Structural Failure and Plasticity

Structural Failure and Plasticity

Elsevier Science lnternet Homepage http://www.elsevier.nl (Europe) http://www.elsevier.com (America) http://www.elsevier.co.jp (Asia) Consult the Elsevier homepage for full catalogue information on all books, journals and electronic products and services.

Elsevier Titles of Related Interest VOYIADJIS & KATTAN Advances in Damage Mechanics: Metals and Matrix Composites ISBN: 008-0436013

Related Journals Free specimen copy gladly sent on request: Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK Engineering Fracture Mechanics European Journal of Mechanics - A/Solids European Journal of Mechanics - A/Solids with European Journal of Mechanics - B/Fluids (Combined Subscription) International Journal of Fatigue International Journal of Impact Engineering International Journal of Non-Linear Mechanics International Journal of Pressure Vessels and Piping International Journal of Solids and Structures Journal of the Mechanics and Physics of Solids Mechanics Research Communications Theoretical and Applied Fracture Mechanics Tribology International Wear

To Contact the Publisher Elsevier Science welcomes enquiries concerning publishing proposals: books, journal special issues, conference proceedings etc. All formats and media can be considered. Should you have a publishing proposal you wish to discuss, please contact, without obligation, the publisher responsible for Elsevier's engineering mechanics publishing programme: Ms Nicola Garvey Publishing Editor Elsevier Science Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 IGB, UK

Phone Fax E-mail

+44 (0) 1865 843879 +44 (0) 1865 843920 n.garvey @elsevier.co.uk

General enquiries, including placing orders, should be directed to Elsevier's Regional Sales Offices - please access the Elsevier homepage for full contact details (homepage details at the top of this page).

Structural Failure and Plasticity Proceedings of The Seventh Intemational Symposium on Structural Failure and Plasticity (IMPLAST 2000) 4-6 October 2000, Melboume, Australia

Edited by

X.L. Zhao and R.H. Grzebieta Department of Civil Engineering, Monash University, Claywn, VIC 3168, Australia

2000

PERGAMON An Imprint of Elsevier Science Amsterdam- Lausanne- New Y o r k - Oxford- Shannon- Singapore - Tokyo

ELSEVIER SCIENCE Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK

9 2000 Elsevier Science Ltd. All rights reserved.

This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 i DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'.. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WlP 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Seience Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

First edition 2000 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

ISBN: 0 08 043875 X

The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Transferred to digital printing 2005 Printed and bound by Antony Rowe Ltd, Eastbourne

Preface The IMPLAST series of symposiums began in 1973 at the Indian Institute of Technology, in New Delhi, India. The theme of the symposia has been in the large deformation of materials and structures subjected to quasistatic, medium and high rates of loading. This symposium is the seventh in the series and the first time it has been held outside of India. Australia was chosen as a venue because of the strong bond that exists between the two countries and also because of the research work currently being carried out in the field of impact mechanics, crashworthiness and plastic deformation of structures. Delegates from more than 20 different countries from 5 continents have come together to present and discuss results from numerous studies in the field of impact and plasticity. What better place to hold such an international event at the start of the new Millennium than in a young nation, where a week before the symposium another inspiring international event was held, the Olympic Games. Most of the symposia have been run under the careful guidance of Professor N. K. Gupta. A very warm and generous host, Professor Gupta always encourages international researchers to visit his institute to motivate and inspire his fellow researchers. He has also encouraged a large number of famous international researchers to attend and speak at the various IMPLAST symposia so as to enhance the exchange of ideas and results among the world's applied mechanics fraternity. Professor Gupta's keynote lecture provides an overview of the contribution the IMPLAST series has made to impact and plasticity engineering and provides further insights into the plastic deformation of tubes and frusta. Melbourne has traditionally had a very strong research school in thin-walled structures. This evolved as a result of investigations into the West Gate bridge failure by the distinguished Professor Noel Murray. His contribution to engineering was honoured at the symposium dinner. This is also why this symposium had a distinct focus on thin-walled structures by the other two keynote presenters. Professor Rhode's paper provides an excellent overview of buckling of plated and thin walled structures, whereas Professor Usami's paper on plastic deformation of thin-walled structures under cyclic loading provides us with some valuable information on how such structures behave during seismic events. As mankind continues to push back the boundaries and begins to explore other worlds and the ocean depths, a thorough understanding of how structures behave when subjected to extremes in temperature, pressure, and high loading rates will be essential. This symposium provides the perfect forum for presenting research into structures subjected to such extreme loads. There were a large number of papers presented under topics of impact, blast and shock loading, indicating a strong research interest in high rates of loading. Similarly new topics have been added to the traditional symposia list such as fire loading, earthquake loading, and fatigue and connection failures. It is clear now that fundamental knowledge of plastic deformation of structures subjected to various extreme loads is coming of age. When the planning of this symposium began a large number of distinguished researchers agreed to join the International Scientific Committee to assist with technical content. The editors are honoured and grateful to all the members for their assistance. The editors were also saddened by the recent passing away of Dr. Raymond Woodward from Australia and Dr. Dusan Kecman from Cranfield, IYK. The international applied mechanics community holds Dr. Woodward's contributions to understanding material behaviour subjected to impact loads and in particular penetration mechanics, and Dr. Kecman's contributions to crashworthiness standards for vehicles and in particular bus rollovers, in high regard. As quiet achievers who encouraged scientific endeavour, their contributions and company will be sadly missed. The editors would also like to thank the organising committee for their assistance. They would also like to thank in particular the reviewers of the papers listed in the proceedings. Each full paper was peer reviewed by at least two experts in the field. The editors are most grateful to them for giving up their valuable time. Finally the editors would also like to warmly thank each of the delegates for preparing their papers, attending the conference and helping make the IMPLAST series a success. Raphael Grzebieta Xiao-Ling Zhao

V1

International Scientific Advisory Committee W. Abramowicz N. Burman S. F. Chen W. F. Chen E. C. Chirwa P. U. Deshpandey P. Grundy N. K. Gupta G. J. Hancock N. Ishikawa N. Jones D. Kecman t

C, W. Kim S. Kitipornchai T. Krauthammer J. Lindner H. A. Lupker Y. W. Mai P. Makelainen N. W. Murray G. Nurick J. A. Packer A. K. Rao R. G. Redwood S. R. Reid J. Rhodes J. Rondal G. Sedlacek G. S. Sekhon H. Schmidt N. E. Shanmugam Z. Y. Shen K. Sonoda G. Thierauf T. Usami J. Wardenier R. L. Woodward *

T. X. Yu R. Zandonini

Impact Design, Europe Department of Defence, DSTO Xi'an University of Architecture and Technology University of Hawaii, Manoa Bolton Institute Ministry of Defence Monash University Indian Institute of Technology The University of Sydney National Defence Academy University of Liverpool Cranfield Impact Centre Yonsei University University of Queensland The Pennsylvania State University Technical University of Berlin TNO Crash-Safety Research Centre The University of Sydney Helsinki University of Technology Monash University University of Cape Town University of Toronto Engineering Staff College of India McGill University UMIST University of Strathclyde University of Liege RWTH, Aachen Dept. of Applied Mechanics Universitat Gesamthochschule Essen National University of Singapore Tongji University Osaka City University Universitat Gesamthochschule Essen University of Nagoya Delft University of Technology Department of Defence, DSTO Hong Kong University of Science & Technology University of Trento

Poland Australia P. R. China USA UK h~dia Australia India Australia Japan UK UK South Korea Australia USA Germany The Netherlands Australia Finland Australia South Africa Canada India Canada UK UK Belgium Germany India Germany Singapore P. R. China Japan Germany Japan The Netherlands Australia P. R. China Italy

vii

Local O r g a n i s i n g C o m m i t t e e

Chairman Co-Chairman Symposium Manager Founding Chairman

Raphael H. Grzebieta Department of Civil Engineering, Monash University Xiao-Ling Zhao Department of Civil Engineering, Monash University Irene Thavarajah Office of Continuing Education, Monash University N. K. Gupta India Institute of Technology, India

Members: R. AI-Mahaidi G. Burkitt D. Saunders P. Dayawansa

G. X. Lu S. Richardson L. Pham A. Potts B. Wang L. Wilson

Department of Civil Engineering, Monash University VicRoads Department of Defence, DSTO Department of Mechanical Engineering, Monash University Swinburne University of Technology ATEA CSIRO Building, Construction & Engineering Australian Marine and Offshore Group Brunel University, Uxbridge, UK Australian Institute of Steel Construction

viii

Reviewers A. Abel A. Afaghi-Khatibi R. AI-Mahaidi M. Attard A. Baker J. Barrados Cardosa L. Beai A. Beasley I. Bennetts P. Berry M. Boutros E. Breil R. Q. Bridge N. Burman S. L. Chan W. F. Chen Y. Cheng S. Cimpoeru E. C. Chirwa P. Ciancy M. Clarke C. Clifton G. Davies P. Dayawansa I. Donald J. Eftis C.J. Flockhart H.B. Ge J. Ghojel J. Giercrak Y. Goto J.R. Griffiths R. H. Grzebieta P. Grundy N. K. Gupta L. Hammond B. K. Han L. H. Han G. J. Hancock H. Hansson N. Haritos S. Herion I. Herzberg W.P. Hu N. Ishikawa N. Jones

C. W. Kim H. Kitoh V. Kodur T. Krauthammer Y. Kurobane R. Lapovok L.A. Louca G. Lu S. J. Maddox M. Mahendran Y. W. Mai Y. Maki

The University of Sydney The University of Sydney Monash University The University of New South Wales DSTO Instituto Superior Tecnico Queensland University of Technology University of Tasmania Victoria University of Technology University of Western Sydney University of Western Australia University of Southern Queensland University of Western Sydney DSTO The Hong Kong Polytechnic University University of Hawaii, Manoa The University of Sydney DSTO Bolton Institute Victoria University of Technology The University of Sydney HERA Nottingham University Monash University Monash University University of Texas, E1 Paso DSTO Nagoya University Monash University Technical University of Wroclaw Nagoya Institute of Technology CSIRO Monash University Monash University Indian Institute of Technology DSTO Hong-ik University Hanbin University of Civil Engineering The University of Sydney Defence Research Establishment The University of Melbourne Universtiy of Karlsruhe Monash University DSTO National Defence Academy University of Liverpool Institute of Automobile Technology Osaka City University Institute for Research in Construction The Pennsylvania State University Kumamoto Institute of Technology CSIRO Imperial College, London Swinburne University of Technology TWI University of Queensland The University of Sydney Hosei University

Australia Australia Australia Australia Australia Portugal Australia Australia Australia Australia Australia Australia Australia Australia P.R. China USA Australia Australia UK Australia Australia New Zealand UK A ustralia Australia USA Australia Japan Australia Poland Japan Australia Australia Australia India Australia South Korea P. R. China Australia Sweden Australia Germany Australia Australia Japan UK South Korea Japan Canada USA Japan Australia UK Australia UK Australia Australia Japan

ix

P. Makelainen J. Marco I. Marshall R. Meichers P. Mendis T. Mori A. Mouritz N. W. Murray W. Muzykiewicz N.T. Nguyen G. Nurick J. A. Packer A. W. Page N. Page J. Papangelis H. Pasternak Y. L. Pi K.W. Poh A. Potts J. Price B. V. Rangan A. K. Rao K. Rasmussen G. Rechnitzer A. Resnyanski C. A. Rogers A. Ruys G. Sanjayan Z. Y. Shen D. R. Sherman D. Shu L. Sironic S. Sloan I. Smith K. Sonoda G. Stevins N. Stokes N. Stranghoener M. Takla G. Taplin J. G. T eng P. Thomson F. Tin-Loi C. Tingvall N. Trahair B. Uy B. Wang C. Wang K. Weynand T. Wilkinson B. Wong M. Xie Y. L. Xu Y. B. Yang G. Yiannakipoulas T. X. Yu R. Zandonini Q. Zhang X. L. Zhao R. Zou

Helsinki University of Technology DSTO Monash University The University of Newcastle The University of Melbourne Hosei University DSTO Monash University The University of Mining & Metallurgy The University of Sydney University of Cape Town University of Toronto The University of Newcastle The University of Newcastle The University of Sydney BTV Cottbus The University of New South Wales Victoria University of Technology Australian Marine & Offshore Group Monash University Curtin University of Technology Engineering Staff College The University of Sydney Monash University DSTO McGill University The University of Sydney Monash University Tonji University University of Wisconsin Nanyan Technical University Monash University University of Newcastle University of New Brunswick Osaka City University The University of Sydney CSIRO HRA Ingenieurgesellschaft mbH RMIT Monash University The Hong Kong Polytechnic University Monash University University of New South Wales Monash University The University of Sydney The University of New South Wales Brunel University DSTO RWTH, Aachen The University of Sydney Monash University Victoria University of Technology The Hong Kong Polytechnic University National Taiwan University DSTO Hong Kong University of Science & Technology University of Trento University of Western Sydney Monash University Monash University

Finland Australia Australia Australia Australia Japan Australia Australia Poland Australia South Africa Canada Australia Australia Australia Germany Australia Australia Australia Australia Australia India Australia Australia Australia Canada Australia Australia P. R. China USA Singapore Australia Australia Canada Japan Australia Australia Germany Australia Australia P. R. China Australia Australia Australia Australia Australia UK Australia Germany Australia Australia Australia P. R. China Taiwan Australia P. R. China Italy Australia Australia Australia

This Page Intentionally Left Blank

CONTENTS Preface International Scientific Advisory Committee

vi

Local Organising Committee

vii

Reviewers

viii

Keynote Papers IMPLAST Symposia and Large Deformations- A Perspective N. K. Gupta Buckling of Thin Plates and Thin-Plate Members- Some Points of Interest J. Rhodes

21

Failure Predictions of Thin-Walled Steel Structures under Cyclic Loading T. Usami and H.B. Ge

43

Impact Loading On the Criteria for Cracking and Rupture of Ductile Plates under Impact Loading N. Jones and C. Jones

55

Dynamic Behavior of Elastic-Plastic Beam-on-Foundation under Impact or Pulse Loading X.W. Chen, T.X. Yu and Y.Z. Chen

61

Load Deformation of Thin Tubular Beam under Impact Load N. lshikawa, Y. Kajita, K. Takemoto and O. Fukuchi

67

Normal Impact of Spherical Balls on Metallic Plates P.U. Deshpande and N.K. Gupta

73

Impact Performance and Safety of Steel Highway Guard Fences Y. ltoh, C. Liu and S. Suzuki

79

Impact Behavior of Shear Failure Type RC Beams T. Ando, N. Kishi, H. Mikami and K.G. Matsuoka

87

Nonlinear Dynamic Response Design and Control Optimization of Flexible Mechanical Systems under Impact Loading J. Barradas Cardoso, P.P. Moita and J.A. Castro

93

Influence of Impact Loads on the Behavior at Alternative Bending over Pulleys of Steel Wires G. Crespo

99

xii Dynamic Actions on Highway Bridge Decks due to an Irregular Pavement Surface

103

J. G.S. da Silva

Elastic-Viscoplastic-Microdamage Modeling to Simulate Hypervelocity Projectile-Target Impact and Damage

109

J. Efiis, C. Carrasco and R. Osegueda

Experimental and Numerical Studies of Projectile Perforation in Concrete Targets

115

H. Hansson and L. ~g&rdh

Prototype Impact Tests on Ultimate Impact Resistance of PC Rock-Shed

121

N. KishL H. Konno, K. Ikeda and K.G. Matsuoka

Penetration Equations for the Impact of 7.62 mm Ball Projectile against Composite Material Sheets of an Aircraft

127

P. Kumar, R.A. Goel and KS. Sethi

High Strength Concrete Beams Subjected to Impact Load- Some Experimental Results 133 J. Magnusson, H. Hansson and L. ~gttrdh

Impact Response of a Laminated Cylindrical Composite Shell Panel

139

P. Mahajan, K.S. Krishnamurthy and R.K. Mittal

Dynamic Testing of Energy Absorber System for Aircraft Arrester

145

K.K. Malik, P. K. Khosla, P.H. Pande and R. K. Verma

Characteristics of Crater Formed under Ultra-high Velocity Impact

151

S. Pazhanivel and V.K. Sharma

Diagnostic Techniques for High Speed Events

157

V.S. Sethi and S.S. Sachdeva

Shock Test and Stress Analysis of a Heavy Metal Forge

165

Y. M. Wu, B. Samali, J. C. Li and S. Bakoss

Blast/Shock Loading Air Blast Simulations using Multi-Material Eulerian/Lagrangian Techniques

173

J. Marco

Damage Evaluation of Structures Subjected to the Effects of Underground Explosions

179

R. Kumari, H. Lal, M.S. Bola and KS. Sethi

An Investigation of Structures subjected to Blast Loads incorporating an Equation of State to Model the Material Behaviour of the Explosive

185

W.P. Grobbelaar and G.N. Nurick

An UNDEX Response Validation Methodology J.L. 0 'Daniel T. Krauthammer, K.L. Koudela and L.H. Strait

195

xiii

The Effects of Local Cavitation and Diffraction on the Underwater Shock Response of an Air-backed 2D Plate Structures with Large Deflections L.C. Hammond and C.J. Flockhart

201

Ductile Failure of Welded Connections to Corrugated Firewalls subjected to Blast Loading L.A. Louca and J. Friis

209

Design Criteria for Blast Tolerant Bulkheads 1. Raymond, M. Chowdhury and D. Kelly

217

The Ballistic Impact of Hybrid Armour Systems H.H. Billon

223

Large Scale Blast Analysis of Reinforced Concrete with Advanced Constitutive Models on High Performance Computers K.T. Danielson, M.D. Adley, S.A. Akers and P.P. Papados

229

Fracture Mechanism of Pre-split Blasting A. K Dyskin and A.N. Galybin

235

Evaluation of Energy Absorption System for Intense Shock Mitigation LJ.L. JaggL R. Kumari, H. Lal and V.S. Sethi

241

Blast Damage Effects of an Explosion of 5 ton TNT Kept in Storage Magazine H. Lal, R.K. Verma, M.S. Bola and V.S. Sethi

247

Evaluation of Damage and TNT Equivalent of Ammunition, Explosive and Pyrotechnics P. Buri, M.M. Verma and H. Lal

255

New Approach to Street Architecture to Reduce the Effects of Blast Waves in Urban Environments E. H. Mahmoud and J. G. Hetherington

261

Generation and Measurement of High Stresses under Shock Hugoniots S.S. Sachdeva, H. Lal, M.S. Bola and V.S.Sethi

267

Dynamic Response of Model Reactor Structure subjected to Internal Blast Loads A.K. Sharma, V.S. Sethi and P. Chellapandi

275

Spallation of Explosively Clad Metal Plates V.K. Sharma, II. Srivastava and D.R. Kaushik

281

Stress and Strain Magnification Effects in Structural Joints under Shock Loading G. Szuladzinski

287

Numerical Methods in Underwater Shock Simulations H.H. Tran and J. Marco

295

xiv Reinforced Masonry Walls under Blast Loading C. Mayrhofer

301

Crashworthiness

Plastic Collapse Mechanisms of Lifeguards for the Class 465 EMU Bogies E.C. Chirwa, E.J. Searancke, A. Hoe and S.M.P. Wong

311

Application of Multibody Dynamics for Simulating Vehicle Impacts on Steel Safety Guardrails G. Sedlecek, C. Kammel, U.J. Gefller and D. Neuenhaus

319

A Large-Deflection Design Technique for the Collapse and Roll-over Analysis of Thin-Walled Tubular Frames S.J. Cimpoeru, N. W. Murray and R.H. Grzebieta

325

A Method of Estimating Velocity in a Car Crash K. Fujiwara

333

Dynamic Characteristics of Bicycle Helmets S.K. Hui and T.X. Yu

339

Crash and High Velocity Impact Simulation Methodologies for Aircraft Structures C.M. Kindervater, A. Johnson, D. Kohlgrfiber and M. Liitzenburger

345

Design for Crash Safety in Mine Shafts G.J. Krige, W. van S c h a l ~ k and M.M. Khan

353

Comparison of Different Car Front Structures under Nonaxial Impacts M. KrOger

361

Impact Attenuation of Frontal Protection Systems in Passenger Vehicles P. Bignell, D. Thambiratnam and F. Bullen

367

Tubular/SheU Structures

Unified Theory for Collapse of Thin Rectangular Tubes under Compression C.W. Kim, B.K. Han and C.H. Jeong

375

Stress-Strain Relationship for Confined Concrete in Various Shapes of Concrete-Filled Steel Columns K.A.S. Susantha, H.B. Ge and T. Usami

383

Experimental Behaviour of Internally-Pressurized Cone-Cylinder Intersections Y. Zhao and J. G. Teng

389

FEM Analysis of Buckling of Thin-Walled Tubes under Dynamic Loading B. Wang and G. Lu

395

XV

Axial Crushing of Aluminum Columns with Aluminum Foam Filler A.G. Hanssen, M. Langseth and O.S. Hopperstad

401

Failure Mechanism and Behavior of Thin-Walled Reinforced Concrete Barrels under Lateral Loading M.A. lssa, M.A. Issa and R.H. Bryant

407

Crushing Behaviour of Composite Domes and Conical Shells under Axial Compression N.K. Gupta, R. Velmurugan and M.S. Palanichamy

413

The Influence of Residual Stresses in the Vicinity of Circumferential Weld-Induced Imperfections on the Buckling of Silos and Tanks M. Pitcher and R.Q. Bridge

419

Improved Marshall Strut Element to Predict the Ultimate Strength of Braced Tubular Steel Offshore Structures K. Srirengan and P. IV. Marshall

425

The Aseismatic Behaviour of High Strength Concrete Filled Steel Tube Z. Wang and Y.H. Zhen Stub-Column Failure Test of Welded Box Steel Section under Axial Compressive Loading Y.C. Zhang, J.J. Zhang, IV. Y. Zhang and D.S. Li

431

437

Strength and Ductility of Concrete Filled Double Skin Square Hollow Sections X.L. Zhao and R.H. Grzebieta

443

The Quasi-Static Piercing of Square Tubes G. Lu and J. Zhang

451

The Splitting of Square Tubes G. Lu, T.X. Yu and X. Huang

457

Strength and Ductility of Concrete-Filled Circular Compact Steel Tubes under Large Deformation Pure Bending M. Elchalakani, X.L. Zhao and R.H. Grzebieta

463

Connections

Finite Element Modelling of Bolted Flange Connections J.J. Cao, J.A. Packer and S. Du Experimental Behaviour of Moment Connections between Concrete Filled Steel Tubes and Structural Steel Framing Beams J. Beutel, N. Perera and D. Thambiratnam Strength and Ductility of Bolted Connections in Normal and High Strength Steels A. Aalberg and P.K. Larsen

473

479

487

xvi Evaluation of Beam-to-Column Connections with Weld Defects based on CTOD Design Curve Approach

495

K. Azuma, Y. Kurobane and Y. Makino

Simulation of Fracture Failure of Steel Beam-to-Column Connections

501

Y. Chen, Z.D. Jiang and Y.J. Zhang

Failure Analysis of Bolted Steel Flanges

507

P. Schaumann and M. Seidel

Ultimate Capacity of Bolted Semi-Rigid Connections to the Column Minor Axis

513

L.R.O. de Lima, P.C.G. da S. Vellasco and S.A.L. de Andrade

Buckling The Effects of Fabrication on the Buckling of Thin-Walled Steel Box Sections

521

M. Pircher, M.D. 0 'Shea and R.Q. Bridge

Inelastic Dynamic Instabilities of Steel Columns

527

T. Yabuki, Y. Arizumi, C. Gentile and L. W. Lu

Plastic Buckling of Circular Sandwich Plates

533

S. C. Shrivastava

Buckling Instability of a Curved-Straight Pipe Configuration Conveying Fluid

539

A.M. Al-dumaily

Axial Crushing of Frusta between Two Parallel Plates

545

A.A.A. Alghamdi, A.A.N. AljawL T.M.-N. Abu-Mansour and R.A.A. Mazi

Strength Analysis of Buckled Thin-Walled Composite Cylindrical Shell with Hydrostatic Loading

551

J. Brauns

Imperfection Sensitivity Function in Dynamic Response and Failure of 1-D Plastic Structures

557

F.L. Chen and T.X. Yu

Straightening Effects of Steel I-beams Failed by Lateral-Torsional Buckling

563

M. Kubo and N. Sugiyama

Ductility/Constitutive Models Investigation of Damage Accumulation using Equal Channel Angular Extrusion/Drawing

571

R. Lapovok, R. Cottam and R. Deam

A Simplified Constitutive Model for Steel Material under Cyclic Loading Conditions S. Murakami, S. Nara, Y. Shimazu and T. Konishi

579

xvii Acceleration Waves and Dynamic Material Instability in Constitutive Relations for Finite Deformation P.B. B~da and G. B~da

585

Plastic Deformation and Creep of Polymer Concrete with Polybutadiene Matrix O. Figovsky, D. Beilin and dr. Potapov

591

Study of Influence of Loading Method on Results of the Split Hopkinson Bar Test A.D. Resnyansky

597

Enhanced Ductility of Copper under Large Strain Rates D.R. Saroha, G. Singh and M.S. Bola

603

Kinematics of Large Deformations and Objective Eulerian Rates A. Meyers, O. Bruhns and H. Xiao

609

A Study of the Large Deformation Mechanisms of Weft-Knitted Thermoplastic Textile Composites P. Xue, T.X. Yu and X.M. Tao

615

Fire Loading Nonlinear Analysis of Three-Dimensional Steel Truss in Fire P. Fedczuk and W. Skowrohski Modelling of Plastic Strength of Composite Tubular Members under Elevated Temperature Conditions M.B. Wong, ,1.1. Ghojel and N.L. Patterson The Experimental and Theoretical Behaviour of Composite Floor Slabs during a Fire C. G. Bailey

623

629

635

Thermal Contact Resistance at the Concrete/Steel Interface of Concrete-Filled Steel Columns J.I. Ghojel

641

Mathematical Model for the Prediction of Temperature Response of Steel Columns Filled with Concrete and Exposed to Fires J.I. Ghojel

647

Non-elastic Load Capacity of Compressed Steel Truss Member during Fire G. Ginda and W. Skowrohski Fire Resistance of Concrete Filled Steel Tubular Beam-Columns in China State of the Art L.H. Han and X.L. Zhao

653

659

xviii

Earthquake Loading Experimental Study on Steel Bridge Piers with Inner Cruciform Plates subjected to Cyclic Lateral Loads K. Iwatsubo, T. Yamao, T. Yamamuro and M. Ogushi

667

Evaluation of Steel RoofDiaphragrn Side-Lap Connections subjected to Seismic Loading C.A. Rogers and R. Tremblay

673

Low Cycle Fatigue of Concrete Filled Steel Tube Members K. Tateishi, T. Saitoh and K. Muramta The Importance of Further Studies on the Capacity Evaluation of Concrete-Filled Steel Tubes under Large Deformation Cyclic Loading C. Lee, R.H. Grzebieta and X.L. Zhao Design of Large Bridge over the Matchesta River in Seismic Zone A. Likverman, G. Shestoperov and V. Seliverstov

679

685

691

Fracture/Fatigue Tensile Fracture Behaviour of Thin G550 Sheet Steels C.A. Rogers and G.J. Hancock

699

Fatigue Strength Properties of Stainless Clad Steel T. Mori

705

Testing of Welded T-Joint with Fatigue Cracks and Comparison with Failure Assessment Diagram T. lwashita, Y. Makino, K. Azuma and Y. Kurobane

711

Crack Surface Contact under Alternating Plasticity C.H. Wang and L.R.F. Rose

717

Modelling of the Cyclic Ratchetting and Mean Stress Relaxation Behaviour of Materials Exhibiting Transient Cyclic Sot~ening W. Hu and C.H. Wang

723

Influence of Specimen and Maximum Aggregate Size on Concrete Brittle Fracture M.A. Issa, M.S. Islam, M.A. Issa and A. Chudnovsky

729

Fatigue Design of Welded Very Thin-Walled Tube-to-Plate Joints using the Classification Method F.R. Mashiri, X.L. Zhao and P. Grundy

735

Cosserat and Non-local Continuum Models for Problems of Wave Propagation in Fractured Materials E. Pasternak and H.B. Miihlhaus

741

xix Dynamic Tensile Deformation and Fracture of Metal Cylinders at High Strain Rate M. Singh, H.R. Suneja, M.S. Bola and S. Prakash

747

Energy Balance in Dynamic Brittle Rock Failure B. G. Tarasov

753

Stress Intensity Factors for Tubular T-Joints with a Curved Surface Crack B. Wang, S.T. Lie and Z.H. Xiang

759

Effect of the Environment and Corrosion on the Fatigue Life of a Simulated Aircraft Structural Joint S. Russo, P.K. Sharp, R. Dhamari, T.B. Mills, B.R.W. Hinton, K. Shankar and G. Clark

765

Numerical Simulation

Plastic Instability Simulation of Steel in Tension S. Okazawa and T. Usami

775

Several Practical Criteria for Nonlinear Dynamic Stability of Lattice Structures Z.-Y. Shen, Z.-X. Li and C.-G. Deng

781

Snap-Through Analysis of Toggle Frame using the Software Package, NIDA, by 1 Element per Member S.L. Chan and J.X. Gu Second-Order Inelastic Analysis of Steel Gable Frames Comprising Tapered Members G.Q. Li and J.J. Li

787

795

A Parallel Three-Dimensional Elasto-Plastic Finite Element Analysis in a Workstation Cluster Environment Z. Ding, S. Kalyanasundaram, L. Grosz, S. Roberts and M. Cardew-Hall

801

Limit Analysis of Cylindrical Shells subjected to Ring LoadA Comparative Study between Analytical and Numerical Solutions J.R.Q. Franco and F.B. Barros

807

Finite Element Simulation of Deep Drawing of Laminated Steel Y.F. Kwan and M. Takla

813

Analytical Solution for Semi-Infinite Body subjected to 3D Moving Heat Source and its Application in Weld Pool Simulation N. T. Nguyen

819

Pseudorigidity Method (PRM) for Solving the Problem of Limit Equilibrium of Rigid-Plastic Constructions Y. Routman

827

Damage Identification and Restoration of Space Frame using Genetic Algorithm C.W. Shen, X.B. Tang and H.H. Sun

833

XX

Simulation of the Hysteretic Behavior of RC Columns with Footings F.F. Sun, Z.Y. Shen and X.L. Gu

839

An Analytical Method for Analysis of Curved Pair Members tied with Struts H. lshihara, T. Yamao and 1. Hirai

845

Numerical Analysis and Simulation for Cold Extrusion S.X. Zhang, B.K. Chen and H.H. Sun

851

General Structures

Experimental Analysis on Key Components of Steel Storage Pallet Racking Systems N. Baldassino, C. Bernuzzi and R. Zandonini Response of Large Space Building Floors to Dynamic Loads which Suddenly Move to a New Position S. W. Alisjahbana

859

865

Effects of Cables on the Behavior of I-Section Arches Y.L. Guo and J.S. Ju

871

Shakedown of Three Layered Pavements S.H. Shiau and H.S. Yu

877

Laser Application to Surface Deformation and Material Failure S.H. Slivinsky, P. Kugler, H. Drude and R. Schwarze

883

Author Index

889

Keynote Papers

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

IMPLAST SYMPOSIA AND LARGE DEFORMATIONSPERSPECTIVE

A

N.K. Gupta Department of Applied Mechanics, Indian Institute of Technology, Delhi, New Delhi 110 016, India

IMPLAST symposia have come of age, and are valued by the IMPLAST fraternity. A brief history of IMPLAST and its growth since the first in the series was held in 1973, and the contributions that these symposia have made to the subject in general, are reviewed. Over the years there has been a phenomenal growth in the analytical, numerical, and experimental methods for the study of large deformation problems. Mechanics of large deformations, however, is yet not fully understood, and experimental observations are of help in providing plausible explanations, realistic assumptions, and parameters for the analysis of the phenomenon. In the second part of this paper, some observations in large deformation experiments which I hope would be of interest are presented. 1. A JOURNEY THROUGH TIME It is a great honour to be invited to deliver this lecture to IMPLAST-2000, organised by Prof. R. H. Grzebieta and Dr. X.L. Zhao of Monash University. It is the seventh symposium in this series, and in fact the first one outside India- all the previous six were held at the Indian Institute of Technology, Delhi (liT, Delhi). The first of the series was held in November 1973, and its main theme was "Stress Waves in Solids". It all began in 1971 when some like-minded people, encouraged by Prof. B. Karunes of the Department of Applied Mechanics, IIT, Delhi, met. They thought that it would be both expedient and essential for scientists from laboratories, industry and academia to group together and seek motivation in becoming aware of the technical developments in the areas of large deformations at low medium and high velocities of impact. This motivation was essentially conceived in national (Indian) context and a beginning was sought in providing a forum wherein the scientists from various organisations would come together and share their findings on a specific topic within the realm of Plasticity and Impact Mechanics. It was also envisaged that we would invite some renowned scientists from abroad who would share their perception and familiarise us with the latest developments in the subject. We all felt the need to point towards another dimension of curriculum development such that the student body was enabled to keep pace with the theory and practice thus substantiated. Efforts were also made for sensitising and activating hard core industry and state enterprises to make use of the research produced, in the aegis towards real life applications.

During that period some of our own research on large deformation problems involved analysis based on stress wave phenomenon. We also got in touch with scientists working in the area, particularly those in Indian defence laboratories where a lot of experimental work was being done on high speed impact problems. Stress waves in solids, particularly for large deformation problems, seemed almost the obvious choice for the theme of the first symposium that was held on November 1 and 2 in 1973. About eighty scientists from defence and other laboratories, and academic institutions in India attended the symposium; and, Prof. H. Kolsky from Brown University was the key invitee. It was also the beginning of an era in which our defence scientists began participating in an open platform to share their experiences, hitherto kept in closets, mostly. The symposium had its impact on the national scene, and the scientists working in the area became familiar with others having similar interests. Experimental techniques, and analytical as well as numerical methods were progressing at a rapid rate. Every institute, not being able to afford in house all the facilities and the expertise, greatly benefited from the co-operation between scientists, which definitely got a big boost because of this coming together. A modest beginning was made - a nucleus of a fraternity was thus formed. Several analytical, numerical, and experimental studies involving large deformations under low, medium, and high velocity impact were presented [1] in the symposium. Discussions among the participants and those with Prof. Kolsky were quite stimulating. The experience was so good that in the valedictory meeting, it was decided that we would have such meetings biennially on some specialised themes within the general area of static and dynamic plasto-mechanics of large deformations. In 1975, the second symposium was held under the title "Large Deformations in Solids". Here again, the problems dealt with were those of large deformations at speeds ranging from a few mm/s to km/s [2]. The current state of research in India was reviewed as scientists from defence laboratories as well as those from other institutes presented the problems of their immediate interest and disclosed their inadequacies. This provided a basis for some of very fruitful co-operation between the laboratories and the academic institutes. During the above period, much progress was made in improving our own experimental facilities at IIT Delhi, and several of us had increased interaction with senior scientists outside India as well. I too spent six months in 1977 at the University of Cambridge, U.K., with Professor W. Johnson. That was my first stay abroad and it was the first time that I attended an international conference outside India. This provided me with an opportunity of meeting several other scientists in the area. On my return I found Professor Karunes very sick, and in June, 1978, he sadly passed a w a y - and that was the end of our ten-year long memorable and fruitful association. On 17-19 Dec., 1978, we had the third symposium which was dedicated to the memory of Late Prof. B. Karunes. This symposium saw a great change in both objectives of the meeting and the level and quality of participation. We had 130 scientists participating from India and about 25 from abroad. An excellent account of the state of research in large deformation mechanics was given by Prof. W. Johnson [3] and this provided great stimulation and impetus for us. Prof. Th. Lehmann's paper on "Some aspects of coupling effects in thermo-plasticity"[4] and Prof. E.T. Onat's presentation on "Why (and how) should one use a tensor to describe the internal state and orientation of deforming material?", [5] provided an excellent exposition of the constitutive behaviour

of materials. The symposium proceedings was entitled "Large Deformations" [6], and it contained papers dealing with analytical, numerical and experimental studies. Though it took more than two years to come out after the symposium was held, the papers eventually were much improved because of the discussions that took place during the symposium. Those four days of being together generated a bond between all the participating scientists and the IMPLAST fraternity became international. Prof. Johnson encouraged such activity and himself attended all the three IMPLAST symposia held at liT Delhi, thereafter. In the mean time several other congresses came to be organised by societies such as the Indian Society of Theoretical and Applied Mechanics and particularly the Indian Society of Mechanical Engineering, which was formed in 1977. Several of us became office bearers of such societies and were responsible for organising their functions that included the annual conferences. IMPLAST seemed to have merged into these, and was not held as a separate event in the eighties. However, the interaction that started between the scientists during the earlier three meetings continued. The conferences of the societies were very broad in scope and it was quite natural that we began to feel the need for putting in special efforts in exchanging our ideas and reviving our own forum for disseminating our research in the area of plasticity and impact mechanics. The fourth event, as a consequence, was thus organised on Nov. 7-13, 1990. In this meeting each day was devoted to an aspect of large deformations. Keynote lectures presented by Prof. W. Johnson, Prof. N. Jones and Prof. S.R. Reid, amongst others, set the ball rolling in different sessions [7]. A special feature of this meeting was that time was found to discuss some already published research papers on each important aspect. The exercise turned out to be very interesting and extensive exchange of ideas took place leading to suggestions for possible procedures in studying various problems, which were of current interest to many of us. The experience of being together for a few days created fresh bonds and it was decided that the IMPLAST meetings would be held henceforth every third year. IMPLAST'93 was the fifth symposium and it was held on Dec. 11-14, 1993 with the title Plasticity and Impact Mechanics. This coincided with the 10th anniversary of the start-up of the International Journal of Impact Engineering. "Unfinished military history, Plate cutting, and Heat lines" was the title of Prof. Johnson's keynote lecture [8]. In the first part of the lecture, he talked of some historical facets related to Benjamin Robins and his stay in India in the middle of the 18th century. Attention was drawn to the fact that the historical facet is now almost totally neglected by universities; students are not afforded the opportunity to read and learn about men such as themselves, to gain insight into how they faced their life and its specific issues in previous generations. Other keynote lectures [9] included those of Prof. C.R. Calladine [10], Prof. N. Jones [11], Prof. Kozo Ikegami [12], and Prof. N.W. Murray [13]. Prof. Murray's presence in the symposium brought several of us close to him and to Australia. His personal charm and concern for others led to lasting friendships, which we all cherish. The participation in IMPLAST'93 and also in IMPLAST'96, held on 11-14 Dec., 1996, was truly international with scientists participating from various countries including Australia, Canada, France, Germany, India, Japan, the Netherlands, Russia, Singapore, South Africa, UAE, UK, and USA. These symposia dealt with mechanics of large deformation and failure of structures and components when subjected to low, medium, and high velocity impact. Different

materials considered included metals, composites, concrete, wood, and ice. Basic principles, experiments, and formulations presented dealt with important problems such as formulation of constitutive equations including high temperatures and strain rates; analysis of large deformations and failures in structures subjected to excessive dynamic loading; design for survivability and control for collision damage in aircraft, ships, trains, and road vehicles; and determination of ballistic response of armours and structures to high velocity impact and explosion. Keynote lectures in IMPLAST'96 [14] were delivered by Prof. W. Johnson [15],Prof. O.T. Bruhns [16] and Prof. N. Jones [17]. Prof. W Johnson had his 75 th birth anniversary in 1996, and in IMPLAST'96 we had a special function to felicitate him for the contributions he has made to various facets of plasticity and impact engineering. In the valedictory meeting of IMPLAST'96, Prof. Grzebieta kindly offered to hold IMPLAST'2000 in Australia, which was more than readily agreed by all the participants. This is thus the first meeting of the series outside India and of course so well organised. With the phenomenal growth of the multinationals, and global transactions opening up of the geographic space beyond India, this symposium in itself, is the starting point for yet another dimension of the IMPLAST. I do hope that all the participants will enjoy being together during the symposium, and would look forward to being together every three years in future too. Over a period of three decades, from a modest beginning, essentially conceived to be a national endeavour, IMPLAST has grown to be a valued international event. Prof. R.H. Grzebieta and Dr. X.L. Zhao have done a magnificent job in organising this symposium over three days in Melbourne just after the Olympics have concluded. I am sure we all are enjoying our stay here. I express my gratitude to them for this - and for all the efforts that they and their colleagues have put in to make it such a grand and memorable affair. 2. LARGE DEFORMATIONS - A PERSPECTIVE Mechanics of large deformation is inherently a complex phenomenon. What makes it more complex is its dependence on various parameters like strain rate, inertia, history of loading, annealing and thermal processes, and geometry. Simple formulations that describe large deformations and bring together various facets affecting deformation are not available. There is a lack of understanding of the mechanics of the large deformation phenomenon. Structured experiments are essential to be able to study the phenomenon and be able to understand the effects thereon of various parameters of the situation. Motivated by the needs of defence, desire for better safety measures against disasters, industrial applications, and academic interests, great improvements have been made in analytical, numerical, and experimental methods for the solution of such problems. However, many problems relating to the deformation modes and their dependence on various parameters, remain unresolved. Our experiments at IIT Delhi, for understanding the mechanics of large deformations, over the last four decades have been an attempt to study the phenomenon in its varied aspects and to propose simple solutions based on the mechanics observed. In what follows, typical observations in some large deformation experiments, which are of interest, are presented in a hope that plausible explanation for these having been found, would help in understanding the large deformation

phenomenon. Obviously, I have not tried to exhaustively dwell in explaining the phenomenon, some of which can be seen in the references cited.

2.1. Necking in Simple Tension The tensile deformation and the corresponding influence of specimen size, particularly in relation to the instability condition leading to the onset of necking [18, 19] is an interesting phenomenon. The classical treatment of instability suggests that a neck would appear in a round specimen of a strain hardening material at the peak of the load displacement relation. This criterion, as revealed by past studies appears to be valid for time independent material behaviour and also when slenderness ratio is quite high. Studies have shown that the appearance of necking is delayed well beyond the point of maximum load due to both strain rate sensitivity and decrease in slenderness ratio. It is, however, evident from the existing literature that tensile deformation, particularly after

2.00t

(o) 3230 I 20

1.00

LO=10Omrn,do=30rnrn

0~ I

~~'"~'~

to0.20

C~,~

"~

~~

O.lO

]D

09/-,

d

(b) b

c

z

Q- 10 0

I

10

j,,

~

,

I

rn)2Orn30~ 8( ,045

0s

80 80100120150180220d. ~(MNm_2)._.

Fig. l. (a) Load - deformation curve for an aluminium specimen; and (b) stress - plastic strain data superposed for several aluminium specimens the onset of necking, has not been amenable to simple analytical solution. Presented below are some results of simple uniaxial tension tests on specimens of different original diameters and slenderness ratios, which reveal some very interesting features. These are of relevance to the understanding of (a) stress and load conditions at the onset of nonuniform deformation and its further localization; (b) the manner in which uniform deformation transforms itself into localized necking; and (c) effect of specimen size on the tensile response of a material. A typical load deformation curve for an annealed aluminium specimen is shown in Fig. 1(a), wherein three regions AB, BC and CD have been identified and corresponding extensions during these periods are denoted as 61 , 6 2 , and 8 3 respectively. A typical true stress and plastic strain data based on the current diameter measurements at the neck section, both before and after its formation, is shown in Fig. 1(b), superposed for many specimens. The corresponding three regions herein are marked as ab, be, and cd. Careful experiments on several specimens revealed that (a) 81 is dependent on length of specimen and independent of its diameter; (b) 8 2 is a function of both diameter and

length of the specimen; and (c) ~53 is dependent on original diameter of the specimen and is rather independent of length. Fig. 2 illustrates the nature of these relationships for 45-

40-

/

(o)

30 E ,EE20co 10 / / / / 0

19 -

/ $1=0"225 L~ ,

I 20 r

~

,

50

,

100

,

150 200

(b)

o< =Lo/d o

eb=0"2011 f15 ~" s =0.3456 E o< =16.67~

/

=

/

9.33-

10 -

/

6.67-',Z~/,

Lo(mm) ' ' "

do(rnm) ~

do(turn)

--~

Fig. 2. Dependence of (a) ~51, (b) ~52, and (c) ~53, on length or diameter of the specimen aluminium specimens. Interestingly no such dependence was apparent in the stressplastic strain data, derived from the continuous measurements of the current specimen diameter, see Fig. 1(b). During a test, if we measure the current diameter at two sections of the specimen - a neck section and a section away from the neck - it is seen that, (a) the diameter reduction is identical at both sections until the specimen extension is 51 , and, therefore, the specimen deformation is uniform until this stage; (b) during the period of extension 8 2 , the reduction in diameter at neck section is faster than the corresponding reduction at the other section, indicating thereby that the deformation during this period begins to be differential over the length of the specimen; and (c) the deformation during the period of extension 6 3 is accompanied by a continuous fall of load, with diameter reducing only at the neck section. Loss of stability is thus seen to be a gradual process that occurs during

_d2.5 "

,-1. '

I--

-

I--~-60~--"11~- 0 100 ,--. (o) _[.6.2 ~-~6C -"~,

5.4

.."

J

I

""---'-1"

--q.lOl'--~ 60

"~1 . ~

D

=I

~1OI------60----~ =.101~---

~----- 60

-f:

-

60"-"--~

-,

(d~1:

Fig. 3. Alumium specimen: (a) untested; (b) when unloaded at a stage of necking; (c) refigured; (d) with neck formed at the specimen end on retesting

the period of extension 82 , and as a consequence a sort of diffused neck extending over the length of the specimen precedes the appearance of a localised neck. With the onset of non-uniform deformations, strain hardening, and the instability stress marking the initiation of necking, begin to be different over the length of the specimen [15]. To illustrate this, a uniform specimen of diameter 9 mm, see Fig. 3 (a), was subjected to simple tension test, and, at some stage after the necking had begun it was unloaded. Neck diameter at this stage was 6.2 mm, Fig. 3 (b). The specimen was then machined to make its diameter uniformly 6.2mm through out. It was then refigured, see Fig. 3 (c), at its mid length, where an artificial constriction was machined to make the minimum diameter 5.4 mm at the mid length i.e., the mid length area of the specimen was reduced by 25% as compared to the rest. This specimen was again subjected to the tension test until a new neck appeared. It is important to note that this neck appeared at the specimen end and not at its middle where the area of cross-section was reduced by 25%.

2.2. Barrelling in Simple Compression When a short cylindrical specimen is subjected to a uniaxial compression between two overhanging rigid platens, forces of friction generated at the interface of platen and specimen end face begin to constrain its deformation there. Consequently, lateral expansion of the specimen, at any time, is maximum at its equatorial section and minimum at the end sections. This gives rise to barrelling of the free cylindrical surface, the extent of which depends in a complex manner on factors including interface friction, strain hardening characteristics of the material, and history of loading. Use of conical dies or intermittent lubrication and machining the specimen, when barrelling became evident, was carried out earlier to offset the effects of friction and obtain uniform deformation. Several studies have discussed qualitatively the effect of lubricant (which creates conditions of low interface friction) on the barrel profile, and obtained bollarding with P.T.F.E. sheet used as a lubricant. The friction conditions, however, are generally not very well understood. Several experiments conducted on various metallic materials reveal that specimen deformation at some stage of barrelling begins to be accompanied by the rolling or folding of the material from the cylindrical surface to the end faces of the specimen. The end face thus after this stage consists of the original end face surrounded by a ring of rolled material. Initiation of this rolling process is accompanied by a sharp rise in the load deformation curve. It is, therefore, important to identify the precise stage at which the rolling begins, its extent and the changes it would induce in the load deformation behaviour. Cylindrical specimens used in the above tests [20] were marked by drawing concentric circles on the end faces and parallel circles along the height at different intervals. During a test, the diameter at the equatorial plane, the current height of the specimen, current diameters of concentric circles marked on the end faces (by interrupting the test) and the current diameter of the end face (which includes the ring of rolled materials) were measured. Several specimens of different diameters and slenderness ratios were tested [21 ]. Fig. 4 shows a plot between e a and e h , where e a is strain at the equatorial plane based on diameter measurement and e h is the strain based on height measurement. A linear

10 relation in Fig. 4 gives a sensible approximation considering the complexities involved in the phenomenon. Its use affords great amount of simplification in the otherwise complex DO=50 mm

~

.

