einea [1] presents the results of full-scale test of prototype sandwich panel under transverse loading in a vertical position. Nijhawan [2] measured experimentally the interface shear force and designed the shear connectors. Eiena et al. [3] used the plastic composite diagonal elements to implement in sandwich panel as shear connector for increasing the thermal insulation of this system. Through this study the behavior of 3D panels in combination to steel moment frame was investigated, the fracture mechanism of concrete wythes and the adequacy of steel bars designed based on ACI 318-95 and procedure of PCI design handbook [4].

2

Theoretical study

2.1 Ductility capacity The term “ductility” in seismic design is used to mean the ability of structure to undergo large amplitude cyclic deformation in the inelastic range without a substantial reduction in strength. The ductility is calculated by various types, which reflects structural behavior. The displacement ductility capacity µ is defined as:

µ=

∆ max ∆y

where max ∆ max is the maximum displacement and ∆ y is the displacement at yield

3

Experimental program

Four 3D walls are provided to be combined with portal steel frames. These specimens are considered to represent the critical and structural elements with a WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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rectangular cross section and are experimentally subjected to low cycle horizontal loading regimes. The type of failure and behavior of such structural elements are investigated in details.

Figure 2:

3D panel with steel frame.

3.1 Wall details The wall specimens have 1200 mm height, 640 mm width, 140 mm constant thickness including 40mm shotcrete in each side and 60 mm of expanded polystyrene core. The welded wire fabric is consisted of a cold rolling of steel bar with final outside diameter of 3.5 mm in accordance with ASTM A82 and automatic welding process with accordance of ASTM A185. The yield and ultimate strength of drawn and annealed wires are 4000 and 5200 MPa, respectively. The shotcrete used for all specimens is used from Portland cement (II), river sand with maximum 8 mm diameter, drinkable water. The (W/C) is about 0.5 and the mix is made of 400 kg cement, 1750 kg sand and 180 kg water for a unit cubic meter shotcrete. Compression tests are carried out on (150*150*150 mm) standard cubes and provided cores from shotcrete. Tables 1 and 2 are shown results of the compression strength tests. 3.2 Frame details Two IPE120 as columns and one IPE120 as beam are used for constructing of flexural steel frame. The beam to column and column to base plate connections are rigid based on Iran steel structure design code. Table 1:

Geometry and compressive strength of standard specimens.

Specimen No.

Wall Dimensions (cm)

Specific Gravity (kg/m3)

Max. app lied force (tone)

035 036 037 038

120*64 120*64 120*64 120*64

2330 2310 2290 2280

87 91 82 85

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Cube compressive strength (bars) 387 404 364 378

708 High Performance Structures and Materials III Table 2:

Geometry and compressive strength of cylindrical cores.

Core No.

Core Dimensions (Diameter*length)

Specific gravity (kg/m3)

Max. applied force (kg)

1 2 3 4 5

107.6*54.4 100.4*54.4 99.5*54.4 108.3*54.4 103.3*54.4

2150 2170 2230 2100 2250

5350 5100 5250 5400 5200

Table 3: Specimen No.

1 2

Specimen

Dimensions

Tensile strength.

Specific gravity (kg/m3)

Max. applied force (kg)

Slump (cm)

2300 2320

3700 3500

8 8

15*30 15*30

(a) side view Figure 3:

Core Compressive strength (kg/cm2) 230.2 219.4 225.9 232.3 223.7

Cylinder

tensile strength

(kg/cm2) 52 50

(b) top view Details of steel frame.

3.3 Loading history and test procedure To simulate loading sequences that might be expected to occur during earthquake, simplified types of horizontal cyclic load history are adopted. Since no standard cyclic test procedures has been introduced for testing of such system, the horizontal load is applied at a quasi-static rate in displacement controlled cycles with different patterns which corresponded to three major states, namely WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

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cracking state, yielding state and ultimate state. The applied displacement is started from 0.5 mm to 5 mm in 10 cycles. In the second phase of loading, the increment of displacement is increased to 1 mm and after 19 cycles in displacement of 14 mm the displacement increased to 2 mm and after 23 cycle the increment increased to 4 mm and in 30 cycle system was failed. Linear transducer of types LVDT is used to measure and monitor the horizontal displacements at top, mid height and bottom of specimens. The measured values of applied load and displacement are recorded by a computer data logger capable of measurement to sensitivity ranges of 0.1 N, 0.001 mm, respectively. 40

30

20

Displacement (mm)

10

0 0

5

10

15

20

25

30

-10

-20

-30

-40

-50 Cycle Number

Figure 4:

Figure 5:

Applied successive cyclic displacement.