,7"40

OIL~--L / 0.4 ""

(~h

'

o.4F J or

0

3" (a)

.2' " ~

20

~P-" P'"

/.. --

12 L

(b)

"/

L/,",,

. J " In _ 0.8i"" ~hO 4

/..

Do =20ram " "

o 0.4 o.B ~.2

--

/ /oo,o..

~

~~

V, .......

0.4 0.8 1.2 1.6 2.0 2.4 2.8,0 0.40.g 1.2 1.6 2.0

6a Ea Fig. 4. Height strain versus area strain plots for different diameter specimens of (a) aluminium, (b) steel problem. Nature of variation of the current diameter of the end face (including the rolled material), the current diameter of the original end face, and other concentric circles with the nominal stress is illustrated in Fig. 5 for 30 mm diameter specimens of height to diameter ratios varying from 0.5 to 3.

H(,o D

Ca) ~

:~~

.0

!~'~ 1 .0

(6)

!

Hobo

~----~~2.0

,.o

~I} ~,-~

Holdo

2.O

" 2.0

o ~

/////),

~ 2~i ~. 81012182022

Ho/Do

-3.0 r - ~ 0 . 5

,~,,~

~0

.Concentric circtes 28 32 36 40 44 48 52 56

Current diameter

Fig. 5. Nominal stress versus current diameters of concentric circles and the end face including rolled ring. It is of interest to note that the rolling of material as indicated by the point of bifurcation of the two curves in Fig. 5, appears to coincide with the point of inflexion in the corresponding load compression graph. This clearly indicates that the rising load gradient is direct consequence of the initiation of rolling phenomenon. It is also seen that the width of the ring of rolled material increases with the increase in the slenderness ratio of the specimen. It is interesting to note that all other conditions remaining the same, higher the slenderness ratio, lower is the increase in diameter of the original end face, and that

most of the increase in the end face diameter is due to rolling of material from the free surface of the specimen to its end faces, see Fig. 5 (c). 2.3. Collapse of Thin Metal Shells The plasto-mechanics of structural elements like tubes of circular and non-circular cross-sections, spherical shells, and conical frusta, have received considerable attention during the last four decades. Their application in the design of devices for absorbing kinetic energy in situations of a crash or an accident is common. Various factors that determine the efficiency of performance of the energy absorbers, and their selection criteria have been discussed in detail in [22]. The axial collapse mechanisms of thinwalled tubes of circular, square or rectangular sections under static or dynamic loading in particular, have been studied by various investigators in the past [23-25]. Here we present some experimental observations, which are of interest. 2.3.1. Axial Crushing of Round Tubes Axially crushed thin walled tubes are perhaps the most investigated structural elements. Their progressive collapse is either axisymmetric due to local axial and radial buckling or diamond due to local circumferential buckling They provide an efficient way of absorbing the kinetic energy of impact. Therefore, study of plasto-mechanics of their post-collapse deformation has received considerable attention. Most of the solutions available in literature pertain to the concertina mode of collapse and the analysis for diamond mode of collapse is almost non-existent. The analytical approaches of analysis of tubes and frusta for axisymmetric folding have so far been by either considering straight folds or curved folds of circular curvature or their combination. In the straight fold models, the energy absorption in bending is assumed to be concentrated at the location of hinges. In many of these studies, the folding has been assumed to be either total outside or total inside. It has, however, been observed in experiments on cylindrical tubes that the folds are partly inside and partly outside. Plausible factors contributing to such folding include consideration of the influence of variation in stress-strain behaviour of the material in tension and compression [26]. The problem in these modelling techniques is in the estimation of the peak load at which the folding starts. The average load obtained analytically on the basis of the formation of independent folds can not be compared with the experimental results because the folds do not form independently. In most of the analytical studies, only two modes of deformation viz. bending and circumferential deformations have been incorporated. Experiments on round tubes of materials like aluminium and mild steel of different sizes and aspect ratios have shown that their mode of deformation remains quite insensitive when tested under quasi-static or drop hammer loading. It is however seen [27] that the size of the specimen, annealing processes, and the presence of any discontinuity like a circular hole influence the mode of deformation very much. The experiments on both as-received and annealed tubes of aluminium and mild steel, reveal that the progressive collapse mode is concertina, diamond, or mixed depending on their state of work hardening, subsequent annealing process and the geometry of the tube. For tubes of d/t ratios between 10 and 40, it is found that a highly cold worked as-received aluminium tube deforms in diamond mode and when annealed, it deforms in a ring mode. On the other hand, as-received strain-hardened steel tubes deform in concertina mode and

12 on annealing, they deform in diamond mode, see Fig. 6; this behaviour is exactly opposite to that of aluminium tubes.

Fig. 6. Deformed shape of the 52.6 mm diameter steel tube in (a) annealed; and (b) asreceived state The corresponding stress-strain curves of the respective materials reveal that their slope at the onset of plastic deformation is much higher in the case of aluminium when annealed and in the case of steel in as-received condition. An experimental study has been carried out in which two diametrically opposite holes were drilled in the tubes of various dimensions of aluminium and mild steel. The diameters of these holes in different tests were varied. It has been observed that the collapse begins at the location of holes if the diameter of hole is greater than a minimum value. It was seen in experiments that these tubes did not buckle in the Euler mode, even for lengths that were much larger than the buckling length of tubes without holes. Figure 7 shows typical load-deformation curves for aluminium tubes of D = 36 mm. It was seen that the tubes without holes collapsed in the Euler mode for L/D = 5, while the tubes with holes did not collapse in the Euler mode even for L/D = 10. The D/t in this case was 22, and it may be seen that holes afford the possibility of increasing the critical overall buckling length by more than 100%. Deformed shapes of a typical aluminium specimen of D/t = 36 mm, L/D = 3 are shown in Fig. 8 at six different stages of the test. The hole diameter is 9 mm in this case. Typical deformed shape of an aluminium tube with two opposite holes is shown in Fig. 9.

201- / ~ / \

~

OL 0

.....

\

, 10

W i t h hole

X._

L=9

, , - 7-" 20 30 - 40 C o m p r e s s i o n (ram)

'"

'

T 50

Fig. 7. Load deformation curve for tubes with and without holes.

"'

, 6c

13

Fig. 10 Deformed shapes of aluminium tubes of.49.4 mm diameter with two opposite holes in one, two and three planes respectively from right. The holes also reduce the first peak load and this reduction is dependent on the hole diameter. The holes may be in one or more planes mad their number is to be optimized. Typical deformed shapes of tubes with two opposite holes in one, two or three planes are shown in Fig. 10.

14 2.3.2. Axial Crushing of Frusta

One of the major advantage in using a frusta as compared to cylindrical tubes as energy absorbing device is that it minimizes the chances of collapse by buckling in Euler mode. Another significant feature of frusta is its increasing collapse load with progression of crushing excepting for large semi-apical angles (> 60 ~ for which reverse bending takes place at some later stages of collapse and that causes fall in load. In experiments, frusta have been found to normally fail by diamond mode excepting those of very low and very high semi-apical angles. As the frusta of low semi-apical angles may fail in concertina mode that is perhaps why many of the studies available in literature seem to be for frusta of low semi-apical angles. The frusta of semi-apical angles up to about 30 ~ are found to begin yielding with an axisymmetric ring, and thereafter these collapse progressively by multi lobe diamond fold mechanism [28]. In case of frusta of semi-apical angles of about 45 ~ and above, plastic buckling is initiated at the smaller end by a rolling plastic hinge resulting in the formation of an inverted frusta. Some typical load-deformation curves of frusta collapsing due to the movement of rolling plastic hinge are shown in Fig. 11, wherein it is seen that the

2,I 18

_q tK

--'dr =180"5

/

.

{;[~N"~J(/~-

0'

0

l

10

-

.

.

I

.

t

20 30 Compression (ram)

61.9 ...... I

40

I

5O

Fig. 11 Load-Compression curves of frusta collapsing due to the movement of rolling plastic hinge structural component shows the ability to sustain the load at the nmximum level with the progress of compression. Figure 12(a) and (b) show deformed shapes of the frusta of semi-apical angle 30 ~ and 45 ~ respectively. The former collapse in diamond mode while the latter collapse by the movement of rolling plastic hinge. Similarity of the collapse mode of the large angle frusta with that of hemispherical shells under axial loading [29] can easily be seen; collapse in these is initiated by inward dimpling and it progresses with the formation of the rolling hinge. In the case of these frusta with low t/d values, in the later stages of compression, stationary plastic hinges are also formed. In all the cases of frusta of semi-apical angles 65 ~ at a certain stage of compression, a reverse bending occurs from the larger end associated with a rolling plastic hinge.

15

Fig. 12 Deformed shapes of frusta of semi-apical angle (a) 30~ and (b) 45 ~

2.4. Collapse Behaviour of Composite Shells Composite thin walled shells, such as tubes, frusta, hemispherical shells, and domes are potential candidates for their use as energy absorbing elements in crashworthiness applications in aircraft and other transport vehicles due to their high specific energy absorbing capacity and the stroke efficiency. The main advantage is that designers have greater flexibility in tailoring the material to meet the specific requirements of loading and changing environment. Their failure mechanism however is highly complicated and rather difficult to analyse. This includes fracture in fibers, in the matrix, and in the fibermatrix interface in tension, compression, and shear. Experimental and theoretical studies on axial compression of empty and foam filled cylinders and cones [30-31 ], reveal that once the matrix crack is formed, it is followed by the breaking of the fibres due to hoop strain. This leads to the formation of petals with fibres bending both inside and outside the mean radius of the shell; nature of this petal formation depends on the material and size of the shell. Axial loading experiments on the composite hemi-spherical shells reveal that their collapse is mainly due to the fracture zones initiated along the meridian and circumferential directions; the latter form at certain regularly decreasing intervals (dl) depending on the radius, thickness and the co-latitude angle ~ of the shell. Shells of lower thickness are found to collapse by fragmentation, but those of higher thickness collapse by inward splaying [32]. The progressive collapse of a dome observed along the meridian and circumferential directions is shown in Fig. 13. The zones of collapse along the circumferential direction are formed successively with the progress of collapse. Mean collapse load of the composite hemi-spherical shells is influenced more by their thickness than their radius. The lateral collapse of GFRE tubes of varying D/t ratios occurs by the formation of four longitudinal fracture lines. For tubes with D/t greater than 10, all fracture lines are located at 90 ~ phase angle. Tubes of D/t less than 10, however, fail by the formation of two fracture lines close to the contact lines of tube with flat platens and the remaining

16 two are located at about 60~ angle; see Fig. 14 [33]. Also, the zones between these close fracture lines are subjected to heavy delamination. Formation of two close fracture lines Too. f-ptate

]--[

l J_ tdh T

.

Z t.

"

~ ~ Bottom ptote

Z~--~,~

c,ne of actuat fracture ercurnferentiat direction) Assumed zone

/ x~/ ./~,1 ] Zone of fracture in the ~' ~ ' . ~ ' x y ~2/y/~'mer'dian direction

Fig. 13 Schematic diagram showing the formation of fracture zones in domes. is due to delamination occuring in the region of the first two fracture lines. It is due to this reason that a considerable portion of the fiat load-deformation curve is obtained for tubes of smaller D/t ratio. Tubes of smaller D/t ratio may also undergo progressive fracture. When D/t is small, variation of strain across the thickness is large, and thus leading to progressive fracture rather than sudden fracture. On the other hand for tubes with large D/t ratio, enough strain gets developed simultaneously in all layers, which is sufficient for causing sudden fracture over the total thickness of the tube. In the case of

A

~

._,'IX

2

,

~

. . . . . .

I~

~

"-. X

~,

/~B=AB=b

_-4--- -I-

Fig. 14 Lateral collapse model for composite tubes with D/t < 10

17 the random orientation of fibres in the tube, stresses developed in the fibres crossing the fracture lines are different. Only those fibres get fractured in which fracture strength is exceeded and the rest remain unfractured. Delamination occurs when the bond strength between the fibres is exceeded. In that case fibres do not get fractured because the delamination causes relief of stress in the fibres. It is due to this reason that the tube does not get separated along fracture lines like brittle material and significant recovery of deformation is observed in experiments. Recovery after failure, however, is not important because energy absorption potential of recovered GFRE tubes is very small as seen from their load-deformation curves. 2.5. Impact of Projectile on Plates Comprehensive surveys of the mechanics of penetration and perforation of projectiles into the targets have been published by Backman and Goldsmith [34], Zukas [35], and Corbett et.al. [36] covering the major experimental and analytical works done in the field. The first formulae to be developed predicted the penetration depths into semi-infinite targets when struck normally by a projectile. The advent of battleship armour in the 19th century led to the development of equations predicting the depth of penetration of finite thickness armour plating. Even to this day these formulae and others like them are being used extensively by impact engineers. In recent years appreciable advances have been made in the analytical approach to the problem of impact with the models gradually becoming more and more sophisticated and more accurate. However, these, too have relied heavily, and indeed still do, on experimental data to justify certain assumptions made and to supply various parameters for the models. A commonly used measure of a target's ability to withstand projectile impact is its "Ballistic Limit Velocity (BLV)" simply known as "Ballistic Limit" and much work has been carried out by researchers to enable estimates of this parameter. Another useful term I '

1000 -

(a)

"Fin

E 800

t = a 10 m m o 12

I

A16

I

o MiLd steel o Atuminium

o 20

120l-|

10o_ 9

._~

u 600 O

(b)

25 --Computed

7

_~ 0 -

>

-6 400

~ 4a h5 g 2o

:3

I/1

200

Z

I0

20 30 40 59 Thickness of ptote (ram)

60

70

O0

,

,I,

10

I

J

i ,

i,,

I

.J

20 30 40 50 60 70 AngLe of obl.iquity

Fig. 15 (a) Residual velocity variation for the impact of projectiles on plates of different materials, and (b) Velocity drop with the angle of obliquity for MS plates. Incident velocity is 820 rn/s.

18 is "Ballistic Limit Thickness (BLT)" [37], which is the minimum thickness of plate required for a projectile of known weight and velocity to prevent any perforation. Figure 15 (a) shows a typical residual velocity variation for the impact of projectiles on plates of different material and thickness for 820 m/s incident velocity. The relationship between the velocity drop and the angle of obliquity is shown in Fig. 15 (b) for MS plates of various thicknesses. Armour steels although the oldest of armour materials, are still considered satisfactory material in dealing with ballistic protection. A basic requirement of armour steel is that it should have high hardness; but it seems that there is no simple correlation between hardness and resistance to perforation, as measured by a structure's ballistic limit. Increasing thickness of the monolithic homogeneous armour beyond a limit begins to present constraints of weight, manufacture, and cost. This has led to the consideration of possible targets made of layered plates of metals, non-metals and their combinations for improving the efficiency of the armour as well as for achieving the required thickness conveniently. It has also been noted that an efficient combination is a hard front face to break up the projectile and a ductile rear face to absorb the projectile's kinetic energy. Many of the available studies pertain to the behaviour of layered targets of the same material. It is seen that for relatively thick plates (with t > t*/4, where t* is the ballistic limit thickness) in two layers, the residual velocities are comparable to those for single plates of the same total thickness. However, when the plates are thin, (t < t*/4), the layered combinations in contact gives higher residual velocity. For spaced targets, the residual velocity is higher than for the plates in contact. For two-layered targets of MS, when the total thickness is greater than t* and the thickness of each layer is less than t*, the projectile gets embedded when the front layer is thinner than the rear layer. However, when the front layer is thicker, one encounters an interesting phenomenon; the projectile penetrates up to a certain depth and then rebounds back, presumably due to a stress wave effect. When a projectile perforates a target at an oblique angle of incidence, it is observed in experiments that it does not come out of the rear side in the same straight path, but tends to turn towards or away from the normal to the plate. This deviation depends on the angle at which it strikes the plate, its material, and the thickness of the plate. When the projectile is fired at an angle greater than the angle for the ballistic limit, a stage comes when the projectile penetrates the plate and comes out of it from the impacted side itself. 3. CONCLUDING REMARKS I have presented above some examples of experimental observations in their pristine form in an attempt to draw attention to the basic complexities of the large deformation phenomena. I have tried not to obscure these by theory or mathematics. There are, however, many issues, concerning the delineation of the mechanics of large deformation under various loading and boundary conditions; numerical methods and analytical solutions; and material constitutive behaviour, which need attention. Three days of the symposium will address many important aspects of relevance to these issues. I conclude by observing that IMPLAST-2000 has been prodigiously successful in bringing us together from all parts of the world. I am sure, we shall carry fond memories of the days spent in Melboume. We all are conscious of the immense efforts required to organise a successful conference of this magnitude; Prof. Grzebieta and Dr. Zhao have

19 no doubt done a fabulous job. I thank them both personally for giving me this opportunity and wish you all an enjoyable and fruitful stay. REFERENCES 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.

B. Karunes and N.K. Gupta (eds.), Stress Waves in Solids, Proc. of the Symposium held at liT Delhi, N. Delhi, India, 1973. B. Kanmes, N.K. Gupta (eds.), Large Deformations, Proc. of the Symposium held at liT Delhi, N. Delhi, India, 1975. W. Johnson, Large Deformations, Proc. of the Symposium held at IIT Delhi, N. Delhi, India, (1978) 1. Th. Lehmann, Large Deformations, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1978) 37. E.T. Onat, Large Deformations, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1978) 164. N.K. Gupta and S. Sengupta (eds.), Large Deformations, Proc. of the Symposium held at liT Delhi, N. Delhi, India, 1978. N.K. Gupta (ed.), Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, 1990. W. Johnson, Plasticity and Impact Mechanics, Proc. of the Symposium held at IIT Delhi, N. Delhi, India, (1993) 1. N.K. Gupta (ed.), Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, 1993. C.R. Calladine, Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1993) 71. N. Jones, Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1993) 29. Kozo Ikegami, Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1993) 52. N.W. Murray, Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1993), 197. N.K. Gupta (ed.), Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, 1996. W. Johnson, Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1996) 1. O.T. Bruhns, Plasticity and Impact Mechanics, Proc. of the Symposium held at liT Delhi, N. Delhi, India, (1996) 37. N. Jones, Plasticity and Impact Mechanics, Proc. of the Symposium held at IIT Delhi, N. Delhi, India, (1996) 21. N.K. Gupta and B.P. Ambasht, Mechanics of Materials, 1 (1982) 219. N.K. Gupta and B. Karunes, Int. J. of Mech. Sci., 21 (1979) 387. N.K. Gupta and C.B. Shah, Proc. of Symposium on Large Deformation, (1978) 146. N.K. Gupta and C.B. Shah, Machine Tool Design and Research, 26 (1986) 137. W. Johnson and S.R. Reid, Applied Mechanics Reviews, 31 (1978) 277. J.M. Alexander, Q. J. Mech. Appl. Math., 13 (1960) 10. W. Abramowicz, N. Jones, Int. J. Impact Engng, 2 (1984) 263. N.K. Gupta and R. Velmurugan, Int. J. Solids & Structures, 34 (1997) 2611.

20 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

N.K. Gupta and H. Abbas, Int. J. Impact Engng., Communicated. N.K. Gupta and S.K. Gupta, Mechanical Sci., 35 (1993) 597. N.K. Gupta, G.L.E. Prasad and S.K. Gupta, I. J. Crash, 2 (1997) 349. N.K. Gupta, G.L.E. Prasad and S.K. Gupta, Thin Walled Str., 34 (1999) 21. N.K. Gupta, R. Velmurugan and S.K. Gupta, J. of Composite Materials, 31 (1997) 1262. N.K. Gupta and R. Velmurugan, Int. J. of Composite Materials, 33 (1999) 567. N.K. Gupta and G.L.E. Prasad, Int. J. of Impact Engg., 22 (1999) 757. N.K. Gupta and H. Abbas, Int. J. of Impact Engng. 24 (2000) 329. M.E. Backman and W. Goldsmith, Int. J. of Engineering Science, 16 (1978) 1. Zukas, J.A., High Velocity Impact Dynamics, John Wiley and Sons, 1990. G.G. Corbett, S.R. Reid and W. Johnson, Int. J. o Impact Engng., 18 (1996) 141. N.K. Gupta and V. Madhu, Int. J. of Impact Engg., 19 (1987) 395.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

21

B u c k l i n g of Thin Plates and T h i n - P l a t e M e m b e r s - S o m e points of interest J Rhodes

Department of Mechanical Engineering University of Strathclyde Glasgow Scotland, UK.

A brief examination of some of the research on the post-buckling elastic and plastic behaviour of plates and plate structures is outlined. This field is so wide ranging that only a very superficial examination has been carded out, and the writer has concentrated on some specific aspects of the general field of study.

1. INTRODUCTION When Euler produced the first paper on the buckling of columns in 1744 this constituted, to quote Salvadori [ 1] " a solution in search of a topic" since with the materials and structures current at that time nothing buckled. Indeed, for quite some time thereafter the problem of buckling was theoretical pie-in-the-sky. This did not remain the case forever, and nowadays a knowledge of buckling and its effects are basic requirements for engineers. In the case of plate structures probably the first references to buckling arose during the mid 19th century. Walker [2] told of a series of tests carded out in the laboratories of University College London on box beams of a variety of cross sections in connection with a projected suspension railway bridge across the Menai Straits by Robert Stephenson. The tests showed that in a number of cases failure was due to the phenomenon now known as local buckling.

The first theoretical examination of plate buckling was by Bryan [3] who obtained a solution to the problem of a simply supported plate under uniform compression in 1891. Since then numerous researchers have investigated local instability in plates under a wide variety of loading and boundary conditions using many different methods of analysis. There has been a number of excellent text books which have described the main results of these investigations, for example [4]-[5], and the reader is referred to these textbooks for a general study of plate instability. In this paper attention will be focussed mainly on the effects of buckling on subsequent plate behaviour.

22 2. POST-BUCKLING BEHAVIOUR OF PLATES AND SECTIONS. 2.1. Plate behaviour at and after buckling.

When a compressed plate buckles it develops out of plane ripples, or buckles, along its length. This behaviour is illustrated in Figure 1 for a thin-walled section in which local buckling is present in all the plate elements. In the elastic range the buckled portions of the plate shed load, and become ineffective in resisting further loading, while in the portions of plate close to supports the out of plane buckling is diminished, and these parts have post-buckling reserves of strength and stiffness. The plate as a whole sustains increases in load after buckling, but the axial stiffness reduces. This effect is demonstrated in Figure 2, where point A is the buckling point. For a plate without imperfections the post-buckling axial stiffness drops immediately upon buckling, and thereafter reduces still further as loading increases. Also because of the highly redistributed stress system the maximum stress grows at an increased rate after buckling, ensuring earlier failure than if the plate remained unbuckled.

Perfect plate " Load P

rfect _

plate

I

End displacement u Figure 1. Locally buckled thin-walled section

Figure 2. L o a d - End displacement path.

As the load increases the stresses also increase. The consequences of this are inevitably detrimental to the plate continuing to fulfil its function, but the way in which the plate fails depends very much on the material from which the plate is made. Fibrous composites, for example, have a wide variety of failure possibilities. In this paper such possibilities will be disregarded, and research into ductile material only will be considered.

2.2. Von Karman Large Deflection Equations

The post-buckling behaviour of thin plates is governed by two simultaneous non-linear differential equations originally set up by von Karman [6] and modified some time later by Marguerre [7] to take account of the presence of initial imperfections. These equations may be written in terms of deflections w, initial imperfections Wo and stress function F as follows:-

23

34F

O~4F

~+2 3x 4

o14w ~ X4

o~x2B y2

+2

- q+D[ - D

+

o~4 w

03 X 2 t~ y 2

[(

3 4F 3 y4 = E ~ oxO y) - Ox 2 0 y z

+

a4w

(1)

=

t~ y 4

c9~F 3 2 ( w + w o) + 3 2Fc92(w+wo) 1 - 2 3xO y2 03xO y Ox 2 By2 J

3zF r

o3y 2

[,c)xOy ) ..I Ox z c?y z

tgx2

(2)

The first of these equations, sometimes called the "Compatibility Equation", ensures that in an elastic plate the in-plane and out-of-plane displacements are compatible. The second equation is based on equilibrium principles, and is sometimes termed the "Equilibrium Equation". Exact solution of these equations is only possible for the simplest loading and support conditions, and in the earliest days of plate postbuckling analysis recourse was made to empirical equations and to significantly simplified analysis to examine plate behaviour.

2.3. Empirical Equations Early research into the post-buckling behaviour of thin-plates was carried out largely in the aircraft industry. In 1930 a large series of compression tests on plates of various materials and having a wide variation in plate width was carried out by Schuman and Back [8]. The plates were simply supported on all edges, and the tests indicated that for plates wide enough to buckle locally before failure the ultimate load which could be carried did not increase in proportion to the width. Indeed beyond a certain width the ultimate load was insensitive to variation in actual width. Over the next few years a number of theoretical investigations were carried out to examine this phenomenon, and in 1932 the first effective width expression was developed by von Karmen et al [9]. This expression states that for a plate of actual width "b" an effective width "b," can be used in the evaluation of the load carrying capacity. Von Karman's effective width expression can be written in terms of the critical stress crcR and yield stress o'r as follows:-

b

where

Vo'r

K ~z2E t 2 CrcR = 12(1_v2) b2

(3)

(4)

In the case of a simply supported plate 1(=4 and for a steel plate with E = 205 N/IIII/I 2, v--0.3 and (rr =280 N/ram 2 the effective width at failure is 51.4t where t is the plate thickness regardless of the plate actual width.

24 It should be mentioned that in the evaluation of the effective width expression it was ensured that the buckle half wavelength in the plate assumed a value which would produce the minimum effective width. Von Karman's effective width expression was found to be conservative and reasonably accurate for thin plates for which the critical stress is very much less than the yield stress. In the case of plates in which the critical stress and yield stress are similar there is a great deal of scatter, imperfections cause substantial reduction in the load capacity and equation (1) is nonconservative. To overcome this, Winter [10] later modified yon Karman's equation to:-

b

VO'e)

(5)

The second term within the brackets modifies yon Karman's equation mainly at the point where the yield stress and applied edge stress are similar. This expression was used in the AISI specification for cold-formed steel members [ 11] until it was modified again (The 0.25 term was changed to 0.218, or 0.22 in some design codes) and in its latest form is probably the best known and most widely used expression from which plate post-buckling strength can be determined. This equation is used in many National design specifications, and in International specifications such as Eurocode 3 [12]. In the determination of the compressive capacity of a cross section the effective widths for all plate elements of the cross section are computed at the yield condition and then summed to evaluate the total effective area of the section. This is then multiplied by the yield stress to provide a value for the squash load of a strut taking local buckling into account. In the application of this approach each plate element is considered separately, although some design codes take some account of interaction between elements via the critical stress. In the 1940s and for some time later an alternative method of approach, based largely on testing, was developed. A number of investigators, e.g. Heimefl [13], Schuette [14], Chilver [15] derived empirical equations governing the load capacity of different short strut sections. Some of these are as follows:-

/0.2 Heimerl [ 13]

am'x = 0.769 ~

t, a , )

a~

f

for Z and C sections

(6)

/0.2

O'm~x = 0.794 O'CR

for H Sections

(7)

25 (

Schuette [14]

or.= =

\0.25

0.8" [

Chilver [15]

r

t, a , ) [

(rtmx =

or,

(8)

for aluminium channels

(9)

,~1/3

= 0.863 | O'c, |

a,

for Z , C and H sections

-%113

0.736 ""/c'~'/

for steel channels

(10)

t a, )

The fact that all of these expressions have factors less than unity signifies that for members in which yield and local buckling theoretically occurred simultaneously the experimental results were less than the theoretical buckling, or yield, load due to imperfections. It is interesting to note that equations (4) - (8), empirically derived for sections, have smaller indices than 0.5, derived for individual plate elements. Figure 3 shows a comparison of the "Plate effectiveness" (i.e. either Effective width/full width or Maximum stress/yield stress) given by each equation. As may be observed the values given by Eqn. (5), i.e. the effective width curve are less than those given by the curves based on complete section strength. This could perhaps be taken to suggest that the curve from Eqn. (5) is rather conservative. This is not borne out by Figure 4, however, which plots a comparison with the effective width/full width ratio for the tests which had originally been used to establish this effective width equation. Note that in Figure 4 the abscissa is the square root of that in Figure 3. -------- Eqn (6) Eqn (7)

1.2 r

...... - Eqn (8) ....

.g'_

Eqn (9)

--------- Eqn (10)

0.8

- = = - = - Eqn (5)

~ 0.6 JlB~

~

~lll~l

IIIIBIIIB

~IRI~

~lllllll

iiiiiill

mlllll i allll ii

N 0.4 0.2 0 0

2

4

6

8

10

Ratio of Yield Stress to buckling stress

Figure 3. Variation of Effectiveness with Ratio of Yield Stress to Buckling Stress

26 1.2 ....

0.8 ,.Q

Eqn (5) 9 Experiments

0.6 0.4 III

0.2 F

9

o 0

1

2

3

4

5

6

7

V((YY/(YCR) Figure 4. Comparison of Winter' effective width expression with experiments The main reason why the effective width curve for individual elements gives lower values than an effectiveness curve derived on the basis of a complete section is the fact that in a section some elements are participating fully in buckling while others are not. In a cross section some elements initiate buckling, while other elements restrain the buckling elements. The elements which initiate buckling lose effectiveness readily, while the restraining elements remain highly effective until the compression reaches a stage at which these elements would buckle naturally. This is illustrated in Figure 5, from Ref [ 16]. In the box section under examination the thinner walls buckle first, with high restraint from the thicker elements which have much lower deflections than the thin elements initially and the axial stiffness of the box is only reduced to about 80%-85% of its initial value due to buckling. When the end displacement reaches the value at which the thicker elements would naturally buckle as simply-supported elements then these begin to participate fully in the buckling of the section and the axial stiffness drops sharply to well under half of its original value. The effective width approach as used in design codes such as the AISI code [ 11] cater for this differential behaviour of different elements in a cross section, as each element is analysed individually, while the complete section approach cannot take the individual variances in sections into account unless different formulae are used for different sections. It appears that the complete section approach, which has taken second place to the effective width method

2.4. Elastic Plate Analysis Just after Von Karman produced the first effective width equation, Cox [ 17] performed an approximate energy analysis of plate post-buckling behaviour. Cox's approach considered that the membrane strain in the loaded plate was constant in the direction of load. The approximations effectively lead to neglect of the effects of shearing stresses in a plate and the method postulated by Cox became known to later researchers as the "lower bound" method, as the plate post buckling stiffness was generally underestimated due to the neglect of some of

27 160tl 120~- 8 0 -

t;,i~k;/ p,ot.

---/----i~,it

/,0-

1:ft.: 1: I 2

line |0| gives ~ corresponding to buckling of

.~ ;/~ I f _

if edges were simply supported

jr

0-

I

I"

10

0

'

20 u

-

i

.......

30

~0

b l-*P--t,

'~ ,,-, P

=

Pb

30q /

,77

~

.'I / I", '."l ..)' ..... , f ........ t" q

/

,'P"

,= 2 0 -

/,,/

." 0 ;-0

i 1"

"

..-'.---"

'|

II

i

i

2

3

i

~

'"i 5

W

tl

Figure 5. Buckling of a Box Section with sides of unequal thickness the strain energy. This method is not a bound of any kind, but provides a simple approach to the approximate analysis of plate post buckling behaviour. After this early pioneering work Cox went on to produce extremely important theoretical findings in plate post-buckling research including the explanation of the reason for snap changes in buckle mode etc. In the years immediately following Cox's first at)proximate analysis a number of researchers produced variations on this approach until the first rigorous solution of the plate post-buckling problem was carried out by Marguerre in 1937 [18]. Marguerre's approach was to postulate an approximate deflected form for the plate, determine the corresponding stress function by solving the compatibility equation (1), and employ the Principle of Minimum Potential Energy, rather than the equilibrium equation to furnish the final solution. Researchers in later years very often used a similar type of approach, i.e. combining an exact solution of the compatibility equation with either evaluation and minimisation of the Potential Energy, or an approximate solution (for example using Galerkin's method ) of the equilibrium equation. With the development of rigorous solutions to plate problems came the recognition of the importance of boundary conditions. While deflection and edge slope conditions were quite

28 obvious and were well appreciated because of their applicability in the examination of initial buckling, rigorous examinations of post-buckling behavieur required also a knowledge of the in-plane loading and deformation conditions. These are highly dependant on the type of construction under consideration. In many bridge, ship and aeroplane structures, where a multiplicity of plates are aligned in much the same plane, the in-plane displacements of adjacent plates at their junctions is such that displacements normal to the plate edge are either zero, or are constant along the plate. Perhaps the most widely applicable condition here is that the plate edges can move outward or inward, but must remain straight. In light structural members, where each plate element is orientated at an angle to the adjacent element any tendency for the edge of a plate element to move in plane is generally not resisted adequately by the adjacent element and so waving of the edges of such elements is probable in the postbuckling range. A detailed examination of the boundary conditions applicable to plate elements is given by Bentham in Ref [ 19] With regard to out of plane displacement conditions, situations in which the plate edges are held straight in-plane also tend to induce conditions approaching simple support, or fully fixed, conditions e.g. for bridge, plate and ship type plates while plate elements of thin-walled structural sections in general have some intermediate degree of restraint on rotation of the unloaded edges. In the years immediately following the second word war a number of investigators improved the knowledge of plate post-buckling behaviour. Among notable research presentations Levy produced the first "exact" solution, in series form, to yon Karman's equations [20], Hemp [21 ] examined simply supported and fixed edge plates under uniform compression. Cox [22] investigated in depth the effects of in-plane edge on plate behaviour, and obtained a solution to the problem of sudden "snap" transition from one buckled wavelength to another, a phenomenon which had previously been observed experimentally. Hu et. al. [23] and Coan [24] studied the effects of imperfections. Yamaki produced perhaps the most comprehensive analysis up to that date in 1959 [25], [26], examining plates with combinations of simply supported and fully fixed boundary conditions, with unloaded edges either free to wave inplane or constrained in-plane. An investigation by Stein [27] in 1951 is worthy of special mention. Stein used the perturbation approach in which the solution is obtained in terms of the power series expansion of a "perturbation parameter". The parameter used by Stein was :P

a=------l+

wo

----

(11)

where P is the applied load and PcR is the critical load to cause buckling A complete picture of the plate behaviour could be derived in terms of a power series of this parameter. The first two terms of this power series could effectively detail the plate post buckling behaviour well into the far post buckling range. Essentially this meant that by obtaining analytical solutions at two specific points, one of which could be the buckling point, and utilising the pertubation approach a picture of the complete post-buckling range of behaviour of identical plates with any magnitude of imperecfion could be produced.

29 Walker used this approach in 1969 [28] to obtain explicit solutions for square simply supported plates. The results were used in the 1975 edition of the UK specification for the design of cold-formed steel specimens. Williams and Walker [29] extended this study to deal with a wide variety of plate geometries and boundary conditions, and tables of coefficients obtained from a finite difference analysis were given from which the reader could analyse the plate of his choice. It is only a short step to go from this position to fitting expressions to the coefficients so that by solving simple equations the coefficients governing rectangular plates of arbitrary buckle half wavelength and arbitrary boundary restraint conditions can be determined. In Ref [30] slightly modified forms of explicit expression, obtained on the basis of a Marguerre type analysis allied to the perturbation technique, are presented. The explicit expressions are in the following forms:P]

)

P/

Pc

0" m

O'eR

-

= ( q - 1 ) t ~ + c 20:2

(12)

=

1)a + c,

(13)

= csCt + c6Ct2

(14)

(c, -

with ct as defined in Eqn. (11), CrcRas defined in Eqn. (4), O'm the maximum membrane stress and e and ecR being the average and critical strains in the plate loaded direction. Expressions for the coefficients Cl to c6 for plates free to wave in-plane on the unloaded edges with varying buckle half wavelengths and rotational restraints on the unloaded edges are given in Appendix 1. The rotational restraint coefficient, R, has a value such that Mb R = ----OD

(15)

where M is the moment per unit length opposing rotation of a plate unloaded edges, 0 is the rotation of the unloaded edges, b is the plate width and D the plate flexural rigidity factor. These formulae gave fairly simple yet accurate representation of the behaviour of plates with any buckle half wavelength, any magnitude of initial imperfection and any degree of restraint on edge rotation within the limits of plate large deflection theory. The slight modifications which were incorporated into the explicit expressions were made to eliminate the possibility of ill conditioning affecting the postulated behaviour in the far post-buckling range, and these equations give results in close agreement with existing theory in comparable cases. Load-out of plane deflection curves and load compression curves for simply supported square plates are shown for illustration in Figures 6 and 7.

30 w0=0

/t

'.~-0.2 0.4 ' - ' ~ 0.6 ~0.8

~ ' ~ 1.0

P

PcR

l

~

el 1 Simply supported

unloadal edges 0

1

2

3

w/!

Figure 6. Load --out of plane deflection curves for square plates

-~- - 0 0.2 0.4 0 6 t I -/-/~o's

3

P

|

0

Simply supported

2

4

6

8

Figure 7. Load end displacement curves for square plates. Figure 8 shows, in the case of .perfect plates for clarity, the variation of load with axial compression into the far post buckling range for plates of a variety of buckle half wavelengths

31

~.

e--I 0.9 0.8

10

0.7

0.6 0.:5 ~. &

0.4 P

yon Kannan

Simple support on unloaded edges

0

lO

20

30

40

50

Figure 8. Load-compression behaviour in the far post-buckling range From this figure it is obvious that as the compressive strain increases the buckle half wavelength for minimum load decreases, although not by as much as the von Karman expression suggests. The von Karman effective width expression is shown here, and it can be seen to be a little more conservative than the lowest of the perturbation curves, but is fairly close to the lower envelope of these curves. In recent years elastic plate postbuckling analysis has been extended substantially by the computer, by virtue of finite element and finite strip approaches. There have many of these approaches presented in journals and conferences in recent years, and some sample references are [31]-[35].

3. ELASTO- PLASTIC ANALYSIS Investigators who have studied the elastic postbuckling behaviour of plates and plate structures have often suggested that failure occurs more or less coincidentally with first membrane yield in compression. This hypothesis has held up over the years mainly because of two facts, namely (1) - It is simple and (2) - It accurately portrays the situation. However, although the failure load can be accurately obtained in many cases by this hypothesis, the deformation behaviour of plates and plate structures at and after failure cannot be evaluated accurately for ductile materials by elastic theory. Because of this, in any case in which the failure and post failure behaviour of a structure is required then generally plastic behaviour must be taken into consideration.

32 In the design codes for cold-formed steel sections it was assumed for many years that the ultimate load which could be carried by light gauge members was that which caused first yield to occur, and first yield was taken as the failure criterion for cold-formed beams. In the writer's PhD research [36] he observed that tensile yield could be accommodated quite safely so long as the compressive stresses were elastic. This has now come to be recognised, and design taking account of tensile yield is allowed in several light gauge steel design codes. The situation where compressive yield occurs in a thin-walled member is much more complicated, however. Probably the first elasto-plastic plate post-buckling analysis was carded out by Mayers and Budiansky [37] in 1955. The accuracy of their method of approach depended upon the accuracy with which they could postulate expressions for three different displacements simultaneously, and this prevented them from determining a condition in which the applied load reached a maximum value. The writers therefore considered that collapse would have occurred when the unit shortening, or average edge strain attained a value of 1%, and took the load at this point as the collapse load. The loads so evaluated were greater than those obtained in experiments. A substantial amount of research into elasto-plastic plate behaviour was carried out in the 1960s at Cambridge University, e.g. [38], [39]. Perhaps the major work here was that of Graves-Smith who examined the interaction of local and column buckling in a landmark paper which also used a rather rigorous plasticity analysis [40]. This paper was the forerunner of numerous papers in the 1970s on elasto-plastie plate behaviour, for example by Moxham [41], Frieze et. al. [42], Rogers and Dwight [43], Little [44], Crisfield [45] to mention only a few. Most of the work was highly computer-orientated, using finite difference and finite element approaches. There were a number of attempts made to obtain simplified analysis of plate elasto-plastic behaviour. One of these, due to the writer [46] will be briefly detailed here. It had been found by Botman and Besselling in the 1950s [47] that derivation of an effective width for plates using elastic analysis gave good predictions of failure when applied to plates with non-linear behaviour, e.g. aluminium. It was therefore interesting to investigate whether effective widths determined in terms of strains or plate shortening using elastic analysis and then using these together with the elasto-plastic stress strain law would give a realistic assessment of the behaviour. As it happens, such an approach gives an extremely accurate assessment of the actual behaviour. It was found that the simple approach gave results in very good agreement with computer predictions and/or experimental findings for a wide variety of plate conditions. Figures 9 to 11 show comparisons of the results of the simple analysis and those of elastoplastic computer analysis, or experimental findings as appropriate. Figure 9 shows results of the simple approach compared to those of Frieze and Dowling for simply supported plates with the unloaded edges constrained to remain straight. The agreement is excellent. In Figure 10 the simple approach predictions are compared with the experimental results of Moxham again showing excellent agreement. It is noteworthy that the approximate results seem to be equally good for cases in which the theoretical buckling strain is greater than the yield strain as it is for cases when initial buckling is elastic. This suggests that elastic buckling analysis can be used in the post-yield range for plates, with strains substituted for stresses. It is of course true that for purely elastic plates the buckling strain is independent on the modulus of

33 (DIO

0"8

P/Py

v'(ovlE) - 1'037

(blt) v'(oyIE) - 2-074

0-6

Simply suPoortecl square plates Unloaclecl eOges constratneO to rema, n straegnt

0.4

wolf - 0 0 9 4

,L . . . . . .

0.2

0

1

Present method Frieze et al 2

~JEy

3

Figure 9. Comparison of approximate elasto-plastic analysis with Frieze et. al. !

.,m, .':"

-.

.0.8

...(_b/t)~ay/E= 1.59 0-61, P/PY ~

/

/ |

0'2l

i/

0

/

/

I

l,~rX;:,

I

x,-~...

"---"

"~'~"~~'_-~

T (blt)~/OY/E= 2"12

simol.VSUOl:)oneclplates.....

Stres.s-lree on unloaoea eoges vo.;,.*o-O

_

Present m e t h ~

0:s

..... ;

,:s

~Ey

~

2.~

Figure 10. Comparison of approximate elasto-plastic analysis with experiments

34

--15-7

0-6 b -21-9 t

O-S J~

Pv Present method Experiment (Rogers & Dwight) 0.2

(~s

1

1.5

~Y

Figure 11 Comparison of approximate elasto-plastic analysis with experiments on outstand elements elasticity, but theories which do not presume linear elasticity of the material do not result in the same simple finding. This result is therefore most interesting. The simple approach also applies to flange elements, or unstiffened elements, or outstand elements. Figure 11 illustrates the simple analysis for simply supported- free plates with length to width ratio of 8:1 and three different plate width to thickness ratios in comparison with the experimental results of Rogers and Dwight. Here again the simple analysis is equally good for plates in which yield precedes buckling as it is for plates which buckle elastically. For these plates the simple analysis was much closer to the experimental results than was the numerical analysis of the authors. In the case of plates with a free edge loaded by compressive stresses which have their maximum nominal values at the free edge local buckling causes the stresses near the free edge to shed and the stress variation in this region can be complex. Here again, however, the use of an elastically derived effective width for these elements yields a simple evaluation of strength. It can be shown that a simple von Kannan effective width type analysis for unstiffened elements yields an effective width at yield varying as the cube root of the critical strain ecR divided by the yield strain, er. Using the formula

35 /"

b__, =

b where

ece-

-xl/3

(16)

\er)

with

12(l-v2) ~-2

3.4

K = 2+

for the channels considered

h ) l+h

values of the effective flange width b~ at failure can be determined using an elastic/perfectly plastic stress strain law and applying simple elastic or elasto-plastic beam theory as appropriate. The comparison of failure moments calculated in this way with experiments [48] shows good correlation as is illustrated in Figure 12. It is of interest here that steel plates with compressed free edges and having a width to thickness ratio of up to 30 can be seen to display some post-elastic capacity, while these plates with width-thickness ratios of around 15 show experimentally fully plastic capacity. It is worthy of note that these were cold-formed steel channels, and design codes for cold-formed steel do not in general suggest anything like the capacities found here. The channels examined had flange width to web width ratio, h, varying from 0.25 to 1, and the upper solid curve is the theoretical curve corresponding to h=0.25 while the lower curve is the theoretical curve for h=l. In the range of b2/t < 30 the failure load tends towards the fully plastic load, and it is noteworthy that for these sections the shape factor is of the order of 1.8.

Mull

"i

~t

\o-

tO

/

I

~'-i

b2/ /~M

/

I Elasto-plastic range

b~

b2 is flange width h = b2/bl

Elastic range

~

. o o

- --~b

zo

30

40

---

sO

eO

7o

80

9o

b2/t Figure 12. Variation of ultimate moment with flange width/thickness ratio for plain channels bent such that the flange free edges are in compression

36 4.

PLASTIC MECHANISM ANALYSIS

The growth in the use of plastic mechanism analysis to examine failure and post-failure behaviour of thin-walled members has been substantial over the past three decades or so. In 1960 Pugsley and McCaulay [49] and Alexander [50] examined cylindrical columns using mechanisms, and cylinders have since been subjected to intensive research with regard to axial crushing, e.g. [51]- [54]. Ben Kato [55], in 1965, was the first author to the writer's knowledge to apply mechanism theory to investigate axially compressed plate elements. The main aim of his work was to derive knowledge of limiting width to thickness ratios of plate elements below which the full plastic capacity could be ensured without buckling. From the early 1970s an explosion in the development of the mechanism approach ensued. This was influenced in no small way by the work of Murray e.g. [56], [57] who published extensively on the use of plastic mechanisms in thin walled beams, stiffened panels etc. Murray summarised the research to date in 1984 [58]. It is not within the scope of this paper, nor the capability of the writer, to give an exhaustive account of plastic mechanism analyses. These now have been used to study the behaviour of Civil, Mechanical, Offshore, Automobile, Train and Aircraft structures, and within these fields mechanism analysis has been applied to such a wide variety of problems that to attempt any comprehensive coverage can not be contemplated within this paper. Instead, a brief mention of the mechanism analyses which have been carried out at the University of Strathclyde in recent years will be made, on the grounds that very little of the research at this University on plastic mechanisms has been published other than in the form of Research Theses.

4.1. Research at Strathclyde University, UK There have been a few M.Phil research projects carried out over the past 15 years or so dealing with the static and dynamic impact behaviour of transversely loaded beams. The work of these projects has not been published. Sin [59] examined a variety of problems involving plate and beam behaviour using mechanism analysis. Included among these were the collapse behaviour of channels in bending, and "refined mechanism analysis of plates". In an endeavour to produce a mechanism approach which could differentiate between different types of plate in-plane and out of plane boundary conditions Sin took account of membrane yield and bending yield lines in plates and produced results quite close to those discussed earlier in Figures 9-11 on the basis of mechanism analysis. Wong [60] studied static and dynamic axial crushing behaviour of closed hat sections. To aid his research Wong was largely responsible for the design and build of an impact test rig which could hurl a 60 kg mass at a specimen with a velocity of up to 60 miles per hour. Wong carried out about 500 static and dynamic crushing tests. Setiyono [61] used mechanism analysis to study the crippling behaviour in thin-walled beams.

37 Lim [62] examined the behaviour of plain and lipped channel and Z section beams using mechanism approaches. Lim's main aim here was to examine statically indeterminate beams when the cross section slenderness was such that local buckling could occur either before plasticity had started, or when moment redistribution was ongoing. There are two other PhD projects under way on side-impact absorbers at the present time.