The crack patterns and failure mechanism in specimen at late stages of cyclic loading.

3.4 Experimental results Table 4 shows the results of applied displacement in type 1 specimen during cyclic loading. Up to 0.4 mm displacement, the panel behaves in elastic zone and the first crack is occurred at the location of 200 mm below the crest, where the connector reinforcements are discontinued. In early stages of loading, the cracks are minor and their direction is mostly horizontal. By increasing applied displacement level, the cracks propagate in both sides of panel. Figure 5 WIT Transactions on The Built Environment, Vol 85, © 2006 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

710 High Performance Structures and Materials III represents the shear cracks pattern at near final stages. It is seen that, the cracks are more visualized and many of them are opened. The direction and propagation of cracks in panels are shown in figure 5. The first crack is located above the base anchor bars when the applied displacement reaches to 3 mm. The main cracks in these types of wall panels are at the bottom area and it continues up to one third of panel height. It is seen that the direction of crack at the area near the panel edges are mostly horizontal and it obliged to 45˚ in the middle of panel. Table 4:

Cycle No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Experimental results in cyclic successive loading for combined system. Horizontal Displacement in both sides (mm) forth back 0.270 0.123 0.616 0.347 0.966 0.672 1.323 1.061 1.703 1.470 2.075 1.878 2.450 2.312 2.848 2.733 3.323 3.171 4.015 4.029 4.797 4.907 5.576 5.771 6.274 6.653 6.986 7.415 7.723 8.140 8.455 8.972 9.103 9.787 9.793 10.563 11.087 12.019 12.299 13.404 13.914 15.174 15.787 17.439 17.624 20.469 21.316 24.748 24.728 29.096 28.025 33.483 31.395 38.939 34.315 43.566

Reaction Forces in Both sides (kN) back forth

Panel Stiffness in both sides (kN/m) back

forth

15.144 27.891 39.463 50.134 60.301 69.308 77.705 85.582 92.177 104.830 116.267 126.511 135.015 143.335 153.12 162.463 169.196 176.249 187.332 194.126 194.889 186.294 172.539 150.83 125.381 114.359 104.619 87.384

55884 45237 40847 37871 35397 33387 31713 30042 28468 26106 24235 22688 21520 20516 19825 19214 18586 17996 16895 15783 14006 11800 9790 7075 5070 4080 3332 2546

136762 83804 63561 50860 43414 38511 34540 31671 29432 26847 247.76 23284 21971 21158 20509 19686 18892 18228 17144 15992 14351 12252 8505 6179 4916 3697 2994 2582

16.808 29.113 42.745 53.966 63.828 72.346 79.857 86.559 93.353 108.191 121.580 134.373 146.174 156.906 166.95 176.630 184.905 192.553 206.064 214.369 217.773 213.666 174.111 152.937 143.06 123.793 116.588 112.512

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Based on ASTM standard method, the applied displacement and their corresponding reaction forces are plotted in a load deflection hystersis curves. Their values are listed in table 4. 250

80

200

60

150

50 0 -60

-40

-20

-50

0

20

40

Reaction Force (KN)

Reaction Force (KN)

40

100 20 0 -80

-60

-40

-20

0

20

40

60

80

-20

-100 -40

-150 -60

-200

-80

-250

Displacement (mm)

Displacement (mm)

(a) combined system Figure 6:

4

(b) steel frame

Load deflection Hystersis energy loops due to successive applied displacement.