4.2 Some remarks regarding inclined plastic hinges. Around 1980, on first studying mechanism analysis, some particular points raised the writer's interest. One of these concerned the general capability of mechanism theory to consider the finer points of plate behaviour, i.e., as mentioned previously, the differences in behaviour of plates with different in-plane boundary conditions was not immediately amenable to calculation using the methods available. Some of the work in Sin's thesis studied this, with some degree of success, but as there still remains some work to be done on this nothing has been published to date. A second, and related, topic concerns the question of the moment capacity of inclined hinges. The writer examined the inclined hinge shown in Figure 13 in the early 1980's, using the von Mises yield criterion to get the following expression for the moment per unit length on the inclined hinge:-

Ilia/ ] '

17,

1 -4[.NooJ sin27 (4 - 3 sin 7)

where No is the yield stress resultant. N ~

I",,

!

",,,

N

Figure 13. Inclined hinge in axially loaded plate.

38 Although this expression was used by Sin [59], Wong [60] and Lira [62] in their PhD theses, and has also been adopted by some colleagues in joint research in Poland, it has not until now been compared with other yield line analyses for inclined hinges. In addition, since its publication until recent times has been limited to PhD theses its existence has not been noticed. It is perhaps an appropriate time to give this expression an airing. In preparing this paper a substantial amount of theoretical and experimental work carried out on the yield capacity of inclined hinges has been brought to the writer's attention. Of particular interest here are the mechanism analyses of of Zhao and Hancock [63]-[65] and the further work of Zhao, Lip and Gzebieta [66]. In ref [64] expressions for the moment capacity of inclined yield lines were derived, and checked against a series of experiments in [65]. The expressions derived could only be solved iteratively, although simplified versions were also produced by curve fitting. Figure 14 shows a comparison of the results given by Equation 17 with those of Ref [64] for hinges at angles, y, of 0, 30 ~ 60 ~ and 80~ Since Equation 17 is based on the von Mises yield criterion, and this is generally considered the most accurate then only the von Mises results from [64] have been shown. These were obtained simply by measurement from the relevant figure in [64] and apologies are made for any unintended errors in reproduction of these results. Also, since Equation 17 was derived for a moment per unit length of hinge then to compare directly with [64] the non-dimensional moment values from Equation 17 must be multiplied by cos y.

1.2

Solid lines give values obtained using Eqn. (17) and symbols alongside each line are values from Ref [64]

1

3,=30~

t-(9

E o 0.8 E r

~60 ~

t--

._o 0.6 C 0

E "? 0.4 c-

y=80 ~

o

z

0.2 ~"

0

*

.

0.2

0.4

0.6

0.8

1

N/No Figure 14. Comparison of inclined hinge moment capacities

39 As may be seen from Figure 14 the differences between the predicted moment capacities is not great, although there are differences. For zero hinge inclination it seems that both approaches give identical results- in Equation 17 the denominator becomes unity. For other values of hinge inclination there are some differences with Ref [64] values being slightly greater at low values of axial loading, with Equation 17 values being slightly greater at high axial loading for some hinge inclinations. As to which of these two particular approaches is the more accurate it is not easy to say, the differences are not substantial. There are some basic differences in the analytical reasoning behind the two approaches, but the end results do not seem very different. The behaviour of inclined hinges is also discussed thoroughly in Ref [67] which reviews much of the work prior to 1990 5.

CONCLUDING

COMMENTS

This paper was intended to give a brief summary of plates and plate structures in the elastic and plastic range from the writer's particular viewpoint, without attempting to be in any way comprehensive. It has quickly become obvious that a such a summary of this rapidly expanding field is certain to omit vast quantities of extremely important research. Some of the main researchers in the field have either not been mentioned, or only mentioned in passing, although their contributions to the field have been extremely substantial, and the writer apologises for the necessary omission of many important names and works.

REFERENCES 1 2 3 4 5 6

M.G. Salvadori. Buckling, Buckling ...Buckled. Introductory Speech at 1986 Annual Technical Meeting of the Structural Stability Research Council A.C. Walker. A Brief Review of Plate Buckling Research. Behaviour of Thin-Walled Structures. Eds J Rhodes and J Spence. Elsevier, 1984 G . H . Bryan. On the stability of a plane plate with thrusts in its own plane with applications to the "buckling" of the sides of a ship. Proc London Math. $oc. 22,1981 S.P.Timoshenko and J. M. Gere. Theory of Elastic Stability.McGraw-Hill, 1961 P.S. Bulson. The Stability of Flat Plates, Chatto and Windus, London, 1970. T. von Karman. Festigheitsprobleme im Maschinenbau. Encyclopaedie der

Mathematischen Wissenschaften, 4, p349, 1910 K. Marguerre Zur theorie der gekreummter platte grosser formaenderung. Proc fifth Int. Congress for Applied Mechanics. Cambridge, 1938 8. L. Schuman and G Back., Strength of rectangular flat plates under edge compression, NA CA Rep. No.356, 1930. 9. von Karman, E. E. Sechler and L. H. Donnel., Strength of thin plates in compression, Trans. ASME, 54, 1932. 10. G. Winter, Strength of thin steel compression flanges, Cornell Univ. Eng. Exp. Stn, Reprint No.32, 1947. 11. American Iron and Steel Institute Specification For The Design of Cold Formed Steel Structural Members, AISI, New York, 1996

7

40 12. CEN ENV 1993-1_3:1996. Eurocode3:Design of Steel Structures - Part 1.3:General Rules-supplementary rules for cold-formed thin gauge members and sheeting 13. G. J.Heimerl. Determination of plate compressive strength, NACA Tech.Note No.1480, 1947. 14. F. H. Schuette, Observations on the maximum average stress of flat plates buckled in edge compression, NACA Tech. Note No.1625, 1947. 15. A. H. Chilver, The maximum strength of the thin-walled strut, Civil Engineering, 48, 1953. 16 J. Rhodes. Secondary local buckling in thin-walled sections. Acta Technica Academiae Hungaricae, 87, 1978 17. Cox, H. L., Buckling of thin plates in compression, ARC R & M No.1554, 1934. 18. K. Marguerre, The apparent width of the plate in compression, NACA TA No.833, 1937. 19. J. P. Benthem The reduction in stiffness of combinations of rectangular plates in compression after exceeding the buckling load Nat. Aero Research Inst, Amsterdam, NLL- TRS 539, 1959. 20. S. Levy, Bending of rectangular plates with large deflections, NACA Ret No. 737, 1942 21. W. S. Hemp, The buckling of a fiat rectangular plate in compression and it behaviour after buckling, ARC R & M No.2041, 1945. 22. Cox, H. L., The theory of flat panels buckled in compression, ARC R & A No.2178, 1945. 23. P. C. Hu,, E. F. Lundquist and S. B. Batdorf, Effect of small deviations from flatness on effective width and buckling of plates in compression, NACA TA No. 1124, 1946. 24 J.M. Coan, Large deflection theory for plates with small initial curvature loaded in edge compression, Trans. ASME, 73, 1951. 25. N. Yamaki, The post-buckling behaviour of rectangular plates with smt initial curvature loaded in edge compression, J. of App. Mech. 26, 1959. 26. N. Yamaki, The post-buckling behaviour of rectangular plates with smt initial curvature loaded in edge compression -(Continued), J. of App. Mech. 27, 1960. 27 M. Stein, Loads and deformations in buckled rectangular plates, NASA Teci Rep. R-40, 1959. 28 A.C. Walker. The posr-buckling behaviour of simply supported square plates. Aero Quarterly, XX, 1969 29 D.G. Williams and A. C. Walker. Explicit solutions for the design of initially deformed plates subject to compression. Proc I. C. E, 59, 1975 30 J Rhodes. Microcomputer design analysis of plate post-buckling behaviour. Jnl of Strain Analysis, 21, 1986 31. S Sridharan and T. R. Grave Smith. Postbuckling analysis with finite strips.Proc. ASCE, 107, EM5, 1981. 32. G.J. Hancock, A. J. Davids, P. W. Key, S. C. W. Lau and K. J. Rasmussen Recent developments in the buckling and nonlinear analysis of thin-walled structural members. Thin-Walled Structures 9, 1990. - The N. W. Murray Symposium. 33 S. Wang and D. J. Dawe. Spline FSM post-buckling analysis of shear deformable rectangular laminates. Thin- Walled Structures, 34, 1999 34 Y.K. Cheung, F. T. K. Au and D. Y. Zheng. Nonlinear vibrations of thin plates by spline finite strip method. Thin-walled structures, 32, 1998 35 K.S. Sivakumaran and N Abdel Rahman. A finite element analysis model for the behaviour of cold formed steel members. Thin-walled structures, 31, 1998

41 36 J. Rhodes. The nonlinear behaviour of thin-walled beams subjected to pure moment loading. Phi) Thesis, University of Strathclyde, Glasgow, 1969. 37 J Mayers and B Budiansky. Analysis of the behaviour of simply supported flat plates compressed beyond the buckling load into the plastic range. NACA TN No 3886, 1955 38 A.T. Ratcliffe. The strength of plates in compression. PhD Thesis, Cambridge, 1966 39 J. B. Dwight and K. E. Moxham Welded steel plates in compression.. The Structural Engineer, 47, 1969 40 T.R. Graves Smith The ultimate strength of locally buckled columns of arbitrary length. Thin-walled steel constructions. Symposium at University College, Swansea, 1967 41. K. F. Moxham. Theoretical determination of the strength of welded steel plates under in plane compression. Cambridge University, Report CU ED/C-Struct~R65, 1971 42. P. A. Frieze, P> J Dowling and R. F. Hobbs. Ultimate load behaviour of plates in compression. Steel plated structures. Crosby Lockwood Staples, London, 197Z 43. N. A. Rogers and J. B. DWIGHT Outstand strength. Steel plated structures. Crosby Lockwood Staples, London, 1977. 44. G. H. Little, Rapid analysis of plate collapse by live energy minimisation, Int. J. Mech. Sci, 19, 1977. 45. M. A. Crisfield. Ivanov's yield criterion for thin plates and shells using finite element, Transport and Road Research Laboratory, Rep. LR919, Crowthorne, 1979. 46 J. Rhodes On the approximate prediction of elasto-plastic plate behaviour., Proc. Inst. Civ. Engrs., 71, 1981. 47. M. Botman and J. F. Besseling. The effective width in the plastic range of flat plates under compression. NIL, Amsterdam, Report 5,445, 1954 48. J Rhodes. Research into the mechanical behaviour of cold formed sections and drafting of design rules. Report to the ECSC, 1987 49 S. A. Pugsley and M Macaulay. The large scale crumpling of thin cylindrical columns. Quart. J. Mech. And Appl. Math., XIII, Part 1, 1960 50 J.M. Alexander. An approximate analysis of the collapse of thin cylindrical shells under axial loading. Quart. J. Mech and App. Math, XIII, Part 1, 1960 51 A. Andronicou and A. C. Walker. A plastic collapse mechanism for cylinders under axial end compression. Jnl of Constr. Steel Research, 1, 1981 52 R . S . Birch and N. Jones. Dynamic and static axial crushing of axially stiffened cylindrical shells. Thin-Walled Structures 9, 1990. The N. W. Murray Symposium. 53. R. H. Grzebieta Research into failure Mechanisms of some thin-walled round tubes. Plasticity and Impact Mechanics. Ed N. K. Gupta. New Age International (P) Ltd., 1998 54 N.K. Gupta and R. Velmumgan. Axi-symmetric axial collapse of round tubes Plasticity and Impact Mechanics. Ed N. K. Gupta. New Age International (P) Ltd., 1998 55 B Kato Buckling strength of plates in the elastic range. IABSE, 25, 1965. 56 N.W. Murray. Buckling of stiffened panels loaded axially and in bending. The Structural Engineer, 51, 1973 57 A.C. Walker and N. W. Murray. A plastic collapse mechanism for compressed plates. IABSE, 35, 1975. 58 N.W. Murray. Introduction to the theory of thin-walled structures. Clarendon Press~ Oxford, 1984. 59. K.W. Sin The collapse behaviour of thin-walled sections. PhD Thesis, University of Strathclyde, Glasgow, 1985

42 H . F . Wong. Dynamic and static crushing of closed hat section members. PhD Thesis,

60

University of Strathclyde, Glasgow, 1993. H. Setiyono. Web crippling of cold formed plain channel steel section beams. PhD

61

Thesis, University of Strathclyde, Glasgow, 1994 T . H . Lim Some plasticity studies relating to thin-walled beams. PhD Thesis, University

62

of Strathclyde Glasgow, 1995 63. X. -L. Zhao and G. J. Hancock. Plastic mechanism analysis of T-joints in RHS subject to combined bending and concentrated force University of Sydney. School of Civil and

Mining Engineering. Research Report No. R763, 1993 64. X. -L. Zhao and G. J. Hancock. A theoretical analysis of the plastic moment capacity of an inclined yield line under axial force. Thin-Walled Structures, 15, 1993 65. X. -L Zhao and G. J. Hancock. Experimental verification of the theory of plastic moment capacity of an inclined yield line under axial load. Thin-Walled Structures, 15, 1993 66. X. -L. Zhao, E. O. T. Lip and R. H. Grzebieta. Plastic Mechanism analysis using newly derived yield line theory. First Australian Congress on Applied Mechanics, Melbourne,

1996 67. R. H. Grzebieta. On the equilibrium approach for predicting the crush response of mild steel structures. Ph.D Thesis, Monash University, 1990. APPENDIX.

Table

1.

Table of buckling coefficients K and postbuckling coefficients for use with Eqns. 4 and 11-15 Coeff

Simply supported plates

Cl

3 + 1.1e 3

Cls =

0.22

C2s

(e

C3

-

0.07) 2 +

1 + 13e 4

c6

C2r = (e - 0.2)2 + 0.07

GF

---

2.44 + 13.25 e l+5e 3

C4s = 0.54 + 0.08 2

0.15 C4v = ""5- + 0.1

C~s = L64 + 2.35e 2 + 0.255e4

Csr = 1.2 + 3.6e 2 + 0.3e 4

e

c5

1 + 1.375e 3

++

0.088

0.06

3 + 50.6e 4 C3s =

Intermediate conditions

2.44 + 1.7 e 3 elF m

1 + 0.673e 3

C2

C4

Fully fixed plates

C,~ =0.21e2 + 044/e 2 -0055

c,

Ks = 2

+ e 2 + l/e 2

C3 "--

C3s -(0.08 +0.5e) R C3r 1 - (0.08 + 05e) R

c4= C~ --(O.175+O075e) R C4v 1 - (0.175+ 0075e)R

e -

C~

C6r =0.4e 2 + 0.08e 4 - 0.15

K r =2A9+5.139e 2 +0.975/e 2

11 Cls - 0.094R/C-at

C2 = C2s - (0.071 e 2 )R C2u 1 O.07Rle 2

C6 "-

IK

1 - 0.094R

Css - 0 . 2 R Csv 1 - 0.2R

C6s - 0.2 e: R C6r

r= g

1 - 0.2e 2 R

-QRtG

1-QR

Q = 0.1e / (0.152 + e)

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

43

Failure predictions of thin-walled steel structures under cyclic loading Tsutomu Usami and Hanbin Ge Department of Civil Engineering, Nagoya University, Nagoya 464-8603, Japan This paper deals with failure predictions of thin-walled steel structures subjected to cyclic loading. To trace inelastic behavior of steel structures, an elastoplastic large displacement analysis using a modified two-surface plasticity constitutive model for the material, is carried out. Discussions of the buckling mode, displacement and strain at failure are made in detail. Empirical formulas for predicting the ultimate strength and ductility are also presented. 1. INTRODUCTION The basic philosophy in the seismic design of a structure is to ensure that the supply capacity is no less than the demand anticipated by a specified earthquake motion. To this end, sufficient capacity in regards to strength, ductility and/or absorbed energy should be provided by the structure. It is well known that thin-walled steel structures are susceptible to local and overall interaction buckling because their sections are characterized by a large width/radius to wall thickness ratio. Their failure behavior can be investigated through an experimental procedure. However, such an approach is far from covering various structural profiles and related parameters. On the other hand, when an analytical approach is considered as an alternative, it is no doubt that a precise analysis is needed. This paper presents failure predictions of thin-walled steel structures under cyclic loading. Elastoplastic large displacement cyclic analysis is carried out to study inelastic behavior and obtain the supply capacity. Cantilever-type steel columns either with a box section stiffened by longitudinal stiffeners or with a pipe section are chosen as numerical examples. To simulate the cyclic behavior of steel with good accuracy, a modified two-surface plasticity model developed at Nagoya University (Shen et al. 1995) is employed. To gain a good understanding of inelastic behavior, numerical results of the local buckling mode, deformed configuration and stress progression are presented in detail. Finally, formulas for predicting the strength and ductility at failure, which are based on the extensive parametric analysis, are briefly summarized.

2. STRESS-STRAIN RELATIONSHIP FOR DIFFERENT MATERIAL MODELS When we conduct a cyclic elastoplastic analysis of a steel structure, the use of a constitutive model that can accurately predict the stress-strain behavior of the structural steel under cyclic loading is quite important. At the present time, available constitutive models, which have been implemented in many software packages for structural analysis, are the isotropic hardening (IH) and kinematic hardening (KH) models that use the von Mises yield criterion. Moreover, the equivalent stress and equivalent strain relationships assumed for structural steel in most of the past studies employing such classical plasticity models are typical bilinear or multilinear curves, as shown in Fig. 1. Cyclic characteristics of these models under uniaxial loading are illustrated in Fig. 2. They are compared with that of a modified two-surface plasticity model (2SM) developed at Nagoya University

44 2 (Shen et al. 1995). The equivalent stress and equivalent strain curve predicted by the 2SM is also shown in Fig. 1. It is worth noting that the 1.5 modified two-surface model was developed on the basis of a large number of cyclic experiments of the material and its validity has been verified for various types of steel structures. As is seen from Fig. 2, cyclic 0.5 . . . . . . . 2SM behavior predicted by classical models, except -----.-- Multilinear for the IH model with the multilinear stress-strain 0 I , , I I Bilinear curve, is completely different from that of the 0 50 I00 2SM model (Fig. 2(a), (b) and (c)). Although eiEy the IH model with a multilinear stress-strain Figure 1. Equivalent stresscurve and the 2SM model seem to give close equivalent strain curves predictions, the former overestimates the stress during the first two cycles. The main reasons why classical models simulate the cyclic behavior so differently from actual behavior are: (1) no yield plateau is included in the bilinear stress-strain curve; (2) Bauschinger effect, reduction of the elastic range of the unloading curve and cyclic hardening effect can not be taken into account.

f

;3 2

2SM .......

I

'

l

'

I

'~

B-KH

16.......__ '

I

d

12

o

J

,,.e

-1 I

,

I

-80

i

,

-40

0

I

t

40

I

i

80

"

2SM

M-KH

r I

~

1

i

I

'

I

'

.

r .

,

o

--"9

t,'" ; . . . .r. .................. . .. . ~ i I -80 -40

,

.

,,JJ

I

,

I

-40

9

.I

0 e/Ey

" - -

";llt ,

0

I

i

I

40

i

"

80

120

- - - -

....... 9

M-IH

I'

,

,

21

,~ , ~ ~ _ ~ ~

2

.

I

1

'

~ -

'

.

,

r,,l

.

,

--=

..e

1/

-80

""

i

.,,'' ,,,''' ..

3

-o

.

-120

l "7

..... I I ;l

~

o

,

ellEy

.

~

t

o

tiSy 3

,

I l l .... : t t ,- . . . . . . . . . .... . .

-16 -120

120

1

...........

-4 -8

-3 -120

2SMiI..

~

. 4 --

, I

s 1

40

I

3

,

80

D

120

i '

~d-'d~l

-120

i

-80

-

_._

I

-40

I

0

40

i

I

80

,

120

t/ey

Figure 2. Stress and strain relationships of material models under uniaxial cyclic loading 3. CYCLIC BEHAVIOR OF STIFFENED BOX-SECTION S T E E L COLUMNS 3.1. Hysteresis curves To show the effects of these plasticity models on the cyclic behavior of steel structures,

45 two tested column specimens with stiffened box sections were analyzed using the M-IH, M-KH, and 2SM models. Fig. 3 shows the lateral load-lateral displacement curves of a thin-walled column B 14 with Rf= 0.56 and ~- = 0.26 (Nishikawa et al. 1996). Here, R/is the width-thickness ratio parameter, ~- is the column slenderness ratio parameter, and their definitions are given as follows. b I12(1-1p2) 4n2/g 2

~ /~

2

I ' I I

Rf ----t

(1)

I I I I I i ], BI 4

3

,,'"

~o =

~/ , z~ ~

_

"'

,'"

-2 -10

"

~"

.

[[

-8

i I I -a -4

2

''

-2

~o =

4

' i'i'

6

"i'

I'LKD.IO

-' . ~ ~

.'-'-"~e~-,

"~c

-3 i I I I , I I I a I J -10 -8 -a -4 -2 0 2 8/8y

L

3

Rf=0.35

I,

,-

--

r,,ck

I! ,

i ,

, i

4

, i

6

M-lit 8 10

' ] KD-10

-

-"9; - ~ . - ~ - ,

1

-1

'

(a) . . . . . . . . . . . . .

M-IH 8 10

liB14

i

,

;~'i,'- ~""

.,

-

I 0 2 818y

I '

~

2 --

.]

-

.

..

--'i

{b),

"

i i

-2

[ I

I I I i

-10

-~ --

-8

-6

-4

-2

0

iest M-KH

2

4

6

8

10

-10

-8

-6

-4

-2

818y 2 ~

.

I I i

' i 'l

3

i I[ BI4 -

_ -

-10

. "

-1

, I , I , I , -8

-6

-4

-2

i 0

M-KH 2

4

6

8

10

818y

1 -

-2

. Test

-2 - ( b ) -3 i I I I I, I i I i

, I 0

2

Rr=0.35

] i I i

' l'

l '1 KD-lO

"

l _

"

-1

crack-

~

j

"

2SM 4

6

8

-3 10

618y Figure 3. Predicted hysteresis curves of a box-section steel column: B 14

, I ,I,

-10

-8

-6

I i I , -4

-2

, II 0

2

2SM 4

6

8

10

816y Figure 4. Predicted hysteresis curves of a box-section steel column: KD-10

46

~ : 2hr7r1~~

H (2)

Failure Point

|emn@mn@gmmee~ll II

in which, txy- yield stress; E - Young's modulus; v - Poisson's ratio; b - flange :. : Envelope Curve plate width, t - plate thickness; n number of subpanels separated by stiffeners; h - column height; and r radius of gyration of the cross section. The dashed line denotes the analytical ~m ~95 results, while the solid line represents the test result. These figures show that the Figure 5. Definition of failure point strengths predicted by the three material models at each reversal point are very close to the test results. From this point of view, the 2SM model does not display any advantage compared with M-IH and M-KH models. However, the hysteresis loops of M-IH become quite "fat" near the peak point due to the omission of the Bauschinger effect. In the case of the M-KH model, the analytical result at the post-buckling stage overestimates the experimental result because the M-KH model does not consider the reduction of the material property's elastic range for the unloading curve (Figs. 1 and 2). On the other hand, the 2SM analysis can predict the test result over the whole range with reasonably good accuracy. Figure 4 compares the hysteresis curves of the analyses and the test of specimen KD-10 where Rf- 0.35. This specimen is a thick-walled column. In the case of M-IH [see Fig. 4(a)], the hysteresis loops of the analysis are much "fatter" than that of the test. The maximum strength and post-buckling capacity are largely overestimated. The reason for this is that it is very difficult for local buckling of such a thick-walled column to occur. On the other hand, the M-KH model [see Fig. 4(b)] gives a lower prediction than the test result. Fig. 4(c) shows the comparison of the 2SM analysis with the test result. It is observed that except for the final loop, the analysis result coincides precisely with the test result. As reported by Nakamura et al. (1996), this specimen suddenly lost its load carrying capacity in the final loop due to a crack occurring on the tension side at the base.

3.2. Progression of local buckling Before we proceed to discuss the progression of buckled deformation during the process of cyclic loading, definition of the failure point that is considered to be the ultimate state of a structure's capacity is first described here. Fig. 5 shows a lateral load-lateral displacement curve which represents the envelope curve of the hysteretic curve. Usually, a point where the load is reduced to 95% of the maximum load (H95)is defined as the failure point. Hence, special attention should be paid to the deformation and stress level at or around this point. Figure 6 shows the local buckling configuration of a column where Rf- 0.35 and ~ -~ 0.35 at states of Hmaxand H95,which are obtained from the cyclic analysis using the 2SM model. At HI,L~,as shown in Fig. 6(a), the compressive flange plate deforms slightly inward and the stiffener buckles out of plane. This observation indicates that local buckling is initiated before the maximum load is reached. On the other hand, Fig. 6(b) exhibits obvious local buckling deformation in both the flange plates and stiffeners corresponding to H95, namely at failure point. To investigate quantitatively the buckled deformation, the lateral load-inward displacement curves at Point A in flange and Point B in stiffener (see Fig. 6(b)), where maximum deformation has occurred, are shown in Fig. 7. The inward displacement A is normalized against the plate thickness t. Values of A/t at the failure point are 2.6 at Point A and 1.7 at Point B. It should be noted that the value of A/t is related to the main structural parameters including Rf and i-, and a further study of the correlation is needed. Such a relationship would be useful in practice because the residual strength of damaged structures can be estimated by measuring plate deformation. m

47

!t !

!

W Jl ~ P

~ aim w

(a) at Hm,,x (b) at H95 Figure 6. Buckling modes of a stiffened box-section column 2

F'

J'lmax

9

I

I

"

I

"__ . . ". , [ J,,~.,~ I '

1

Hy

m -1

-1

-2[ Pos,ition: " ...." .... I, . q .2 Position: Point B in stiffener -6 .5 -4 -3 .2 -1 0 0 2 4 (a) A/t (b) A/t Figure 7. Deformation progression in a stiffened box-section column

6

4. C Y C L I C BEHAVIOR OF PIPE-SECTION STEEL COLUMNS 4.1. Hysteresis curves Figure 8 compares the hysteresis curves of a pipe section column for the test and analyses for models 2SM, B-IH and B-KH. The column has a slenderness ratio parameter of ~ - 0.26 and a radius-thickness ratio parameter of Rt - 0.11. Rt is defined as r, = a ,

Oc,

o, o

O)

E 2t

where D and t are the diameter and the thickness of the pipe section, respectively. The curves of the nondimensionalized lateral load versus lateral displacement from both the test and 2SM analysis are shown in Fig. 8(a). The shape of the hysteresis loops from the 2SM analysis agrees with the experimental result at both the peak and post-buckling stage. Figs. 8(b) and 8(c) show the corresponding lateral load-lateral displacement hysteresis curves obtained by using the B-IH and B-KH material models compared with the experimental result. The following phenomena can be observed: (1) The load carrying capacity at each half-cycle is overestimated by the B-IH model; and (2) The computed hysteresis curve by the B-KH model is in good agreement with the experimental result when the horizontal displacement lies within 48y. Beyond that, the analytical curve deviates from the experimental curve and overestimates the load-carrying capacity. These differences are the result of drawback of the B-IH and B-KH material models. The Bauschinger effect is

48 neglected in the B-IH model, and in the B-KH model the size of the elastic range for the unloading curve is assumed constant. This differs from the actual behavior of structural steel, especially during the large plastic deformation range. Moreover, both the B-KH and B-IH models can not properly model the yield plateau and fail to consider accurately the effect of cyclic strain hardening. In contrast, the 2SM takes into account the aforementioned important cyclic characteristics of structural steel. Therefore, the analysis using the 2SM can predict accurately the cyclic behavior of a pipe-section steel column.

4.2. Progression of local buckling

,1.! . . . .

, a,:?.,l

I o

-I

(a) -2 2

pl

9

l.

.

~

.

.

.

.

.

.

Rt=O.l I

~ , -....' ~.-~.. ~.=o.z.~

I o

-I -

"~. . . .

"";-~.:

~.er~.7~'7

."

Figure 9 compares the buckling modes between the test and the analyses, 2 P l [. . . . . . . . I Rt=0.11 respectively. It is observed that the 1 ' 0..~/:~ -. I ~--o.z6; buckling modes predicted by the 2SM and B-KH analyses are quite similar to that of the tested specimen. At the 0 commencement of local buckling, the length of the buckle is limited to an extremely small area in both the -1(c) - ' " ~ longitudinal and circumferential directions. With an increase in loading -I0 -5 0 S IO cycles, this buckling wave that was an 8183 outward displacement will be transmitted Figure 8. Predicted hystereticcurves of rapidly in the circumferential direction, a pipe-section column and eventually an elephant-foot buckling mode is formed. This phenomenon matches well with the actual mode of the steel bridge piers failed in the Hyogoken-Nanbu earthquake. However, the extent of deformation of the B-KH analysis is smaller than those of both the test and 2SM analysis. On the other hand, the buckling mode predicted by the B-IH model greatly differs from that of the test. The position of the local buckle shifts upward. One possible reason why this occurs is due to the exaggerated expansion of elastic range of the B-IH model, as stated previously. Comparison of the buckling modes in Fig. 9 indicates that the 2SM model can duplicate the buckling mode of the test with satisfactory accuracy, whereas both the B-IH and B-KH models predict unlikely buckling modes. _ Figure 11 illustrates buckled deformation of a thin-walled column where RI - 0.11 and - 0.3 at H,,,,x, H95 and other points, as noted in Fig. 10(a) for monotonic loading and Fig. 10(b) for cyclic loading. It is observed that maximum deformation occurs at Point A in both the monotonic and cyclic loading, but the outward displacement w/t corresponding to H,n,,x is about 0.5 and 1.0, respectively. When the load has decreased to 1-195 in the monotonic loading, the value of w/t at Point A increases to 1.3. In the case of cyclic loading, a loading point corresponding to H95 is not available in the hysteresis loops, so two points at Hs5 and 0.89Hy (Hy is yield load) are chosen to show deformation progression. As is seen in Fig. 11 (b), the outward displacement w/t at Point A reaches approximately 2.0 at Hss, and 3.0 at 0.89Hy, respectively. Figure 12 shows how the stress progresses at Point A under two types of loading

49 programs. Plots (a) and (b) represent the inner surface and outer surface, respectively. Maximum axial strain at the failure point is around 50 times of Ey (Ey--" 0.00141 for this column). Computed results of a thick-walled column where R r - 0.05 and it - 0.3 are shown in Figs. 13 to 15. It can be observed that the maximum axial strain at failure point (H95) is about 150 times Ey (Ey = 0.00114 for this column). Thus, such an analysis needs an accurate plasticity model that can simulate cyclic behavior in a large strain range. m

5. S T R E N G T H TILITY

AND

DUC-

Based on extensive elastoplastic large displacement analyses using the 2SM model, some of empirical formulas have been proposed by authors (Gao et al. 1998, Figure 9. Buckling modes of a pipe-section column Usami and Ge 1998) to determine the strength and ductility of steel columns subjected to cyclic lateral loading and a constant axial load. The columns are composed of stiffened box-sections or pipe-sections. These equations are expressed as functions of the main structural parameters such as Rf (or R3, it, and P/Py. Here, P is the axial load, and Py is the squash load of the cross section. For stiffened box-section columns, the ultimate strength (Hmax/Hy)and ductility (Sm/Sy, 895~y) can be calculated from the following equations: Hma x

0.10

H,

(Rf;t-~')~

~

=

S,,,

= ~

S, •95

t~,

0.22

+1.06

(4)

+ 1.20

(5)

Rs ~-~Z, '

=

0.25

(1 + PlPy)Ry~~

'

+ 2.31

(6)

in which X,' is the stiffener's slenderness ratio parameter (Usami and Ge 1998), which is defined as

50 2

,

D--891mm, t-8.41mm, h : 4 3 9 0 m m

9

,,,t

|

1

9

,

,

9

9

9

9

i

*

9

- - -Hmax

1.5

1

gl

. o.sE/

t

~.~/iss0.89H,1

0 -1 ~

7.=0.30

0

P/Py:O.15

0 (a)

2

4

6

8

10

-2

i

"

-5

.

.

,

,

0

,,.

, 5

. . .

.

"

.

!

8~5y (b) 8~y Figure 10. Lateral load-lateral displacement curves of a thin'walled pipe'section column &~k" &

0.1

[ - - ' O - - - a t Hmax I - - - ~ - - a t H95

---O----at Hmax - & - at Hss - - - O - - - a t 0.89Hy

0.1

x: Distance from the base w: O u t w a r d displacement

"~ 0.05

~0.05 Point A

Point A

...,41' ~

O=

0

0.5 1 1.5 0 (a) wit (b) wit Figure 11. Deformation progression in a thin-walled pipe-section column :

9

,

9

9

|

,

9

,

,

i-

9

' 9

l .-

9

: Hnmx Failure P o i ~ ::::::::::::::::::::::::::::::::::::

1

" Hnmx .

..:,,

::::::::::::::::::::::::::::::::::::::

" H95 ~ . . . . .

- 1

-1

(a)

1

"::

1

"

-2 Position: Point -150 -100

-_

. . . . .

A, Inner surface

-50

0

-2

~.. . . . . .

Monotonic

Position: Point A, Oute'~r

-40 -20 0 (b) e]Ey Figure 12. Stress progression in a thin-walled pipe'section column

20

E~Ey

1 al~-~ (7)

where Q is the local buckling strength of plate panels given by

51

2 ~

,t=16.8mm, hf4390mm

1

9

2 ..

.

,

. ' '. . . .

_It,.,] "

9

1.5 !

Hmax

Rtffi0.05 L-0~0

0 ~ 10

20

'-

.98

"

-1

P/Py=0.15

0

9

|;

0.5

94

-2

"

30

t

.

.

,

,

-10

0

10

(a) ~Y (b) ~y Figure 13. Lateral load-lateral displacement curves of a thick-walled pipe-section c o l u m n

'&

! --4J---at Hmax "--&---at H9s x: Distance from the base w: Outward displacement

0.1

~0.05

0.1

~0.05

Point A

0

1

(a)

2

0

wit

1

Co)

2 w/t

Figure 14. D e f o r m a t i o n p r o g r e s s i o n in a thick-walled pipe-section column

2.''1 ;:::::H:-.:m:x: ~::.:::::.....-::-..........~jiure_ Point. .

.

.

.

I

"

' '

'

i

.

.

.

.

.

2

.

. H;~,~==I~, ,,~TI ~- ~

1

-2 Position: Point A, Inner surface[ -300

-200

(a)

-100

~[l~y

. ,0

-1 .

_?..c

-2 -100

-50

(b)

. /

t ~

:

Point A, Outer surface 0

50

100

e/ey

F i g u r e 15. S t r e s s p r o g r e s s i o n in a thick-walled pipe-section c o l u m n

Q=~y1 [P-~/p2-4R, ]

(8)

= 1.33R f "~ 0.868

(9)

52 and ct is aspect ratio of flange plate (= a/b, a is flange length), rs - radius of gyration of the T-shape cross section which consists of one longitudinal stiffener and the adjacent subpanels. In the case of the pipe-section columns, the proposed equations are given by Hn~ x - - - 0.02 ~ + Hy (Rt~-) ~

6m _

~,

~9._.ff_5 =

~y

1.10

(10)

2 3

(11)

1 3(R, ~'-~ ) ~

0.24

(12)

(1 + P I ey)2/3-~/3R,

6. CONCLUSIONS Elastoplastic large displacement analysis was carried out to predict the failure of steel structures under cyclic loading. The cyclic characteristics of classical isotropic and kinematic hardening plasticity models as well as a modified two-surface model were investigated, and their application to failure prediction of steel structures were presented. Moreover, local buckling, deformation and stress progressions were discussed. Comparisons of analytical and experimental results showed that accurate failure predictions require an accurate plasticity model. REFERENCES

Gao S. B., Usami T., and Ge H. B. (1998). Ductility evaluation of steel bridge piers with pipe sections. J. Engrg. Mech., ASCE, 124(3), 260-267. Nakagawa, T., Yasunami, H., Kobayashi, Y., Hashimoto, O., Mizutani, S., and Moriwaki, K. (1996). Evaluation of Strength and Deformation for Box Section Steel Piers by Finite Element Analysis. Proc. of The 1st Conference on Hyogoken-Nanbu Great Earthquake, JSCE, 599-604. Nishikawa K., Murakoshi J., Takahashi M., Okamoto T., Ikeda S., and Morishita H. (1999). Experimental study on strength and ductility of steel portal frame pier. J. Struct. Engrg., JSCE, 45A, 235-244 (in Japanese). Shen C., Mamaghani IHP, Mizyno E., and Usami T. (1995). Cyclic behavior of structural steels. 11: theory. J. Engrg., Mech., ASCE, 121, 1165-1172. Usami T., and Ge H. B. (1998). Cyclic behavior of thin-walled steel structures - numerical analysis. Thin-walled structures, Vol. 32, 41-80.

Impact Loading

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

55

On the criteria for cracking and rupture o f ductile plates under impact loading Norman Jones and Caroline Jones Impact Research Centre, Department of Engineering (Mechanical Engineering) The University of Liverpool, Liverpool L69 3GH, U.K.

ABSTRACT Some recent studies which have been undertaken into the behaviour of circular plates subjected to impact loads which produce large inelastic strains and material failure are discussed in this article. The experimental data yields the threshold conditions for failure. In order to predict the quantities which might be used to construct a failure criterion, the threshold conditions are examined using a numerical finite-element code, without activating any failure algorithms. It turns out that critical values of the rupture strain and the strain energy density are both promising failure criteria worthy of further study.

1. I N T R O D U C T I O N

Maximum values of dynamic loadings which cause structural failure are required for economical and safe designs. This article is concerned with the failure of ductile structures which are subjected to dynamic loads causing large inelastic strains. Structural instability and other types of structural failure are not considered. Rigid-plastic methods of analysis [1] are often used to predict the response of structural members when subjected to sufficiently severe dynamic loads. These methods predict various features of the response including the permanent displacement profile of a structure, but they assume that the idealised material has an unlimited ductility. Nevertheless, these methods have been developed further to explore the failure of structures and have identified three modes of failure in impulsively loaded beams [1 ]. The first failure mode is called mode I and relates to the large permanent ductile deformations which are produced without any material failure. For larger impulsive loads, a mode II material failure might occur when the uniaxial rupture strain of the material is exceeded. At still higher blast loads, a transverse shear failure might develop and is known as a mode III failure. This approach has been used by several groups to examine the dynamic inelastic failure of other structures subjected to impact and blast loadings, as discussed in Reference [2], and has been successful in highlighting the principal response characteristics, in identifying the major parameters and is a useful aid for interpreting experimental data. Numerical schemes such as finite-element codes are used extensively in modem structural design. However, a numerical code requires a universal failure criterion, but even the dynamic inelastic failure criterion for a simple beam is unknown [3] and, moreover, the failure mode depends on the kind of dynamic loading. Many numerical calculations assume that a dynamic inelastic failure occurs when the maximum equivalent strain reaches the corresponding value at

56 failure in a static uniaxial tensile test. This simplification ignores any change of the rupture strain with strain rate or any variation with the hydrostatic stress [4]. The dynamic response of a fully clamped ductile metal beam struck by a mass having a sufficient initial kinetic energy to produce large inelastic strains and material failure was studied in Reference [5]. This experimental arrangement was selected because it is quite straightforward and easy to control. Initially, the beams suffered large inelastic deformations without failure (mode I), but, as the initial kinetic energy was increased, cracking was first observed, then complete failure occurred for sufficiently large impact energies. Thus, the dynamic loading conditions associated with the failure threshold could be established. The experimental test beams were made from mild steel and the static and dynamic tensile properties were obtained using specimens cut from the same block of material over the range of strain rates observed in the beam impact tests. A more comprehensive programme of tests was conducted recently on fully clamped beams struck by a mass travelling with an initial impact velocity which causes large inelastic deformations and material failure [6]. The beam test geometry, dynamic material properties and the impact loading details, for the threshold conditions, were then employed in the ABAQUS finite-element numerical code. No failure criterion was implemented in the implicit finiteelement code. This enabled the parameters required for a variety of failure criteria to be assessed at the initiation of cracking or at the threshold of failure, indicated by the experimental results, without the numerical code being prejudiced towards any particular failure criterion. However, the numerical values for various quantities such as the equivalent strain, strain energy density, shear stress, etc., can be obtained at the threshold of failure and, therefore, can be used to suggest a criterion which controls failure. This partnership between experimental results and accurate numerical predictions is essential because it is very difficult, if not impossible, to record all of the detailed information from experimental tests at the initiation of failure for structures subjected to dynamic loads which produce large inelastic deformations and swains. It transpires from a careful comparison between the experimental results [5,6] and the numerical predictions [3,6] that the maximum membrane force in a beam cross-section and the uniaxial tensile rupture strain appear to be the most promising criteria for a tensile failure, while the maximum Tresca stress or maximum Mises stress and the maximum plastic strain energy density are worthy of further study for predicting a shear failure. The results of a recent experimental test programme on the impact loading of fully clamped mild steel circular plates are discussed in the next section. Numerical calculations for the plates using the ABAQUS finite-element code are discussed in Section 3. This article is completed with a discussion and conclusions in Sections 4 and 5, respectively.

2. FAILURE OF PLATES DUE TO IMPACT LOADS A beam might be considered as a one-dimensional structure from a global, or design, perspective, although local three-dimensional affects are important at the failure site. The impact failure of plates, which are nominally two-dimemional structures, is examined in this section. A considerable body of theoretical work and several empirical equations have been published on the behaviour and perforation of ductile plates struck by missiles [7]. Nevertheless, the field remains active with articles being published currently on high velocity perforation producing adiabatic shearing effects etc. and on low velocity impacts causing large transverse displacements which induce membrane forces in a plate before failure. The material failures in

57 these studies occur in the plate immediately underneath the striking mass. It is difficult to develop a universal failure criterion on the basis of the behaviour in this highly localised region, even for a low velocity impact [8]. Tests have been conducted recently on 203.2 mm diameter (D) fully clamped circular plates struck by masses which produce large inelastic deformations and material failure. It was the objective of these tests to promote failure around the outer boundary of a plate and away from the complicated behaviour at the impact site. Static and dynamic material properties were obtained from tests which were conducted on specimens cut from the same mild steel plate. In order to achieve material failure at the plate boundary, an indenter having a rounded nose was constructed with the same profile as that used for the beam tests discussed in Reference [6], but generated as a volume of revolution. The centreline of the impacter having a main cylindrical body with a diameter D/5 is located at a distance of D/8 from the plate boundary and seven different failure modes were identified for 1.6 turn and 3 mm thick plates. These failure modes lie within the four categories: Mode Mode Mode Mode

I: II: III: IV:

large permanent ductile deformations of a plate, through-thickness failure in a plate underneath the impactor, through-thickness failure at the clamped edge of a plate, and through-thickness failure at the clamped edge of a plate and limited failure underneath the impactor.

Photographs of typical mode I to IV failure-s in 1.6 mm thick circular mild steel plates are shown in Figures 1(a)-(d) for specimens IBP15 (4.50 m/s), IBP17 (8.23 m/s), IPB 14 (11.00 m/s) and IPB3 (14.45 m/s), respectively. A laser-Doppler velocimeter was used during the impact tests in order to record the velocitytime histories of the indenter from which the temporal variations of the impact forces in Figure 2 were obtained. Faster rise times of specimens IPB14 and IPB3 for mode III and mode IV failures, respectively, are due to the higher impact velocities of 11.00 and 14.45 m/s in Figure 2. The slowest rise time is associated with test specimen IPB15 which undergoes large ductile deformations, or a mode I response, without any material failure. The largest force is associated with a mode II failure 0BP17) and is likely due to the development of large membrane forces before a local through-thickness failure occurs underneath the indenter. The lower maximum forces associated with mode III and mode IV failures are probably because the length of tom plating at the boundaries in Figures l(c) and (d) prevents the development of localised membrane forces as large as those for a mode II failure. It is interesting to note that the pulses (i.e., areas under the curves in Figure 2) are similar for those specimens exhibiting modes II (137 Ns), III (147 Ns) and IV (130 Ns) failures. The pulse of 74 Ns for the mode I case in Figure 2 is much lower became the impact velocity of 4.50 m/s is well below the mode II threshold value of 8.17 m/s, approximately. On the other hand, the input energies are 147 J, 497 J, 878 J and 1513 J for the mode I to IV cases in Figures 1 and 2. The impact forces of the four test specimens in Figures 1 and 2 are replotted in Figure 3 with the impactor displacement as the abscissa. The areas under these curves give the external work of 140 J, 494 J, 875 J and 1304 J for modes I to IV, which, with the exception of the mode IV value, are very similar to the values noted previously.

58

Figure 1. Photographs depicting examples of plate specimens that exhibit the failure modes (a) I, (b) II, (c) 1II and (d) IV under low velocity impact conditions 100

A

60

a)

o

40

u.

20

z

0

IPB17 (11)

IPB3

80

IPB14 (111)

IPB15 (I) -20

0

1.0

2.0

3.0

Time (ms)

Figure 2. Force-time histories for the plate specimens in Figure 1.

59 100 -

E]15 (i)

f~

80-

-.

A

Z _~e O O t_ O It-

60 4020

O.

d,

IPB17 (11) B14 (111)

30v)

L x

I"

|

0

10

l

20

30

40

50

Displacement of Impactor (mm)

Figure 3. A comparison of force-displacement histories for the plate in Figure 1.

3. NUMERICAL FINITE-ELEMENT STUDIES ON THE IMPACT RESPONSE OF BEAMS AND PLATES The finite-element code ABAQUS has been used to examine the experimental studies on beams and circular plates subjected to impact loads which are reported in the previous two sections. It was observed that good agreement between the numerical predictions and the experimental results were obtained when the plates suffered large permanent transverse displacements without any material failure (mode I response). This comparison serves as a calibration of the ABAQUS finite-element code for this particular plate impact problem and offers a degree of confidence in the numerical predictions. It appears from these preliminary numerical calculations for the circular plate specimens in Figures 1 to 3 and the recent tests on thicker plate specimens (3.0 ram) that the strain energy density and the rupture strain criteria are the most suitable for predicting material failure. However, other criteria are being explored currently and furt.her details will be published in due course.

4.

DISCUSSION

The preliminary conclusions of the current research programme suggest that critical values of the uniaxial tensile rupture strain and the plastic strain energy density are both promising criteria for the material failure of mild steel beams and plates which are subjected to impact loadings producing large inelastic strains. However, additional studies, particularly experimental ones, are required to clarify the role of the triaxiality (ratio of hydrostatic and yield stresses) and strain rate on the uniaxial rupture strain criterion. The various observations have been made for structures made from mild steel. Clearly, the failure criteria for structures made from other materials might be different. In fact, the impact behaviour of flat aluminium alloy beams have been examined in Reference [9] and it was observed that, in contradistinction to the flat steel beams, failure occurred either at the

60 support or at the impact location. Moreover, all of the failures for the beams with enlarged ends occurred at the supports. 5.

CONCLUSIONS

The experimental results in this article for circular plates subjected to impact loads, which produce large inelastic strains and material failure, together with the experimental data on the static and dynamic properties of the materials over a range of strain rates, can be used as benchmark studies for the calibration of numerical codes and the development of dynamic inelastic failure criteria. In the present work, the predictions of various quantities given by the ABAQUS finite-element code are calculated for the impact conditions at the threshold of failure according to the experimental test results. The numerical conditions do not activate any failure criteria which might be available in the computer code in order not to prejudice any conclusions. However, the inelastic material behaviour observed in the experimental work was incorporated in the numerical scheme. This partnership between experimental results and numerical predictions suggest that critical values of the uniaxial tensile rupture strain and the plastic strain energy density are both promising criteria for the material failure of mild steel beams and plates which are subjected to impact loadings producing large inelastic strains. However, further studies are necessary to examine the accuracy of these failure criteria for other types of dynamic loadings, different structures and other materials.

ACKNOWLEDGMENTS The authors are grateful to EPSRC for their support of this study under grant number GR/J 699998 and to Mrs. M. White for her secretarial assistance. REFERENCES ~

2. .

~

.

N. Jones, Structural Impact, Cambridge University Press, paperback edition, 1997. N. Jones, Dynamic inelastic failure of structures, Trans. Japanese Society of Mechanical Engineers, 63( 616), 2485-2495, 1997. J. Yu and N. Jones, Numerical simulation of impact loaded steel beams and the failure criteria, Int. J. Solids and Structures, 34(30), 3977-4004, 1997. M. Alves and N. Jones, Influence of hydrostatic stress on failure of axisymmetric notched specimens, J. of the Mechanics and Physics of Solids, 47(3), 643-667, 1999. J. Yu and N. Jones, Further experimental investigations on the failure of clamped beams under impact Loads", Int. J. Solids and Structures, 27(9), 1113-1137, 1991. N. Jones and C. Jones, Dynamic inelastic failure of beams and plates, Impact response of Materials Structures, Eds. V. P. W. Shim, S. Tanimura and C. T. Lim, Oxford, pp. 3747, 1999. G. G. Corbett, S. R. Reid and W. Johnson, Impact loading of plates and shells by freeflying projectiles: A review, Int. J. Impact Engineering, 18(2), 141-230, 1996. N. Jones, S. B. Kim and Q. M. Li, Response and failure analysis of ductile circular plates struck by a mass, Trans. ASME, J. Pressure Vessel Technology, 119(3), 332-342, 1997. J. Liu and N. Jones, Experimental investigation of clamped beams struck transversely by a mass, lnt. J. Impact Engineering, 6(4), 303-335, 1987.

Structural Failure and Plasticity (IMPLAST2000)

Editors:X.L.Zhaoand R.H.Grzebieta 9 2000ElsevierScienceLtd.All rightsreserved.

61

D y n a m i c Behavior o f Elastic-Plastic B e a m - o n - F o u n d a t i o n under Impact or Pulse Loading* X. W. Chen a"b, T. X. Yu b and Y. Z. Chen a alnstitute of Structural Mechanics, China Academy of Engineering Physics, Mianyang City, Sichuan, 621900, China bDepartment of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong A mass-spring model is used to analyze elastic-plastic BoFs under impact or pulse loading. A general analytical method for elastic-plastic BoFs under dynamic loading is proposed. The elastic, perfectly plastic BoFs under dynamic loading undergo various deformation scenarios, merely depending on a few dimensionless parameters. Two peculiar phenomena, i.e. "plastic hinge migration" in the beam and the successive propagation of plastic deformation in the foundation, as early explored in case of static loading [1 ], also occur in the dynamic cases.

1. Introduction The analysis of beam-on-foundation (BoF) systems has a large variety of engineering applications. Besides being directly applied to the case of actual foundation-supported structures or networks of beams, BoF is also used to form a simple but useful analogy to the structures consisting of thin-walled cylindrical shells. Most theoretical models and analytical methods of BoFs are mainly based on either elastic or rigid, perfectly plastic idealization. For example, Yu and Stronge [2] analyzed the dynamic response of a rigid, perfectly plastic BoF (RPB/RPF) subjected to a rigid-mass impact. Elasticity often plays a significant role not only in the elastic stage of the structure response, but also in altering the deformation history and the energy dissipation partitioning in the structure compared with the prediction of a rigid-plastic model. The incorporation of elasticity will make an essential step in developing more advanced model for BoFs. Chen and Yu [1] developed a discrete spring model and analyzed the static behavior of elastic-plastic BoFs. By using a similar mass-spring model, this paper is mainly aimed to analyze the dynamic behavior of elastic-plastic BoFs under impact or pulse loading.

2. Model and Formulation Consider an elastic-plastic beam of finite length L and density per unit length o, resting on an elastic-plastic foundation of Winkler-type. A rigid-mass G (or a pulse) impacts the BoF at * The project supportedby the NationalNatural ScienceFoundationof China under the ContractNo. 19672059

62 Qi+!

oad I bar

1

i-I i

node 1 2

Ill i-

i i+l (a)

n-I

Q,-]

Mi

A

1'

_ ~ )

y

.L4' T Mi.i

,;ii ..... Qi., * Qi Pi

(i-l)th bar n-I

Fi

ith node

i ith bar

n (b)

Fig. 1 The mass-spring model of an elastic-plastic BoF system any position with an initial velocity v0. Due to the limitation of Winkler model, i.e. the independence of individual spring elements, shear and the mass of foundation are not considered. In addition to the assumption of small deflection, the effect of axial force and strain-rate are neglected. It is also assumed that the foundation and the beam must always be in contact. A mass-spring model is proposed and all the springs representing the beam and the foundation are supposed to be bilinear ones. As shown in Fig. 1, the original beam is first discretized into (n-l) elements of equal length, while the mass of each element is assumed to be concentrated at its two ends. Hence the model consists of n flexible nodes with lumped mass p L / n , which are connected by (n-l) massless rigid links of length L/(n-1). The flexural deformation is represented by the relative rotation between adjacent rigid links and resisted by an elastic-plastic rotational spring, which reflects the flexural rigidity of the beam. At each node, an elastic-plastic spring linearly acting along the transverse direction of the beam is added to represent the effect of Winkler-type foundation. Using matrix notations, {F} =

(FI,...,Fn) T, {W} = (Wi,...,Wn) T, {Q} = (QI,...,Qn_I) T,

{m} =

(ml,...,mn) T, {(I~}= ((I~l,...,r

T, {~IJ}=

(1)

(~zJi,...,Vn) T

where {F} and {w} are the reaction force and the deflection of the foundation springs, respectively. {Q} and {M} represent the shear force and the bending moment of the beam springs, respectively. {~} and {q'} denote the absolute rotation of each link and the relative rotation angle between the adjacent links, respectively. To non-dimensionalize the formulation, define {f}-"

L{F}/My, {w}= {W}/L, {q}= L{Q}/My, {m}--{M}/My, {~}= {~}/q2y, {W} = {~}/~y,

rr = Z x/ o Z / M y , x = t / r r , (

)= d / d'r,, f y

= L Fy / M y , Wy = Wy / L ,

(2)

9I t ? l a x / r r m a x ~, = -~, 1"1= UeB /t.,eF = MrqJy* /(VrWr ), e0 = UOK/My = ,L(~ + ,a/n)Vo2 2My

where My and ~I,~ denote the maximum elastic bending moment of beam and the maximum elastic relative rotation angle of beam springs, respectively. Fr and wr are the yielding force

63 and the maximum elastic deflection of foundation springs, respectively, x is the mass ratio of the colliding body to the beam. 11 is the ratio of the maximum elastic deformation energies dissipated in the beam and foundation, e0 is the ratio of the initial kinetic energy u ~ to the maximum elastic bending moment of the beam. The governing equations of an elastic-plastic BoF can be formulated as

(3)

{W}= -n(n - I)B-I(AAT ~m}- nB-I {f }

{q} = -(,.-OA ~ {m}. {q.}= -(,.-O'r

{V} = -A{~.} = (.-OA,r {w}/~;

(4)

where/1 and B are both matrixes 1

"~

En n~

Ao l

and B -

0 + nVna)Enaxna

(5) E nxn

-- 1 nx(n-l)

in which E is a unitary matrix, ns/n and n,,/n denote the relative loading position and the loading width on the BoF, respectively. If only pulse loading is applied, B should be a unitary matrix, w~ is related to the compliance of the beam w~ = q'__L,~r - MrL n E1

(6)

Non-dimensional quantities of BoF can be formulated as k

~--(,.-0' L3K k =~ My

,

~r

~

kWy / ~lz ,

n-

(7)

L2KVey t 2Fy9 wr f r _ r Wy . . . . . . My My L k, k

~, . . . .

,

where K is the stiffness of Wirdder foundation. By the definition, k and ~ represent the nondimensional relative rigidity and limit load of elastic-plastic BoF, respectively. The non-dimensional constitutive laws of the bilinear springs of BoF can be described as

(8)

{m}= G{,V}+ Z, {I}- Z{w}+ where G, Z, H and D are all diagonal matrices and the corresponding parameters are

Gu

= ~1, elastic loading & unloading Lcz, plastic loading

0, elastic loading & reloading Z~i = ~ (1-or), plastic loading [-(1-(x)(qs:-1),

unloading

(9)

64

k., elastic loading & unloading Hii = k.f~, plastic loading

f O, elastic loading & reloading plasticloading

Oi~= (l-13)fr,

(10)

where a and fl are the hardening parameters of springs of beam and foundation, respectively. Thus the Eq. (3) can be finally rewritten as (11)

The non-dimensional initial conditions at x = 0 can be written as wi = O, i = l, ..., n

and

{ f~i = O, i = l, ..., ns, ns + na + l, ..., n ,.

fvj -

(12)

,

--~-+ k , j = ns + l, ..., ns + n a

The solution can be obtained by Runge-Kutta method. Calculations have confirmed that when n is sufficiently large, say n>_60, the solution is almost independent of n. An examination of the equations indicates that only a few non-dimensional parameters, i.e., a , 13, k, ~, 11, e0 and ~, (~, =0 for pulse loading), together with ns/n and na/n which related to the location of impact or pulse loading, are needed to characterize the dynamic behavior of a bilinear BoF system. 3. M a s s I m p a c t

on elastic, perfectly

plastic BoFs

Considering an elastic, perfectly plastic BoF, i.e. ~ =0 and 13=0, subjected to a mass impact at the mid-span of the beam ((ns +0.5na)/n=0.5). For the initial impact, we assume ;~ = 1, e0 =0.1 and na/n= 5/81 = 0 . 0 6 2 . Thus only k, ~, and rl are needed to characterize the dynamic behavior of the elastic, perfectly plastic BoF. Based on the quasi-static analysis [ 1], in general, the following conditions are required. For EB/EPF, k should be small enough and n >> 1 ; for EPB/EF, k should be large but 11<< 1 ; for EPB/EPF, k should be sufficiently large. ,/ is used to distinguish "short" or "long" beams, i.e. if v > 64, the elastic-plastic BoF is regarded as a "long" one; while if v < 64, the BoF is regarded as a "short" one. Thus, with various combinations of k, ,~ and rl, three typical deformation scenarios will be examined below. Scenario 1. EB/EPF (k - 4000, y - 8, n = 100 )

After an initial elastic vibration stage, a plastic region is initiated under the impact position and then expands outwards in the foundation, as shown in Fig. 2, whilst the beam remains in elastic bending. The velocity of the beam reduces abruptly due to the energy dissipation in the plastic deformation of the foundation. Fig. 3 demonstrates the evolution of the elastic-plastic

65 Non-dimensional lencjth x 0.25 0.5 0..7~ ................

-3 0

._o

~

,_,

~-_ _ ~.

_- _- - : -

_

9: _ . . . ' _

__

- . _

r 19

E=

3 ~

0

6

~

t=O 003

.~9~"

0.06

I11 t,--

._o I9

I

.

.

.

.

.

.

.

.

I

19

EE tO

z

Z " O

_, .

.

.

f

'

0

,

.

Fig. 2 Deformed shapes of EB/EPF

'~ I

i

"l

0.25 0.5 0.75 1 Non-dimensional length x

Fig. 3 The evolution of E-P boundary in the foundation of EB/EPF

boundary in the foundation of EB/EPF with time. The BoF collapses when the foundation deforms plastically along the whole length. -- 100, rI = 0.1 ) Fig. 4 shows the bending moment distributions of the beam at different time. After initial mass impact, as shown in Fig. 5, the stationary plastic hinge occurs at the sides of impact position. The stationary hinge may change alternatively between positive and negative yields due to the vibration of the beam after impact. It also disappears for a while during the transition. The migration of plastic hinge only takes place before the bending moment changes sign at the stationary hinge position. Under quasi-static loading, as shown in [1], the reverse plastic hinge migrates continuously and is limited in a very small distance until collapse. Differently, under impact loading, the second or third plastic hinges appear and migrate skippingly in the beam, and they can also change between positive and negative yields. As the foundation deforms elastically, only a little energy dissipates in the plastic yielding of beam. The BoF finally re-experiences elastic vibration. Scenario

2. E P B / E F ( k = 10,000, y

Scenario

3. E P B / E P F

( k = 80,000, ~ = 100,11 = 2 )

Fig. 6 depicts the evolution of the elastic-plastic boundary in the foundation and the plastic hinge in the beam. Actually the BoF undergoes a local yielding in the foundation, accompanied by the occurrence of a stationary plastic hinge in the beam. Like that of EPB/EF, the stationary hinge can change between positive and negative yielding, and it may E

.o~

1 T'~

Iil~

t=O.O09 --+- t=0.0228 =

"

~, 9 ~

0.1 o.o7s

•

.......

Non-dimensional

length x

"

o "-

~

c

o

eZ

0.025 9m = l

| ..Q

-1 Fig. 4 Bending moment diagrams of EPB/EF

9m=-I

0

, , 7,,=1,=

0

0.25

.~

, ,

0.5

,

,

0.75

1

Fig. 5 The evolution of plastic hinges in the beam of EPB/EF

66 --

-~ 9

0.1

0.075

i ~ W/Wy>1

o

"~ *"

m

.==

~ / / /

0.025

w/wy= 1 = hinge(m=1) hinge (m=-l)

._~Non_dimensiona?ie~gOl~ 0897

z

0 0.25 0.5 0.75 1 Non-dimensionallength x

Fig. 6 The evolution of E-P boundary and plastic hinges in EPB/EPF

---- t=0.0153 ....§ ....t=0.0429 --o .... t=0.0663

3 Fig. 7 Deformed shape of EPB/EPF

disappear during the transition. The plastic deformation region becomes smaller with the increase of time and finally disappears. That means, the plastic deformation is replaced by the elastic unloading. As shown in Fig. 7, the maximum plastic deflection of foundation in the first yielding stage is obviously larger than that in the second plastic deformation stage. The reverse (upward) deflection always behaves as elastic during the vibration of the BoF. Finally the elastic recovery occurs completely and the BoF re-experiences an elastic vibration. In the quasi-static analysis [1], two collapse mechanisms exist, i.e. plastic yielding of foundation and "rigid-body" rotation of beam. Differently in dynamic analysis, only one collapse mechanism is observed, i.e. the entire yielding of foundation. Actually, the dynamic behavior of an elastic, perfectly plastic BoF is determined not only by its own characteristics k, ~( and 11, but also by the impact loading parameters, especially by ~. and e0. These factors strongly affect the final phase of BoF after impact, meanwhile result in the transition of various scenarios in impact. More details are discussed in a successive paper. 4.

Conclusions

1. A mass-spring model is developed to analyze dynamic response of elastic-plastic BoF subjected to impact or pulse loading. 2. Any bilinear BoF under impact or pulse loading can be characterized by a few dimensionless parameters, e.g. ~, p, k, ~,, 11, e0, X, ns/n and na/n. 3. The migration of plastic hinge in the beam and the propagation of plastic region in the foundation, are demonstrated in the dynamic analysis of elastic, perfectly plastic BoF subjected to rigid-mass impact. References

[1 ] X.W. Chen and T.X. Yu, Elastic-plastic beam-on-foundation under quasi-static loading, Int. J. Mech. Sci., in press. [2] T.X. Yu and W.J. Stronge, Int. J. Impact Engng., 9 (1990) 115. [3] E. Manoach and D. Karagiozova, Computers & Structures, 45, 3 (1992) 605-612.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

67

Load deformation of thin tubular b e a m under impact load Nobutaka Ishikawaa, Yukihide Kajita a, Kensuke Takemotoa and Osamu Fukuchi b Department of Civil Engineering, National Defense Academy, Yokosuka, 239-8686, Japan b Nippon Kokan Light Steel Co. Ltd., Tokyo, 103-0012, Japan

a

The aim of this study is to examine the local deformation of thin tubular beam subjected to impact load. First, the weight dropping type impact test was performed for the thin tubular beams. The modified Ellinas formula that expresses the relation between the load and local deformation of steel pipe is validated comparing with the experimental results. Second, the impact test was carried out for the thin tubular beam reinforced by tie bolt. It should be noticed that this newly devised technique is remarkably effective in order to control the local deformation of thin tubular beam under heavy impact load. 1. INTRODUCTION In recent years, many steel pipe check dams as shown in Photograph 1 have been constructed as the protective structure against debris flow in the mountainous area in Japan. These structures can absorb the kinetic energy of huge rocks in the debris flow by the local deformation of steel pipe and the structural deformation. Herein, the authors propose a new type of steel pipe check dam in which thin tubular beams are equipped as the impact energy absorbers against huge rocks in front of main check dam structure as shown in Figure 1. The aim of this study is to examine the local deformation of the thin tubular beam under impact load as a shock absorbing system. Many studies have been so far devoted to the tubular beam under impact loading. For instance, N.Jones et.ah[1,2] have investigated the lateral impact response of fully clamped pipelines from the viewpoints of the theoretical and experimental approaches. C.P.Ellinas et.al.[3] have proposed a formula that expresses the load~local deformation relation of tubular member. T.Hoshikawa et.al.[4] have also proposed a modified Ellinas formula considering the strain rate effect. In this study, the weight dropping type impact test was first performed in order to confirm the validity of the modified EUinas formula for thin tubular beams with different span length. Second, the impact test was carried out for the short thin tubular beam reinforced by the tie bolt in order to reduce the local deformation. Finally, experimental results are compared with the modified Ellinas formula and are examined on the effects of span length and tie bolt. 2. OUTLINE OF EXPERIMENT 2.1 Experimental apparatus Weight dropping type of impact loading apparatus as shown in Figure 2 was used. The impact load is applied by dropping the weight (W--1.78kN) with spherical shape of diameter 22cm under the different dropping heights (velocities). The specimen is simply supported at

68

Figure 2. Weight-dropping type impact apparatus

Figure 3. Local deformation profile

both ends and the concentrated load is applied vertically at the center of it. 2.2 Measurement The impact load is obtained by multiplying the mass of the weight by acceleration measured by the accelerator attached to the weight. The upper displacement at the loading point of the beam means the total displacement ( 6 r ) (the sum of local (6L) and beam (6 s ) deformations), which is found by integrating the value of the acceleration twice. The beam displacement (6 B) was measured by the laser type displacement sensor. It is found from Figure 3 that the following equation holds during the deformation of pipe. 6 r + 6 D ffi6 L + 6 B + 6 0 • D + 6

B

where, 6 0 : the deformed diameter which is m e a s u r e d by slide calipers. the local deformation 6~ can be obtained as follows : 6L = D - 60

(1) Therefore,

(2)

Consequently, the ratio ( a ) of local deformation (6L) and diameter ( D ) is found by measuring the deformed diameter (6 D) as follows"

69 200

1000

.

/

.

.

.

.

.

.

200 A ,..L,.

~,.

.

.

.

.

200

.

.

.

.

.

.

.

.

1400

.

.

.

.

.

.

.

.

.

.

A

.

.

.

unit (mm)

. j . 200

~

f

.........................

267 4 .

200A

r

.

.

.

.

.

.

.

.

.

.

1800

.

.

.

A

.

.

.

.

.3

i .....

.200

~

fl

..........................

267 4 A

2.3

&

Figure 4. Specimen (SPCC, o3,-170Mpa)

0.

i20 .

.

A

000 .

200 o A

.

1 ..... v .........:

~

Figure 5. Specimen reinforced by tie bolt (M10) a - - t$- -L - 1 - - - t$ o D D

(3)

2.3 Specimen The SPCC (cold-reduced carbon steel sheet, ISO3574-1986) pipe with diameter (D)-thickness (t) ratio (D/t=ll6) was used as about 1/3~ 1/4 scale model of the actual shock absorbing pipe. The three kinds of span length L=I.0, 1.4, 1.8m were selected as shown in Figure 4 in order to examine the effect of span length on the impact local deformation. The newly devised specimen reinforced by tie bolt was also made for the short span (L=l.0m) in order to control the local deformation as shown in Figure 5. 2.4 Dropping weight The dropping weight (W=l.78kN) was chosen as about 1/3.3 scale of the maximum rock weight 20kN in the debris flow by using the Froude scale law. 2.5 Determination of kinetic energy Herein, it is assumed that the external kinetic energy of dropping weight is absorbed by only local deformation of the pipe beam. C.P.Ellinas et.al.[3] have proposed the static load "~local deformation relation of a tubular beam. For the thick tubular beam (D/t<40), T.Hoshikawa et.al.[4] modified it to the dynamic load~local deformation relation by considering the strain-rate effect as follows 9

where, P 9dynamic local load, 6 L local deformation, D 9diameter of pipe, t 9thickness of pipe, o yd (--1.20 y) 9dynamic yield stress considering the strain-rate (~ --10~ effect, "

70 O y 9static yield stress, K 9constant value obtained by the test as follows [4] 9

i

K --- 161

TM

[185

(D o < 3.5D)

20 18! 161

~

(5)

( 0 o > 3.50)

~beam

,o,a, '

~ 10 ~' 8 6

,//

4

where, Do " diameter of impacting body. Therefore, the absorbing energy by the local deformation can be found by integrating Equation (4) as follows:

U =f:"Pd6 L ..1KtTyat2 aL---~SD 1.8

(6)

2 0

10

,

,

,

r

,

,

20

30

40

50

60

70

80

5 (ram) (a) Ho.4=74.5cm (V---382cm/s) 20 18 16 14

.-

--12 ~ 10

where, U" absorbing energy by local deformation of pipe. When the kinetic energy of dropping weight is assumed to be absorbed by the energy due to only local deformation, the former energy is equal to the latter one as follows:

loca~.~.~f.. ,'p

14 '12-

local f

J

" ~

.~" ~t'~otal

6 4 .

0

2/ .

10 20 30

.

.

40 50

.

60 70

-

_

80 90

5 (ram) (b) 1-Io.5=106.5cm (V=457cm/s)

Ea - l m v 2

--U

where, E,,

9kinetic energy corresponding to the local deformation ratio a , m 9dropping

(7)

Figure 6. Load - deformation curve (L--1.4m) Effect of dropping height

weight mass, Va : impact velocity corresponding to a . Therefore, it is found that the external kinetic energy can be expressed as the function of a ~ 6L/D. For instance, E0.3 is defined as the kinetic energy so that the value of a =0.3 may be found by changing the dropping height (velocity) at the constant weight W=l.78kN. Therefore, the relation between kinetic energy and dropping height (velocity) is found by using Equations (6) and (7) as shown in Table 1. Herein, the kinetic energy is determined so that the local deformation ratio a may be occurred in the range of a =0.3~0.6. It is desired, however, from the viewpoint of design purpose that the value of a may be occurred in the range less than 0.5 ( a ~--0.5). Table 1. Relation between kinetic energy and dropping height (velocity) ct E~ (kN" cm) Ha (cm) V~ (cm/sec) 0.3 75.8 42.6 ...... 289 0.4 132.9 74.7 382 0.5 189.9 . . . . . 106.7 ..... 457 0.6 274.7 154.3 549

71

0.8 0.7 'A @ 0.6 0.5 0.4 0.3 0.2 0.1 / 0.0 0

I__im ,id9in -for.a

L=l.0m(L/D=3.73) L=l'4m(L/D=5"24) L=l.8m(L/D=6.73)

~

~

modifiedEllinasformula 9 /

0.8 0.7

0.6_

/

/~ usualpipe

0.50.40.3 i i i

I I

Eo3 i Eo.4 I Eo.sl .

1

50

.

.

,

[

,

250

Figure 7. ct"~E,~ relation 9 Effect of span length

I

I

0.1 0.0

Eo.o

100 150 200 Ect (kN-era)

~~----,

300

0

50

I

100 150 200 Ect (kN cm)

250

300

Figure 8. ct "~E,~ relation 9Effect of tie bolt

Photo~aoh 2. Effect of tie bolt (E0.5)

3. EXPERIMENTAL RESULTS AND CONSIDERATIONS 3.1 Load~"deformation curve Figure 6(a) and (b) illustrate the load "- deformation curves at the dropping height H0.4=74.5cm and Ho.5=106.5cm in case of the span length L=l.4m. It is recognized that the beam deformation becomes large as the increase of the dropping height, although the local deformation is not so changed.

3.2 Comparison with the modified Ellinas formula Figure 7 shows the relation of a "~Ec~ comparing with the modified Ellinas formula. It is found that the modified Ellinas formula estimates relatively large local deformations (safe evaluation) in the ranges of small kinetic energy and long span length. It should be noted, however, that the experimental value c~ becomes sometimes larger (critical evaluation) than the modified Ellinas formula in the range of large kinetic energy and short span length. Therefore, the local deformation should be controlled by reinforcing by the tie bolt in case of short span length (L=l.0m).

72 3.3 Effect of span length It should be also noted from Figure 7 that the value of a decreases generally as the span length increases. This tendency may be the reason why the beam deformation is increased and, as such, the local deformation will be reduced as the span length increases. 3.4 Effect of tie bolt Figure 8 illustrates the c~"~E~ relation comparing the pipe reinforced by tie bolt with the usual pipe in case of short span length (L=l.0m). It is found that the local deformation is about 20% reduced by reinforcing by tie bolt. Photograph 2 shows the deformation profile taken from the end side of the beam when the impact load is applied at the center of span in case of kinetic energy E~=0.5. It is noted that the reinforcement due to tie bolt is remarkably effective for the control of the local deformation. 4. CONCLUSIONS The following conclusions are drawn from this study. 1. It was noted that the beam deformation increases as the increase of dropping height, although the local deformation was not so changed in case of long span beam. 2. It was recognized that the modified Ellinas formula can approximately estimate the ratio of local deformation of thin tubular beam in the range of kinetic energy E~ =0.3~-0.4 and the long span length. 3. It is noted, however, that the local deformation of short span exceeds the modified Ellinas formula in case of E~ =0.5. 4. It was found that the local deformation of short span was reduced about 20% by reinforcing with the tie bolt. Therefore, the short span pipe with tie bolt will be effective for the control of local deformation under heavy impact load. The effect of tie bolt will be required to examine from the analytical point of view in the future. REFERENCES 1. N.Jones and W.Q.Shen, A Theoretical Study of the Lateral Impact of Fully Clamped Pipelines, Proc. Instn. Mech. Engrs. Vol.206, pp.129-146, (1992). 2. N.Jones, S.E.Birch, R.S.Birch and M.Brown, An Experimental Study on the Lateral Impact of Fully Clamped Mild Steel Pipes, Proc. Instn. Mech. Engrs. Vol.206, pp.111-127, (1992). 3. C.P.Ellinas and A.C.Walker, Damage on Tubular Bracing Member, IABSE, Colloquium, Copenhagen, Vol.42, (1983). 4. T.Hoshikawa, N.Ishikawa, H.Hikosaka and S.Abe, Impact Response Displacement of Fixed Steel Pipe Beam Considering Local Deformation and Strain Rate Effect, Proc. JSCE, No.513fl-31, pp.101-115, April, (1995).

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

73

Normal impact of spherical balls on metallic plates b P.U. Deshpande a and N.K. Gupta a

b

Defence Research & Development Organization, Sena Bhavan, New Delhi 110 011, India Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

Experiments involving the normal impact of spherical metallic balls of Tungsten Carbide, and Mild Steel on thin metallic plates of Mild Steel, Armour Steel, and Duralumin have been reported. Based on these test results, a semi-empirical model for the prediction of the ballistic limit and residual velocity has been developed. The results obtained from the model have been compared with the experiments and these match well.

1. INTRODUCTION Comprehensive surveys of the Literature on mechanics of projectile impact, available up to the time of their publication, have been published by Backman and Goldsmith [1], Zukas [2], and Corbett et.al. [3] These cover major experimental and analytical studies that deal with the penetration and perforation of plate targets when impacted by projectiles. The first formulae that were developed [1, 4] predict the penetration depths into semiinfinite targets when struck normally by a projectile. The advent of battleship armour in the 19th century led to the development of equations predicting the depth of penetration of finite thickness armour plating [5]. Even to this day these formulae and others like them are being used extensively by impact engineers. One of the major works dealing with the perforation of plates by spherical balls is by Goldsmith and Finnegan [6] who have reported a series of tests in which Aluminium and Mild Steel plates were struck normally by hard steel spheres, travelling between 150 and 2700 rn/s covering sub-ordnance, ordnance, and ultra-ordnance ranges. A commonly used measure of the target's ability to withstand projectile impact is its "ballistic limit" and much work has been carried out by researchers to estimates this parameter. In general terms the ballistic limit of a structure is the greatest projectile velocity the structure can withstand without perforation. Another term "ballistic thickness" was introduced by Gupta and Madhu [7] This is defined as the minimum thickness of plates required for the projectile of known weight and velocity to prevent any perforation. In recent years appreciable advances have been made in the analytical approach to the problem of penetration, with the models gradually becoming more and more sophisticated and accurate. However, these, too have relied heavily, and indeed still do, on experimental data to justify certain assumptions made and to supply various parameters for the models. It shows

74 the continuing importance of experimental data in improving the understanding of perforation process. The present paper is an attempt in this direction for developing a model for prediction of ballistic limit and residual velocity in the case of the normal impact of a spherical ball on a metallic plate. 2. EXPRIMENTS Experiments were conducted in DRDO laboratory wherein Tungsten Carbide (TC) and Mild Steel (MS) spherical balls were fired on Mild Steel, Armour steel (AR), and Duralumin (DU) plates using a standard fragment launching gun. The velocity of the impacting balls were within the ordnance range. Result of about 100 tests are summarised in table 1. All plates were square in shape and were held at their four comers onto a specially designed fixture with the help of C-Clamps. The impact and residual velocities of the balls were measured with the help of foil and counter method. There was no significant deformation of the balls after perforation of the plates. The sequential record of the perforation of 8 mm thick MS plate by 16 mm MS spherical ball weighing 16.5 g at an impact velocity of 651.5 m/s is shown in Fig. 1.

Weight W (g)

Ball Dia d (mm)

TABLE 1 Experimental Data Plate Thick t (mm) .

Striking Vel. Vs (m/s) .

No. N~ of Data

Tungsten Carbid e Ball, Mild Steel Plate ,Tensile Strength s 430 MPa 1.832-1.930 6.0 04.0 700-1050 1.690-1.875 6.0 06.0 700-1000 1.730-1.780 6.0 08.0 900-1200 1.865 6.0 10.0 1150-1150 Mild Steel Ball, Mild Steel Plate, Tensile Strength =430 MPa 1.050 6.0 04.0 800-1100 16.400-16.420 16.0 06.0 500-1000 16.430-16.600 16.0 08.0 650-1100 16.590-16.620 16.0 10.0 650-1100 Mild Steel Ball, Armour Plate, Tensile Strength =430 MPa 16.360 16.0 10.0 806-1018 16.500 16.0 12.0 942-1061 01.028 06.3 04.0 661-721 01.028 06.3 06.0 1127-1167 Mild Steel Ball, Duralumin Plate, Tensile Strength =430 MPa 01.028 06.3 10.0 99-0231 Overall Range 1.028-16.62 6.0-16.0 4.0-12.0 99-1200

4 4 4 1

15 16 9 5 14 91

75

Fig.1

Sequential Perforation of MS Plate by MS Ball at 651.5 m/s (Thickness of Plate = 8 ram; Dia. of baU = 16 mm; Weight of BaU = 16.5 g)

3. A N A L Y S I S Most of the previously derived expressions for the prediction of residual velocity can be written in the form [8]

Vr =or

(Vsp-

q

(1)

76 where, Vs, V r, and V b are the striking, residual and ballistic limit velocities respectively in m/s, p and q are the parameters varying fi-otn one author to another, and

1

ot =

(2)

where, W is the weight of tile ball, Wp is tile weight of the target ino jetted in rront of the ball, and 13 is a parametric constant. A dimensionally ho.nogeneous relation used in Ihe prese.t work fi>r the prediction of ballistic limit velocily is:

I (t Y Vb - +#-p-tCi) where, t is the thickness of the target plale, d is the diamcler of Ihe Imlls, p is the density of ball material, and ft is the tensile strength of the target plate. The values of tile five parametric constants involved in the model 13, Y, p, q, and r have been estimated from the regression analysis of the test data derived from more than 100 experiments. The value of these constants is found Iv be IT = 6.6,1' = 1.6, p = 0.8, q = 1.54, and r - 0.64.

800 '0 "~ 600

.....

....

/

~

A MS/AR x ~

O

>

t3

.~

A

0 0

400 0

4~ o 9~

200

4••x

o

............... T" 0

T--

200

400

O l ~

600

l ~ i d m l Vekxzity

Fig. 2. Predicted Versus Observed Residual Velocity (m/s)

800

77 0

0

,-

............

i

o

. . . . . . . . . . -=~-. . . .

600

~-

[............'.........d=6. t 4

o

d=6, t,=6

~

~4oo

'

I

K

l 0 d'16, t ' 6 i / ," : ....................d..--'16, t=6 I / a : /

500

- - - - - - d--6. l=6 ~9 d='6, t*8 . . . . . . d'~6. t,=8

,..600

8

~

o d--6, I-4 ------- d=6. t'-4

i ~ d2)o,~'sLl

'I/ ""I/

I ...... d~~ 1"8 ~ i [---

d.--16, t-.lO~a : I / "

~3oo

400

/

"o

. .r

200

t00 0

.... ~ . . . . . . . . .

-~...:.,

0

300

_ :: . . . . . .

:::::::::::::::::::::

.

600

~- - = - . : : : = : = = 7

900

0

---~

0

1200

300

60,0

9(K~

1200

Striking Velocity (n~s)

Striking Velocity On/s)

Fig. 4 M S Ball, MS Plate

Fig. 3 T C Ball, hiS Plate

250

t...............; .............d;~i6,~=i6

1

i .......... a~z6, ~,,10 :

u d=16, 1=12, [ .~ .................d--16, t=12 ! a d.~6.3, t=~4

I! . . . . . .

"~" =

---

200

i J

d_-6.3, t ~

r

I .~r

t~ ,,tq

d'~6,.3, 1,,,.6 ] ]'1;t,

o

t50

_9,0

,i

5O

0

I .....

0

300

600

900

Striking Vdo~ty (m/s) Fig. 5 MS Ball, Armour Plate

1200

0

=

........ ~'

......... : ~ :

3O0

--*-

~. . . . . . . . . .

600

i-- ...........................

90O

Striking Velocity (u4x) Fig. 6 MS Ball, DU Plate

12O0

78 4. CONCLUSIONS The paper presents result of experiments in which mild steel, armour steel and duralumin plates were subjected to normal impact by tungsten carbide or mild steel balls. A model has also been developed to predict the residual velocity of the spherical balls impacting metallic plates with a velocity up to 1200 m/s. Results computed from this model match the experiments well. The model is dimensionally homogenous and is applicable to various combinations of impacting balls and target plates considered. REFERENCES

1. Backman, M.E. and Goldsmith, W., The mechanics of Penetration of Projectiles into Targets, Int. J. of Engineering Science, 1978, 16, 1-99. 2. Zukas, J.A., High Velocity Impact Dynamics, John Wiley and Sons, 1990. 3. Corbett, G.G., Reid, S.R. and Johnson, W., Impact Loading of Plates and Shells by Free Flying Projectiles: a Review, Int. J. oflmpact Engineering, 1996, 18, 141-230. 4. Young, C.W., Depth Prediction for Earth Penetrating Projectiles, Proc. ASCE, 1969, 95, SM3,803-817. 5. Johnson, W., Some Conspicuous Aspects of the Century of Rapid Changes in Battleship Armours ca 1845-1945, lnt. J. lmpact Engineering, 1988, 7, 261-284. 6. Goldsmith, W., and Finnegan, S.A., Penetration and Perforation Process in Metal Targets at and Above Ballistic Limits, Int. J. Mech. Sci., 1971, 13,843-866. 7. Gupta, N.K., and Madhu, V., An Experimental Study of Normal and Oblique Impact of Hard-Core Projectile on Single and Layered Plates, Int. d. Impact Engineering, 1997, 19, pp. 395-414. 8. Lambert, J.P., and Jonas, G.H., Ballistic Research Laboratory, BRL-R1852(ADA021389), 1976. 7. Gupta, N.K., and Madhu, V., An Experimental Study of Normal and Oblique Impact of Hard-Core Projectile on Single and Layered Plates, Int. J. Impact Engineering, 1997, 19, pp. 395-414. 8. Lambert, J.P., and Jonas, G.H., Ballistic Research Laboratory, BRL-R1852(ADA021389), 1976.

Structural Failure and Plasticity (IMPLAST 2000)

Editors:X.L.Zhaoand R.H.Grzebieta 9 2000ElsevierScienceLtd.All rightsreserved.

79

Impact performance and safety of steel highway guard fences Yoshito Itoha, Chunlu Liua and Shinya Suzukib "~Center for Integrat~ Research in Science and Engineering, Nagoya University, Fum-cho, Chilcasa-ku, Nagoya 464-8603, Japan bD~almaent of Civil Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

In the recent years, a lot of obvious changes with the traffic safety and reliability of highways have happened along with the improvement of the road network and vehicle capacities. These changes consist of the increases of the traffic stxxxt, the large-scale vehicles and heavy trucks, the improvement of vehicular performances, and the height of the center of gravity of tracks. Therefore, to take into consideration these changes into the design and construction of new highway guard fences, the design specifications of guard fences were re-examined and the revised specifications were implemented from April 1, 1999 in Japan. However, because of the huge consumption in time and cost to test the performances of full-scale guard fences in the field, some assumptions are adopted while modifying the design specifications. Numerical analyses are still necessary to confirm the impact performance and safety of new types of steel highway guard fences for the design of new highway guard fences. Fuaher, it is also very important to study such issues of the existing guard fences that were design under the old specifications and are taking effect in the field. In this study, FEM models are developed for trucks and guard fences to reenact their behaviors. The validity of these models is demonswated through numerical examples. The solution approach is carried out using nonlinear dynamic analysis software of stmctuw,s in three dimensions and the calculation results ale compared with the full-scale experimental data_

1. INTRODUCTION With the improvement of the road network and vehicle capacities, the vehicles have taken a more imtmrtant role in the freight transport. In Japan, the change of the allowable weight of tracks from 20 tf to 25 tf from November 1994 increases the percentage of heavy trucks and the height of the gravity center of the tracks. Accordingly, from both the function and safety viewpoints, these changes challenge the conventional transportation infrastructures such as roads, bridges, and guard fences. Furthermore, the increases of the traffic speed, the large-scale vehicles and heavy tracks, the improvement of vehicular performances, and the height of the center of gravity of tracks also challenge the design and analysis of highway guard fences. Therefore, to take into consideration these changes into the design and construction of new highway guard fences, the design SlXX:ificationsof guard fences in Japan were re-examined and the revised SlXX:ifications were implemented from April 1, 1999 to replace the former design guideline published in 1972tq. In USA, the nationally recommended procedures for the safety performance evaluation of highway fea..aues comprise of three factors: the structural adequacy, occupant risk, and vehicle trajectoryt2]. However, because of the huge consumption in time and cost to test the performances

80 of full-scale guard fences in the field, some assumptions are adopted while modifying the design specifications. Numerical analyses are necessary to confirm the impact performance and safety of new types of steel highway guard fences for the design of new highway guard fences. Furthermore, it is also very important to study such issues of the existing guard fences that were designed under the old guidelines and are taking effect in the field. Several approaches in this field have been carried out on the impact simulation between vehicles and roadside safety hardware [3' 4], a finite element computer simulation for the vehicle impact with a roadside crash cushion t~, and a procedure for identifying the critical impact points for longitudinal barrierst6]. Because of the huge consumption of time and cost, it is difficult in the field to measure the collision performances of the full-scale guard fences for various cases. In this research, by taking the advantages of both computer software and hardware, the collision impact process between the heavy trucks and the guard fences is simulated based on the presented numerical calculation models for both the heavy tracks and guard fences. A nonlinear, dynamic, three-dimensional finitemlement code LS-DYNA3D is capable for simulating the vehicle impact onto the guard fences['0. The analysis results are further compared with the full-scale experimental results using a real truck in order to demonstrate the approach presented in this research.

2. FEM ANALYSIS MODELS OF GUARD FENCF~ AND TRUCKS 2.1FEM malysis modd of guard fmces This re.seamh focuses on the collision impact of heavy tracks with a high ~ onto the guard fences at the two sides of roads and bridges. The angle between the truck movement direction and the guard fence plane is an important pamnrder to determine the impact force and displacement in addition to the track speed, the track weight, the height of the gravity center of the track, the guard fence, the curb, and others. Figure 1 shows the basic collision analysis components including a moving vehicle, the guard fence, the impact speed and the impact angle in this research. The codes of columns and beams are also given in this figure.

Figure 1. Collision features

Figure 2. Analysis model of guard fence (mm)

In 1992, a structural model was presented for the steel bridge guard fences for the purpose of the full-scale experiment carried out in the Public Work Research Institute of Japan tsl. In the present ~ h , an FEM analytical model based on the shell elements was formulated for the structural components of the steel bridge guard fences and the application procedure was presented in the previous research t91.Figure 2 shows the cross section of a highway bridge guard fence and the FEM model of the guard fences in three-

81 dimensions. The fence column is made of the H-type steel whose web and flange are 150 mm wide and 9 mm thick, and 150 mm wide and 9 mm thick, respectively. Both the main beam and sub-beam are of pipe sections. The pipe diameter and thickness of the main beam are 165 mm and 7 mm, restxx:tively. The pipe diameter and thickness of the steel sub-beam are 140 mm and 4 mm, respectively. The span of the beams over two contiguous columns is 1500 mm. The Young's modulus of steel is 206 GPa, and the Young's modulus of concrete is 24.4 GPa. The Possoin's ratios of steel and concrete are 0.3 and 1/6, mstxx:tively. The shear moduli of steel and concrete are 88 GPa and 10.5 GPa, respectively. The yield stress and initial swain hardening of steel are 235 MPa and 4.12 GPa, restxx:tively. The strain hardening of steel starts from 0.0014. The concrete volume modulus is 12.18 GPa. The concrete compressive and tensile strengths are 23.52 MPa and 2.29 MPa, respectively. The steel is assumed to be an isotropic elasto-plastic material following the von Mises yielding condition. The swain hardening and strain velocity are taken into consideration the stress-strain relationship. The concrete constructed in the curb is assumed as a general elasto-plastic material. This means that the concrete is in the general elasto-plastic condition while the concrete in the compressive side reaches the yield point and only the cut-off stress is available once the tensile stress increases to the tensile strength. The boundary condition at the concrete curb is considered as a fixed end. 2.2 FEM analytical model of trucks In this research, the tracks whose weights are 25 tf are studied by modeling the truck frame, engine, driving room, cargo, tiers and so on. The structure of the 25 tf track is similar to the 20 tf truck except the strengthened flan~ and the loading capacity of the vehicle axles. As shown in Figure 3, the track is modeled according to the ladder-type track frame whose two side members are of channel sections so that some facilities such as the fuel tanks and pipelines can be attached inside the side members. The thickness of the side member is 8 mm, and the yield stress is 295 MPa. The general elasto-plastic stress-strain relationship is adopted. The solid element with the same shape and volume is modeled for the engine and the transmission, and their weights are adjusted according to the practical vehicles. The tiers, wheels, and gears of a truck influence its behaviors during the collision impact significantly. The connection of the tier and the wheel is assumed to be a rotation joint so that the movement of the wheel can be simulated. A constant value of 0.45 is used for the friction coefficient between the tier and the road pavement. The driving room and other small portions are also modeled for the purpose of the numerical calculation.

Figure 3. Truck frame model

Figure 4. Truck structural FEM model

82 Figure 4 represents the presented FEM model of a heavy truck that will be used in the following of this paper. In this model, the numbers of nodes and elements are 3532 and 3904, restxx:tively. The Young's modulus of steel is 206 GPa, while that of aluminum is 70 GPa. The Possoin's ratios of steel and aluminum are 0.30 and 0.34, respectively. The shear moduli of steel and aluminum are 88 and 26 GPa, ~ v e l y . In the case of guard fences, the steel is assumed to be an isotropic elasto-plastic material following the von Mises yielding condition, and the sires-strain relationship is perfectly elasto-plastic. The aluminum used for the cargo body is assumed in a multi-piece linear stress-swain relationshiptl~

3. PARAMETRIC STUDII~ 3.1 Effects of Strain ~ and Slrain Veka:ity Paramelric study is first carried out to check the effects of the strain hardening and strain -velocity on the displacement of the guard fence c o u n t s . It is assumed that the strain hardening starts from 0.0014 and the initial strain hardening modulus is 4.01 GPa (2% of the Young's Modulus). On the other hand, the yield stress usually increases with the increase of the strain velocity. The scaling relation of the yield stress is used in this research to investigate the effects of the strain velocity. Figure 5 shows the displacement of a column with time in four combined cases by considering the strain hardening and strain velocity or not. In this calculation, the track weight, collision speed and collision angle are 14 tf, 80 km/h and 15~ respectively. According to the displ~,.ement tracks as shown in this figure, the effects of the swain hardening and strain velocity on the maximum response displacement and the residual displacement are very obvious. The displacements follow the similar tracks with time if one of the swain hardening and the strain velocity is considered and the other is eliminated. It should also be noticed that at about 0.5 second after the collision impact the displacement increases rapidly within a very short time at all cases. The experimental results,are close to the results obtained by considering both the strain hardening and the strain velocity. Therefore, these two factors will be taken into account in the following part of this paper. f"

'

'

'

"

'

' " '

'

~ is01-l-

~ o u t v e l o c i t y_..,...,..-.-.-IF-~ r I Withouthardening/Withvelocity

no0

'

'

i

i

I

I 0.5

i

i

i

'

....

'

'

.

.

.

.

' ' /

] P10(1-2-8model) -I

~' ~ _

i

Time (sec) Figure 5. Effects of strain hardening and velocity

I I

.

.

.

.

.

.

.

.

.

.

d

. . . . .

~"

N~xpefimentalmaximumdisplacement ,

'"

L ,ooI- ............. ~ ......... b - - - ~ . . -

F- --~f Experimentalresidualdisplacement I 0

"'

[ Experimentalmaximum ~ displacement " .

N~--'-WI~ h~den'in~bVith-v-eI-'ocTty-" ~

.......

O_

;~/'

houtvelocity

i F~ ......':-~ ~:a" s0~ ~

'

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

:1.6mode!) .

.

.

.

.

.

.

.

.

L / \ P10(8-8-32 model, ~ .__~"~/ Experimentalrelsidualdisplacement , _O--

0

0.5

I

Time (sec) Figure 6. Effects of mesh sizes

3.2 Effects of Mesh Sizes Further study is carried out to dexermine the appropriate mesh sizes by following the tracks of the displacement of the bridge guard fence with time. The calculation results are compared with the

83 experimental values by adjusting the mesh sizes of the column web, the column flange, and the horizontal beam pipe. Three cases, 1-2-8 model, 4-4-16 model and 8-8-32 model, are studied. The three numbers of each model represent the classified portions of the column web, column flange and beam pipe, lr.SlXX:tively.The numbers of FEM nodes of these three models are 5739, 10404, and 28125, respectively. Their elements are 5158, 9574, and 27045, re~qx~ively. The calculation results are shown in Fig. 6 as well as the detected values from the actual experiment for the case when the truck weight, collision speed and collision angle are 14 tf, 80 kmha and 15~ respectively. This figure shows the displacement of only one column C 10 whose position can be recognized from Fig. 1. According to the displacement curves in Fig. 6, the residual response displacements in cases of 44-16 model and 8-8-32 model are very large at 0.5 second after the collision impact (about 40%). The tracks in these two cases are almost same within the first 0.5 second. The final displacement is about 10% less than the maximum value in all cases. Compared to the maximum and residual displacements from the ex~riment, the 4-4-16 model contributes very good agreements. Therefore, this model will be adopted in the following analysis.