Discussion

Due to the significant stiffness of wall panels, up to 217kN, the majority of produced energy in cyclic loading is absorbed by the action of combination of wall and frame. It was shown, [5], the bare shotcrete wall could resist only about 20 kN. The corresponding displacement of total system is about 15.2 mm. It should be noted that all the combined systems are essentially subjected to the intended in-plane action of successive displacement where the out-of-plane movement is prevented, figure 7. Measured maximum value of the vertical displacement at the top of the wall is found to be insignificant since its value is about 0.1% of the maximum wall displacement for all specimens.

Figure 7:

Lateral supports for out-of-plane movement.

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712 High Performance Structures and Materials III 4.1 Cracking For all specimens, cracks are initially formed near the bottom part of the tensile zone of the wall when only about 10% of the final horizontal deformation capacity is applied. The significant inclined cracks initiated at the tension zone of the wall during compressive reversal displacements. These cracks continue to penetrate deeply into the centre of the wall towards the compression zone. 4.1.1 Strength reduction Table 5 indicates horizontal load-carrying capacity of specimens 035 and 036, respectively. It is seen from these tables that the cyclic displacement regime employed in all specimens appeared to have had an insignificant effect on the ultimate strength of the walls. 4.1.2 Stiffness and deformations Variation of lateral deflection in successive cyclic loading and also average of stiffness against horizontal reaction forces are illustrated in table 4. The displacement relating to the first occurred cracking is about 10%. The horizontal load versus top displacement is shown in figure 8, indicates a distinctly nonlinear deformation response. In terms of stiffness reduction, it should be stated that the stiffness is gradually decreased for both, fig. 8. 4.1.3 Ductility The capacity of the structural elements to deform beyond yield or elastic limit with minimum loss of strength and stiffness depends upon their ductility. The load-displacement envelope curve obtained from hystersis loops is sketched in figure 7. The maximum displacement corresponding to 80% of ultimate load is introduced as ∆max. The displacement at the first yielding in panel, ∆y, is determined at the intersection of two lines. The initial tangent of envelope curve with horizontal line passes at the level of 80% of Pu, [6]. It is described in figure 8. 250

Reaction Force (kN)

200

150

100

50

0 0

10

20

30

40

Displacement (mm)

Figure 8:

Load deflection envelope curve.

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50

High Performance Structures and Materials III

Table 5:

The ductility values for wall specimens based on ∆max /∆y.

Specimen without gap

Wall panels

5

713

∆max

∆y

µ

035

21.32

2.44

8.73

036

22.07

2.58

8.55

Conclusions

The current work describes the cooperation between 3D panels as infill wall and traditional steel frames. The following conclusions remarks are raised based on previous work by the first author [5]: - For all specimens, the plastic hinge is formed at the extreme fiber of the wall section and at the vicinity at the base, above the foundation. The distribution and propagation of cracks show that 3D sandwich panels with limited height, behaves in shear performance. The observed horizontal crack at the base prior to ultimate state may be due to sliding at the vertical reinforcement. This caused considerable reduction in the strength, stiffness and energy dissipation of the specimens. - Externally reinforcing 3D wall panels, which basically behaves in shear manner, enhances more ductility in performance design approach and increases substantially load carrying capacity of system. Finally, this study clarifies benefits of using lightweight prefabricated panels as a strengthening method for existing of steel frame building and confirms feasibility resistance of such compound system.

Acknowledgement The authors gratefully acknowledge the financial support from Pre-fabricated sandwich Panel firm, SAP company in Tehran for present work

References [1] [2] [3] [4]

Salmon, D.C. and Einea, A., Partially composite sandwich panel deflections, Journal of structural engineering, Vol. 121, No.4, pp. 778783, 1995. Nijhawan, J.C., Insulated wall panels interface shear transfer, PCI journal: 98-101, 1998. Einea, A., Todros, M.K., Salmon, D.C. and Culp, T.D., Culp, T.D. A new structurally and thermally efficient sandwich panel system, PCI journal, 39(4):90-101, 1994. PCI design handbook-precast and prestressed concrete, 3rd edition, precast/prestressed concrete institute, Chicago, 1985.