4. NUMERICAL IMPACT ANALYSF.S OF GUARD F E N C ~ Further study is carried out to demonstrate the presented models by comparing the calculated results with the actual experimental results in the case of collisions between the heavy truck and guard fences. In both the experiment and calculation, the impact speed is 80km/h and the impact angle is 15~ The weight of the track is 14 ft. The impact performances of the guard fence after 0.1 and 0.5 seconds of the vehicle collision are shown in Fig. 7 (a). Figure 7 (b), (c), and (d) represents the calculated results and the experimental results of the displacement responses for the columns at the top, the main beams, and the sub beams, respectively. In Figs 7 (b), (c), and (d), the horizontal and vertical axes represent the time (s) and displacement (mm), respectively. The responses of several fence column tops in the form of displacement are shown in Fig. 7 (b) in terms of different types of iines. The maximum and residual response displacements of the column C10 are 95 mm and 85 mm, respectively. In the practical vehicle experiment, these two values are 97 mm and 84 rnm, respectively. It is obvious that the calculation results are quite near to the experimental results. Figures 7 (c) and (d) show the displacement curves of several main Ix:ams and sub-beams with time, ~vely. The calculation value of the main beam B10 at the central section is 99 mm, about 30% higher than the detected value of 76 mm from the practical experiment. However, the calculated displacement value of 105 mm of the sub-beam $9 at the central section is less than the experimental value of 130 mm. 5. CONCLUSIONS The following conclusions can be stated from this research: (1) It is possible to simulate the collision process and to visualize the movement of the track and the performances of bridge guard fences due to the collision impact of heavy tracks based on the FEM models for trucks and guard fences.

84

200

150

I

'

'

'

i

I

'"'

'

'

'

/'

Experimental residual displacement

Experimental residual displacement

S9 100

9

-

.

.

.

SI0

.

I00 ..........

I

~i

--1-,

0 c)

_ _ . ~ _ - 2. . . . ,

,

J

I

0.5

,

,

I

I

o-

I

~ 0

!

ss

Sl2

~ 0.5

I

d)

Figure7. Displacement responses and comparisons of guard fences

(2) Parametric studies show that both the strain hardening and the strain velocity should be considered, and the mesh sizes also effect the acxaaacy of calculation. (3) The performances of heavy trucks during the collision impact obtained from this research are very consistent with the actual experimental results. This research can be extended in several ways. Energy absorption of each guard fence component needs further emphasize in the revision of the present design ~ifications. The performances of passengers within and after the collision impact need research in detail. It is also invaluable to study the performances of guard fences under a continuous collision. Further rese,a~h is also needed to study the performances of guard fences and concrete curbs simultaneously.

85 REFERENC~ 1. Design spec~cations of guardfences, Japan Road Association, Tokyo, 1999 (in Jalmnese). 2. Recommended Procedures ]br the Safety P e r f o ~ e Evaluation of Highway Features, NCHRP Report 350, Transportation Research Board, Washington, 1993. 3. Wekezer, J., Oskard, M., Logan, R. and Zywicz, E., Vehicle Impact Simulation, Journal of Transportation Engineering, ASCE, 119:4, 1993, 598-617. 4. Reid, J., Sicking, D., Paulsen, G., Design and Analysis of Approach Terminal Sections Using Simulation, Journal of Transportation Engineering, ASCE, 122(5), 1996, 399-405. 5. Miller, P. and Camey, J., Computer Simulations of Roadside Crash Cushion Impacts, Journal of Transportation Engineering, ASCE, 123:5, 1997, 370-376. 6. Reid, J., Sicking, D., and Bligh, R., Critical Impact Point for Longitudinal Barriers, Journal of Transportation Engineering, ASCE, 124:1, 1998, 65-72. 7. Hallquist J., LS-DYNA3D Theoretical Manual, Livermore Software Technology Corporation, LSTC Report 1018, University of California, 1991. 8. A study on the Steel Guard Fences, Research Report No. 74, Public Works Research Institute, Tsukuba 1992 (in Japanese). 9. Itoh Y., Moil, M. and Liu C., Numerical Analysis on High Capacity Steel Camrd Fences subjected to Vehicle Collision Impact, The Fourth International Conference on Steel and Aluminium Structures, Espoo, Finland, 53-60, 1999. 10. Itoh, Y., Ohno, T. and Liu, C., Behavior of Steel Piers subjected to Vehicle Collection Impact, The Fourth International Conference on Steel and Aluminium Structures, Estx~, Finland, 821-828, 1999.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

87

I m p a c t b e h a v i o r of shear failure t y p e R C b e a m s Tomohiro ANDO a, Norimitsu KISHI a, Hiroshi MIKAMI b, and Ken-ichi G. MATSUOKA a aDepartment of Civil Engineering, Muroran Institute of Technology, 27-1 Mizumoto, Muroran, 050-8585 Japan b Technical Research Institute, Mitsui Construction, Co. Ltd., 518-1 Komaki, Nagareyama, 270-0132 Japan

In this study, in order to establish a rational impact resistant design procedure of shear failure type Reinforced Concrete (RC) beams, weight falling impact tests are performed. Twelve simply supported rectangular RC beams without shear rebars are used for these experiments. All the RC beams are of 150 mm width and 250 mm depth in cross section, in which rebar and shear span ratios are taken as variables. Impact load is surcharged onto the midspan of RC beams by freely dropping a 300 kg steel weight. Here, iterative and single loading methods are applied to investigate the effect of loading method on impact behavior of the beams. From these experimental results, it is seen that the impact resistant design for shear failure type RC beams may be rationally performed by using static shear capacity with some safety margin. 1. I N T R O D U C T I O N

In order to enhance the safety margin of RC structures against impact load such as rock sheds, check dams, and nuclear power plants, many researchers have been studying the impact resistance of RC members (i.e., beam, slab, and column) experimentally and analytically[l, 2, 3]. Consequently, it becomes clear that the impact resistance of bending failure type RC beams may be estimated by using static bending capacity, and the impact resistant design for the beams may be also rationally performed based on the relationships among maximum reaction force, input and absorbed energy, and residual deflection[4]. However, the impact resistance of shear failure type RC members has not been adequately understood yet even regarding beams. From this point of view, the impact behavior of shear failure type RC beams without shear rebars is experimentally discussed in this paper. 2. E X P E R I M E N T A L OVERVIEW

The static design values of twelve RC beams used in this study are listed in

88 Table 1 List of static design values of twelve RC beams Impact Rebar Shear span Static shear Static bending Shear-bending capacity ratio ratio capacity capacity Specimen velocity ratio a(=V,,,c/P,,,c) aid Vu,~ (kN) P,,,c (kN) v (m/s) Pt 0.42 A24'I, S 1-3, 3 2.4 163.1 0.63 A36-I,S 1-3, 3 0.018 3.6 68.8 108.7 0.84 A48-I,S 1-3, 3 (A) 4.8 81.5 0.67 B24-I, S I - 2 , 2 2.4 78.4 1.00 B36-I,S I - 3 , 3 0.008 3.6 52.3 52.2 1.33 B48-I, S 1 - 3, 3 ( B ) 4.8 39. 2

Figure 1. Dimensions of RC beams

Photo 1. Experimental set-up

Table 1. Nominal name of each beam is designated with reference to main rebar type (A or B), shear span ratio a/d (2.4, 3.6 or 4.8, here, a: shear span; d: effective depth), and loading method (I: iterative loading or S" single loading). The static shear and bending capacity V~,c, P~,c are calculated using conventional prediction equations[5]. According to the equations, all the RC beams except B48 beam will be collapsed with shear failure mode under static loading, since those shear-bending capacity ratio as (= V,,,c/P,,,c ) are smaller than 1.0. General view of RC beams used here is shown in Figure 1 which is of rectangular cross section of 150 x 250 mm in size and their clear span length is varied from 1.0 to 2.0 m long. Here, two kinds of deformed rebar are used: A type( 19 m m in diameter); and B type( 13 m m in diameter). At commencement of the experiment, the average concrete compressive strength and yielding stress of rebars are approximately 32 MPa and 390 MPa, respectively. Each RC beam is simply supported and is pinched on its top and bottom surface at a point 200 m m inside from the ends as shown in Photo 1. Impact force is loaded onto the mid-span of beam by means of a freely falling method using a 300 kg steel weight. Here, two types of loading method are applied:

89

zx-

zX

(a) A~4-I

(d) B24-I

(b) As6-X

(~) Bs6-I

/x

/X (r

A48-I

(f) B 4 8 - I

Figure 2. Crack patterns of six RC beams after Rerative loading iterative loading with 1 m / s initial and incremental impact velocity until RC beam is collapsed; and single loading with the same impact velocity to the final one in the iterative loading case. It is assumed that RC beams have been collapsed when a severe diagonal crack was developed from the loading point to supporting point (see Figure 2(a),,~(e)). On the other hand, in case of bending failure type RC beams, it is assumed that RC beams have been collapsed when cumulated residual deflection reached one-fiftieth of span length[4]. Here, only B48 beam has been collapsed with bending failure mode (see Figure 2(f)). In this study, weight impact force, reaction force, and the mid-span deflection (hereafter, deflection) are continuously recorded by using wide-band analog data recorders. The maximum measurable frequencies of the load cells and LVDT are 4 kHz and 915 Hz, respectively. All these analog data are converted into digital ones with 100 ps sampling time. 3. E X P E R I M E N T A L R E S U L T S A N D D I S C U S S I O N S 3.1. Characteristics of response waves

Time histories of impact and reaction force P, R and the deflection ~ for A36 and B48 beams are shown in Figure 3. Here, reaction force is evaluated summing up the values both supporting points. From Figure 3(a), it is seen that the duration and maximum value of impact force are similar to those of reaction force in each impact test. And these two forces are excited with only one half-sine wave having about 10 ms duration in cases of impact velocity V = 1, 2 m/s. The duration is a little longer than the fundamental natural period of the beam which is about 7 ms. In case of V = 3 m/s, the second wave is subsequently generated after the half-sine wave. On the other hand, the deflection wave behaves depending upon the magnitude of impact velocity. In cases of V = 1, 2 m/s, the deflection wave behaves as a damped free vibration. Whereas, in case of V - 3 m/s, it faintly vibrates with some drift after unloading. From Figure 3(b), it is seen that at the beginning of impact in each impact velocity, even though distribution characteristics of impact and reaction force waves are different each other, the duration and maximum values of the forces are almost similar. Namely, the impact force is excited with two hMf-sine

90

9 Iterative loading (kN) impact force, P

150

lm/s

1SO

75 0-

.

.

.

.

.

.

.

-75

7S

,,

0

f%v ~_

__ ~,,

/%

,

-75 1SO TS --

150

2m/s 75

o

I\

-75 150 7s

~

0

3m/s

-Single 5~ling Reaction force, R

(kN)

-75 150 750

q

6 3

(ram) Deflection, 8

0

r,~,

__ . . . .

12

,

, ,

6

'

_A

0

-

'11~r

.

-%

20

,.o 60 ti..(m,)

80 _TSo-

20

L __,,

-3

40

timelm.)

80 -1~'0

60

:..J.

r-

.

.

40

.

.

80

120

160

(~) As~

- - - - - - Iterative loading 150 lm/s

(kN) Impact force, P

1SO 75

75 . . . . .

Single loading (kN) Reaction force, R

1SO

150 T5

......

-75 1SO 75

, ,,

JIt-~.~

-

1501 3m/s 7 5 - -

-%

- _

20

,

4o 6o e..(m.)

o-

~'

]'~

.

.

.

C-

.

.

.

.

.

....

S

~3

12

. ,

o

-75

Oi..

3

'-----. ......

-TS

0

6

.

-75

2m/s 75

(ram) Deflection, 8

-~

,

,

_Ts[L 80 0

I 2o

i -I 4o 60 t~.e(m.)

-6

~ '/"

,,

24 ,

_

80

(b) B4S

Figure

3.

Time histories of impact

force, reaction

force

and deflection

waves: a main wave having comparably long duration; and an incidental wave at the beginning of impact having extremely short duration and two times bigger amplitude than that of the main wave. On the other hand, reaction force is excited with almost only one half-sine wave. The deflection wave behaves as a damped free vibration in spite of the magnitude of impact velocity. The vibration period is gradually prolonged with the impact velocity V being increased, as well as duration of the impact and reaction force. This is due to the progress of damage of R C beam.

Comparing the results obtained from iterative and single loading tests for A36 beam with impact velocity V ffi 3 m/s, it is seen that the m a x i m u m impact and reaction forces in single loading test are slightly bigger than those in iterative loading one. Whereas, the m a x i m u m deflection in single loading test is smaller than that in iterative loading one. However, in case of the B48 beam with bending failure mode, the m a x i m u m values of each force and deflec-

91

15~ I

100f~ 0~-

R-a 6

"12

Deflection,

150~,, 100[

= -O.o" 5 0 ~ "

. . . .

!

, 'v=;./, 1

V= 2m/$ _

o" 50~

"

l

P-a

eL

,

i

' Vf''lm/s-I

'

,

,

18

8' (mm) ,

t

24

18

24

0

6

12

18

i

|

V= 3m/s

Deflection, 8 (mm)

24

(a) A36-I

i '

-

V= 1m/$

--- :P-a , :R-8

i2

Deflection, 8 (mm)

i

t~--~ i'2" 1'8 24 Deflection, 8 (mm)

""

'

....

t~"1

'

"

i

I

1

, _

12

18

Deflection, 8 (mm)

24

g"

6 12 18 24 Deflection, a (mm)

(b) B48-I Figure 4. Hysteresis loops P-6, R-6 for A36/B48 -I beams tion are almost the same irrespective of loading method, respectively. 3.2. Hysteresis loops of P-6 and R-6 Hysteresis loops of impact force - deflection P-~ and reaction force - deflection R-6 for A36/B48 -I beams are shown in Figure 4. From this figure, it can be seen that absorption energy estimated integrating a looped area is increased with increment of the impact velocity, and the distributions of P-6 and R-$ loops are similar to each other except the initial hysteresis in B48 beam. In case of A36 beam, it is seen that both P-6 and R-6 loops can be drawn as a triangular form. Because taking deflection as abscissa, impact and reaction forces are increased monotonically at the beginning and then they are reduced after reaching the maximum value. On the other hand, in case of B48 beam, since 1) maximum reaction force is generated at the impact force being decreased, 2) magnitude of the force is kept almost constant irrespective of the deflection increasing, a n d 3 ) the force is decreased according to decreasing of the deflection, the R-6 loop may be drawn as a parallelogram.

3.3. Distribution of dynamic response ratio Maximum dynamic reaction force - static capacity ratio R~d/P~ (Rffid: maximum reaction force, P,~" static capacity) at failure in each type RC beam is shown in Figure 5. From this figure, it is seen that the ratios for all beams except B48 beam are almost equal to unity irrespective of loading method. However, the ratio for B48 beam is bigger than unity because the beam is failured with bending mode[4]. It suggests that the dynamic capacity of RC beams governed with shear failure mode under static loading is almost equal

92

d

2.0

ii I

9

II

I

e i

1.5 o a , n -B m

t~ ~e

em

E m

a

~

1.0 L

l

0.5 r | 0.0 a:

. 9

- 0

. ~

W

0 : iterative loading O: si~l. lo,~ing I 9 9 ' A24 A36 B24 A48 B36 B48

(0.42) (O.i$) (O.i7) (0.114) {1.OO) (1.$3)

Spedmen Figure S. Distribution of dynamic response ratio

R~d/P~,,

to the static shear capacity. 4. CONCLUSIONS From this experimental study on impact resistant behavior of shear failure type RC beams without shear rebars, following results are obtained: 1) Both impact and reaction force waves behave similarly to each other and are excited with almost a half-sine wave; 2)The mid-span deflection wave of RC beams with no diagonal cracks developed behaves as a damped free vibration. However, after RC beams suffering severe damage due to diagonal cracks developing, it faintly vibrates with some drift; 3) The distributions of impact force - deflection and reaction force - deflection hysteresis loops are similar to each other; 4) The maximum impact and reaction forces at the ultimate state are almost equal to the static shear capacity; and 5) Shear failure type RC beams without shear rebars subjected to an impact load may be designed by using static shear capacity with some safety margin.

REFERENCES 1. N. Kishi, K. G. Matsuoka, H. Mikami and Y. Goto, Proc. of the 2nd Asiapacific conference on shock & impact loads in structures, (1997) 213. 2. M. Kobayashi, M. Sato, N. Kishi and A. Miyoshi, Proc. of the 2nd Asiapacific conference on shock & impact loads in structures, (1997) 221. 3. N. Kishi, M. Sato, H. Mikami and K. G. Matsuoka, Proc. of the 6th East Asia-Pacific Conference on Structural Engrg. & Construction, (1998) 973. 4. T. Ando, N. Kishi, H. Mikami, M.Sato and K. G. Matsuoka, Proc. of the 7th East Asia-Pacific Conference on Structural Engrg. & Construction, (1999)1075. 5. JSCE, Japan Concrete Standard, 1996, in Japanese.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

Nonlinear dynamic response design a n d control mechanical systems under impact loading

93

optimization

o f flexible

J. Barradas Cardoso, P.P. Moita and J.A. Castro Instituto Superior T6cnico, Universidade T6cnica de Lisboa Av. Rovisco Pais, 1049-001 Lisboa, Portugal

A design and control sensitivity analysis and multicriteria optimization formulation is derived for flexible mechanical systems. This formulation is implemented into an optimum design code and it is applied to a nonlinear impact absorber and to a flexible vehicle chassis with suspension. Structural dimensions as well as lumped damping and stiffness characteristics plus control driven forces, are the decision variables. The dynamic response and its sensitivity are discretized via space and time finite elements. Nonlinear programming and optimality criteria are used for the optimization process.

1. INTRODUCTION Structures and flexible mechanical systems, by one hand, as well as optimal design and optimal control, by another hand, have been traditionally treated with separated formulations. However, as the theory and methods of nonlinear structural analysis have progressed, there is no more distinction between flexible mechanical systems and structures. Also, in the last decade there has been the integration of optimal design and optimal control problems [ 1-5]. This paper presents an integrated methodology for optimal design and control of nonlinear flexible mechanical systems. In order to implement it, one uses: (i) a nonlinear structural finite element technique to model large displacements, referring all the quantities to an inertial frame and using stress and strain measures that are invariant with the rigid body motion; (ii) a conceptual unification of time variant and time invariant design parameters, by including the design space into the control space and considering the design variables as control variables not depending on time. By using time integrals through all the derivations, then the design and control problems are unified. Both types of variables are designated here as design variables. A bound formulation is applied to handle the multicriteria problem. The systems are modeled by space-time finite elements and the solution of the dynamic equations is obtained either by global integration or step-by-step. The aim of the Design Sensitivity Analysis (DSA) is the calculation of the gradients of performance measures w.r.t, the design variables. It represents an important tool for design improvement and it is a necessary stage within the optimization process. A general overview of the DSA problems and methods of nonlinear structural mechanics is given elsewhere [6]. Both the direct differential method and the adjoint system method are applied in this work, the latter one for global integration system response, the

94 former one for the step-by-step response. The response analysis and corresponding DSA are implemented in the interactive optimal design code OPTIMISE in order to use optimality criteria or nonlinear programming optimization runs.

2. RESPONSE ANALYSIS AND DESIGN SENSITIVITY ANALYSIS The virtual work dynamic equilibrium equation of the system at the time t is given as ~iW = ~ ( f - 6 u - pu . 6 u - S . 6 6 ) d V + ~ T . ~ u dF = 0

(1)

where all the quantities are referred to the undeformed configuration, 8 represents variation of the state fields, '.' refers to the standard tensor product, upper dot '.' refers to the material time derivative, p is the mass density at time t = 0, u is the displacement, S is the 2nd Piola stress measure, c is the Green strain tensor, f is the body force, T is the surface traction, V is the u_nderformed volume of the body, and F is the surface of the body. Considering now a general performance measure defined in the space-time domain as q~ = j'{j"G(S,e,u,u ,u ,b) dV(b)+ j"g (T ,u,u ,u ,b) dF(b)}dt

(2)

the DSA problem is to derive the total variation 8 qJ = 8 qJ + 6 W w.r.t, the design b, 8 and 15 representing respectively the explicit and implicit design variations. In order to formulate adjoint structure or direct differentiation methods, write Eq. (1) as W a= ~ (Ou "u a + S'e a- feu a) dV - ~Tou a dF = 0

(3)

where ca replaces 6s after substitution of 6u by u a, and define an extended 'action' functional A = ~F- JW adt

(4)

The basic idea of the direct differentiation method is to satisfy equilibrium after design variation. Then, auxiliary fields ua are determined by the equation 8 W a=0

or

8W a=-SW a

(5)

and used to determine 6 ~. The basic idea of introducing an adjoint structure is to replace the implicit design variations of the state fields by explicit design variations and auxiliary fields to be determined by imposing the 'action' functional A to vanish [4,5], 8 A=0

(6)

and stating the total design variation of the functional q~ as m

8 qJ = 8 A

(7)

DSA of dynamic response is path-dependent. The selection of the DSA method is based on the number of active constraints and design variables and on the response time integration

95 method. For step-by-step integration, we have selected the direct differential approach due to its easier implementation. For the at-once integration, the adjoint approach has been chosen because the number of constraints is smaller than the number of design variables.

3. DISCRETE MODEL The dynamic analysis and sensitivity analysis responses are discretized by a space-time nonlinear finite element model. Design sensitivities are calculated at the element level and assembled in order to get the DSA model for the entire system. For the space discretization, several structural elements have been implemented together with their corresponding sensitivity models [5], using hermitean and isoparametric interpolation. After space discretization we have the governing matrix equation as M tO + t c t~j + tK tU = tR

(8)

For temporal modelling, we considered finite elements of dimension At, selecting hermitean cubic elements to model displacements, velocities and accelerations, and quadratic lagrangean elements to model the excitations, extending the algorithm given in [7] to the case of nonlinear systems. By one hand the time derivative of Eq. (8) is taken, and by another hand, Eq. (8) is integrated once and twice using average values of stiffness tK and damping tc in At. These four equations combine to give the dynamic time-element equation as D e z e = R e,

z e = ( t , t+Atz)'

tz = (tu, tO, tQ )

(9)

Eq.(9) may be solved step-by-step, or assembled as Dz = R to be solved globally, i.e, at-once. In this case, the 2n time boundary conditions, where n is the number of space degrees of freedom, are imposed by transferring the corresponding coluns of the assembled matrix D to the right-hand side of the equation Dz = R after multiplying the vector U c of those conditions, resulting KU=R-DcU c (10) This is a nonlinear equation where K is a nonsymmetric matrix dependent on the response U. It has to be solved iteratively. Concerning to DSA, application of Eq.(5) to Eq.(10) or of Eq.(6) to Eq.(4), gives respectively, for the direct differentiation and the adjoint system approaches, n

.

.

~

w

m

K 5 U = 5 R - DcSU c -5 F,

6 9 6 9 + (6~/5U)6 U

(11)

i~T U a= (6W/5U) T,

g qJ = 6 W.+ (U a )T (6 R - iic~U c - ~ F)

(12)

---

where F = lDz is the vector of internal forces and K results of derivating D w.r.t, z and again imposing the time boundary conditions. In Eqs.(11) and (12), R and U c are respectively to be considered the driven forces and initial condition control variables.

96 ~,

K=Kox]u[ -1

0.4 0.2 ,--, 0.0

M

--=.

r

~%~~~

"~ -0.2

C=Co• -1

~ -0.4

u

-

-0.6

0.02

o

-0.01

I

5

10

-0.02 -0.03

D e s i g n w r t K 0 a n d 120

t

0.260 0.259

v

-0.04

0

t[s] Figure 3. Absorber optimal control-Case I.

el

c3' I '0*'%--k3

c2

c5' I,k4L'%"- k5

c4

14~176! Beam Element 1

5 t[s]

10

Figure 4. Absorber optimal control-Case II

kl

I e

~

0.266 7 0.265 0.264 -~'o0.263 ~ 0.262 rj 0.261 ~

0.01

,-'h

-

lO.O

Figure 2. Optimal designs for the impact absorber

Figure 1. Nonlinear impact absorber.

o.oo

-

-..,, ----- :

- - = ~DesignwrtKOandCO + Control

-0.8

o

. . . . . .

6 ~ 7

k2

k4

[

eBeam Element 2 e

Figure 5. Model of vehicle chassis with suspension

97 4. N U M E R I C A L E X A M P L E S

4.1. Nonlinear Impact Absorber The system shown in Fig.1 may represent a landing gear for an aircraft impacting the ground at a certain velocity v = 1 at t = 0. The problem is to find the spring and damping coefficients K0 and Co, and the control force [P(t), 0 _< t <12 s], satisfying a displacement constraint lul -< 1, w i t h v - 11 = 2, s u c h that: (I) minimizes the maximum absolute acceleration lal of the mass M = 1, with starting decision variables at K0 = Co = 0.5 and P(t) = 0; (II) minimizes the bicriteria {Co, E = ~(100 U2+ U2+ 100 p2) dtl where E is a performance integral widely used in optimal control [3]. Starting decision variables are K0 - C o = 0.597 and P(t) = 0, and starting objective upper bound value is 20. In Case I, the optimum design was obtained by sequential quadratic programming as K0 = 0.00667, Co - 6.96, and the optimal control P(t) is shown in Fig.3. Starting and optimal acceleration responses are shown in Fig.2 for different levels of the optimization process. The responses were obtained by at-once integration. In Case II, K0 - C o - 1.57 and P as shown in Fig.4, are the optimum decision variables, and the optimum value for the performance E is 61.1, this value coinciding with the optimum objective upper bound. Responses were obtained by step-by-step integration. Optimality criteria were used in this case for the optimization run.

4.2. Flexible vehicle chassis with suspension Figure 5 shows a vehicle which suspension is kinematically excited by the ground. The tires are represented by springs k4 = k5 = 26.275 N/m and by shock absorbers c4 = c5 = 875.8 Ns/m. The chassis body is partially distributed as beam elements with p = 7800 kg/cubic meter, E = 200 GPa and partially as concentrated masses. The relationship 2nd moment of area/area for the chassis cross-section is considered as I/A = 0.01. Other masses are ml = 131.6, m3 = m4 = 43.6 kg. The distance between wheels is 3.048 m and the vehicle moves at 87.78 Km/h. Starting design is {A = 0.1 m 2, kl = 1.75e4, k2 = k3 = 5.26e4 N/m, ca = 1.75e3, c2 = c3 = 4.38e3 Ns/m} and starting control is [P(t) = 0 N, 0 _< t _< ls)], where P is the actuator control force at the passenger seat. It is intended to minimize the maximum absolute acceleration such that the relative vertical displacement between masses is not larger than 0.05m. Two optimization cases were run corresponding to a different ground conditions, both a sinusoidal obstacle of height equal to 0.1016m and: (I) wave length equal to 0.1016m; (II) wave length equal to 24.384m. In Case I, optimal design was obtained as A = 1.9562 m 2 with no changes on the other variables, and optimal control is represented in Fig.6. In Case II, optimal design was obtained as {0.35579, 6.94e4, 6.31e4, 5.02e4, 1.e6, 1.34e4, 1.e2} and optimal control is represented in Fig.7. Starting and optimal accelerations, respectively for Cases I and II, are shown in Figs.8 and 9, for different levels of the optimization process.

5. C O N C L U S I O N S A unified methodology for optimal design and control of flexible mechanical systems has been derived and implemented successfully with three main ingredients: nonlinear structural formulation; inclusion of the design space into the control space; using time integrals throughout all the derivations.

98 5.064N

3.0FA]4 2.0F_.~

0.0E-r

. 1.0F_Xl4

-

-a

--O 0.0E-~ ~ - 1.0F_Xl4

0

r,.) -1.0F_~3

-3.0F__,~

~

0.5

10

,./

-1.5FA)3

-4.0F_~

Figure 7. Vehicle optimal control-Case II.

Figure 6. Vehicle optimal control-Case I.

2.E-05

1.0E-05 ----------

0.E+00

-~"-l.0E-05 E

/

-

o

O

r j -2.(F_X]4

0.0E+00

~

eq ct~

-2.0E-05

-2.E-05 -4.E-05

-3.0E-05 -4.0E-05

-6.E-05 t[s]

t[s] Starting Design Optimal Design wrt Area . . . . . . Optimal Design wrt Area, K's and C's . . . . Optimal Design wrt Area, K's and C's + Control Figure 8. Vehicle optimal acceleration-Case I Figure 9. Vehicle optimal acceleration-Case II REFERENCES

1. M. Salama, J. Garba, L. Demsetz and F. Udwadia, Simultaneous Optimization of Controlled Structures, Computational Mechanics, 3 (1988) 275. 2. C.H. Tseng and J.S. Arora, Optimum Design of Systems for Dynamics and Controls using Sequential Quadratic Programming, AIAA Journal, 27 (1989) 1793. 3. J. Arora and T. Lin, A Study of Augmented Lagrangean Methods for Simultaneous Optimization of Controls and Structures, 4th AIAA/Air Force/NASA/OAI Symposium, U.S.A. (1992). 4. J.B. Cardoso and J.S. Arora, Design Sensitivity Analysis of Nonlinear Dynamic Response of Structural and Mechanical Systems, Structural Optimization, 4 (1992) 37. 5. J.B. Cardoso, L. Sousa, J.A. Castro and A. Valido, Optimal Design of Nonlinear Structures and Mechanical Systems, Engineering Optimization, 29 (1997) 277. 6. M. Kleiber, H. Antunez, T.D. Hien and P. Kowalczyk, Parameter Sensitivity in Nonlinear Mechanics, John Wiley & Sons, 1997. 7. M.Gellert, A New Algorithm for Integration of Dynamic Systems, Comp.&Struct., 9 (1978) 401.

Structural Failure and Plasticity (IMPLAST 2000)

Editors:X.L.Zhaoand R.H. Grzebieta 9 2000ElsevierScienceLtd.All rightsreserved.

99

Influence o f impact loads on the behavior at alternative bending over pulleys o f steel wires Crespo, Germfin Universidad Sim6n Bolivar, Departamento de Mecfinica, Caracas, VENEZUELA. Fax: 58-29064062 / 9621697. Email: gcrespo@,usb.ve.

Stranded steel wires used for cable manufacturing which must be operated on pulleys, are subject to a complex state of stress where the following factors take part: axial tension, torsion, bending, crushing, and the dynamic friction or gall. Recent investigations have a tendency toward cable constructions that increase their fatigue-proofed useful life while continuously passing upon the pulleys. This paper determines the influence of the impact loads in tension over the wires in order to quantify its effect over the other stresses. Equivalent loads of up to the 25% of the breaking load of 3.15ram diameter wires were applied by free fall weights, while maintaining the wire in bending position, with relations of 30, 45 and 60, of pulley diameters to the diameter of the wire. A relation was obtained that predicts the reduction of the fatigue-proofed useful life, in comparison to the no impact situation, for various values of dynamic load. Likewise, it was observed that the predominant breakage continues to be due to fatigue, even when fractures were obtained due to the effect of the cyclic impact at less than 3000 load cycles.

1. INTRODUCTION Cables manufactured with steel wire which must bend over pulleys or chains, are subject to a state of complex stress produced by various bending effects, gall and plastic deformation in determined places, which provoke the finite life of those cables even when the tension loading they endure is not higher than the 25% of the breakage load. Some studies have theoretically quantL~ied the effect of bending (Bahke 1980a), (Freyre 1981) or experimentally the combined effect of the stresses which do not result from the constant axial tension (Oplati~ 1990), (Crespo 1995). Further to the above, in some cases the cable must stand sudden forces applications due to abrupt changes of speed or failure of the brake systems. In this study, an analysis is made of the effect of the impact loads applied over the individual wires which form the steel cables, with the purpose of trying to quantify the stress produced by this situation.

100 2. EXPERIMENTAL DEVELOPMENT Round wires of 3.15 mm diameter, used for cable manufacturing for cable cars (full locked coil rope), were tested applying in a cyclic manner loads equivalent to the 25% of the breaking load of the wire, while it is bent over a pulley, with a relation of pulley diameter to wire diameter (D/d), of 30, 45 and 60. The wires are ofpearlitic steel, with 0,79% of Carbon, 0.76% of Manganese, 0.22% of Silicium, 0.003% of Phosphorous and 0.007% of Sulphur, stranded starting from a rod of 6.35 mm of diameter until attaining a condition where the maximum resistance to the tension was 1960 MPa. For conducting the tests, equipment was used as shown in Figure 1.

Force

I

Rigid stop

IlL

Piston

Fall

Pulley

Weight Figure 1. Diagram of the testing equipment used for applying the cyclic impact loads. The pneumatic piston of Figure 1 raises the weight equivalent at 25% of the breaking load of the wire being tested, until it runs a length of 70 ram, which is equivalent, approximately, to 22.5 times the wire diameter. The air pressure is cut and the weight is set free to fall until the wire-splicing clamp makes contact with the rigid stop. Under these conditions, the fallingspeed of the weight at the moment of making contact with the stop is approximately 5 m/s, speed which is used in the major part of the load elevation systems, elevators and cable cars. The operation is repeated until attaining the wire breakage. (Test "A'3 To differentiate and quantify the effect of the pure simple bending over the wire part that is bent and straightened in a cyclic manner, the same equipment shown in Figure 1 was used, however maintaining the tension without impact constant, and lifting and lowering the "weight at a frequency of 0.25 Hz until producing the wire breakage. (Test "B') For quantifying the effect of the impact load, similar wires were tested at tension fatigue using a universal testing machine, and applying different values of maximum load for each test at a 1 Hz frequency, maintaining the minimum tension at 25% of the breaking load, until obtaining the wire breakage in approximately the same number of load cycles than when using the impact test. (Test "C")

101 3. RESULTS AND DISCUSSION Following the order of the experimental development the results are as follows: For the Test "A" (impact condition): Conditiony. a weight of only the 25% of the breaking load and an average of 5 tests per condition with variation of +l 5% among extreme values. Results: Total failure of wires produced around load cycles of 1200, 1800 and 5300, for the relations D/d of 30, 45 and 60, respectively. For the Test "B" (pure simple bending): Conditions'. an average of 5 tests per condition with variation o f + l 1% among extreme values. Results: Wire breakage produced around load cycles of 135000, 224000 and 722000, for the relations D/d of 30, 45 and 60, respectively. For the Test "C" (tension fatigue): the Figure 2 indicates the life to fatigue of wires tested in tension-tension with a minimum load at 25% of the breaking load, as function of the maximum applied stress expressed in percentage of the static breaking load of the wire. 1000000

~

i

l

100000 ~

o..

~"

10000 D/d=60

100o "O m

100 20

30

40

50

60

70

Maximum load as percentage of the breaking load

Figure 2. Cycles of life to breakage in the tension-tension test of the wire, maintaining constant the minimum load at 25% of the breaking load, as function of the maximum load applied.

In the Figure 2, the arrow indicates the equivalent condition for the Test "A" (impact condition, 5300 load cycles), with a relation D/d equal to 60. As it can be appreciated, the effect of bending plus the impact is equivalent to increasing the load over the wire with a 27% of the breaking load. The equivalence showed in the Figure 2 for the Test "B"(pure simple bending, 722000 load cycles), means an increase of 5% of the breaking load (with the same relation D/d of 60). So, only the impact effect must be between an 80 and a 90% of the increase mentioned for the Test "A" (27%). On the other hand, if the number of life cycles is plotted as function of the D/d relation, it can be observed that an increase of this relation seems not to improve much, the working condition of the wire. (See Figure 3). If we take into account that the D/d relation in the full locked coil rope systems for cable cars is larger than 90, we could arrive to the conclusion that tests made at the laboratory would seem to reproduce with certain similarity the actual situations in the use of steel cables, although, the contact effects between wires is not being

102 considered in the present study. No influence of the pulley gorge radius was found, neither effects of superficial friction, which could alter the obtained results. Likewise, wire rotation was not detected during the tests. Although the fracture was produced at a relative low number of cycles under the condition of impact plus bending, the fractography revealed a predominant morphology of fatigue more than static load, with cracking growth in radial sense, followed by a change in the sense of the wiredrawing. Numerous secondary cracks were appreciated. The fact that the influence of simple bending results with only 20% of the cyclic total load applied (or equivalent to a 5% of the breaking load as additional amount to the basic load), seems to prove that the effect of the cyclic bending stress over wire cables is not the principal magnitude provoking the failure of wires due to fatigue, but the principal effect is the gall or local dynamic compression with relative movement (also called dynamic friction).

6~ 4~ f 50 i

~n L 30

20

[-

-

0

2000

4000

60O0

Cycles o f ~ to bmaka~ Figure 3. Influence of the D/d relation over the cycles of life to breakage for the condition of impact plus bending.

4. CONCLUSIONS The results of this study show that the effect of the cyclic impact on wires that are bent over pulleys represents an increase of a 108% of the stress applied by the basic tension, of which between an 80 and a 90% correspond to the impact itself and around a 20% to the simple cyclic bending of the wire over the pulley. Increasing the relation between the diameter of the pulley and that of the wire (D/d), would not seem to help much over the behavior to fatigue for values larger than 60.

REFERENCES 1. Bahke, E., (1980), "Bases para la resistencia de cadenas y cables de alambre', Partes I, II, Alambre, 30, 3, pp 142-150 y 30, 4, pp. 181-190. 2. Feyrer, K., (1981), "Evahmci6n estadistica de los resultados obtenidos en ensayos a flexi6n de cables de alambre'; Partes I, II, Almnbre, 31, 3, pp. 112-116 y 31, 4, pp. 167172. 3. Oplatka, G., (1990), "Relation between the number and distribution of wire breaks and residual breaking force", Wire, 40, 2, pp. 206-209. 4. Crespo, G., (1995), "Modelaje y redisefto de cables tipo cetrado (full locked coil rope) para sistemas telef&icos con contrapeso'" Anales del COBEM-CIDIM/95, CDROM.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

103

Dynamical Actions on Highway Bridge Decks Due to an Irregular Pavement Surface J.G.S. da Silva Mechanical Engineering Department, Faculty of Engineering, UERJ E-mail: [email protected] An analysis methodology is proposed to evaluate the dynamical effects, displacements and stresses, on highway bridge decks, due to vehicles crossing on the rough pavement surfaces defined by a probabilistic model. To this purpose, the methodology is developed to evaluate the vehicle-structure response under a full probabilistic formulation, running in the frequency domain. The mathematical model assumes a finite element representation of the beam like deck and the vehicle simulation uses concentrated parameters of mass, stiffness and damping. The deck surface roughness is defined by a well-known power spectral density of road pavement profiles. The moving load is formed by an infinite succession of equally spaced vehicles moving with constant velocity. Only steady-state response is considered. Response data are produced on concrete box girder elements assembled as a simply supported beam with overhangs. Results of a study are presented to verify the extension of the dynamical effects on highway bridge decks, due to vehicles crossing on the rough pavement surface defined by a probabilistic model.

1. I N T R O D U C T I O N Since the middle 80's the scientific community in recognition of the major importarice of this subject has started a continuous effort on the study of the dynamic effects on bridge superstructures due to vehicular traffic on irregular pavement surfaces. The problem remains a major one for regions where the road maintenance is not effective leading to a premature deterioration of the bridge superstructures and their bearings [1,2,3,4]. Several studies have been published since then and it was made evident that the effects due to the interaction of the vehicles with an irregular pavement surface can be much more important than those produced only by the smooth movement of the vehicles. [4,5,6,7,8,9,10]. In design practice, most of the technical recommendations use a dynamical load factor applied to the vehicle static effects to take into account all the dynamical actions. In addition to these points, field reports say that some bridges have been submitted to excitation levels, under usual traffic conditions, which have deteriorated their service conditions and structure durability, this can be an indication of under conservative design criteria. So, it is desirable to have the problem parameters quantitatively evaluated to better estimate their participation in the structure disruption. To this purpose, the analysis methodology considers a vehicle-bridge mathematical model which includes the interaction between their respective dynamical properties and is developed

104 to evaluate the vehicle-structure response under a full probabilistic formulation, running in the frequency domain [4,6,10]. The deck surface roughness is defined by a well-known power spectral density of road pavement profiles [ 11]. The moving load is formed by an infinite train of similar vehicles regularly spaced and running at constant velocity, in such way to obtain steady-state mean maximum response quantities of the vehicle-deck system, which are necessary to a fatigue analysis of the deck material [4,12]. The analysis methodology use a frequency domain transfer on the pavement spectral densities, to obtain directly the bridge response spectral densities, followed by an integration of these functions over a suitable frequency band to get the mean square values of the vehiclebridge response quantities [4,6,10]. Response data are produced on reinforced concrete beam decks, continuous on several supports, with overhangs and with constant box cross section. A parametric study is developed to verify the extension of the dynamical effects, displacement and stresses, on highway bridge decks, due to vehicles crossing on the rough pavement surface defined by a probabilistic model. The conclusions include qualitative and quantitative aspects of the problem and the attitudes concerned with bridge design and maintenance.

2. P A R A M E T R I C

STUDY

2.1. Structure and Moving Load Model The work is carried out on a computer code, developed in previous studies [4,6,10], which follows an analysis methodology to evaluate the vehicle-structure response under a full probabilistic formulation, running in the frequency domain. The structure model includes the vehicle-deck dynamical properties interaction and the moving load is modelled as an infinite series of equal vehicles regularly spaced and moving at constant speed, in such way to determine the maximum displacements and stresses in the steady-state response of the bridge deck [ 12]. 2.2. Vehicle Properties and Data The individual vehicle of the series weighs 450kN and is modelled as a two degrees of freedom system. The vehicles keep a regular spacing, 1, and run at a constant velocity, u; from test to test The single vehicle natural frequencies, in rigid base vertical motion, corresponding to the vehicle suspended mass and non suspended mass degrees of freedom, are made equal to 3Hz and 20Hz, respectively The relative damping coefficient of the vehicle vibration mode with predominant movement of the vehicle suspended mass is assumed to be ~=0.1 [4] 2.3. Surface Irregularities The definition of the irregular pavement surface follows a non-determimstic model based on the pavement roughness power spectral density The pavement quality control parameter is chosen using grades from excellent to very poor surface according to [ 11 ]. 2.4. Structural System To verify the influence of redundant spans and of overhangs on the amplification factors due the excitation produced by the interaction of the vehicle wheels with the irregular

105 pavement surface, the infinite train of vehicles is applied to continuous beam decks modelled as shown on Figure 1. The basic span, L, is varied from L=24.0m to L=36.0m. A finite element model associated with a reinforced concrete highway bridge, is chosen to consist of a straight box-girder beam, continuous over several supports, with overhangs and constant cross section. Figure 1 presents the finite element model of the beam with the following properties: I=3.98m 4, E=3xl071d',l/m 2, distributed mass ~ =9200kg/m and damping factor, ~=0.03, for the lowest mode frequency with predominant deck displacements. mAl Am2A ..,_

A 'qp'w'

~qpqlF

1 2 3-7-/~

4 I ~ = 0.2.~_

r

9-- 11134 11135 A v

A

A Ip'~

A

A A A ~ ~ - ~

9

.Le •O.83L

T

Jh, O ~ ~lh ~ ~ ~ A ~ J~ ~ A ill A d& ~ A ~ qJ, q p q l F ~ q p q l c V q p q p / ~ , , q p q P q p q P q P ~ q P q P q l F

~,

20

14 _l_

A

Jl&

A A "'.,IP'~v

31 26

L

A

....

T

A

39 36

_!_

Le = 0.83L

t-

._~b =0 ~ S L

T

-L

Figure 1. Model of continuous beam deck with overhangs. Table 1 shows the variation of beam fundamental frequency with the changing of the span and overhang length and with the variation of the beam finite element length. This last parameter is important since the vehicle spacing is chosen to be twice its value. If it were to be shorter then more vehicles can be arranged on the deck Table 1 Continuous beam natural frequency varied with bridge geometry. Middle Span

End Span

Overhangs

Section Spacing

Unloaded Deck

Loaded Deck

L(m)

Le(m)

Lb(m)

Ls(m)

Lowest Mode Frequency Hz)

Lowest Mode Frequency Hz)

24.00 30.00 36.00

20.00 25.00 30.00

6.00 7.50 9.00

2.00 2.50 3.00

9.12 7.43 5.20

9.22 7.78 5.86

2.5. Analysis Procedure The equations of motion follow a standard procedure in which the effect of the pavement roughness is introduced considering that, for the vehicle, it acts as a base motion. The power spectral density of the vehicle-pavement interaction force is transferred through the frequency domain to obtain the vehicle-deck system response function [4,6,10]. A proportional damping matrix is adopted that includes a submatrix related to the vehicle degrees of freedom and another to the bridge nodal velocities. The effect of the rough pavement is introduced in the motion equations as a load vector analogous to which would be considered if the vehicle were subjected to a base movement equal to the irregular pavement profile [4,6,10]. By making use of the Wiener-Kintchine theorem, the second order moments of the response displacements of the vehicle-bridge system can be obtained. To keep these response quantities in the same magnitude of those obtained by a time domain approach with the same structural model [4,5,8,9], this last integration has to be taken only over a narrow band of the spectrum.

106 The criteria to establish this band width and its locations are also proposed [4,6,10]. From this point on, the second order moments of the other response quantities are reached by similar and suitable procedures [4,6,10]. The mean maximum displacement and stress values of the response of the vehicle-bridge system are evaluated based on the standard deviation obtained by a statistical analysis. The results compare very well adjusted to those of the previous study mentioned above [4,5,8,9]. The following parameters are then varied: the bridge main span, keeping the same relative overhang length; the traversing velocity and the road surface quality grade, With the study of situation of simply supported decks is now complete the case of straight continuous decks over a varying number of supports, with and without overhangs was then considered [4,5,6,8,9,10]. 2.6. Presentation of Results The response of the vehicle-bridge system has been obtained considering excitation produced by the interaction of the vehicle wheels with the irregular pavement surface. Deck displacements and stresses have been computed. Tables 2 to 4 illustrate the variation of the mean maximum values of displacements for three typical sections of the continuous beam model, presented in Figure 1. These tables present the results in terms of the medium amplification factor, FA, mean maximum displacements, By, and second order moments of the response displacements, E[v2]. These response displacements of the vehicle-bridge system are determined at the vehicle critical velocity which is equal to l.fol, where I is taken as twice the beam element length and f01 is the single vehicle natural frequency, in rigid base vertical motion, corresponding to the vehicle suspended mass degree of freedom. The medium amplification factor, FA, represents the ratio between ~t+. and the statical effects, vest. For a better understanding vest is defined as the maximum displacement at the corresponding model section due to static action of the weight of the vehicle train, as it crosses the bridge.

Table 2 Mean maximum displacements. Pavement quality: very good. Section S 1. Middle Span F--A= l'tv ~tv L(m) Vest (cm) 24.0 1.15 0.16 30.0 1.22 0.33 36.0 1.10 0.51 Table 3 Mean maximum .displacements. Pavement quality: very Middle Span F---A= lay Vest .... L(m) ..... 24.0 0.85 30.0 0.90 36.0 0.86

E[v~] (cm2)B 0.028 0.120 0.270

good. Section $9. tXv (era) 0.11 0.23 0.37

E[v:] (cm2)[] 0.013 0.054 0.140

107 Table 4 Mean maximum, displacemems. Paveme m qua!ityl m e d i ~ . Section $20. Middle Span F---A=~tv Vtv L(m) 24.0 30.0 36.0

Vest 3.20 3.40 3.28

(cm) 0.52 1.00 1.76

Z[v~] (cm~)~ 0.272 1.024 3.328

2.7. Discussion of Results The main objective of this work is the evaluation of the magnitude of the dynamical effects, due to excitation produced by the interaction of vehicle wheels with an irregular pavement surface, relatively to the static values. These FA values are higher than those obtained for the simply supported decks and have to be considered with some caution because they may not precisely represent the value of the ratio of the dynamic value to the design static value computed as recommended by many codes, if loading of alternate spans is allowed. The roughness of an excellent pavement surface produces response quantities, displacements and stresses, to the bridge deck, that can reach magnitudes to the same order as those due to the static effect of a train of vehicles long enough to load all the bridge (see Tables 2 to 3). The displacements and stresses increase drastically with the decrease of the pavement surface quality. For medium quality pavement surface some of these effects can be from 2 to 4 times higher (see Table 4). These magnitudes are several times greater than that due to load mobility alone [ 12,13]. The large number of cases studied suggest a similar condition applies to all the response quantities, either displacements or stresses.

3. C O N C L U S I O N S In all cases studied, it was verified that for typical vehicle velocities, the dynamical effects on highway bridge decks due exclusively to the interaction of the vehicle suspension flexibility with a smooth pavement surface represent a significant component of the vehiclebridge system response. They can be as high as 90% of the static effects of the moving load. The effects due to the interaction of the vehicles with an irregular pavement surface are much more important than those by considered the load mobility [ 12,13]. In some cases these effects are larger than those due to the presence of the vehicles statically. These dynamical effects increase drastically with the decrease of the pavement surface quality. The problem is still more serious if one considers lower quality pavement surfaces, irregularity amplitude larger than 1.40cm; for medium quality pavement surface some of these effects can be from 2 to 4 times higher and for the worst case, (very poor quality surfaces), the problem becomes very serious [4]. As a consequence, two main pieces of advice are offered to structural and maintenance engineers: i) considering a linear elastic analysis, special attention needs be given to the load factor magnitudes which multiply the static action of the moving loads to account for their dynamical effects; ii) the maintenance effort needs be sufficient to insure a low roughness pavement surface - irregularity amplitudes not higher than 1.40cm.

108 4. A C K N O W L E D G E M E N T S The author acknowledges gratefully the financial support for this work provided by FAPERJ, Funda~;~o de Auxilio/t Pesquisa do Estado do Rio de Janeiro, and CNPq, Conselho Nacional de Pesquisa.

REFERENCES 1. F.N.M. Ramalho, "A~6es Dinfimicas em Pontes Rodovifirias com Defeitos na Pista", X CILAMCE, Iberian Latin-American Conference on Computational Methods in Engineering, (1989). 2. G. Sedlacek and St. Drosner, "Dynamik bei B~cken", Baustatik Baupraxis, Universit~t Hannover, (1990.) 3. T.L. Wang, D. Huang and M. Shahawi, "Dynamic Response of Multigirder Bridges", ASCE, J. Struc. Engr., vol. 118, pp. 2222-2238, (1992). 4. J. G. S. Silva, "An~lise Dinfimica N~o-Deterministica de Tabuleiros de Pontes Rodovi/Lrias com Irregularidades Superficiais", D.Sc. Thesis, Departamento de Engenharia Civil, PUC-Rio, (1996). 5. J. G. S. Silva and J. L. Roehl, "Dynamical Analysis of Bridge Decks with Irregular Pavement Surface Defined by a Profile Spectral Density", XVII CILAMCE, Iberian Latin-American Conference on Computational Methods in Engineering, (1996). 6. J . G . S . Silva and J. L. Roehl, "Probabilistic Formulation for the Analysis of Bridge Decks with Irregular Pavement Surface", VII DINAME, International Conference on Dynamic Problems in Mechanics, (1997). 7. Cal~ada, R.; Cunha, A., "Stochastic Modeling of the Dynamic Behavior of Bridges under Traffic Loads", IV WCCM, Fourth World Congress on Computational Mechanics, (1998). 8. J.G.S. Silva and J. L. Roehl, "A~;0es D i ~ i c a s em Tabuleiros de Pontes Rodovi~irias Provenientes das Irregularidades do Pavimento", 1II SIMMEC, 30 Simp6sio Mineiro de Mecfinica Computacional, In portuguese, (1998). 9. J. G. S. Silva, "Dynamical Load Factor for Highway Bridge Decks with Pavement Irregularities", IABSE Symposium, Structures for the Future - The Search for Quality, Rio de Janeiro, Brazil, (1999) 10. J. G. S. Silva and J. L. Roehl, "Analysis of Highway Bridge Decks with Irregular Pavement Surface Using a Full Probabilistic Formulation", ECCM-99, European Conference on Computational Mechanics, Munich, Germany, (1999). 11. Braun, H., "Untersuchungen Von Fahrbahnunebenheiten und Anwendungen der Ergebnisse", Von tier Fak~ltat for Maschinenbau und Elektrotechnik der Technischen Universit~t Carolo-Wilhelmina zu Braunschweig, Dissertation, (1969). 12. J. G. S. Silva and J. L. Roehl, "Dynamical Actions on Bridge Decks due to a Train of Vehicles", IV WCCM, Fourth World Congress on Computational Mechanics, (1998). 13. R. Carneiro and J. L. Roehl, "A Model for the Analysis of Moving Load Effects on Highway Bridge Girders", RBCM, Journal of the Brazilian Society of Mechanical Sciences, vol. 12-1, 29-44, (1990).

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

109

Eiastic-viscoplastic-microdamage modeling to simulate hypervelocity projectile-target impact and damage J. Eftis, C. Carrasco and R. Osegueda FAST Center for Structural Integrity of Aerospace Systems University of Texas at El Paso, TX, 79968-0516, USA Abstract - Constitutive-microdamage equations are developed that are capable of describing the full range of thermomechanical material behavior, microdamage evolution and fracture associated with hypervelocity projectile-target impact. 1. INTRODUCTION Because of the current limits of laboratory gas gun technology in propelling other than pellet sized projectiles at hypervelocities (e.g. 6.35 mm spherical projectile at 4.9 km/s), laboratory hypervelocity impact experiments with projectiles that are four or five times as large cannot now be performed. To investigate hypervelocity impact events that are related to space sciences, such as ballistic missile interception, or the protection of orbiting space vehicles from orbiting space debris, it is necessary to rely almost entirely upon numerical simulations. This requirement places heavy responsibility upon the material constitutive model used in the computer codes. Constitutive models used for this purpose must have the capability of describing accurately the full range of thermomechanical deformation and fracture of projectile and target following hypervelocity impact. A set of material constitutive-microdamage equations that have this capability are presented below in abbreviated manner in deference to space limitations. 2. RATE OF ELASTIC DEFORMATION The rate of deformation tensor D is expressed as the additive sum of the elastic rate D e and the viscoplastic rate D p. Since there are large volume strains associated with high shock compression, D' can be decomposed into deviatoric and spherical components and expressed as the sum of the rates of small scale shear strain and large scale volume strain. The elastic shear strains are constrained to small order because of the limitations imposed by onset of plastic yield. Thus the deviatoric component is linear with respect to stress, as in small strain elasticity, while the spherical component is not. Accordingly

De = i T , + ~ -1s v 1

(1)

2/.t

T is the Cauchy stress tensor, a~, = (~3)trT is the mean stress, (..V.)represents the Jaumann rate applied to T and ~m, P and K are the isotropic shear and bulk moduli. All polycrystalline materials contain microvoids or microcracks. Since hypervelocity impact can produce large temperature increases, the corresponding dynamic fracture processes will be ductile, caused essentially by nucleation, growth and joining together of microvoids. Therefore the measure of ductile material microdamage can be described by the microvoid volume fraction

110 = '"/,., where Vv is the void volume for a material element having volume v. The microdamage is treated as a scalar valued continuous point function of position and time ~: = ~ (x, t). The volume strain ~ = (1 - ffvo)= (1 - ~/~p), where v0 , Po and v, p are the reference and current values of the volume and density. The equation relating the mean stress to volume strain and temperature (' equation of state') may be shown to have the Gruniesen form [ 1]. Thus during shock compression and release

r,

L.

+[yo

During dilatation the initial shock wave speed reduces to a tensile wave traveling at the bulk sound speed c o . This is equivalent to setting s = 0 in (2). Therefore the dilatational mean stress is given by

or.= K W~- 89 { Yo(~_-~)+ao(~._~_;}]+ )2 [to + ao(~_-~)]E(W,T), W<0

(3)

In these expressions E is the internal energy per unit volume, T is the absolute temperature, c ois the bulk sound speed, 2'o is the Gnmiesen constant, a 0 and s are material constants the latter of which is obtained from shock compression data. Large volume strains (or high pressures), large temperature variations and material microdamage alter the values of the elastic moduli which, accordingly, for high compression can be expressed as K = K ~[ (1- W'(~1] s+;(s~-"y~

"=

('i :s~)'"

[1 -a s (T~o- 1)] ~[ )4P~ L 4+/ ~3Ko'J] o

1- k.T,. - TOJ

(1-d~)(l- 6K~ + 8,Uo 2/'t~ ~

(4) (5)

where Ko, ixo are the moduli values at the reference state ~ = 0, T = TO and ~,= go = 10-3 _ lO-' -_--0 for the ideal solid without microdefects (i.e., the values determined by routine laboratory tests). T.n is the melt temperature. The first bracketed term expresses shock compression hardening of the elastic moduli, and represents the variation of the mean stress with respect to volume strain (or density). The second reflects the softening of the moduli at elevated temperatures [2], while the third softens the moduli with microdamage [3]. For zero volume strain and dilatation, ~ < 0, K and ~ are given by eqs. [4] and [5] with the parameter s = 0. The increase of the melt temperature with large volume reduction is expressed by of. [4]. Tm = Tm0eXp(2a, i//)(ij.u

I//> 0

Tin0 is the melt temperature at zero volume strain and a~ is a material parameter.

(6)

lll

3. RATE OF VISCOPLASTIC DEFORMATION The spherical and deviatoric decomposition of the rate of viscoplastic deformation Do = V~(trDP)l+ D'p allows for easy expression of the fact that plastic volume change stems only from the compressibility of the microvoids. Designating eo =-In ('-r162 as the logarithmic void volume strain, it follows that

l(trDP) I

I( W,, Ii

~

I,

where the dot signifies time rate. The deviatoric rate of viscoplastic deformation has the general form proposed by Perzyna [5], and may be given explicit expression as

D'=

J'2

[Ty ~ (D.)+ h (~.)l~ f~ (T)f, ~r

-1

Im

T'

(8)

1 S ~ T' is the deviatoric stress, J2t = ~ tr(T'. T') is its second invariant. Ty = )/4~ Ty,, where TyS~ is the uniaxial yield stress at the reference temperature and quasi-static strain rate. The dimensionless strain-rate hardening function

(.o(D'):I+ r In [D' [, D' : D ~ o ,

(9)

where D O= 1.0 s ~ . Isotropic strain hardening is described by a nonlinear concave power form

h6')= ~=l~Pl"~

(10)

with equivalent plastic strain

'[(trDP) 2 - t r ( D e ' D p ) ] 9

eP=~22dr310I~),~ildt',

II D.=-~

The temperature and material damage softening functions

[ ( )] ( T-To

fs(Y)= 1-c3(.T, _T ~

~

1, T > To , c3= - 1 , T < T o ,

[(

~-r f4(~)= 1-(~.~_~.o)3 ,

(11)

(12)

where fs(T~) = 0 and f4 (~F) = 0 ~ D p ~ oo, represent the melt and local fracture conditions respectively. For high shock compression at high strain-rates, e.g., 10~ -10 ~ s -', the material viscosity varies with pressure (or volume strain), temperature and strain rate, where higher shock pressures increase temperature and strain rate and correspondingly lower material viscosity. A material viscosity-temperature-volume strain relation has been obtained for Copper from limited high stain-rate impact data [3, 6-9], coupled with the assumption that the lower bound value for the viscosity at melt temperatures that increase with high compression, nevertheless remains constant at rl~ ~ 10-2 Poise.

[ l 01I TT0 /1

q = q0exp In

Tm(~)_ To

13,

112 The designations m~, m 2, m 3, c~, c 2, r/0, r/m, ~:0,~:F appearing above are material parameters. Values for all of the material parameters for high purity Copper are tabulated below.

4. MICRODAMAGE MODEL Micro-degradation of polycrystalline metals is generally complex. However for dynamic loading that induces high tensile mean stress and temperatures that are not low, microdamage is essentially ductile, appearing as microvoids nucleating at grain boundaries, inclusions and other defect sites. Fracture comes about as nucleated microvoids, as well as pre-existing ones, grow and join together forming micro-cracks that subsequently connect with other voids progressively forming macrocracks and fragments. For high strain-rate microdamage, the stress-temperature driven nucleation rate process is considered most important [5, 10-12], whereby

m, [ (m,l<,-<,~l)_l ]

6N = ( 1 _ r

exp

kT

'

<,<<,,.

(14)

o N is the tensile nucleation mean stress threshold, k is the Boltzmaan constant and m4, m 5are material parameters. The model that simulates voids by a hollow sphere has been employed for pore collapse analysis [ 13], and later generalized for low shock impact studies [3, 5, 14, 15]. For hypervelocity shock impact, inertial and temperature effects are important for the micromechanics of void growth. A most general calculation of the void growth rate ~G requires solution of the differential equation r A(~)-~2~+B(~)

(15)

+TI(T)F(~,~0) ~-

where

k

x=-~--

(16)

~o ; ' A ( ~ ) = ( l - ~ ) h '

B(~)= ~ [

1 (~-)~-l)- .~-(~-'-'-1)}, F(~,~o)=~4 ll?- ~ T;. In

I (~, r

--"

f,," (1 + x)" X

ldx

(T), ,

x

g(~:) e :~ ~

l~ -1 , ~"1 ~-o

(17)

(18) (19)

and

(r = (1_r

'

[32 = r162

) 9

(20)

The void nucleation rate ~N given by eq. (14), when added to the solution of the set ofeqs. (15)(20) for the void growth rate ~G, provides the void volume fraction growth rate ~ (x, t ) at any

113

point x of the solid at any instant t. Time integration of ~ will then give the void volume fraction ~ ( x, t ) appearing in the constitutive equations shown in Sections 2 and 3. 5. T E M P E R A T U R E AND I N T E R N A L E N E R G Y The entire process from shock compression to fracture takes place adiabatically. Thus with the absence of any external heat sources, the instantaneous temperature and internal energy can be obtained by time integration of the following energy rate balance equations: pocvT=[ o m ! ~+(1-V)CO tr (T.DP)

(21)

for the temperature, and 1~=1o m I(l.-~-)+tr(X.D" ) + t r ( T . D

p)

(22)

for the internal energy. In these expressions em is given by eqs. (2 or 3), D" = ( ~ , ) T ' and D p is the sum of equations (7) and (8). 6. N U M E R I C A L C A L C U L A T I O N S Material parameter values for high purity copper are shown below. Space limitations prevent discussion of how they may be determined, or from where they were adopted from the work of others. Table 1 OFHC Copper Material Parameters On = 8920 kg/m 3 Ko = 13.2 x 104 MPa go = 4.8 x 104 MPa c o = 3.94 km/s "In = 300 OK Tin.. = 1,360 OK Y0 = 2.04 13= 1.09 .

.

Ti;'] = 180 MPa ~n = 1 0 -3 -

1 0 -4

~F = 0.2 - 0.3 r/, =105P 13,, = 10-2 P s = 1.45

.

a 0 = -3.296

m 4 = 2.07

aj = 1 . 5

a 2 = 0.0038 e I = 0.025

m~ = 70.37 s -! o r~ = 500 MPa k = 1.38 x 10 ~ J/~

C2 = 292 MPa

c v = 376 J/Kg .0 K

mr=2

D " = 1.0 s "i

m 2 = 0.31

co = 0.90

x 1 0 -23 c m 3

~

ro =10 -4 cm a =20

,

,

To begin to test the constitutive model prior to performing projectile-target impact simulations, simpler calculations (made at the time of preparation for the paper) of the pressure, volumestrain and temperature at a representative point behind a passing shock wave are shown below. The calculated and experimental shock pressure volume-strain Hugoniot for copper up to 160 GPa appears in Fig 1.

114 200

6000.

150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t /

50001--

i /

'

--2- .1. . . . . . . . . . .

~ .............

i - J. . . . . . . . . . . . . . . .

40001-=~

loo

...................................

_ ...............

i

or

30001

'

i

f

20001--

so . . . . . . . . . . . . .

-i- ......

~

........

10001 o .............

0

i

0.1

l

0.2 Volume Strain

,/

i~

'

013

Fig. 1. Pressure vs. volume strain. (~ Experimental ref. 16)

0.4

...... /

i

jJ'/

i

50

................... ////

i

i ..................... = i ......................

1 ,

! i

100 Pressure (GPa)

150

200

Fig. 2. Shock pressure-temperature rise (Temperature scale is ~

We note that at 0.3 volume-strain, corresponding to 135 GPa pressure, the melt temperature increases to 4,312 ~ according to the temperature-volume strain model employed. From Figure 2, which illustrates the calculated temperature rise for pressure up to 160 G P a , the calculated temperature at 135 GPa pressure is approximately 4,400 ~ indicating that the shocked compressed material will release into the molten state at this pressure. Because of the very great difficulty of obtaining temperature measurements, directly or indirectly, o f thermal changes that are of nano-second duration, there is no experimental data available that measures high shock pressure- induced temperature fluxuations that can be used for comaprisons with calculated values. ACKNOWLEDGEMENTS This work was supported by Grant HBCU/MI BAA #97-01, of the Ballistic Missile Defense Organization and Project #F49620-97-0536 of the Air Force Office of Scientific Research, with Dr. Arje Nachman of AFOSR serving as Program Manager. REFERENCES

2.

3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

M.H. Rice, R.G., McQueen, J.M. Walsh, Solid State Physics, F. Seitz and D. Turnbul (eds.), Academic Press, New York, 1958. ll.B. Huntington, The Elastic Constants of Crystals, Academic Press, New York, 1958. J.N. Johnson, J. Appl. Phys., 53 (1981) 2812. D.J. Steinberg, S.G. Cochran and M.W. Guinan, J. Appl. Phys., 51 (1980) 1488. P. Perzyna, Int. J. Solids Struct., 22 (1986) 797. L.C. Chhabildas and J.R. Asay, J. Appl. Phys., 50 (1979) 2749. D.E. Grady, Appl. Phys. Lett., 38 (1981) 825. M.M. Carroll and K.T. Kim, J. Appl. Phys., 59 (1986) 1962. D.R. Curran, L. Seaman and D.A. Shockey, Phys. Reports, 147 (1987) 253. R. Raj and M.F. Ashby, Acta Metall., 26 (1978) 993. S.N. Zhurkov, Int. J. Fract. Mech., 1 (1965) 311. L. Seaman, D.R. Curran and W.J. Mufti, J. Appl. Mech., 52 (1985) 593. M.M. Carroll and A.C. Holt, J. Appl. Phys., 46 (1972) 1626. J. Eftis and J.A. Nemes, Int. J. Plasticity, 7 (1991) 275. J. Eftis, lligh-Pressure Shock Compression of Solids If, L. Davison, et. al. (eds.) Springer, New York, 1996. R.G. McQueen, Metallurgy at High Pressures and High Temperature, K. Gschneider et al. (eds.) Gordon and Breach, New York, 1964.

Structural Failure and Plasticity (IMPLAST 2000)

Editors:X.L.Zhaoand R.H. Grzebieta 9 2000ElsevierScienceLtd. All rightsreserved.

115

E x p e r i m e n t a l a n d n u m e r i c a l s t u d i e s o f projectile p e r f o r a t i o n in c o n c r e t e t a r g e t s H. Hansson and L./kgfirdh FOA (Defence Research Establishment) Weapons and Protection Division SE- 147 25 Tumba, Sweden

In the paper measurement results from ballistic tests on fibre reinforced concrete slabs performed at FOA Grindsj6n in April 1999 are presented. The tests were performed with single and double slabs that were 60 mm thick. The projectiles were steel cylinders with the weight of 60 g and fired the velocity range 1400 - 1600 m/s. Results from the experiments were compared with numerical simulations performed with Autodyn 2D. 1. I N T R O D U C T I O N One reason for the interest of steel fibre reinforced concrete is that the increased fracture energy of steel fibre reinforced concrete, might give increased protection against perforation. In [1] results from perforation experiments through concrete slabs with fibre reinforcement and conventional reinforcement are presented. Those experiments were also simulated numerically with LS-DYNA3D [2]. The major difficulties are the description of the material behaviour at the high pressures and strain rates present in this problem. 2. E X P E R I M E N T S 2.1. Specimens and material sample test results The test series consisted of three slabs with the dimensions 1.2• m, poured of a type of concrete that normally is used for bridge decks. The concrete specifications are 440 g/m 3 of Portland cement and a water cement ratio of 0.38. The aggregates is of porphyry red granite with size up to 20 mm and 55 kg "Dramix ZC 30/50" steel fibres has been used per cubic meter concrete. The slabs and 150 mm cubes for material tests were poured in December 1996. In March the unconfined compression strength was determined to 79.5 MPa and the splitting strength determined to 7.4 MPa. From earlier tests of the same type of concrete the modulus of elasticity was taken to be 52.9 GPa and the density 2330 kg/m 3. The projectiles were cylinders with diameter 15 mm, length 43 mm and weight 60 g. They were fabricated from steel SS 14 2541-03 with 0.2-limit of 700 MPa and with failure limit between 900 and 1100 MPa.

116 2.2. Experiment arrangements During the experiments the velocities before impact and after perforation of the target were estimated. In the target, craters on the entrance and exit surfaces were measured and documented. The exit velocity of the projectile was registered with the help of two measurement frames supplied with diodes and a high speed video camera used to verify the position of the projectile. The slabs were fixed in a rig about 4 m from the end of the gun barrel. Two shots were fired against the single slab and two shots against the double slabs, that were fixed together with jackets.

2.3. Results from experiments The projectile from shot no. 96 was not found, and the projectile from shot no. 99 was deformed aider perforation. Crater diameters and hole diameters are presented in table 1. Detailed views from the craters for shot no. 97 are presented in figures 1 to 2. 3. NUMERICAL SIMULATIONS The numerical simulations were performed with Autodyn 2D version 3.0.12 Four sets of simulations were performed with velocities from the tests.

3.1 Material model for the target The RHT concrete model developed at "Ernst Mach Institut" (EMI) in Germany was used for the target. The model and material parameters for concrete is described in [3], and partly also in [4]. The material model is a three invariant model for the definition of the elastic limit surface, failure surface and remaining strength surface for the crushed material, see figure 3. The model also accounts for the loading rate and the limitation of the elastic behaviour with a cap. The damage in the material grows due to increase strain after the stress in the has reached Table 1. Crater dimensions in the concrete slabs (ram). First slal; No. Target Entrance FIole Exit crater diameter crater #96 Singleslab 160 80 #97 Single slab 160 75 170 #98 Double slab 220 65 190 #99 Double slab 190 55 180 .

--

9

i

.

.

iiiii

.

.

i

ii

_

i

m

_

Figure 1. Entrance hole for shot #97.

i

.

.

Second slab Entrance mad Exit hole diameter crater

75 45

240 210 i

i

iii

Figure 2. Exit hole for shot #97.

117

the failure surface The stress is then interpolated from the strength values for the undamaged material and the completely damaged material. The used Equation of State (EOS) is the "P-o~model". Explanation is published by Herrmann [5] and is also described in the Autodyn theory manual [6]. A polynomial EOS gives the behaviour of the compacted solid material. The material parameters in table 2 are chosen from estimations and comparisons with other concrete types [3, 4]. The uniaxial compression strength for cubes was used as the compression strength for the concrete. The tension strength of the concrete is assumed to be 80% of the splitting strength. With Q~ equal to zero, the effect of the third invariant is calculated according to William-Warnke [7]. To obtain numerically robust results, erosion strains of 300 - 400 % were chosen.

3.2. Material model for the projectile The projectile was simulated with the material model Johnson & Cook material model [6, 8]. The mechanical properties are based on data for steel 4340, and an erosion strain of 300-400 %. There were also simulations performed with modified data from uniaxial material tests with the SS 14 2541-3. The used material parameters are given in table 3.

Figure 3. Meridians and surfaces for elastic-, failure- and remaining strengths. From [3]. Table 2. Used values of th e material parameters fo r RHT-model concrete mode I. Parameter Value Parameter Values Parameter Initial p 2.45 g/cm 3 cv 640 J/kgK PREFAC;F p porous 2.44 g/cm 3 G 17.2 TENSRAT c porous 2561 m/s f'c 80 MPa COMPRAT p crush 80 MPa f'df'c 0.073 D1 p lock 1000 MPa A fail 0 D2 n 1 B fail 1.6 EFMIN Solid EOS Polynomial N fail 0.61 B fric p solid 2.68 g/cm 3 QI 0 N fric A~ 85 GPa Q2 0.6805 SOFTFLAG A2 - 171 GPa BQ 0.0105 SHRATD A3 208 GPa R1 0.007 CAPFLAG TI 85 GPa R2 0.007 Bulk strain Tref 300 K RLIMYr 1 Erosion strain i

i

i

rr

i

Values 2 0.6 0.3 0.04 1 0.01 0.8 0.85 2 0.13 1 3 3

118 Table 3. Used material data for the steels 4340 and SS 14 2541-3 4340 steel SS 14 2541-3 steel 7.83 g/cm 3 7.83 g/cm 3 Reference density 159 GPa 159 GPa Bulk modulus 300 K 300 K Reference temperature 477 J/kgK 477 J/kgK Specific heat 81.8 GPa 81.8 GPa Shear modulus 792 MPa 600 MPa Yield stress 510 MPa 850 MPa Hardening constant 0.26 0.30 Hardening exponent 0.014 0.014 Strain rate constant 1.03 1.03 Thermal softening exponent 1793 K 1793 K Meitin[~ temperatur e ,

l,

,l

,

,,,

3.3. Geometric models The simulations were performed using rotation symmetry for the models. The target slab is 60 mm thick and a circular plate with a diameter of 500 mm is simulated. The element size in the target is gradually increased from the centre and outwards. To reduce the reflections at the boundary, a transmission boundary condition was used at the boundary around the plate. The target that consisted of two slabs fixed closed together was simulated both as joined plates and with a distance between the surfaces to observe the differences. Further an element grid with approximately half the element sides was used for improved simulations. For these simulations the target plates consisted of 60x100 elements and the projectiles of t6x44 elements, this gives approximately one mm's element side in the centre of the target. 4. RESULTS FROM SIMULATIONS AND COMPARISONS WITH EXPERIMENTS The results from the simulations and the comparisons with the experiments are presented below, see table 4. After perforation the length and diameters of projectiles from shot 97 and 98 was measured and compared to the simulated results, see also table 4. In figure 4 the projectiles from shot #97 and #98 are shown, representing shots against single and double slabs. In figure 5 the corresponding simulated projectiles after the test are shown. The damaged concrete zone for the models FRAG6E and FRAG7E are shown in figure 6.

Figure 4. Projectile after shot #97 (left) and #98 (right).

Figure 5. Projectile from simulation FRAG6E (left) and FRAG7E (right).

119

Figure 6. D a m a g e levels in the concrete targets after perforation, for models F R A G 6 E F R A G 7 E . T h e scale for the d a m a g e level of the concrete are s h o w n to the right.

and

Table 4. Results from simulations and experiments. i

Experiments NO.

,..,.,.,

.,..,

Simulations '"'

Vlmpac VExit

,

,.

,

,.,,, ....

,,,

,,,,,,,

i

,

t m/s

m/s

96

1637

1120

......

97

1505

1080

41

28

23

FRAG2B l FRAG6E 3 F R A G 6 F 3'4

990 1066 1051

55 46 47

98

1557

455

35

20

26

FRAG3B l FRAG3C 2 FRAG7E 3 F R A G 7 F 3'4 FRAG4B ! FRAG4C 2 FRAG8E 3 F R A G 8 F 3"4

618 598 570 563 487 498 530 538

51 51 39 40 54 55 46 48

1421

418

,, ,,,,

32

......

,

O mm

,.,

MExit g

99

L mm

,,,,,,

Identity

WExit m/s

MExit g

F R A G 1B' FRAG5E 3 F R A G 5 F 3"4

i112 1175 1160

51 42 44

26.5 -

23.7 -

20.7

25.0

f

,J

i

L mm

O' mm

-

,

C o m m e n t s : ') Coarse e l e m e n t gridl 2) Joined slabsl 3 ) E r o s i o n strain increased to 4 0 0 % for proiectile and target. 4 ) E s t i m a t e d material data for SS 14 2541-03 has been used, all other simulations with 4340 steel.

120 5. S U M M A R Y The simulations were performed with AUTODYN2D version 3.0.12 with the RHT concrete model. The results from the simulations were compared to the ballistic tests. It is shown that exit velocities of the projectile were simulated with good agreement for the single slab and fairly good agreement for the double slab. The exit velocities for the double slabs were higher in the simulation, this indicate that the concrete model may be to soft. The residual mass of the projectile was in all cases overestimated, this may be explained by the very high erosion strain used for the steel. A more suitable value would have been 150 to 200 %. The deviation can be explained by uncertainties regarding the material data for concrete and steel. However, the use of different steels in the numerical models did not have any significant influence on the result. The concrete had probably higher strength values at the time for the ballistic tests compared to the test results, because the material tests were performed one year earlier. On the other hand the compression strength determined from cubes is probably to high, this is caused by the friction constraint of the concrete at the load surfaces. In the used material model the steel fibres are not considered, thus the effect of the increased fracture energy can not yet be simulated. Therefor, it remains to perform corresponding ballistic tests of slabs without steel fibres. The reason for this is to determine the difference in behaviour for concrete with and without fibre reinforcement. There is also a need for better characterisation of the material behaviour of the used concrete at high pressures and strain rates.

REFERENCES 1. A. Bryntse, Perforation of steel fragments at 1500 m/s through steel fibre and conventionally reinforced concrete slabs, FOA-D-97-00304-311--SE, Tumba, 1997. (In swedish) 2. L. ,~,gS.rdh and L. Laine, 3D FE-Simulation of High Velocity Fragment Perforation of Steel Fibre Reinforced Concrete Slabs, Proc. Int. Symp. honouring Mr Arnfinn Jenssen, Trondheim, 1998. 3. W. Riedel, Ein makroskopishe, modulares Betonmodell fiar Hydrocodes mit Verfestigung, Sch~idigung, Entfestigung Drei-Invarientenabh~gigkeit und Kappe, EMI bericht 7/98, Freiburg 1998. 4. W. Riedel, K. Thoma, S. Hiermaier and Schmolinske E, Penetration of Reinforced Concrete by BETA-B-500. Numerical Analysis using a New Macroscopic Concrete Model for Hydrocodes, Proc. 9~ Int. Symp. IEMS, Berlin, 1999, pp 315 - 322. 5. W. Herrmann, Constitutive Equation for the Dynamic Compaction of Ductile Porous Materials, J. of App. Physics, vol 40, No 6, 1969, pp. 2490-2499. 6. Autodyn Theory Manual, Century Dynamics Ltd, Horsham 1999. 7. K. J. Willam and E. P. Warnke, Constitutive Model for the Triaxial Behaviour of Concrete, Sem. on Concrete Structure Subjected to Triaxial Stresses. IABSE Proc. 19, Italy, 1975. 8. G. R. Johnson and W. H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures, Proc. 7th. Int. Symp. on Ballistics, Holland, 1983, pp. 541-547.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

121

Prototype Impact Tests on Ultimate Impact Resistance of PC Rock-Shed N.Kishi a,

H.Konno

b, K . I k e d a b , and K.G.Matsuokaa

aCivil Engineering, Muroran Institute of Technology, Muroran, 050-8585 Japan bStructural Laboratory, Research Institute of Hokkaido Development Bureau, Sapporo, 062-8602 Japan In order to confirm the ultimate impact strength of Prestressed Concrete (PC) rock-sheds constructed on a highway in Japan, prototype impact tests are conducted by using two types of PC rock-shed frames: inverted L-shaped type and fully rigid frame type (frame type). These both types of PC rockshed frames are of the same dimensions which are: span length = 10.1 m, upper flange width = 150 cm, and column height = 4.65 m. Impact tests are performed by 3.0/5.0 ton steel weight iteratively and freely falling onto the center of each frame covered with 90 cm thick sand cushion. The results obtained from this study are as follows: 1) the frame type has more than 1.7 times impact resistant capacity than the inverted L-shaped one with reference to a falling weight potential energy; 2) the frame type can rationally disperse the sectional forces over the whole structure; and 3) these inverted L-shaped type and frame type PC rock-sheds designed based on allowable stress design procedure have more than four and five times margin for collapse, respectively. 1. I N T R O D U C T I O N

In Japan, there are many rock-sheds over mountainous and coastal roads with cliffs. So far, three types of the structure have been constructed which are Reinforced Concrete (RC), Prestressed Concrete (PC), and steel type. In future, PC rock-sheds are expected to be the most popular type because of good quality control and low labor cost due to prefabrication works. In PC rock-sheds, an inverted L-shaped type is usually applied of which column base and the top of mountainous side-waU (hereafter, side-wall) are hinged and the girder is rigidly connected to column. The structure is a first-order statically indeterminate structure. However, in order to attain economy and to increase number of redundancy for safety against an unexpected bigger impact load than specified one in design, a structural type having several redundancies should be chosen. In this paper, in order to investigate ultimate impact resistance of inverted L-shaped type and frame type PC rock-shed, steel weight falling impact tests

122

iI

steel weight

(3/Ston)

surcharged girder G3

J

/SAND

i

....

~r~be

--)

Figure 1 Experimental set-up using full scale PC rock-shed frames with 90 cm thick sand cushion are conducted. Here, impact load is surcharged iteratively and its input energy is gradually increased till the shed reaches the ultimate state. It is assumed that the PC rock-sheds are in the ultimate state when maximum response value of the main girder displacement reaches around 35 mm which is about 1/300th of the main-girder length. The ultimate impact resistance for these types rocksheds is discussed using sectional bending moment and crack pattern developed in girder after the final impact test.

2. EXPERIMENTAL OVERVIEW The both types full scale PC rock-sheds are composed of laterally arranged five frames. A girder in each frame has a simple T-shape cross section, of which dimensions are: height - 110 cm, width of upper flange (hereafter, roofslab) -- 150 cm, and length -- 12 m. The column has a rectangular cross section (110 cm • 50 cm) and is about 4.7 m high. The impact tests are conducted by freely dropping 3.0/5.0 ton steel weight onto the roof-slab of each rock-shed with a 90 cm thick sand cushion as shown in Fig. 1. The dimensions of dropped steel weights are the same in each other and these are: diameter = 100 cm and height of spherical bottom = 17.5 cm. The PC rock-shed used in this experiment is designed based on the Japanese conventional design manual 1) for anti-impact structures against a falling rock under the following conditions: 1) a 1.0 ton mass rock falls on the roof-slab with a 90 cm thick sand cushion from a 30 m height; 2) a 1.17 MN impact load, estimated by using a conventional Hertz's contact theory with Lame's constant of sand cushion )~ = 0.98 MPa, is statically surcharged onto the center of girder; 3) the impact load is dispersed in the area of 45 ~ direction of salad cushion; and 4) concrete compressive strength and allowable tensile stress

123

Table 1 List of experimental cases Notation_ L3-5 L3-20 L3-30 F3-5 F3-10 F3-20 F5-20 F5-30

Structural . type Inverted L-shaped type

Mass of weight (ton) 3

Frame type 5

Falling height (m) 5 20 30 5 10 20 20 30

Note: steel weight is dropped at the mid-span of girder are 54.0 MPa and 2.0 MPa, respectively and its Young's modulus is 34.3 GPa. The PC girders have the dimensions mentioned above and are prestressed using 24 tendons of steel strand with pretensioning system. Introduced prestress in the mid-span's upper and lower fibers of girder are 1.0 MPa tensile stress and 10.7 MPa compressive stress, respectively. The girders and columns of inverted L-shaped type axe connected to the side-wan and the basement, respectively with 2 round bars to keep a pin condition. On the other hand, those for frames type are rigidly connected introducing prestress with 12 and 4 PC tendons, respectively. The girder is rigidly connected to column by prestressing with 4 PC tendons in the outer area of column section and 8 PC tendons over the whole area for inverted L-shaped type and frame type, respectively. The sand layer is compacted by foot stamping at every 20 cm thick layer. The moisture content and relative density of sand after coInpactions are 6.4 % and 36 %, respectively. Experimental cases are listed in Table 1, in which each case is notified symbolizing structural type (L" inverted L-shaped type, F: ~frame type), mass of steel weight (ton), and falling height of steel weight (m). In these experiments, the strains in the upper and lower rebar of girder and in the outer and inner rebar of column are measured and are recorded using wide-band data recorders. These strains are converted into sectional forces using fiber model based on the linear variation concept on sectional strain distribution, in which each material stress-strain relation and axial compressive forces introduced due to prestressing are considered. And girder displacement is also measured to confirm whether the shed model reaches the ultimate state or not.

3. EXPERIMENTAL RESULTS 3.1. Bending Moment Distribution of Inverted L-shaped Type Figure 2 shows the bending moment distributions in cases of L3-5 and L3-30 on inverted L-shaped type. In this figure, column and side-waU are arranged in the left and right hand side, respectively. The bending moment of side-wall

124

I

9

3 MNm

L3-5

O o o o ee~

" 0 0 00000 0

10

~

lS

msec

50

m sec

80

:~176 Co o-11

-:::.....

~176176 0

~

30

io o 1

~ OOOoOO~ ~

25 m n c

msec

I v~.... :o "l I o

9 o o

msec

60

~ 1

: " O o. .~. . . .

Ioo---~---i

99

o o

OOOO~

20 m s e c

40 m s e c

10

m sec

o

msec

,_.....o.~ao

"-~" "'""" ~

L3-30

90

~ffi.

,,

msec

o

1

o li.,oo Q_ goffi.

1o- ..... o....

msec

100

1

msec

Figure 2 Bending moment distribution in cases of Inverted L-shaped type

I3

MNm

o,,

.

A F3-5

O F3-30

[] F5-30

,,G*A

"":':"-~o~

8....-~oo ~o

10 msec

o

25 msec

~

-~176176176176 $0 m s e c

rt

80 msec

0o

- 8 ~~- - - - - e oS --

15 msec

~_

.

, ,~: DO

30 muec

~_ .......

O

20 msec

40

~

-

60 msec

(3

~..

msec

- . . . . .. . . . . . .

O

_.

;,~,~ ~"- 90 msec

~ ~ ~___?.....

70 m s e c

7

-~100

. . . . . . .

msec

Figure 3 Bending moment distribution in cases of frame type is estimated assuming one-way slab with a 1.5 m wide effective length. From this figure, it is seen that the bending moments of girder are almost linearly distributed taking the maximum value at the center (loading point) and zero moment near both ends of girder in spite of the magnitude of a falling weight potential energy. Focusing on the bending moment at the connecting point between girder and column, even though statically a half of mid-span bending moment is occurred at that point, the experimental results are less than one sixth of the mid-span's value. It implies that the erection method applied for inverted L-shaped type does not enable the girder moment to rationally transfer to the column, and the connection should be assumed as not rigid but pin condition in design. The maximum bending moment is occurred near the loading point in case of L3-30 and of 5.1 MNm which is a similar value to the ultimate moment capacity estimated by using fiber model. Then, bending cracks may have been developed near the lower surface of the mid-span of girder.

125

I

3 MNm

9

o

o 0o

~:) A~

-

I:::~ g O,g o t ~ n ^ _oa

~s...l~o 10

msec

25

L3-30

%~

msec

SO

o--

F3-30 FS-30 ..__ t i l . ~

80

msec

i2im.

.... t

m ser

G oo

IS

'-k

.__ -_, . % * _ * . , * = .

msec

30

msec

o

60

Oo

.... 20

.8.~'"

msec

~

-~~176

40

msec

I

..... 7'0

90

msec

o

"I

m sec

el 9,,IL~_*. n h .

msec

Figure 4 Comparison of bending moment distribution between two types

3.2. Bending Moment Distribution of Frame Type The comparison of bending moment distributions among three cases (F3-5, F3-30, and F5-30) for frame type is shown in Fig. 3. From this figure, since the bending moments at both ends of girder, cohmm basement, and the top of side-wall are increased with increment of a falling weight potential energy, the rigidly anchoring effect at the supporting points and rigidly connecting effect of girder to column on moment distribution can be confirmed. Therefore, negative bending moment is occurred at both ends of girder and column basement. The girder bending moment for frame type is of parabolic distribution similarly to that of statically and uniformly surcharged girder. This bending moment distribution mode may be conveyed due to the rigidly anchoring and connecting effects because the distribution in case of inverted L-shaped type is of almost linear as mentioned above. Investigating vibration mode of the whole structure, at the time of 40 msec from beginning, the frame in cases of F3-5 and F3-30 is restored to the original position and has a tendency to move to the rebounding state (negative loading state). However, in case of F5-30, the same situation comes out about 20 msec later than that in cases of F3-5 and F3-30. It means that a fundamental natural vibration period of F5-30 is prolonged corresponding to decreasing the bending stiffness due to the progress of plastic region around the mid-span accompanied with cracks developing. The maximum bending moment is occurred at the mid-span after 20 msec from beginning of which the value is about 5.2 MNm. This value is almost the same with that in case of F3-30. Then, it is seen that the safety margin of PC rock-sheds can be increased by 1.7 times (= (5 ton • 30 m)/(3 ton • 30 m)) by changing structural system from inverted L-shaped type to frame one with reference to a falling weight potential energy.

126 i colum side I

inverted L-shaped type

qr l

wall ddei

I I

| I

i I

'

frame type

I

~

I

J t

I

Figure 5 Crack pattern of girders after final impact test 3.3. Comparison of Dynamic Responses between Two Structural Types Figure 4 shows the bending moment distributions of L3-30 for inverted Lshaped type and F3-30 and F5-30 for frame type. Comparing the results between L3-30 and F3-30 which are the cases with the same input energy, it is seen that the girder moment (F3-30) for frame type is moved to the negative side and the positive bending moment is effectively decreased by rigidly anchoring the supporting points and rigidly connecting girder to column. Figure 5 shows crack patterns developed in each girder of L3-30 and F5-30 after the final impact test. From this figure, in case of inverted L-shaped type (L3-30), bending shear type cracks are developed around the mid-span area. On the other hand, in case of frame type (F5-30), the similar cracks with the case of inverted L-shaped type (F3-30) are developed except two crack lines are diagonally occurred near a quarter span in the wall side. Maximum residual displacement of girder for L3-30 and F5-30 are 1.8 mm and 3.6 m m long, respectively. As mentioned above, it is recognized that the maximum bending moment at the final impact test reaches the same with the ultimate bending capacity estimated using fiber model. However, the severe damage such as compressive failure around loading point and/or tension failure due to rebars and PC tendons yielding has not been found from the crack pattern, and still both frames have retained sufficient residual impact resistant capacity. 4. CONCLUSIONS The results obtained from this experiment can be summarized as follows: 1)the frame type has more than 1.7 times impact resistant capacity than the inverted L-shaped one with reference to a falling weight potential energy; 2)the frame type can rationally disperse the sectional forces over the whole structure; and 3)inverted L-shaped type and frame type designed based on ASD procedure have more than four and five times margin for collapse, respectively.

REFERENCE 1. Japan Road Association, "Manual for anti-impact structures against falling rock", 1983.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

127

Penetration equations for the impact of 7.62mm ball projectile against composite material sheets of an aircraft Pardeep Kumar, R.A Goel and V.S Sethi Terminal Ballistics Research Laboratory Sector 30, Chandigarh, India

Abstract An aircraft is generally exposed to the attack from antiaircraft gun (AAG) or fragmenting type of warheads. But its vulnerability is often evaluated in terms of the attack against 7.62mm BalI/API ammunition. The failure of the target is measured in terms of ballistic limit and the amount of remaining velocity or remaining energy of the projectile. To compute the vulnerability of an aircraft, extensive experimentation was undertaken on various thicknesses of a carbon fiber composite material. Empirical equations for ballistic limit and remaining velocity after perforation have been formulated and presented. 1. INTRODUCTION The ballistic survivability / vulnerability of an aircraft involves the study of the impact behaviour of the threat mechanisms, i.e fragments and small arms projectiles. The impact of a high velocity projectile on a target is a very complex phenomena and may result in partial penetration or complete penetration. A complete description of the dynamics of impacting projectile demands the accounts to be taken about geometry of interacting projectile and target, their elastic, plastic behaviour, etc. An analytical approach is not only quite formidable but does not give realistic results. Therefore, the large amount of work in this field has been experimental in nature. Thor equations can be used to determine the penetration of steel fragments of different shapes against different types of metalic and non-metalie materials. But, for small arms, Thor equations for penetration can not be used against aircraft materials because of the shape factor of the penetrator. Hence extensive experimental work was carried out to study the behaviour of the composite material against the impact of 7.62 mm ball projectile. The equations have been established between impact velocity, critical velocity and remaining velocity. 2. DEVELOPMENT OF MATHEMATICAL MODEL 2.1 DETERMINATION OF BALLISTIC LIMIT The determination of ballistic limit / critical velocity is of prime importance in the design of protective structures, in evaluation of the effectiveness of military armour vehicle, aircraft vulnerability analysis and in any problem area where an impact can cause damage. Two techniques (deterministic or probabilistic) are available to determine the critical velocity. In former, a limit velocity is determined from physical principles (the conservation laws and material constitutive relations), but because of complexity of problem, the choice of probabilistie technique is made. In this technique a statistical approach is employed. In the simplest form, Vs0 is determined by averaging six projectile striking velocities that include

128 the three lowest velocities that resulted in complete penetration and the three highest velocities that resulted in a partial penetration (a spread of 46m/s or less is required between the highest velocity for partial penetration and the lowest velocity for a complete penetration). In United States, three criteria of ballistic limit have commonly been used, i.e., Army, Protection and Navy. Under the Army Ballistic Limit (ABL) criterion, a complete penetration occurs when light is visible through the penetration in the armour or when the nose of the projectile can be seen from the rear of the armour. In the Protection Ballistic Limit (PBL) criterion, a complete penetration occurs whenever a fragment or fragments from either the impacting projectile or the armour are caused to be ejected from the back of the armour with sufficient remaining energy to pierce a thin sheet of aluminium alloy, 0.05 mm to 0.50 mm thick (witness plate) placed parallel and 15.24 cm behind the target. While the Navy Ballistic Limit (NBL) criterion for a complete penetration requires that the projectile or a major portion of a projectile passess through the plate. On the basis of procedure explained in determination of ballistic limit, the ballistic limit for carbon fiber composite panel of different thicknesses namely 4.5, 6.5, 7.5, 8.5 and 10.5 mm against a 7.62 mm Ball projectile at normal obliquity has been experimentally determined. The geometry of the 7.62mm ball projectile is standard having conical shape with CRH ratio 5/10. The mechanical properties of carbon fiber composite material are (i) Specific gravity 1.13 (ii) Tensile strength, psi 120000 (iii) Tensile modulus of elasticity, psi 6000000 A mathematical expression for Vm will depend upon density, thickness, mechanical properties and shock wave velocity in the target and angle of impact, presented area and weight of the projectile. i.e Vso : f(Pt, Ap,W, t, s, a, a )

(1)

where, p, = Density of the target Ap = Presented ~rea of the projectile t = Thickness of the target W = Weight of the projectile s = Strength of the target a = Velocity of sound in the target ot = Angle of impact The dimensional analysis will provide the solution in the following form

n, :,(n,, n,, n,)

(2)

If w e assume It I to be nondimensional group with Vso , II 2 and II 3 with s and a,

ptApt 1-I4 - " 7 , then the experimental simulation of equation (2) gives (PtApt~ ~ Vso= 2106.83[ W J The mechanical properties have been submerged in the values of constants.