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714 High Performance Structures and Materials III [5] [6] [7] [8]

Kabir, M.Z., Jahanpoor, A.R. and Rahbar, M.R., An estimation of ductility and behavior factor of 3D sandwich shotcreted panels subjected to monotonic shear loads, ERES Conference, Seville, Spain, 2003. Uang, C., Establishing R and Cd factors for building seismic provisions, ASCE, Journal of structural Engineering, vol. 177, 1991. Duan L. and Reno, M., Performance based seismic design criteria for bridges, in W.F. Chen (ed), Handbook of structural Engineering, CRC, Press, 1997. Newmark, N.M., and Hall, W.J., Earthquake spectra and design, Earthquake engineering Res. Inst. El Cerrito, Calif., 1982.

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Author Index Abdel-Mooty M. A. N. ............ 419 Aiello R.................................... 449 Akita K. ........................... 351, 359 Al-Jabri K. S. ..................... 93, 381 Alnuaimi A. S. ........................... 93 Aoyama E. ............................... 169 Arasteh A................................. 269 Artemev A. .............................. 103 Avalle M. ................................. 249 Avila J. A................................. 439 Belingardi G. ........................... 201 Bhandari Y............................... 331 Bitar M. A................................ 419 Bossuyt S. ................................ 239 Boufas S................................... 311 Bouquerel J. ............................. 259 Brooks W................................. 481 Caliez M. ................................. 211 Černý R.................................... 409 Chalioris C. E. ......................... 459 Cheng L. .................................. 501 Chew H. B. .............................. 501 Chiandussi G............................ 201 Cizmar D.................................. 429 Cooreman S. ............................ 239 Couroneau N.............................. 73 Crea F. ..................................... 449 Crooks R. ................................. 549 Cruz H...................................... 539 Custódio J. ............................... 539 Cuypers H. ................................. 21 Darling T.................................. 323 Dascotte E................................ 225 De Bolster E............................... 21 De Temmerman N. .................... 41 De Wilde W. P............................... ................... 3, 13, 21, 31, 625, 635 Debacker W. ............................ 625 Degrieck J. ............................... 259 Dénès G. .......................... 301, 311

Dimiduk D. .............................. 331 Dobrila P.................................. 111 Domack M. S........................... 549 Estévez Cimadevila J............... 571 Estévez J. ................................... 63 Euler E. .................................... 225 Farag H. M............................... 419 Frachon A. ............................... 211 Frontera P. ............................... 449 Fujii T. ..................................... 179 Fujita H.................................... 189 Gan M. D. ................................ 511 Geddes B. ................................ 103 Ghosh S. .................................. 331 Gordon V. ................................ 581 Govorunov I. N........................ 343 Gratton M. ............................... 211 Grich W. M.............................. 685 Groeber M. .............................. 331 Gross T. S. ............................... 685 Guo T. F................................... 501 Gu J.......................................... 239 Hago A....................................... 93 Hendrickx H. ........................... 625 Henrotay C............................... 625 Hirayama T. ............................. 189 Hirogaki T. .............................. 169 Hirukawa K. ............................ 351 Hoagland R.............................. 323 Hoechbauer T. ......................... 323 Ibba A. ..................................... 201 Ichihara Y. ............................... 189 Ignatova O. N. ......................... 343 Issa M. E.................................. 419 Jiřičková M.............................. 409 Jourani A. ................................ 369

716 High Performance Structures and Materials III

Kabir M. Z. .............................. 705 Katayama T.............................. 159 Kibriya T.................................. 469 Klanšek U. ....................... 605, 643 Kochuparampil J...................... 311 Kravanja S. .............. 605, 615, 643 Kubo S. .................................... 179 Kukielka K............................... 663 Kukielka L. .............................. 663 Laou E...................................... 301 Latteur P. ................................... 31 Le V. D. ................................... 211 Lecompte D. ............................ 239 Lee J. Y.................................... 121 Lee S. J. ................................... 121 Lentzen S. ................................ 653 Lin B. ....................................... 491 Lu G......................................... 491 Madamba M. C. ....................... 301 Madhkhan M............................ 269 Marchese S. ............................. 449 Martella P. ............................... 249 Martikka H............................... 593 Martín E..................................... 63 Martín Gutiérrez E................... 571 Martinez M. ............................. 103 Martínez-Abella F.................... 401 Matsuoka T. ............................. 189 Merazig H. ............................... 311 Mestrovic D. ............................ 429 Mezghani S. ............................. 369 Mines R. A. W. ........................ 481 Minoshima K. .......................... 131 Misra A. ................................... 323 Mizuta K. ................................. 189 Mňahončáková E. .................... 409 Mollaert M................................. 41 Mori H. .................................... 151 Muñiz Gómez S. ...................... 571 Muntasar A. ..................... 301, 311 Mura I. ..................................... 695 Murakami T. ............................ 159