(3)

129

2.2 DETERMINATION OF REMAINING VELOCITY The empirical relation for remaining velocity versus striking velocity and ballistic limit/critical velocity of 7.62 mm ball projectile against composite panels has been formulated and given below

(v,: - v d ) ": V,=

(4)

(1+ p, t ] p. o)

Where t = Length of the projectile pp = Density of the projectile V, = Striking velocity of the projectile 3 EXPERIMENTAL RESULTS AND DISCUSSION 3.1 BALLISTIC LIMIT Observed and computed values of the ballistic limit have been placed in Table 1. It is evident from the Table that the maximum error between observed and computed values of ballistic limit varies from -0.37 % to 3.40 %. The graph for observed and computed values of ballistic limit verses thickness of plate has also been presented in Fig 1 which shows as thickness of plate is increased, the ballistic limit for computed and observed values is also increasing. 3.2 REMAINING VELOCITY The comparative observed and computed values of the remaining velocity of various thicknesses (4.5, 6.5, 7.5, 8.5 and 10.5) of target panels have been placed in Table 2,3,4,5,6 respectively. It is cleared from these tables that observed and computed values are within a maximum error of 4%. It is also seen as thickness of target increases, the % of error between computed and observed values of remaining velocity also increases.

350 300 E 250 -

--+- Observed Ballistic Limit Computed Ballistic Limit

E 200

o..,.

.J

.o_ 1 5 0 o.m

m

100

-

50 i

i

~

'

"

I

2 4 6 8 Thickness of the plate (mrn)

I

10

Figure 1. Ballistic limit verses thickness of plate

........ i

12

130 4 CONCLUSION It can therefore be inferred from the above paragraphs that the empirical equations formulated for ballistic limit and remaining velocity of composite panels against 7.62 mm ball projectiles are suitable to compute accurate results within an error of maximum 4%. ACKNOWLEDGMENT The Authors are greatly indebted to Sh. V.S Sethi, Director, TBRL for granting permission to publish this work.

Density of plate=l.5 g/cm 3 Presented area of projectile=0.456cm 2 Weight of projectile -9.33 g Density of projectile--11.4 g/cm 3 Length of projectile -2.87 cm Table I S. No

-

Angle of Impact .......... (degree) 1 90 2 3 4 5 -

-

~

:

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1"hickness of Plate(mm) .

.

.

.

.

.

_ ~

.

.

.

.

.

.

10.5 8.5 7.5 6.5 4.5

. . . . . . . . . .

.

observed' Ballistic ' 'Computed Ballistic Limit(m/s) Limit(m/s)

'//~Erro'r

.

305.00 265.00 231.00 206.00 154.00

,

.

.~ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

308.84 255.98 235.29 206.77 155.95 , ...................

__

.

, .

.

.

.

-1.26 3.40 -1.86 -0.37 -1.27 .

, . ~ ,

Table 2 S.No .... 1

2 3 4

5 6 7 g

9 10

Impact .... Thickness Angle of Plate de ree (rqm 9O 4.5

Striking ' Observed ' ComPuted %Egor Velocity Remaining Remaining ..........(m/s) ...................Ve !o cit y( ~ S )_____V_elo_c.it_,y_(_m/s]. 805.68 770.55 774.47 -0.52 862.80 835.54 831.44 0.48 862.00 830.76 830.64 0.01 874.50 840.55 843.09 -0.09 760.85 728.43 729.14 -0.09 970.40 945.35 938.43 0.71 860.46 833.84 829.11 0.55 865.55 838.35 834.18 0.48 867.82 838.40 836.44 0.23 ............ 85_6"47................. 8 2 6 . 5 5 825.13 0.17

131 Table 3 s.do

Angle of Impact (degree)

Thickness of Plate (mm)

90

6.5

1

Striking . . . . . Velocity (m/s)

2 3 4

5 6 7 8

9 10 9

,

839.40 850.30 852.08 810.55 820.45 860.35 838.48 842.65 846.84 815.75 . . . . . . . .

,.

. . . . . . . .

,,

,j

.

.

Observed. . . . . Remaining Velocity

.

.

.

.

799.55 807.64 810.05 765.10 782.20 823.00 797.35 801.55 810.47 773.08 .

.

.

-

,

.

.

.

.

Computed Remaining Velocity

% Error'

790.03 800.64 802.69 761.05 770.99 810.97 789.07 793.25 797.45 766.28

1.19 0.87 0.91 0.53 1.43 1.46 1.04 1.04 1.61 0.88

.

.

.

.

.

.

.

.

.

.

.

.

,

.

.....

Table 4 _

S'l~ O .

.

.

.

Angle of ~ Thici~ness S t r i k i n g Impact of Plate Velocity (degree) (mm) (m/s) , .

1 2 3 4 5 6 7 8 9 10

.

.

.

.

.

90

.

.

.

.

.

.

.

7.5

.

.

.

.

.

.

.

-

.

836.15 840.64 837.54 829.40 824.00 839.50 796.33 815.64 834.55 796.38

_

.J_,

Observed Remaining Velocity

,

.

,,,

i,

Computed Remaining Velocity

(m/s) . . . . . .

789.55 795.66 793.47 777.25 778.45 793.40 748.25 768.69 790.36 746.85

,,,

L,

% Error

(m/s)

....

775.69 780.21 777.09 768.89 763.44 779.07 735.49 755.01 774.08 735.54

1.76 1.94 2.06 1.08 1.93 1.81 1.71 1.78 2.06 1.52

Table $

0

Angleof Impact (degree)

ThickneSs ''~ Striking --' of Plate Velocity (ram) (m/s)

Observed Remaining Velocity (m/s)

90 2 3

4 5

6

. . . . . . 8'.5. . . .

915.85 .... 840.50 845.45 854.84 849.80 843.40

865.73 786.25 793.70 802.25 794.34 792.45

Computed ~ % Error Remaining Velocity ....

(m/s)

846.37 770.54 775.54 785.02 779.94 773.47

.......

2.20 1.99 2.29 2.15 1.81 2.39

132 7 8 L,,

9 10 ,,

,,

,,,

.........

852.35 852.85 865.27 838.43 ,

,,

,,,

800.43 801.25 816.55 783.28 i

ii

|ll

.

i

.

,

,,

782.51 783.01 795.54 768.45

,,,,,,,

,

2.24 2.28 2.57 1.89

,,,,,,

Table 6 S,N o

1

2 3 4 5 6 7 8 9 10

....

Angie 'of '"Thickness Impact of Plate (degree) (mm) 90

Striking Velocity

(m/s)

Observed Remaining Velocity

864.58 816.44 854.80 844.85 861.65 753.55 838.57 836.70 824.44 757.50

795.60 747.55 788.61 778.52 796.85 680.61 767.05 770.45 755.35 685.65

10.5

.....

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,(m/s)

.

.

.

.

.

,,,

,

' Computed Remaining Velocity

(m/s)

770.45 721.06 760.45 750.26 767.47 655.79 743.82 741.90 729.30 659.92

,,,,,,,,,

---

.

% Error

.

.

.

.

.

,,

.

,,

.

,,,

.

.

.

3.16 3.54 3.57 3.62 3.69 3.65 3.03 3.71 3.45 3.75

. . . . . . . . . . . .

Structural Failure and Plasticity (IMPL,,IST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

133

High Strength Concrete Beams Subjected to Impact L o a d - S o m e Experimental Results Johan Magnusson, H~mn Hansson, Lennart Ag~rdh Defence Research Establishment (FOA), Weapons and Protection Division SE-172 90 Stockholm, Sweden In this investigation, high strength reinforced concrete beams have been tested for impact loads. The unconfined compressive strength of the concrete was 112 MPa. Three beams were subjected to quasi-static loading and five beams were subjected to impact loading. A weight with a mass of 718 kg dropped with a striking velocity of 6.7 m./s served as the impact load. Main emphasis was put upon performing accurate measurements for validation of numerical models. 1. INTRODUCTION Accidental loading is often comprised of impacts from collisions, striking objects, explosions etc. The failure process of the target is complex and depends on several loading parameters such as striking mass, velocity, load direction and the interface between striker and target. Also, target material parameters such as strain rate sensitivity and ductility, and structural parameters such as boundary conditions influence the failure process. For design purposes it is necessary to use validated numerical models. This investigation involves experiments of reinforced concrete beams subjected to both quasistatic and dynamic loading. The objectives were to increase our basis for numerical modelling and simulation of high strength concrete structural response and to register the crack initiation and development under slxain rates up to 100 sq . The investigation is completely reported in [1]. Similar tests and simulations have been performed with smaller beams reported in [2] and with normal strength concrete in [3]. 2. EXPERIMENTAL PROCEDURES

2.1. Specimens A total of eight concrete beams were tested, each with dimensions according to Figure 1. The unconfined compressive strength of the concrete was 112 MPa and the splitting strength was 6.7 MPa at the day of testing. Both the compressive strength and the splitting strength were determined from 150 mm cubes. Young's modulus according to CEB-FIP was equal to 44.2 GPa and according to ISO 6784 was equal to 45.3 GPa. Young's moduli were determined on ~100 x 200 mm cylinders. The beams were reinforced with both longitudinal rebars and stirrups. The mean density of the specimens was 2520 kg/m 3.

134 Stirrups d=10Ks500STslS0 1500

100 ~

l,/llll

,//

~k.

Ill

-7~

illi

100_

1500

llIll!!

i

t

.....

'

4200

.i

ELEVATION /

Jl.~

d=12 KsS00ST

~ ' " . P l I'7t

CROSS SECTION

Figure 1. Concrete beams used in the experiments. The reinforcement was of type Ks500ST with a nominal yield strength of 500 MPa. The results of a uniaxial tensile test of one rebar is shown in Figure 2. Three strain gauges were positioned on each rebar at the bottom layer of the beam as shown in Figure 1. The cover in the compression zone was increased to avoid complications in the simulation of the failure process, see Figure 1. 700

~

m

I

5OO

~ 300 200

100 0 0

2

4

6

8

Strain

10

12

14

16

(%)

Figure 2. Stress-strain curve from the uniaxial tensile test of one rebar. (Note: The broken line is an extrapolation based on the test results).

135

2.2. Quasi-Static Load Three beams were tested with a quasi-static load. The beams were simply supported with a span of 4.0 m in the quasi-static tests. The supports were considered to be rigid with negligible vertical displacements. Two uniform line loads were positioned at the third points

Crack indicator

!,.333 j,_.

~

Load (P) Load cell ~........r

3 4000

Hing~

-, "

I

........ i<

Deflection gauge [positioned at mid-s~n

Figure 3. Experimental set-up for the quasi-static tests. of the span. The load displacement rate was 1.0 mm/min for one beam and 2.0 mm/min for two beams. A load cell and a deflection gauge were positioned as shown in Figure 3. A twomillimetre wide painted line was used as a crack indicator. The line was painted on the side and 20 mm above the bottom surface of the beam. A nickel paint was used because of its ability to conduct electricity. By connecting an electric voltage to each end of the line an electric circuit was closed. When a crack was initiated the circuit was disclosed and a registration of the crack initiation was obtained. Furthermore, six strain gauges were positioned on the rebars as shown in Figure 2.

2.3. Dynamic Load Altogether five beams were tested with a dynamic load consisting of a heavy drop weight striking each beam at mid-span. The striker with a mass of 718 kg was dropped from a height of 2.68 m. The striker head struck a steel pad positioned on the beam with a velocity of about 6.7 m/s. The steel pad was 50 mm wide and 30 mm thick. The supports were considered to be rigid. In order to avoid upward displacements of the beam-ends during impact, restraints were used at the supports as illustrated in Figure 4. The span between the supports was 4.0 m. Two accelerometers were positioned on the beam as shown in Figure 4. For the last beam tested an accelerometer was also positioned on the striker head. Six strain gauges were used in the same manner as in the quasi-static tests except for the last beam tested. For this beam two strain gauges were positioned on the concrete surface on both sides of the beam at the same level as the middle gauge on the rebars, and four strain gauges were used on the rebars. Furthermore, the striker velocity was registered with a pulse transducer which was also used for the beam velocity in the last experiment. The same crack indication method was used as in the quasi-static experiments with the difference that the line was painted on the bottom surface of the beam in the dynamic experiments. A contact indicator was used between the striker head and the steel pad in order to register the time of impact. The event was recorded

136

Registrationof strikerveloci~

A [ ~

,/

v [m/s]

Steel vad

v [m/s]

Accelerometers

T

__

m l l m

L

"/

1

,, 4ooo

/

.

.

.

.

.

i .... f

A ELEVATION Figure 4. Experimental set-up for the dynamic tests.

f

f

f

f

f

f

f

A-A

with a high speed film camera with 1000 flames per second. For the last experiment 1540 frames per second was used. 3. RESULTS The load-deflection curves for the beams tested with a quasi-static load are shown in Figure 5. The mean beam deflection at the first crack initiation was found to be 1.5 mm at a crack load of 22.8 kN. From the registrations of the moment of impact in the dynamic experiments, separations between striker and beam were observed. The moment of crack initiation was captured. The maximum registered strain rates in the middle gauges on the rebars varied between 21 s ! and 102 s! for the different tests. The scatter is due to the location of the cracks. A crack in the concrete which appears where a strain gauge is located gives rise to higher registered strains and strain rates. The strain rates were determined on the upper part of the strain versus time curves, i.e., after yielding of the rebars. Strain rates from the middle gauges determined on the strain curvesbefore yielding of the rebars varied between 15 s"1 and 17 s"l. The maximum registered strain rates in the concrete before crack initiation was 6 s"~. The acceleration time history was integrated twice in order to obtain the velocity of the beam and the deflection of the central part of the beam. The beam velocity and the deflection were also determined from the high speed photos. Figure 6 shows a sequence from the high speed photos. The vertical distance between the crosses on the beam was 100 ram. Figure 7 illustrates the deflection versus time curves determined from the high speed photos and the integration of the acceleration curves. The third curve in Figure 7 illustrates the striker displacement time history obtained from the pulse transducer. The deviation of the integration curve is probably due to the integration process. The two other curves separate after about 18 ms which is the moment when the striker hits the shock absorbers (see Figure 4).

137

70 60

~ 50 "~ 40 ~

30 20 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

~

,

,

,

,

,

,

,

,

,

,

10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

0

'

0

20

,

'

'

40

,

'

'

60

'

'

'

80

100

...........

120

140

Deflection (ram)

Figure 5. Load-deflection curves from the quasi-static tests.

Figure 6. Beam #2 during dynamic loading. Time intervals: 0.6, 20, 30 and 50 ms respectively after beginning of deflection.

138 200 ....................,

180

' ..~. ~.--- "; . . . . . . , . .

160 "~ 140 U 120

......

,. . . . . .

I, " ~ r

,

,

0.03

0.04

I - - - - P u l s e trans.

.2o 11)0 g

8O

4O 20 0

0.01

0.02

0.05

0.06

Time (s)

Figure 7. Beam deflection from pulse transducer, high speed photos and integration of acceleration curves.

4. SUMMARY OF RESULTS The following results were obtained: 1. Registered strain rates were larger in the rebars compared to strain rates in the concrete. Strain rates up to 102 sn in the rebars were obtained. 2. Moment of impact and moment of crack initiation were captured. 3. Registrations of beam velocity and striker velocity were successful. 5.

CONCLUSIONS

The failure process during impact loading is rather complex and difficult to register in detail during the short event. In the tests several techniques were used and the experiences from using a high speed film camera and contact-free sensors, accelerometers and strain gauges are reported. As a conclusion the used techniques were applicable for the purpose of the investigation. The displacement history was also estimated from integration of the acceleration history. More efforts could be made to increase the accuracy of this technique. REFERENCES

1. L. Ag~dh, J. Magnusson and H. Hansson, "High Strength Concrete Beams Subjected to Impact Loading. An Experimental Study", FOA-R--99-01187-311--SE, Defence Research Establishment, Stockholm August 1999. 2. L. Ag~dh and L. Laine, "Experiments and FE-modelling of fibre reinforced concrete beams exposed to impact", Proceedings of the 2nd Asian Pacific Conference on Shock and Impact Loads on Structures, Melbourne, pp. 1-8. Nov. 1997. CI-Premier Conference Organisation, Singapore. 3. G. Hughes and D. M. Speirs: "An Investigation of the Beam Impact Problem", C&CA Technical Report No 546, London 1982.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

139

Impact response of a laminated cylindrical composite shell panel Puneet Mahajan, K.S.Krishnamurthy, R.K.Mittal Applied Mechanics Department, Indian Institute of Technology, Delhi, India The response of a laminated cylindrical composite shell panel subjected to impact by a spherical object is analysed using the finite element method. The evolution of the resulting damage in the composite has also been studied. The effects of various problem parameters such as impaetor mass, velocity, shell curvature, thickness, ply stacking on contact force and response are examined. Hertz's contact law is used to calculate the contact force between the impacting mass and the shell. Damage in the form of matrix cracking due to transverse shear or bending and subsequent delaminations are predicted using appropriate failure criteria. The stiffness matrix is modified during each time step to take account of this damage.

1. INTRODUCTION Predicting impact induced damage in fibre reinforced composite laminates has been a topic of intense research due to their "vulnerability to damage in impact, which adversely affects the overall strength and stiffness of the laminate. Accidental dropping of tools or other metallic objects on the surface of the laminates may cause such impact. Often the damage in the form of matrix cracks and delaminatic~ns may be difficult to detect by the naked eye since the damage can be wholly era.bedded within the thickness of the laminate. During an impact a high contact pressure develops within a small area of contact producing high stress concentrations. The study of impact and impact induced damage in composite laminates has been approached by various researchers through experimental, analytical, and numerical techniques. A review by Abrate [1 ] contains a large bibliography on the subject of impact on composites. While experimental methods are indispensable for direct observation, they are inconvenient for parametric studies. Analytical methods though providing deeper insight into the physics of the problem suffer from the drawback in not being able to predict damage. Finite element method is better suited to simulate the physical phenomenon such as damage initiation and propagation through use of appropriate failure criteria can be modelled. A number of researchers have deployed the finite element method for the solution of impact on laminated composite but literature on impact of laminated shells is sparse [2-8]. Chandrasbekara and Sehroder[7] have shown results for a 0/90/90/0 shell using the finite element method with a non-linear shell theory. The authors have used the contact law for loading and unloading given by Tan and Sun [2 ]. In general the analysis of impact and predicting impact induced damage by means of finite element method involves the following steps: a) Determining the contact force between impacting mass and the shell as a function of time. This contact force cannot be determined explicitly as it depends on the relative approach, ix, of the two contacting bodies.

140 b) Applying the contact force to find the transient dynamic response of the impacted structure as a function of position and time (displacements, strains, stresses, etc.) which depends on the mass and velocity of the impactor and the laminate characteristics (geometry, boundary conditions, ply arrangement, elastic properties, etc.). c) Predicting damage in the laminate using appropriate stress failure criteria. 2. METHODOLOGY The equations of dynamic equilibrium in matrix form obtained through the finite element procedure for the shell is given by

[M]{/}+ [~:]{d}: {F}

wh~r~ {F}: [o,...O,F~,O,....o] ~

O)

Here [M] is the mass matrix, [K], the stiffness matrix, {d}, the vector of nodal displacements of the shell, {F} is the force vector, and Fc is the contact force between the shell and the impactor. The force Fc is a distributed force but since the contact area is very small it is assumed as a point load. Applying Hamilton's principle to a rigid impactor results in the following equation of motion:

,,,~, : - F ,

(2) where mi and wi are the impactor mass and acceleration respectively. The initial displacement and velocity of the impactor are wi =0 and ~vi =v0. In order to solve equations (1) and (2) above, the contact force b~tween the laminated shell and the impactor must be known. It is assumed that Hertz's contact law is applicable to the loading phase and the expression given by Tan and Sun[2] can be used for the unloading phase

i.e., f = k a 312

for loading

and

r

)

f = f~ a - a,,,,, t a . -~Zo

for unloading

O)

Here o ~ = wi(t)-ws(t) is the depth of indentation, wi and ws are the displacements of the impactor and displacement of the point on the mid-surface of the shell below the point of contact, am is the indentation when force reaches the maximum value fm and a0 is a constant. The expression for k is

=

+_--

/.

E,

+

(4)

Here, ri, Vi, and F.,i are the radius, the Poisson's ratio, and the foung's modulus of the impactor. E2 is the modulus of elasticity transverse to the fibre direction in the uppermost layer of the composite laminate. The displacement of impactor at time step n+l is determined by Newmark's integration scheme for the differential equation (2), as

r

~,

r ,,t2 ]~ ~

(w,).+,: (w,). +,,,(,,). + t.Tj~w,). -t~J"

~"+'

(5)

141 This expression for displacement of the impactor is substituted into (3), yielding, for the time step n+l,

(Fc).+ ,

=k[q-(w.).+,- fl(F~).+,]

'5

for the loading phase

(Fc )n+ ! -" ki [q - (Ws )n+ 1 - i~(Fc )n+ 1 -/2'0] 2"5

(6)

for the unloading phase.

In these equations , , 8 = At ~ / 4m i and

q = (w,). + A t ( w i ) . + (At 214)(w~).

k, = F . / ( a m - Cto~ 5

(7)

The above equations for the contact force are solved by means of the Newton-Raphson root finding algorithm in conjunction with the solution for the dynamic response of the shell. Since the expression for Fr contains Ws which is not known at the beginning of any time step an iterative solution procedure has to be used. Starting with an aoproximate value of Fe, Ws is found from equation(I). With this value of Ws, Fe is recomputed using (6). The process is repeated till the required accuracy is achieved. Only 3 to 4 iterations were required for convergence. The contact force is now used to calculate acceleration, velocity and displacement of the impactor and applied on the shell for the next time step. The computer program for the dynamic analysis of shells by Hwang[8] deploying the 9 noded isoparametric shell element of Abroad which has 5 degrees of freedom at each node has been modified to work for the impact loading case. 3. IMPACT INDUCED DAMAGE Impact induced failure mechanisms are complex ;,~d predicting and relating the extent of damage with the problem parameters has been attempted by the finite element procedure notably by Choi and Chang[3,4]. The authors have given two failure criteria, one for matrix cracking and another ibr delamination failure. They have also suggested empirical constants, based on impact experiments on composite plates, to modify these failure criteria. It is assumed that these failure criteria are applicable to laminated composite shells as well. The criteria are: a) Matrix cracking criterion:

t,"r)

+

t,"s,J

e u >_ 1

= eM

Failure,

e M <_ 1

NoFailure

(8) if o n' > 0,

"Y="Yt

"Y="Yt

if o n" > 0

b) Delamination criterion Oa

El-.// n S rz ,

2

+

n + 1~..XZ ,, +l S i

n

e D

"+'Y="+iYt

/

2> - 1

+

n +I'~FY yy''

Failure,

if o'rr > 0,

-. e D

en

2

2

< 1

"+'g="+iYc

(9) No Failure if O'rr < 0

142 Details of these two formulae are given in [8]. Whenever the average stresses in any of the plies in a laminate first satisfies the criterion (eM>l) during impact, initial impact damage is predicted. It is assumed that the crack would propagate throughout the thickness of the ply group that contained the cracked ply and initiate delamination at the interfaces with the adjacent ply groups. The two failure criteria are applied at all the points where stresses have been computed in every time step. In order to modify the stiffnesses of the failed laminas, a reduced compliance matrix is used post-failure analysis. The reduced compliance matrix has, along its diagonal, Ex, Ez, Gxz, and Gxy as the only non-zero elements of the matrix. This matrix has been currently incorporated into the computer program with modifications to account for the reduced degrees of freedom. 4. RESULTS The problem solved is a 4-layered Graphite/Epoxy 0/90/90/0 symmetrical laminated composite cylindrical shell panel clamped on all four edges with the following properties: Chord length a=b= 0.254m, thickness = 2.54x10-3 m. The elastic properties of a lamina are: E1=144.8GPa, E2= 9.65GPa, G12 = 7.10 GPa, G~3 = 7.10 GPa, G23 = 5.92 GPa v:2 = 0.30, p= 1389.2 kg/m 3, c~0= 8.03 x 10-2 mm. The composite strengths are as given in [8]. The impactor is a sphere of diameter, 12.7ram and its weight is 0.08 kg. A time step of 9.5 x l 0 .7 sec has been used. 4.1. Parametric studies The effect of impactor's mass and velocity, and the curvature of the shell on the contact force history and the resulting damage( the extent of damage was noted by the number of points where matrix cracking and delaminations occurred, not shown in this paper) were obtained by varying a) the mass by a multiplying factor, keeping the curvature and velocity constant. b) The velocity at 10,20, and 30 m/see c) The curvature by varying the ratio of radius to chord length(a=b) as R/a=5,10,100 d) The ply orientation. The results are depicted in figures 2 to 5. From Fig. 2 , it appears that increase in mass leads to longer contact duration before the first separation of the impactor and the shell. The number points where matrix cracking and delamination occurred increased significantly due to input of larger amount of energy. Matrix cracking failure occurred mostly in the ply furthest from impact location and delamination at a number of points between this and next upper ply. The maximum contact force does not however proportionately increase with the increase in mass. This may be because the shell deflection has also increased which in turn reduced the increase in the approach of the two masses. Figure 3 hows the effect of the curvature on the contact force -time history for a selected mass and velocity. It appears curvature has less influence on the contact force-time in as far as the maximum contact force is concerned. The displacements experienced by the shell are however different. The extent of damage also increased with the decrease of curvature. The decrease in curvature has made the structure more flexible thereby increasing bending strains leading to higher damage in the form of matrix cracking in the layer at the bottom of lamina-stacking. The effect of velocity had on

143

/'1 ~llll

~,::Orr

Fig. 1. Composite shell geometry and impactor. R2 =oo

1250 ,

-.--- .

1000

3 x imp. mass 2 x imp. mass 1 x imp. mass

2oool ~

750 5OO

1000(,

25O 0

IrrTL velocity = 30 m/sec, Irrp. velodty = 20 nYsec, Imp. velocity =10 nYsec

m

400

0

6oo

800

looo

TIME ( X 9.5 E-07 SECONDS ) Figure 2. Effect of impa~or's mass on contar force

2000

. . . . .

2O0

c -;

400

..................

1

= R/a=10 = Rla=100 ~ Rla=5

A

TllVE(Xg.SE~7~) Rgure 3. Effect of irrpactor's velocity on contact force

2000~

1500

nl90190/0 0145145/0

1500

1000

1000

500

0

0 0

500

_ ~

0

-

'

' 200

400

600

800

1000

TIME ( X 9.5E-07 SECONDS ) F i g u r e 4. E f f e c t o f c u r v a t u r e o f s h e l l o n c o n t a c t f o r c e

0

200

400

600

800

1000

TIME( X 9.5 E-07 SECONDS) F i g u r e 5. E f f e c t of p l y o r i e n t a t i o n o n c o n t a c t f o r c e

144 maximum contact force is shown in Fig. 4. To study the effect of ply orientations, the inner two plies were changed to 450 angle with respect to the outer plies and the results are shown Fig. 5. The number of points where matrix cracking failure occurred increased from 4 to 10 at the bottom most ply. This may be due to the fact that the bending stiffness is lower for laminate with 450 plies. 4.2. Other observations

The contact law implementation poses some problem since its difficult to decide when loading ends and unloading begins. We have assumed that a switch over from loading to unloading takes place as soon as the contact force begins to fall for a short time, i.e. local peaks in force time diagram have not been ignored. Different results will be obtained if loading law is used, ignoring the local peaks, till the maximum value is reached. 5. CONCLUSIONS A finite element code to model dynamic behaviour and subsequent damage of a composite shell subject to impact loading is successfully implemented. The stiffness of the failed laminas is modified to account for their lack of contributions in appropriate directions in each time step during impact. It is shown that the degenerated shell element provides sufficient accuracy for use in impact-damage analysis. Parametric studies involving the effect of mass, curvature, ply orientations, etc. on the impact response has been shown. REFERENCES

1. S. Abrate, Impact On Composite Structures, Cambridge University Press, NY, 1998. 2. T.M. Tan and C.T. Sun, J. App. Mech 52, 6-12 (1985). 3. Choi and Chang, J. Composite Materials, 25(1991), 992. 4. Choi and Chang, J. Composite Materials, 26(1992), 2134. 5. R.L. Ramkumar, and Y.R. Thakur, J. Engg. Mat. & Tech., 109 (1987). 6. Christoforou A.P. and Swanson S.R., J. App. Mech, 27(1990) 376. 7. K. Chandrashekara and T. Schroeder, J. Composite Materials, 29(1995).2160. 8. H.C. Hwang, Static and Dynamic Analysis of Plates and Shells, Springer-Verlag, Berlin, 1989.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

145

Dynamic testing of energy absorber system for aircraft arrester K.K. Malik, P IK. Khosla, P.H. Pande and R.K. Verma Terminal Ballistics Research Laboratory, Ministry of Defence, Sector- 30, Chandigarh- 160020, India

Energy absorber is the heart of an aircraft arrester barrier system and is mainly responsible for absorption of the kinetic energy of the trapped aircraft. An innovative methodology has been used to test the rotary energy absorber system independently by simulating the dynamic conditions of sudden loading. A special assembly and rocket motors were used to generate the desired rotary motion. The performance evaluation of the energy absorber under dynamic conditions was observed and related parameters were monitored. The details of conducting tests and their results are presented in this paper.

1. INTRODUCTION An aircraft arrester barrier system is used to engage a fighter aircraft to halt its forward momentum in the event of an aborted take off or landing overrun with minimal damage to the aircraft or injury to the crew. It consists of an engagement system comprising of multiple element net (MEN) to envelop the aircraft during its arrestment, stanchion system to provide support and remote controlled operation for erecting and lowering the MEN assembly, purchase tape and the energy absorbing unit. The momentum of an engaging aircraft is diluted through various other mechanisms such as net engagement, stretching of nylon tape, friction of tyres of aircraft and air drag on it. Yet, the rotary energy absorber device (READ) gradually absorbs the major portion of the kinetic energy of an aircraft during the run out distance. The connection of the READ with the net is made by means of nylon tape. One end of the tape is attached to the net through a tape connector while the other end is wound on the tape drum of the energy absorber. The rotary energy absorber device is tints the heart of an aircraft arrester barrier system, which is mainly responsible for absorption of the kinetic energy of the trapped aircraft. Hence the READ is designed to sustain the sudden loading [2].

2. ENERGY ABSORBER SYSTEM DETAILS The energy absorber is a turbine type rotary hydraulic device causing fluid turbulence, used for absorbing the kinetic energy of the aircraft. It consists of a tape drum and vaned rotor, both splined to a vertical common shaft. The shaft of the rotor is supported by two

146 bearings, one mounted in the housing cover and the other in the bottom of the energy absorbing device. The rotor, which produces fluid turbulence, has nine tapered radial vanes on top and bottom surfaces. The tapered stator vanes are welded to the bottom side of the energy absorber housing cover. There are identically tapered stator vanes welded to the bottom of the housing. These static vanes contribute to impart resisting torque to the rotating shaft. An aircraft, having engaged the MEN, exerts a pull on the nylon tape, which is spiral wrapped on to the tape drum. As the tape drum and the rotor in each energy absorber are connected to the vertical rotor shaft, the tape drum and the rotor rotate as a unit. The waterethylene glycol mixture in the housing is agitated due to the rotation and interaction between the rotor vanes. The kinetic energy of the aircraft is thereby absorbed by the fluid

inside the housing by the work done against the resisting torque of the vane system. During the process of development of the energy absorber, it is mandatory to subject it to the dynamic testing to establish the strength and integrity of its various components against sudden loading. The design requirement of the READ is that it should provide the retarding force to absorb aircraft energy with in 275 m runout distance without any failure. For the performance evaluation of such a system, ideal testing would be to engage an aircraft with specific mass and speed but considering the safety aspects of pilot and aircraft, it would not be possible to perform such test. The alternate approach is to

simulate the conditions of sudden d),tutmic loading on the energy absorber.

3. PEAK FORCE CALCULATIONS During the dynamic test of energy absorber, peak force on the READ was simulated. The theoretical estimation shows that the total torque produced by the set of nine rotor vanes is 9.18 r where co is the rotational speed in rad/sec [1].The rotor can be rotated at the maximum angular velocity achieved during the aircraft engagrnent.The quick angular speed on the energy absorber was achieved by fixing two supporting arms, mutually perpendicular, on the rotor flange and rotating them with the help of suitable number of rocket motors, fixed at ends of each arm. Tmax = 9.18 o2,

where m is angular speed.

(1)

Under the extreme conditions of loading, when an aireratt of maximum mass 30,000 kg engages the system at speed of 275 kmph and with run out distance 275 m, the value of is about 104 rad/sec. Thus for the extreme loading conditions, Thrust required =

9.18 x 104 x 104 = 99291 Nm

Since two rotating couples, mutually perpendicular were planned, the thrust required per couple is 49645.5 Nm. For an arm length of 4 m, force required is 12411 N. Considering rocket motors with an effective thrust of 4120 N, three rocket motors were found adequate at each arm.

147 When the energy absorber system is subjected to rotations, each rocket motor experiences considerable amount of the centrifugal tbrce. This force also needs to be considered while designing the fixture tbr generating torque. Centrifugal tbrce = m r ~:

(2)

The rocket motors with mass 3.0 kg each were used for the test. Thus for radius of 2 m, each rocket motor experienced centrifugal force of 64896 Nm. 4. EXPERIMENTAL PROCEDURE The energy absorber was mounted on a specially laid RCC foundation to sustain the thrust generated due to the ignition of rockets. Figure 1 shows the lowering of the energy absorber in the central cavity of the RCC foundation and Figure 2 is the view of READ fixed with the foundation bolts. The four arms, spaced at 90 ~, were attached to the lower and upper flanges of the rotor to fix the rocket motors. The rocket motors with an effective thrust of 4120 N and burning time of 0.7 sec, were used. The fixtures were designed to withstand the centrifugal forces and the bending forces [3]. Ethylene glycol and water mixture in 6 0 40 ratio was filled in the housing. A temperature sensing element with its digital readout recorded the rise in fluid temperature during the experiment.

Figure 1. The energy absorber being lowered in the cavity of the RCC foundation.

148

Figure 2. View of rotary energy absorber device with pickup coils mounted on the RCC foundation. The energy absorber with the torque generating arms and rocket motors mounted at their ends has been shown in Figure 3. The rotational speed of the rotor of the energy absorber was measured using a magnet mounted on the lower flange and three pick up coils fixed on the foundation, 120 ~ apart. Each time the magnet crossed the coils, a pulse was generated and recorded on a digital storage oscilloscope. A safety enclosure of RCC blocks was erected around the test site. The 'test results have been shown in the Table 1. Table 1. Test results

S.No.

No. of Rockets

Max. RPM

Time taken to Max. RPM (see)

Torque (kN-m)

1.

2

70

......

16.0

2.

8

300

......

64.0

3.

16

492

0.840

128.0

4.

16

491

0.776

128.0

5.

16

488

0.433

128.0

149

Figure 3. Rotary energy absorber device with fixtures for mounting rocket motors.

5. OBSERVATIONS

The energy absorber was subjected to dynamic testing by varying number of rocket motors. The system withstood these tests successfully without any sign of damage or distortions. Though the energy absorber was designed to sustain the dynamic loading of 99.3 kN-m, it could withstand higher thrust of 128.0 kN-m. A temperature rise of 3~ C was recorded inside the fluid. The time taken by the READ from ignition of rockets to the halt was of the order of 65 sec. It included the burning time of 0.7 sec. for the rocket and remaining 64.3 sec. as non-bum time. This time is quite high because towards the end, rotor keeps rotating without much of resistance on the vanes. The RPM recorded and achieved were not quite proportional to the thrust employed. This was attributed to the reduction of thrust of rocket motor due to the air drag on the fixture arms.

6. CONCLUSIONS This method of dynamic testing of energy absorber provides a very reliable and cost effective mechanism to simulate the conditions under which strength and integrity of various parts of the energy absorber system can be tested. Whereas, the major portion of the energy imparted to the system, due to ignition of the rockets, was absorbed by the fluid turbulence inside the chambers of the energy absorber, a portion of it was absorbed due to the air drag on the torque generating arms. Contrast to the time of rotation taken by the energy absorber rotor, in the actual situation it rotates till the whole length of the nylon tape has been pulled out by the engaged aircraft. The energy absorber withstood all the dynamic forces successfully.

150 ACKNOWLEDGMENTS The authors express their thanks to Sh. V.S. Sethi, Director TBRL for his encouragement and kind permission to publish the present paper. The guidance and help rendered by Dr. S.K. Vasudeva is highly acknowledged. The authors also expresses their thanks to Sh. Neeraj Srivastav and Sh. A.K. Tewari for their co-operation in carrying out the experimental trials. REFERENCES [1]

J.R.N. Reddy and A.S. Reddy, "Theoretical Studies and Model Testing of Rotatory Energy Absorbing Device", ADE report No. ADE/Tr/90-86 (a), Jan 1990.

[2]

A.S. Reddy, "Integrated Design of Rotatory Energy Absorbing Device for Launcher", Proceeding of National Conference on Design Engineering.

[3]

N. Jones and J.G. Oliveira, "Dynamic Plastic Response of Circular Plates with Transverse Shear and Rotatory Inertia". J. Applied Mechanics (1980).

[4]

N. Jones and J.G. Oliveira, "The Influence of Rotatory Inertia and Transverse Shear on Dynamic Plastic behavior of Beams". J. Applied Mechanics (1979).

[5]

Stephen Timoshenko, "Strength of Materials". Part I.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

151

Characteristics of Crater Formed under Ultra-High Velocity Impact S.Pazhanivel*, V.K. Sharma Terminal Ballistics Researh Laboratory, Sector-30, Chandigarh -160 020, India. The impact of an ultra-high velocity projectile onto a target produces a shock wave that propagates into both the projectile and the target. Depending upon the amplitude of shock energy the projectile and the target undergo various processes like solid state phase change, melting, vaporisation and explosion. In this paper an attempt has been made to describe the target damage under Ultra-High Velocity Impact (UHVI) conditions. 1. INTRODUCTION Under UHVI, the portions of projectile and the target subjected to the impact energy can melt or vaporise if the fusion or sublimation energies of material exceed. However, the amount of projectile and target that experiences the impact pressure is limited by isentropicpressure release waves that emanate from the free surface of the materials. Thus, target damage under UHVI conditions may involve besides extensive plastic deformation appreciable target vaporisation leading to explosion. In this paper the method of calculating the threshold conditions using semi- empirical equations of state for impact explosion of metals has been discussed and an effort has been made to quantify the crater diameter created by impact explosion of metals. 2. CRITICAL PRESSURE FOR EXPLOSION OF METALS Whenever the initial heating i.e. by work of compression plus shock heating reaches the critical level (heat of sublimation of metal) the metal will explode. The heat of sublimation evidently provides a measure of cohesive energy of substance, since it is the work required to separate the substance into its components of atoms or molecules and place these at infinite distance from one another. Therefore, the critical pressure for explosion metals ' P m ' is given by Pm - 6p/IV[

(1)

as described by M.A.Cook [ 1] Where, c -Cohesive energy of metal p - Density of metal M - Atomic weight of metal Therefore, the threshold compression ratio (x = p/po) for impact explosion of metals have been calculated from Pm= (a~)q(x'/a- In(x) - aq) + cpok(x-l)3/x3

(2)

152 Where, a & k - Metal constants 13- Compressibility of metal at atmosphere pressure po- initial density of metal e- Specific heat of metal x - Compression ratio (p/po) 3. SHOCK WAVE COMPRESSION OF METALS In specifying the thermodynamic state of shocked metals, usually the pressure 'P" and change in internal energy (E-Eo), are specified as a function of the volume compression Xl = (Vo-V)/Vo The measurements of shock front and the shock particle velocities ~ ' and 'u', show that for various strength shocks 'U' and 'u' can be related by the equation for a wide range of values of ' u ' U - - c + su

(3)

Under these conditions, it can be shown immediately from the Rankine-Hugoniot relations that the pressure 'P' generated in a shock of compression 'x~' is c 2 x1

P=

(4) V o (1-SXl) 2

as described by G.B.Benedek [2]. Where, Vo -Specific volume of unshoeked metal & V-Specific volume of shocked metal The increase in internal energy (E-Eo) produced by a compression 'xl' is given by 89c2xl 2 E-Eo =

(5) (1-sxl)2

4. SEMI-EMPERICAL EQUATION OF STATE The theoretical description of hypervelocity impact requires Equation Of State (EOS) that covers a wide range of pressure and temperature Because direct measurements of the EOS are difficult to obtain over much of this range, it is desirable to develop models that do not require extensive experimental data for their calibration In this work a theoretical model has been developed by taking recourse to semi-empirical equation of state to describe the physical phenomenon that usually occurs during UHVI conditions This method also takes into account the elastic interaction of the crystal lattice, thermal vibration of atomic lattice and thermal excitation of electrons, which can not be neglected in any ease for high pressures and temperatures applications The EOS & change in internal energy in additive form A V Bushman et al[3] and L VAI' Tshuler et al[4] P = P.+P,+ Pc E - E o - F~+E, +Ec

are given, as discussed by

153 4.1 Pm E ~ - the heat portion of internal energy, which is the oscillation energy of the particles (atoms) around their equilibrium, position. Heat required for the vaporisation of metals at atmosphere pressure is calculated from QA = C (Tv-To)

(6)

The effect of pressure upon the transformation temperature depends upon the sign of (AV) volume change and enthalpy change (AH). In vaporisation process, the heat of vaporisation is always positive but the volume change may positive or negative. For metallic components crystallising in the close-packed structure, the volume change is positive and so the vaporisation point is increased upon increase in pressure. The increase in vaporisation temperature for a growth pressure 'P' is calculated by (T-TO = Tv (AV) (AP)/L

(7)

Where,

T - Vaporisation temperature of metal under the impact pressure(P) "Iv- Vaporisation temperature at atmosphere pressuregPo) L- Latent heat of sublimation AP -- P-Pc Therefore, when metal subjected to pressure, the heat required ( ~ ) to vaporise the metal is calculated from Pn = 71( c / V ) (T-To+Eo/c)

(s)

En- c (T-To)

(9)

Where, 7r Gruneisen coefficient for lattice To- Initial temperature of metal Eo- Internal energy under normal conditions c- Specific heat of metal 4.2. Pe, Ee are the terms due to thermal excitations of the electrons. By making use of the concept of electronic specific heat coefficient 'b', the pressure and energy can be written as Pe = (1/4) bpo (Vo/V)~ T z

(10)

Ee = (1/2) b (V/Vo)~ T 2

(11)

Where, b- Electronic specific heat coefficient 4.3. P,, Er - are the terms characterising the interactions of atoms at T=0~ represented by a series expansion in the power of the interatomic distance rc~d 1/3

Pc is

154 5 Pc = ~ ai d ]+v3 i=l 5 Ec= 3 V o t e a~/i(d ~a-1) i=l

(12)

(13)

Where, ai - determined from the experimental values of compressibility d = VodN'; Voc- Specific volume at P = 0 & T = 0~ 5. IMPACT VELOCITY The threshold impact velocity for explosion of metals has been calculated from P = poUu

(14)

(ptst- ppSp) ut2 + (ptet + ppep + 2ppspVO ut- (pp%Vl + ppspVi2 ) = 0

(15)

In this equation subscript 't', 'p' represents target and projectile respectively Where, V r Impact velocity of projectile & U-Shock velocity u- particle velocity Solving Equation (15) V~ can be found out. 6. EXPERIMENTATION An experimental study has been carried out to evaluate the effect of impact explosion of metals An aluminium jet (projectile) having tip velocity of 12500 m/sec formed from hollow charge was fired on 20mm thick Rolled Homogeneous Armour(RHA) target Upon the impact, the explosion of aluminum projectile was observed The explosion of projectile resulted to create larger crater diameter than the crater created by normal penetrator (without explosion) on RHA target. The quantitative analysis of crater diameter created by impact e3q~losionvis-a-vis normal penetrator impact is presented in table I. 7. DISCUSSION AND CONCLUSION Threshold conditions for impact explosion of metals (impact pressure 'P', impact veloeityVi, compression ratio 'x') have been calculated by this model and threshold conditions for different metal combinations (aluminium-RoUed Homogeneous Armour; aluminium-aluminium) are presented in the table 2. Experiment conducted (using aluminium projectile and RHA target) has shown the effect of explosion of metal on crater diameter created on the target. The threshold conditions for explosion of metals calculated by the present method has been compared with the other methods in table 3 & 4 and the calculated values are matching with the experimental observations.. The effect of pressure on the vaporisation temperature has been studied. The increase in vaporisation temperatures for the pressures of 219GPA in aluminium and 764Gpa in RHA are 8203~ and 18494~ respectively. The role of heat in the internal energy balance is larger, in the ease of aluminiun and RHA for the above pressures, the thermal energy has become major fraction, amounting to 61% & 70% respectively.

155 Table I 9 Comparison Of Crater Diameter Created By Impact Explosion & Normal Penetrator Impact(without explosion) Projectile

Target

VI (m/see)

Crater diameter assuming no explosion of metals take place . . . . . (mm)

Crater diameter created by Impact explosion (mm)

Al

RHA

12500

56

70

Table 2 : Threshold Conditions For Impact Explosion Of Target Metals Projectile

Target

x=p/po

Al

Al

11818

~ - ~ | ~

Vl (m/sec) 12100

~

~A

E-Eo (MJ/gg) 18.02

4'1

T(~

P (Opa) 219

10450

~ [;~|O]ii~lii

Table 3 9 Comparison Of Energy Required (E-Eo) for Impact Explosion of metals Calculated By The Author With Shockey's Results [5] Metal

.....Author calculation Energy required for explosion metal upon impact (E-Eo) MJ/Kg

.... Shoekey's Results

""AI

Energy caieulated for explosion of metal (QA) MJ/Kg 3.0

Energy required"for explosion of metal upon impact (E-Eo) MJ/K 8 1 5 . 1 5 - 18.12

Iron

2.4

12.00- 24.00

,,

18.02 22.32 ......

Table 4 : Comparison of Threshold conditions (x, Vl) for explosion of metals calculated by the present method with M.A.Cook's method

Projectile .... Target Steel Steel .

.

......

.

AI Iron .

.

M.A.Cook x ....... 1.79 1.79 .

.

.

.

.... VI 13.3 14.70 .

.

"

.

Present method x 1.8:2 1.85 .

.

.

V~ 9.52 12.70

156 ACKNOWLEDGEMENTS The authors sincerely express their gratitude to Mr. V S Sethi, Director TBRL for his keen interest and kind permission to publish this work. REFERENCES 1. Melvin A. Cook, "Mechanism Of Cratering In Ultra-High Velocity Impact",pp 725735,Volume 30,Number 5, Journal Of Applied Physics, May, 1959. 2. G.B.Benedek, '~l'he temperature of shock waves in solids",Gordon Mckay Loboratory, Harvard University, Cambridge, Massachusetts 3. A.V.Bushman, G.I.Kanel', A.L.Ni, V.E.Fortov, "'Intense Dynamic Loading Of Condensed Matter", 1993 Taylor & Francis. 4. L.V.AL' Tshuler, S.B.Kormer, A.A.Bakanova, and R.F.Trunin, "Equation Of State For Aluminum, Copper, and ~ in The High Pressure Re, on", pp. 573-579,Volume 11, Number 3,Soviet Physics JET P, September 1960. 5. D.A. Shockey, D.R. Curran, J.E. Osher and H.H. Chau, "Disintegration Behaviour Of Metal Rods Subjected to Hypervelocity Impact", Int. J. Impact Engg. Vol.5, pp.585593,1987.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

157

Diagnostic techniques for high speed events V.S. Sethi and S.S, Sachdeva

Terminal Ballistics Research Laboratory, Sector 30 Chandigarh- 160020, India Email : root@/lrtbrl.ren, nic. in

The transient events occurring as a result of an explosion are of very short duration~ The measurement of these events require -very accurate and sophisticated instruments having resolution times of the order of fractions of a microsecond or a few nanoseconds. The paper discusses various diagnostic techniques available at TBRL, Chandigarh for studying the salient features of explosion-target interaction The high speed instnnnentation techniques, such as Pin Oscillographic, Fiber optics , Air and Underwater blast, Medium speed photography, Ultra high speed photography and Flash radiography, for measurement of fast events have been described.