Nagy J. B. ................................ 449 Nakagawa H. ........................... 169 Nassira Y. ................................ 705 Nastasi M................................. 323 Nitzsche F................................ 103 Novikov S. A. .......................... 343 Ogawa K.................................. 169 Ohya S. ............................ 351, 359 Okubo K. ................................. 179 Orduña A. ................................ 673 Otero Chans D. ........................ 571 Otero D. ..................................... 63 Pardo Tràfach P. ...................... 559 Peroni L. .................................. 249 Picart D.................................... 211 Ponsaert W........................... 13, 31 Premrov M....................... 111, 615 Procházka P. ............................ 391 Prokopenko Yu. A. .................. 521 Pushkov V. A........................... 343 Radic J. .................................... 429 Rao J. N. .................................. 653 Rahai A. R. .............................. 705 Rebouillat S. ............................ 279 Rivera D................................... 439 Rovira Santa Olaya J. L. .......... 559 Rybinskaya A. A...................... 521 Ryu B. J. .................................. 121 Samyn Ph........................... 13, 635 Sano Y. .................................... 359 Santorinaios M......................... 481 Savochka P. A. ........................ 521 Schmidt R. ............................... 653 Šejnoha M.................................. 83 Shin K. B. ................................ 121 Shorkin V................................. 581 Šilih S. ..................................... 615 Sinitsyn V. A. .......................... 343 Smedley D. .............................. 539 Smits A. ................................... 239

High Performance Structures and Materials III

Sol H. ............................... 225, 239 Steffenino B............................. 279 Stolarski T. A........................... 351 Stopp H. .................................. 529 Strangfeld P. ............................ 529 Sutcliffe C. J. ........................... 481 Swadener J. G. ......................... 323 Swartjes F. H. M...................... 511 Taitokari E. .............................. 593 Takagi H. ................................. 151 Takemura K. ............................ 143 Tanaka H.................................. 359 Tanaka K.......................... 131, 159 Tobe S...................................... 351 Tomioka Y............................... 169 Tsukrov I.................................. 685 Uchic M. .................................. 331 Uno K. ..................................... 159

717

Van Mele T................................ 41 Van Slycken J. ......................... 259 Van Steirteghem J................ 13, 31 Vandenbergh T. ......................... 31 Vantomme J............................. 239 Vázquez J. A.............................. 63 Vázquez-Herrero C.................. 401 Verbeeck B. ....................... 31, 635 Verleysen P.............................. 259 Wagner J. A. ............................ 549 Wang W................................... 511 Wang Y.-C............................... 323 Wastiels J................................... 21 Wuzella G.................................. 53 Yamada H................................ 131 Zahouani H. ............................. 369 Zeman J. .................................... 83 Zhornik V. A. .......................... 521 Žula T. ..................................... 643

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Deployable Structures Analysis and Design C.J. GANTES, National Technical University of Athens, Greece Accessible to practicing structural engineers and graduate students with no previous knowledge of the field, this title formulates and solves the complex engineering design problems with which deployable structures are associated. It also presents the issue of design of snap-through type deployable structures in an organized way which will be of interest to more experienced readers. Series: High Performance Structures and Materials, Vol 2 ISBN: 1-85312-660-8 2001 384pp £132.00/US$198.00/€198.00

“…an excellent introductory text to response control in structural engineering design.” JOURNAL OF STRUCTURAL ENGINEERING

Bridging the gap between conventional structural design and the field of motion based design, this book provides a basis for the preliminary design of motion sensitive structures. The authors present a systematic outline of basic concepts, computational procedures and behavioural issues for structural motion control. ISBN: 1-85312-454-0 1996 440pp £95.00/US$171.00/€142.50

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