1. INTRODUCTION The detonation of an explosive charge converts the original material into gaseous products at very high temperature and pressure. The conversion takes place at a very high speed resulting in the release of high energy. High speed instruments are required for determining the characteristics of detonation and shock waves, the dynamic behavior of structures under intense blast loading and projectile-target interaction involving high strains and large deformations. Various diagnostic methods are employed to record the events occurring at very high speed. In electrical methods, the oscillographic technique is widely used to study the detonation process in explosives, to measure shock Hugoniot parameters in condensed materials and to determine the shape of the detonation wave. In optical methods, fibre optical cables are used to carry the fight signals to recording equipment. In ultra high-speed photography, streak and flaming cameras are used to photograph the transient events. Streak cameras give a continuous record of the event in space and time coordinates whereas framing camera takes discrete photographs. For the flash radiography technique, hard x-rays are used to get the radiographs of the dynamic events at different instants of time or at desired positions in space. The blast associated with the explosion causes damage to structures and installations and is characterized by blast parameters. Free air blast, under ground blast and under water blast are different branches of blast studies where damage criteria is different in each case. A brief description of these techniques with their applications is discussed subsequently. 2. PIN OSCILLOGRAPHIC TECHNIQUE (POT) As the name suggests, in this technique pins of electrical contactor are used as sensors and a high-speed oscilloscope is used as recording equipment. These electrical contactors are in open circuit connected to an R-C network. On arrival of the high an~litude pressure wave, the contactors get closed and R-C network generate very short duration pulses. Each pulse corresponds to the position of respective pin probe. Knowing the distance between two probes

158 and measuring the corresponding time accurately gives the velocity. The following types of studies have been carried out using this technique. 2.1 Determination of velocity of detonation (VOD) and shock Hugoniots Pin probes made up of copper enamelled wire are put at different positions in a cylindrical charge which is simultaneously initiated by a plane wave generator (PWG). By measuring VOD, the other parameters of the explosive can be calculated using the following relations: Pcj = po D 2/(T+ 1) Up-"D/(T+ 1) p = po (T+I)/T C=TD/(T+I) Q = D 2/2(r 1)

(1) (2) (3) (4) (5)

Where Pcj is the detonation pressure corresponding to detonation velocity D, p and po are the densities behind the detonation front and undetonated explosive, respectively. U v is the particle velocity of the explosion products, C is the sound velocity and Q the heat of explosion or chemical energy and ~/= C~ Cv, the ratio of the specific heats - a thermodynamic function. Generally T = 3 is taken for explosion products but its exact value for different explosives can be calculated from the equation given by Kamlet and Short [1] as (6)

),- 0.655 / po+ 0.702 + 1.107po

Table 1 gives the velocities of detonation and pressures Pcj for some of the important explosives. Measuring shock and particle velocities and applying jump conditions [2], other Hugoniot parameters of the materials can be determined. 2.2 Wave shaping studies When an explosive is point initiated, the general shape of the detonation front is spherical. This spherical wave can be modified in a plane wave or a converging wave by employing geometry of two explosive components, multi point initiations or interaction of detonation wave with inert materials. The pin oscillographic technique (POT) is used to determine the shape of the emerging detonation front. A plot between the radial distance of the probe vs. arrival time at the corresponding probe determines the shape of the detonation front. Table 1" Velocities of detonation and pressures of different explosives Explosive

Composition %

Cast .........................Velocity Of

T

Pressure

(C v/C0

(Pcj)

Density

Detonation0~)

gm/cc

kin/see

1.61-1.62

6.9

2.89

19.7

G Pa

TNT

100

Composition-B

RDX-60, TNT-40

1.68

7.8

2.95

25.88

Torpex

RDX-41, TNT-41,

1.81

7.6

3.07

25.69

1.85

7.45

3.1

25.04

Al-18 Pentolite

PETN-50, TNT-50

159 2.3 Other applications Shock attenuation studies (i.e. the decay of shock pressure and shock velocity with the increasing tl-Ackness of the material) can be carried out. These studies help in the development of shock attenuators to be used in various armament stores. The technique can also be used to determine the jet velocity and rate of penetration in the shaped charge studies. Generation of very high pressure of the order of megabars can be generated in targets by the impact of a flying plate propelled by explosives. The velocity of the flying plate and the pressure generated can be measured by this technique [3]. 2.4 Development of new instruments for POT * The R-C network, which generates pulses on making close contact of probes, has been replaced by a digital shock velocity recorder developed by TBRL. Each channel of this equipment senses the event at its input terminal and generates a TTL pulse at its output. The system is not prone to noise and thus avoids spurious triggering of the recorder. . A time multiplexing system has been introduced to avoid the use of numerous cables and sequential mixing of events. 9 Programmable computer based digital transient recorders with sampling rates up to 1 nsec have replaced the old high velocity oscilloscopes.

3. FIBRE OPTICS TECHNIQUE This technique has been developed in which light energy is transmitted instead of electrical signals. Thus this technique is safer than electrical methods which involves current and voltages. Moreover, sensors based on the fibre optics principle can be directly embedded in the explosive or distributed at required points over a 3D geometry of the shaped charge warhead or test sample. Fibre optic cables HFBR 3000(100 micron) and HFBR 3500(1000 micron) have been used in fabrication of these pin sensors. The simplest shock sensor has a small air gap ~ 0.1 to 0.2 mm at the terminal point and is created by using very fine steel capillaries. On arrival of the shock wave, air in the gap is ionised and produces intense light due to the shock heat. The streak record of this type of sensor is shown in fig. l(a) which shows intense line of light corresponding to variable width of air gaps. A small quantity of PETN explosive or argon filled microballoon is placed adjacent to the air gap which acts as shock amplifier. This improves the air ionisation and hence the dynamic optical pulse. Figs 1(b) and 1(e) show these sensors with the response recorded on DSO through shock velocity

Fig-1 (a)

Fig" 1(b)

Fig" 1(c)

Fig" l(d)

Fig" 1(e)

160 recorder and photo diode respectively. Fig 1(d) is a record of fixed air gap fibre optic sensor recorded through a photo diode. Fig l(e) shows the record of a multi-gap pipe sensor developed using perspex spacers and air gaps. 4. BLAST AND DAMAGE STUDIES

Blast studies provide vital information for the design and development of warheads. These studies also play a major role in design and construction of blast resistant structures using innovative concepts of shock absorbing techniques and construction materials. The studies are generally divided into two categories namely air blast and underground blast. In air blast the damage to the target is caused by direct blast from the explosion and subsequent reflection from the rigid mirfaees whereas in underground blast, ground shock and vibrations play important role. In this technique measurement of basic data on blast parameters, transmission of blast wave and interaction of blast with different types of structures and other targets are carried out. 4.1 Measurement of blast parameters

Piezoelectric crystal-baseA blast pressure gauges have been developed at TBRL. Blast gauge having a pile of twelve X-cut quartz crystals as sensing elements is used for blast measurement in the intermediate pressure range of 0.1 to 15.0 kg/cm2 .The sensitivity of the gauge is 100 pC/psi. The blast gauge has streamlined design and produce minimum distortion in the blast flow field around the gauge.. The blast parameters i.e. peak over pressure, positive time duration and impulse of the blast wave are determined at different distances and correlated with the damage to strucUnes. 4.2 Under water blast studies

Under water explosion test facility consists of a tank fabricated from 20mm thick mild steel plates. One third of the tank is embedded in the ground to withstand high pressure. Small spherical charges, up to the weight of 50 gins of explosive are used to carry out the experimental measurements. The pressure transducer is positioned at a required depth and predetermined distance from the point of explosion. Piezoelectric quartz crystal gauges developed by TBRL are used to measure pressures upto lkbar and have sensitivity of 1.5 pC/psi. Tourmaline gauges and PCB gauges are used for measurement of higher ranges of pressure. Digital storage Oscilloscope and progranunable digital transient recorders are used to record the pressure time signatures of the shock wave and bubble pressure pulses. Primary shock wave and secondary shock waves are recorded on microseconds and milliseconds time base respectively. Fig. 2(a) and fig. 2(b) show the pressure -time profile of these pressure waves. Shock energy per unit of the primary shock wave [4] at any radial distance R from the explosion can be estimated from

Fig. 2(a). Primary shock wave (~tsec record)

Fig. 2(b). Secondary shocks (msec record)

161 4nR 2 Es = ~ I p2dt W pw Cw

(7)

Similarly the energy in secondary bubble pulses, Eb can be estimated from the time period of the first bubble oscillation [4] Tb = 1.135 p l~ Ebl/3 / ph5/6

(8)

Here P is the pressure, W the charge weight, Cw the velocity of sound in water, I~ the density of water, Ph is the total hydrostatic pressure at the given charge depth. Underwater technique has been used for studying the following phenomenon 9 Comparison of explosive performance in different types of naval warheads can be carried out 9 Heat of detonation of unknown explosives can be determined. Shock energies in the primary and secondary shocks are estimated. 9 Blast parameters i.e. pressure time duration and impulse of under water explosions can be measured at different distances. 9 Effects of venting of explosion products and optimum depth of explosion for formation of primary and secondary shocks are studied. 5. MEDIUM SPEED PHOTOGRAPHY

Medium speed photographic technique is comprised of FASTEX and HIMAC make cameras with a maximum speed of 16000 pps. The technique is used for studying the high strain rates encountered in shock- structure interaction and projectile- target penetration trials. The technique is used to record strain rates of the order of 10 4 per see and the strain time histories of the loaded structures. The response of a scaled down model of a reactor structure

Fig 3(b) Deformed vessel after trial

Fig : 3(d) Strain- time history

162 subjected to simulated loads of a Hypothetical Core Disruptive Accident (HCDA) was studied. A stainless steel 1.25mm thick right circular cylinder of size 430mm x 370mm was subjected to the load of detonation of 25 gm of pentolite charge kept at its centroid in the fully water filled conditions. The cylinder was rigidly fixed at both ends as shown in fig 3(a). Fig 3(b) shows the view of specimen after the trial. The strain time history induced in the cylinder was recorded using FASTEX camera running at a speed of 2880 pps Fig 3(c) shows the shadow-graph of the expanding cylinder and fig 3(d) the strain time profile of the cylinder. 6. ULTRA HIGH SPEED PHOTOGRAPHY

This is versatile technique employed to record transient events lasting for a few microseconds. In ultra high speed photography, two rotating mirror type of cameras are used for the study of explosive dynamics. The streak cameras, models B&W 770 and Cordin 1360S, take one dimensional continuous photographs. The framing camera model B&W 189 takes two dimensional photographs in sequential order. The minimum resolution time of a streak camera is 10 nanosec and inter frame time of framing camera is 0.81asec.

Fig. 4(a) Streak record showing two slopes

Fig. 4(b) Framing record - AI jet

6.1 Applications This technique is widely used in explosive dynamics and detonics studies, shock wave propagation, hyper velocity impact phenomena and determination of jet characteristics in shaped charge warheads. Typical records of streak and framing cameras are shown in fig. 4(a) and fig. 4(b). Records of streak camera show two slopes in time and space coordinates representing the shock and particle velocities recorded for polypropylene. Framing camera photographs show aluminum jet formed by the collapse of cavity in the dynamic loading of the target material. Tip velocity determined from these records is 5.4 kin/see.

7. FLASH RADIOGRAPHIC TECHNIQUE TBRL is equipped with three channel Flexitron model 730 series X-ray system with operating voltage varied from 150 kV to 300 kV at the maximum output current 1400 A thus giving a peak power of 420 MW. X-rays are generated based upon the principle of field emission. The wavelength of the X-rays emitted at 300 kV is of the order of 0.04/~. High intensity X-ray flashes which are emitted for a fraction of microsecond (0.1 ~t sec) capture the high speed events without causing any aberration. A new addition to the facility is Scandiflash model 450S having four channels with pulse width of 25 nanosec. This technique is very helpful to diagnose the events, which are in contact with the explosive and occur for a very

163 short duration i.e. a few microseconds. Events, which are generally surrounded by explosion products and can not be viewed by ultra high-speed photography, are studied by this technique.

7.1 Applications Some of the studies of interest carded out by flash x-ray radiography are as follows: 9 Hollow charge studies: The studies include the collapse of copper liner, determination of collapse angle I$, formation and particulation of jet, tip velocity of the jet, velocity gradient and interaction of the jet with the target. 9 Explosively formed penetrator (EFP~: It is a projectile of metal/alloy which gets forged under shock loading and it can defeat armour at longer distances. The shape, size and velocity of EFP is determined by flash radiographic technique. 9 Expansion of ~agmenting sheU: Smaller size of fragmenting warhead such as shells and grenades etc are studied to see the pattern, direction, shape and size of the

Fig.5 X-ray radiographs showing expansion of 32 mm fragmenting shell fragments. Fig. 5 shows x-ray radiographic records of 32 mm fragmenting shell at different times and the velocity of expansion determined is 770 m/see. * Scabbing phenomem: When a metal plate or target is shock loaded, some chunks of the metal get detached from the free surface and move with sufficient velocity to completely damage the men/material inside the target. Prirnary and secondary effects in the multiple scabs along with their formation criteria are studied. 9 Wound ballistics: Bullets are fired at different velocities and angle of attacks in gelatin gel to correlate with the damage and cavity formation. Other associated phenomena such as tumbling and retardation of the bullet in gelatin gel can also be studied. 9 ~ h e r aoplieations: There are numerous studies that can be thought of but some important to mention are shock wave studies in opaque media, exploding wire and plasma studies, hyper velocity impact studies by the flying plates.

REFERENCES 1. M.J. Kamlet and J.M. Short, Chemistry of Detonations, VI, A rule for Gamma as a criterion for Choice Among Conflicting Detonation Measurements, Comb and Flame, 38, 221(1980) 2. M.H. Rice, R.G. McQueen and J.M.Walsh, Solid State Physics, 6 (1958) 3. H.S. Yadav, P.V. Kamat and S.G. Sundram, Study of an Explosive-Driven Metal Plate, Propellants, Explosives, Pyrotechniques 11, 16-22 (1986). 4. R.H.Cole, Underwater Explosions, Princeton University Press, Princeton, NJ, 270-285 (1948).

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

165

Shock test and stress analysis o f a h e a v y metal-forge Yimin Wu, Bijan Samali, Jianchun Li, and Steve Bakoss Center for Built Infrastructure Research Faculty of Engineering University of Technology, Sydney P.O. Box 123, Broadway, NSW 2007, Australia

In this paper, the stress field of the upper platen of a Metal-forge was analysed using a piezoelectric accelerometer and strain gauges. The total stress of the upper platen consists of two parts. One is pre-stress, caused by the insertion of the hammer pole into the upper platen (insertion joint), and the other being shock stress, caused by the impact of the upper platen against the forge. It was found that the pre-stress caused by the assembly is a major factor in the initiation of cracks. A three-dimensional elastic finite element program was used to analyse strains of the upper platen. The calculated strains are consistent with the measured strains.

1. INTRODUCTION Some cracks were found in the upper platen of a Metal-Forge which had failed. It was important to investigate this problem in order to reduce the possibility of failure in the future. On-line testing and stress analysis were performed to identity the mechanisms responsible for the failure of the upper platen. The maximum shock power of the Metal-Forge was one hundred ton-metre. Its working principle is shown in Figure 1. Steam energy changes into kinetic energy of upper platen through the cylinder. The piston forces the pole and the upper platen to move. Due to their movement in the opposite direction, the upper and lower platens impact against each other when they hit the forge in this process. The upper and lower platens are subject to a very large shock force.

166 Gas cyl"mder

Pole

i

:

Upperplatent ---A/~ ~ " ' f \~ ~ L e v e r ~..~. ~ ~

\~

Handle

Lowerp l a ~"~'1ii~t [ [~l ~~i~iU,__ !: Oil cylinder -a. i . i

Figure 1.A schematic diagram of the working principle of metal-forge machine 2. TEST METHOD In order to understand the stress state of the upper platen, shock acceleration and strain measurement were performed [1]. 2.1 A c c e l e r a t i o n M e a s u r e m e n t

The block diagram for electronic measurement using a piezoelectric accelerometer is shown in Figure 2. Type YD-12 Piezoelectric Accelerometer

Upper Platen HP9000-320C Computer Workstation

L

F

HP 35665A Dynamic Signal I Analyzer

Type YE5852 Conditioning Amplifier ~r TEAC XR-50C Cassette Data Recorder

Figure 2. Block diagram of electronic measurement with piezoelectric accelerometers

167

2.2 S t r a i n M e a s u r e m e n t The block diagram for electronic measurement with strain gauges is given in Figure 3. Upper Platen And Pole

~

Strain Gauges

HP 9000-320C Computer Workstation Figure 3.

-'~

Circuit

~

DPM-600 Dynamic Strain Amplifier TEAC XR-50C Cassette Data Recorder

HP 35665A Dynamic Signal Analyser

Block diagram of electronic measurement with strain gauges

3. T E S T RESULTS The maximum acceleration and velocity of the upper platen (relative to the ground) are given in Table 1. Four tests were conducted. Impact energy of the platen was varied in an ascending order from test one to test four. Tablel Maximum relative acceleration and velocity of the upper platen Test 1 Test 2 Test 3 Test 4 Maximum Acceleration (m/s 2) Maximum Velocity

78.6

165.8

191.0

572.1

1.69

2.22

2.27

2.92

(m/s)

Point 1

B

Figure 4.

Strain gauges distribution on surface A-A of the upper platen.

168

a(m/~;2) (a)

t(mc)

(b) .

....

--

t(~)

(c) .

Figure 5.

.

.

.

.

.

tlmc}

The variation of the acceleration, velocity and displacement of the upper platen with time

The surface stress distribution in the critical section B-B (Figure 4) is shown in Figure 6. The variation of the acceleration, velocity and displacement of the upper platen with time are shown in Figure 5. The stress distribution at the critical sections B-B and C-C (shown in Figure 6) were analysed. The results for Point 1 at cross section B-B are presented in Table 2.

t (sec)

Figure 6.

Strain distribution on the surface of the critical section B-B.

169 Table 2 Measured and calculated results for point one at cross section B-B Acceleration Shock Pre-stress Total stress Calculated (m/s 2) Stress strain ( MPa ) (MPa) ( MPa ) ~c 78.55• 1.44 24.5 25.9 72.19 165.80x2 3.05 24.5 27.5 152.53 191.00• 3.51 24.5 27.9 175.57 391.40• 7.14 24.5 31.6 359.70 572.10• 10.5 24.5 35.0 525.76

Measured strain l.t6 82.06 161.1 180.6 314.6 579.4

4. D I S C U S I O N S The stress in the upper platen of a given forging press was analysed. The total stress in the upper platen consists of two parts, a pre-stress and a shock stress. One can consider that the pre-stress remains unchanged in the platen after assembly and does not vary with time. According to the specification, the calculated pre-stress is 24.5MPa. From Table 2 it can be seen that the shock stress is only a small fraction of the total stress when the acceleration is relatively low. When the measured acceleration reaches its maximum value of 572.10m/s 2, the shock stress is still less than one half of the pre-stress. The analysis reveals that the pre-stress plays a significant role in the creation and growth of cracks and is the key factor to the upper platen damage. A three-dimensional elastic finite element program was used for the analysis. Considering the symmetry of geometry and load, the finite element analysis of the upper platen takes only one quarter of the platen into consideration. This 88 part is divided into 73 elements and there are 526 nodes. For calculations, the twenty-node equal parameter element is adopted. The calculated strains agree well with the measured ones.

5. CONCLUSIONS The measured maximum acceleration and maximum velocity at test four were 572.1 m/s 2 and 2.92 m/s, respectively. Measured strains agree well with those calculated. Among the stress components contributing to total stress, the prestress is the dominant one. The large pre-stress due to imperfect assembly is one of the major causes for the cracking and damage to the upper platen. An improvement of the connection of the upper platen to the pole should be considered to alleviate the problem.

170 REFERENCES

1. 2.

Kenneth G. McConnell, Vibration Testing, Theory and Practice, John Wiley &Sons, Inc, New York, 1995, p. 9. Anil K. Chopra, Dynamics of Structures, Theory and Applications to Earthquake Engineering, Prentice Hall, Upper Saddle River, New Jersey, 1995.

Blast/Shock Loading

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

173

Air blast simulations using multi-material eulerian/lagrangian techniques John Marco

DSTO, Aeronautical and Maritime Research Laboratory P.O. Box 4331 Melbourne 3001

Numerical finite element techniques are more increasingly being used to simulate air blast scenarios when experimental solutions are not economically possible or could cause safety problems. New techniques have been developed in recent times where by the explosive, air and structure can all be modelled using a combination of multi-materials, Eulerian and Lagrangian methods. An example of this technique using the LSDYNA explicit code will be shown by comparing the results of a field trial on a l m by l m cubic box with the two numerical techniques, the Lagrangian method incorporating externally calculated load curves and the multi-material Eulerian/Lagrangian coupled technique.

1. INTRODUCTION Air blast explosions inside naval warships can cause wide spread and catastrophic damage to the vessel's structure and equipment. Numerical finite element (FE) techniques are increasingly being used to simulate these kinds of loading scenarios when experimental solutions are not economically possible or could cause safety problems. Traditionally, explicit codes using Lagrangian techniques to model the structure and pressure time curves to represent the shock loading have been used. This method provides some insight into the modes of response of the structure, but is limited, in that; the load curves are calculated independent of the response or subsequent failure of the structure. New techniques have been developed in recent times in which the explosive, air and structure can all be modelled using a combination of Multi-Materials, Eulerian and Lagrangian methods. That is, an Eulerian fixed cell system is used to model the air/explosive components and a Lagrangian deformable cell system is used to model the structure. The Lagrangian system is 'coupled' to the Eulerian system within the code packages. The MultiMaterial feature of the code allows more than one material type to be present in an Eulerian cell (ie explosive products and air) and keeps track of each of the volumes for each material during the calculations. The shock wave produced from the explosion interacts with the structure and as it deforms, the pressure loads are redistributed to account for the change in volume and boundary conditions occurring from the resulting deformation of the structure.

174 An example of this technique using the LSDYNA [1] explicit code will be shown by comparing the results of a field trial on a lm by lm cubic box with the two numerical techniques, the Lagrangian method incorporating externally calculated load curves and the multi-material Eulerian/Lagrangian coupled technique. 2. THE FIELD TESTS A series of tests [2] where conducted in which 560 g TNT of explosive charges were placed at the centroid of lm cube steel boxes with 5mm wall thickness, see Figure 1. At the base of each box, a flange 100mm wide and 20mm thick was welded to the structure. The boxes were bolted down onto a concrete platform. Some results from the tests are included in Table 1 showing permanent vertical displacement for three locations on one of the boxes, namely, the centre of a box wall, the centre of an edge between two walls and the comer of the box, see Figure 1 for details. Table 1 Measured and Predicted Values of Vertical Permanent Displacement for an Explosively Deformed Steel Box ~1,2,3j Technique Wall Location (ram) Centre Edge Comer ;rests 118 -25 -6 Lagrangian 140 -42 - 18 Multi-material 115 -25 - 15

Note 1. .

.

FE analysis of permanent deformation values were obtained by extrapolation since simulation times was only 20 ms The centre and edge location values are relative to the comer values, where as, the comer value is relative to its undeformed position A negative value means inward motion whereas, a positive value means outward motion.

3. THE LAGRANGIAN/LOAD CURVE TECHNIQUE This technique models the box structure using one quarter symmetry, see Figure 2. Two dimensional 'shell' elements are used for the box structure and, due to the loading symmetry, fifteen pressure time curves on each one eighth wall panel were used to load all the box walls. These load curves were calculated using the Ray-Tracer program [3]. The code is based upon a source and image technique where an empirical free field explosive source profile is used to compute the pressure time history for the incident wave. At~er detonation of the explosive charge a spherical blast wave is produced which interacts with the nondeforming walls of the structure producing reflected pressure waves. Using a combination of ray tracing techniques to determine a ray path and non linear acoustic addition rules to sum the contribution of all incoming pressure waves at a point, a loading profile was produced for all the fifteen predefined wall locations. The finite element details are shown in Table 2,

175

Edge

Flanged Box

er

Center

Flange Base Figure 1. Box Geometry Charge located at box centroid - 560g TNT

Wall

Fifteen Load Locations

Flange Base

Figure 2. Finite Element Model ~/~Symmetry

176 material properties in Table 3 and the resulting permanent deformation at the three nominated locations are shown in Table 1. 4. THE MULTI-MATERIAL EULERIAN LAGRANGIAN COUPLED TECHNIQUE The multi-material technique models all the components of the scenario, including the box, the explosive and the surrounding air, see Figure 3. Finite element parameters are detailed in Table 2, material properties in Table 3 and the magnitude of the deformation responses for three locations are detailed in Table 1. This technique employs an Eulerian grid system (ie fixed) to model the air and explosive materials. Upon detonation of the explosive, a shock wave propagates into the surrounding air cells. These cells now contain two material types, explosive products and air. The box structure is modelled using a Lagrangian grid system (ie deformable) but is 'coupled' to the Eulerian system. When the approaching shock wave impinges on the box structure the 'coupling' routines transfer load from the Eulerian (ie air/explosive) to the Lagrangian system (ie box) causing it to deform. During this process all of the explosive products remain enclosed within the box structure, unless part of the wall fails and vents the gases. The size of the air model therefore needs to be large enough to surround the peak deformation of the box structure during the simulation. Figure 4 shows a sequence of time deformation plots for the box structure. 5. DISCUSSION The results in Table 1 for the three methods used clearly show that the coupled Eulerian/Lagrangian technique predicts responses similar to those of the tests and better results than the Lagrangian load curve technique. The major difference between the Lagrangian and Multi-Material techniques lies in the size of the finite element models, the preparation and execution run times. The Lagrangian load curve model is about 1/5 the size of the Multi-Material model and takes about 1/20 of the execution time of the Multi-Material model to run. The execution time of the Multi-Material model was about 38 hours on an SGI 1NDIC~: workstation for a 20 ms simulation time. Another factor that needs consideration is the preparation time to get the model up and running. Considerably more effort is required, typically several days for the Lagrangian load curve technique because the 'ray tracer' code needs to be executed first, then pressure data extracted and then formatted for the finite element structural code. This is a time consuming process and is not required if the Multi-Material approach is used. The number of elements required in the finite element model for the air-explosive parts in the multi-material technique needs to be large in order to capture and transmit the shock front. This then implies that the size of the elements for the box need to be similar otherwise numerical leakage of the shock front will occur through the box during the coupling process. Hence similar element sizes are required for the air and box structure.

177

Figure 4. Sequence of Time - Deformation Responses - Multi-Material Method

178 Table 2 Finite Element Para.meters-On.e Quarter Model Technique Component Nodes Elements Lagrangian structure 5876 5636 Multi-material structure 3240 3104 explosive 7681 6000 air 28801 24000

Mass (kg) 129 129 0.14 n/a

Table 3 Material and Equation of State Properties Metal Box Material Properties

Explosive C.harge Material Properties

Elastic Modulus (GPa) 200.0 Plastic Modulus(GPa) 0.05 Poisson's Ratio 0.3 Density(kg/m**3) 7864.0 Static Yield Strength(MPa) 450.0 Dynamic Yield Strength(MPa)600.0

Detonation Velocity(m/s) Chapman-Jouget Pressure(GPa) Density(kg/m**3) Air Material ..Properties Den sity(kg/m* "3)

6930.0 21.0 1630.0 .. 0.1293

6. CONCLUSION The implementation of the Multi-Material technique for solving air blast problems is an effective approach and more efficient than the traditional Lagrangian load curve method. Results from this new technique are comparable with test data and better than the traditional method. The increase in use of CPU time is not a disadvantage as it out weights the reduction in human time required to build and execute the model in the traditional approach. REFERENCES

[1] LS-DYNA USER'S MANUAL, Version 950, May 1999, Livermore Software Technology Corporation, USA. [2] Marco J., et al, Second International LS-DYNA3D Conference, Sept 1994, "Dynamic Deformation Modelling of Box Structures Subjected to an Internal Explosion"

[31

Blast and Structural Workstation Code, Combustion Dynamics, Canada

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

179

Damage evaluation of structures subjected to the effects of underground explosions Rajesh Kumari, Harbans Lal, MS Bola and VS Sethi Terminal Ballistics research laboratory, Sector-30, Chandigarh-160020, India

Abstract The paper presents the analysis of ground shock data and cratering parameters recorded in the instrumented studies of buried explosions of High Explosive charges. In an underground explosion, most of the energy released is irreversibly coupled to the surrounding soil media resulting in the formation of camouflet or crater .A small fraction of explosive energy about 3% results in the generation of strong ground motion in the near region. The cratering parameters and the ground shock coupling strongly depends on the depth of burst. The strong ground motion in the near region of under ground explosion results in vigorous shaking of buildings. The ground shock parameters in the near region i.e. from 2 W 1/3to 15 W ~/3 has been monitored using piezoelectric accelerometers and electrodynamic geophones .W is the explosive yield in Kg of TNT. The ground shock attenuation scaling laws have been determined for a typical alluvial soil. The paper further discusses the interaction of strong ground motion with building Structures. The key parameters of ground motion have been related to the damage. The damage correlation in terms of ground particle acceleration and ground particle velocity has been determined for various levels of damage to different categories of structures. The threshold level of vibration has also been determined for the occupants inside the structures. 1. INTRODUCTION In underground explosion most of the energy released is irreversibly transferred to the soil in the immediate neighbourhood of the explosion. In the near region it results in formation of crater or ~moflet depending on depth of burst. At far off distances the stress level in the shock falls below the elastic limit and it degenerates into a seismic wave. The subject of eratering mechanism and ground shock propagation in underground explosion has been studied by many investigators (1-3). In the earlier analysis, the investigators of U~ has used cube root scaling for cratering data of underground explosion (3). However Murphy and Vortman have quoted that extrapolation of cratering results by cube root scaling is not realistic in the case of high explosive yields and these are found to be in excess by more than 50%. A generalised empirical analysis of cratering data has been presented by Violet (4). Violet reported scaling exponents in terms of yield for eratering parameter and depth of burst as 1/3.4 and 1/3.6 respectively for a typical alluvial soil. The paper presents the analysis of experimental data acquired by conducting a number of trials with explosive weight varying from 8-120 kg with different depth of burst. A close agreement exists between the experimental data and empirical relations given by Violet.

180 The paper also presents the experimental technique for generation of ground shock data within the region of 2 W ~/3to 15 W 1/3 metres from the point of explosion. The ground shock parameters have been measured in terms of peak ground particle acceleration and peak ground particle velocity. A Cube root scaling law has been used to develop the statistical correlation between scaled distance and ground shock in terms of particle velocity and particle acceleration. Damage correlation in terms of ground shock velocity for different categories of structures have been developed. Threshold levels of vibration for occupants and structures have been used for developing safe zone for various types of activities. 2. EXPERIMENTAL SET UP AND OBSERVATIONS 2.1 Instrumentation

The instrumentation system used for capturing the ground motion was comprised of piezoelectric accelerometers, electrodynamic geophones and recorders. PCB make piezoelectric accelerometers were used to monitor the ground particle ~ l e r a t i o n at various sc~ed distances varying from 2 W t/3 to 15 W ~r~.These accelerometers contain quartz as a sensing element which produce electric charge proportional to crystal deformation. The response of the ~ l e r o m e t e r is linear up to 1/5th of its resonance frequency. The PCB accelerometers with built in amplifier used have sensitivity of 50mv/g and resonance frequency of 40 KHz. The velocity transducers consist of a permanent magnet which moves up and down within a coil. The sensitivity of the geophone is 200 mv/cm/sec which is nearly constant above resonance frequency (4.5 Hz). Recording System includes magnetic tape recorder, Digital Storage Oscilloscope, thermal array recorder etc. 2.2 Trial Set up Underground trials were conducted to establish the scaling law for a typical type of soil with characteristics shown in Table 1. These characteristics conform to the alluvial type of soil. Empirical relations for ground motion for various kinds of soil have been reported in the literature (5). High explosive cylindrical charges of TNT with weight varying from 8 to 120 kg were detonated at different depth of burst. The high explosive charges were kept at predetermined depth of burst in vertically drilled bore holes which were later on filled with loose soil. The geophones and accelerometers were tightly coupled with the Table 1 Soil Characteristics BULK DENSITY POROSITY COARSE SAND FINE SAND SILT CLAY MOISTURE CONTENT

1.72 gm/cc 38.32% 8.77% 49.38% 22.37% 19.48% 12.0%

to record the time history of the vertical components of ground particle velocity and ground particle acceleration respectively at different locations.

181 2.3 Ground Motion Parameters

In the close vicinity of the underground explosion the ground particle acceleration is of the order of 104 to 105g. As this shock travels through the surrounding soil, it decay fast into complex ground motion. We have used our instrumentation in the region 3 W ~/3 to 15 W ~/3 metres from the explosion point, where W is the explosive yield in kgs of TNT. At distances greater than 3 W ~ metres, the dominant frequency of the ground motion lies between 0.1 to 30 Hz and the maximum ground panicle acceleration is of the order of 2 g. The ground shock study has been done for depth of burst for optimum ground shock coupling. The ground shock coupling factor for alluvial type of soil approaches unity for a depth of burst of 0.55 W l/3 metre and thereafter remains constant for higher depth of burst (5). The statistical empirical relations fitted in the ground motion data of particle velocity and acceleration versus radial distance are =61.77 (R/W " Vwl/~ a = 14.52 (R/Wl,3)-l.s4 1/3)-1 53

(1) (2)

Where V = Peak ground particle velocity in cm/sec R = Radial scaled distance in metres W= Explosive yield in Kg of TNT a = Peak ground particle acceleration in terms of'g' where g is the acceleration due to gravity Figure 1 & 2 shows the relation of experimentally acquired ground shock data with the empirical relation (1) & (2). In figure 3 & 4 typical records of ground particle velocity and ground particle accelerations have been shown which were recorded when two cylindrical charges of TNT having weight 90 & 30 kg were detonated simultaneously with depth of burst of 2.87m and 1.82m respectively.

2.O

i.5

t

1.0

1.5

x- observed points ;

o s

.5

x-observed points

1.0

0.5

.25 9

0

0.5

1.0

1.5

R/W ''~

Figure 1. Scaled distance ( R / W113) Vs Ground Particle Acceleration (a. W va)

_,

0

.

_.

5

10

15

Scaled distance (R/W ''~)

Figure 2. Scaled distance (R/W ~/3) Vs Ground Particle Velocity

182

Figure 3. Typical records of ground particle velocity at a distance of 40, 75 & 100 m from the point of explosion. X axis 1 cm = 100 msec; Y axis 1 cm = 1cm/sec; M=Magnification Factor

Figure 4. Typical records of ground particle accelerations in the near vicinity of explosion, X axis 1 cm = 10 msec Y axis 1 cm = 20 g, M=Magnification

2.4 Crater Parameters A Number of trials have been conducted to yield the crater of different dia and depth by varying the blast size at different depth of burst. The experimental data of the crater radius and depth have been plotted (Fig 5 & 6) and found to be in close agreement with the empirical relation given by Violet as below:

R, / W

2

=0.61 +0.72(H/W1/36)-O.18(H/W'/36)

TM

- 0.11(H / W';36) 3 R: /W

TM

(3)

=0.177 + 0.63(H/W'/36)-O.20(,H/W1/36):

-0.13(n/W~/36)3

(4)

Where R~, R2 & H are the apparent crater radius, apparent crater depth and depth of burst in metres. W is the explosive yield in kg of TNT. x

9

?

1.O

x-observed points

x-observed points

T

z m m 0.5

apo

0.5

0

0.5

1.0

H I W I/34

1.5

2.0

Figure 5. Scaled crater depth of burst (H/W 1/36 ) vs scaled crater radius

(RIfW'I/3"4)

2.5

0.5

1.0

H I W I:~6

1.5

Figure 6. Scaled crater depth (H/W 1/3"6) vs scaled crater apparent depth (R2/W1/3"4)

183 A high degree of correlation exists between observed and computed values. The correlation coefficient r--0.95 for crater radius & 0.82 for crater apparent depth. The explosions are found to be contained if the depth of burst is increased beyond 2.3 W 1/3.6 metres and underground explosion yields the optimum crater parameters if the depth of burst lies between 0.857 W 1/3.6 and 1.029 W 1/3.6 metres. A 90 kg charge yields a crater of 7..80 metres dia and 1.70 metres depth whereas 30 kg yield a crater of 5.90 metres dia and 1.45 metres depth when detonated simultaneously at optimum depth of burst. 3. DAMAGE CRITERIA FOR BUILDING & HUMAN BEINGS Personnel and buildings can be represented by a spring mass system of single degree of freedom. Shock & vibration response of structure and personnel can be defined in terms of ground particle acceleration, ground particle velocity and displacement. If the ground vibration is of impact type like ground shock induced by an underground explosion then the ground particle velocity defines the damage criteria. If the ground vibration is of steady state type like continuous vibration induced by machinery and the vibration frequency is less than the natural frequency of the structure than the ground particle acceleration defines the damage criteria. If the ground vibration frequency is dominating than displacement will become the damage criterion (6). The possible damage sustained by structure can be divided into three zones i.e. no damage zone, minor damage zone where formation of new cracks and opening of new cracks and major damage zone where serious cracking occurs without the collapse of structure. For a brick structure, in the no damage zone particle velocity should not exceed 5.08 cm/sec. The threshold level for minor and major damage zones are 13.72 ctrgsec and 19.3 em/sec respectively (6). The Human threshold for ground vibration can be divided as just perceptible, clearly perceptible and annoying. If the peak particle velocity lies between 0.254 mrrgsec to 0.762 mm/sec it is just perceptible for human beings. If it is more than 0.76 ram/see but less than 2.5 mm/sec it is clearly perceptible and if it is more than or equal to 2.5 mm/sec it is annoying (6). 4. SAFETY DISTANCES In order to calculate the safety distance, brick structural targets have been subjected to different sizes of underground blasts. Table 2 below shows the peak ground particle velocity at different scaled distance along with damage description for brick structure. Figure 7 shows a view of damaged brick masonry model which was subjected to a blast of 47.5 kg at scaled distance of 1.82 m/kg 1/3

Figure 7. A view of damaged brick masonry model.

184

Table 2

....P.~..~~d.~c!e.ve!~it.y.. .at ...di.'ff~nt..~~..di~.~ .................................................................................................... Scale distance m/kgI~ peakparticle velocity Damagedescription to structure cm/sec > 14 14to5.57

<0.9 0.9-4.1

5.57-3.52

5 . 2 - 13.5

1.82

24.6

1.82

24.6

<0.5

-

No damage Fine hair line cracks in joints & loosening of brick cement Widening of cracks both horizontally & vertically but structure remain intact. Severe damage to brick building but not total collapse Total collapse Crater formation

5. CONCLUSION 1. Ground Shock Parameters follow cube root scading law and ground shock coupling acquires a maximum at a depth of burst of 0.55 w ~ metres 2. For alluvial type of soil unde~rmmd explosions yield the optimum crater parameter if depth of burst lies between 0.857 w t ~ to 1.029wIt3"6metres 3. The explosions were found contained if depth of burst is increased beyond 2.3w 1/3"6 4. Brick structures are safe at scale distance greater than 14 m/kg ~, suffer minor damage at sc~e distance 5.57-3.52 m/kg l~ and if the sc~e distance is equal to or 1.8 m/kg t~ suffer major damage. 5.

ACKNOWLEDGEMENT

The attthors are thankful to the experimental trial team Shri LIL Jaggi, Smt Parmeshwari Sharma, Shri Pankaj Sajen, Shri Jatinder Pal Singh. Authors are also grateful to the DRDO for granting permission to publish this work. The authors acknowledge with thanks the excellent secretarial assistance provided by Mrs Pankajavally. REFERENCE 1. Nordyke, ]Vii), Nuclear Cralers & Preliminary theory of mechanism of explosive crater formation J. Geophysics 66(10) 3439 (1961). 2. Murphy, BF & Vortman, LJ, High explosive crater in desert Alluvium, Tuff & Basalt, J. Geophysics 66(10) 3389 (1961). 3. Test book of air armament Pt I, Chapter 5, "Craters & Earth Movement". 4. Violet CE, A generalised empirical analysis of cratering J. Geophysics 66(10), 3461

(1961).

5. P r Y i n g of symposium on the interaction of Non-nuclear munition with structure, US air force Ac~emy Colorado, May 10-13, 1983, AD-A132 115 6. A mamml for prediction of blast and fiagment loading on s t m ~ , Southwest research Institute San Antonio, Tx. Page (5.60). Report No. DOFJTIC-] 1268, 1980.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

185

An investigation of structures subjected to blast loads incorporating an equation of state to model the material behaviour of the explosive W. P. Grobbelaar and G. N. Nurick Department of Mechanical Engineering, University of Cape Town, Private Bag, Rondebosch 7701, South Africa.

Numerical modelling on blast loaded structures with particular emphasis on the modelling of the explosive is presented. The modelling is carried out using the generalpurpose finite element program ABAQUS/Explicit. The Jones-Wilkins-Lee equation of state is used to model the detonation and subsequent expansion of the plastic explosive. The model is verified using existing experimental data [ 1], which considered the blast loading of thin circular clamped plates. The plates are loaded by detonation of centrally located discs of explosive. For four loading conditions the trends predicted exhibited satisfactory correlation with the experimental results.

1.

INTRODUCTION

During the past few years there have several attempts to model blast loaded structures, as for example those reported in [ 1,2,3,4]. The emphasis of the analyses has been on the prediction of the response of the structures. The predictions compare favourably with experimental results with regard to the displacement profiles of the structures. However, the loading conditions have been modelled based on assumptions of blast time and constant pressure. In the quest to understand the aspects that are dependent on the loading process, for example material behaviour, it is important to accurately model the blast loading. It is the modelling of the loading process in particular that this paper reports. The ability to model the explosive using an equation of state is assessed by considering its interaction with particular structures. Experimental data exists for blast loaded circular plates, conducted by Radford [ 1]. The plate deformations are used to verify the modelling of the explosive. Experiments [1] are simulated using ABAQUS/Explicit, but unlike the methods in [1,2,3,4] of modelling a pressure loading on the structure the explosive is modelled using an equation of state to determine its material behaviour. In this way the explosive material and its interaction with the structure is investigated. This provides an insight into the spatial distribution of the explosive process and eliminates the need to assume a pressure-space-time distribution for the loading. The Jones-Wilkins-Lee equation of state is used to define the material behaviour of the plastic explosive (PEA).

186 2.

EXPERIMENTS USED TO VERIFY THE NUMERICAL MODEL [1]

The experiments investigated blast loading of thin circular steel plates clamped at the boundary. The loading was provided by the detonation of a disc of plastic explosive (PE4) positioned centrally on the steel plate. Radford [1] used steel plates of test diameter 100mm and thickness 1,6mm. The explosive discs had diameters of 18mm, 25mm, 33mm and 40mm. The height of explosive varied to create a range of impulses causing small plate deformations to plate tearing. A schematic of the experimental load setup is shown in Figure 2.1. Detonation point I

I /F~

~ /Expl~

[

Plate ~

"

_3

Clamping rig

Figure 2.1: Schematic of the plate and explosive

3.

MODELLING

Modelling of the circular steel plates An axisymmetric model of the explosive and plate is developed. The explosive and plate are both modelled using 4-noded bilinear continuum elements through the thickness of the structures. The constraining boundary conditions simulated that of the experimental clamped conditions. This is achieved by eliminating all degrees of freedom for the nodes beyond the clamping radius on the back surface of the plate and all the nodes beyond a plate thickness in addition to the clamping radius on the front surface of the plate. 3.1

The material properties used for the steel plates are: Young's Modulus 210 Gpa, Poisson's ratio 0,3, and density 7850 kg m "3. A bilinear curve is used to approximate the plastic behaviour of the steel, with a yield stress of 194 MPa. Strain rate dependence is accounted for by incorporating the Cowper-Symonds formulation, with the commonly used values for steel, D=40,4 s1, n=5 [5].

3.2 Modelling of the explosive using the Jones-Wilkins-Lee equation of state A function of pressure with respect to density (Hugoniot curve) is used to calculate the pressure state achievable by the material behind the shock during detonation. The function relating pressure to the density used for high explosives is the Jones-WilkinsLee (JWL) equation of state. The JWL equation of state models the pressure generated by the release of chemical energy in the explosive. This causes a body, defined only by

187 the equation of state, to have a hydrostatic strength only. The JWL high explosive equation of state is shown in equation 1,

p=All_ R, PoCO~-R,e~+BIl_ P CO..PR2Po ~ -R:-~+cOp2poEmo (1) where P = pressure, p = density, and Em0 are material constants.

/3, = at density beginning of process, A, B, Rl, R2,co

ABAQUS/Explicit[6]

implements the JWL equation of state in the form of a programmed burn. A programmed burn refers to the material remaining inert until the detonation wave reaches the material point in the explosive. The release of energy is not initiated by the shock wave in the material. The burn speed and the geometry of the explosive thus determines the time to initiation of a given material point. The detonation time is computed by ABAQUS using equation 2,

ty = min[tff ~[(X "P - X ~ ) "(x mp- x ~/C) / a (2) where X mp is the position of the material point, Xrqd is the position of the Nth detonation point, tNd is the detonation delay time of the Nth detonation point and Cd is the detonation wave speed of the explosive. In order to avoid discontinuities the burn wave is spread over several elements in the finite element model. To achieve this a burn fraction Fb is computed using equation 3.

(3) where Bs is a constant that controls the width of the burn wave, le is the characteristic length of the element. The default value Bs=2,5 is used and the value for le is calculated internally; it is the same value used to compute the stable time increment. If the time is less than the detonation wave arrival time tmPd,the pressure at the material point is zero, otherwise, the pressure is given by the product Fb and P as determined by the JWL equation of state (equation 1).

4.

RESULTS

The mid-point displacements and plate profiles, amongst other parameters, were recorded by Radford [ 1] and used in this study for purpose of verification. In addition a qualitative investigation of the explosive process is presented.

188

4.1

Qualitative investigation and discussion of the explosion process The expansion of the explosive and its interaction with the plate, as calculated by the model, are shown in Figures 4.1, 4.2 and 4.3. The model shown depicts the 25ram load diameter with a resulting impulse of 7,7 Ns. The figures illustrate three phases of the explosion processPhase I

The expansion of the explosive from time of detonation to interaction with the plate (5/~ s). Phase II Explosive plate interaction (6-30 ~ s), Phase III Expansion of the explosion from time of separation from the plate to the time of plate equilibrium (40-450 ~ s). The expansion of the explosive is thus defined by three parameters: Position of the detonation point, burn speed of the explosive and the geometry of the explosive. In Figure 4.1 it can be seen that the expansion of the explosive starts at the point of detonation (centre of the explosive disc), i.e. on the axis of symmetry. The front of the initiation of expansion spreads outwards from the detonation point at the burn speed of the explosive, Ca = 8600 ms -I [7,8]. As this detonation front passes a material point, the point begins expansion. In order to avoid a discontinuity, the material points ahead of the burn front are made to experience a pressure. This pressure is calculated using the burn fraction, as discussed in section 3.2. Figure 4.1 shows the expansion of the explosive as a result of its internal pressure only. There are no 'outside' forces governing the expansion of the explosive until the explosive interacts with the plate. It can be further seen in Figure 4.1 that the first point of interaction is on the axis of symmetry. This interaction point is dependant on the geometry of the explosive and the detonation point. Since the geometry of the explosive is simple, i.e. a disc, the dominating factor is determining the first point of interaction in the point of detonation. The first stage is naturally the most energetic, and will have a significant influence on the plate profile. Figure 4.1 also shows how the expansion of the explosive is greater in the vertical (2-axis) direction than the radial (1-axis) direction. This is as a result of the overall geometry of the explosive, and the tendency for the explosive to expand in the direction of the longer boundary. At t=2,5 las the explosive is almost symmetrical about the horizontal axis. This could be attributed to the detonation point being only half the explosive height (approximately 2,5mm) above the original horizontal axis of symmetry for the explosive. The extremely high-speed interaction of the gas with the plate can be seen in Figure 4.2. The boundaries of the explosive interact with the plate at speeds up to 4000 ms ~. Note the severe deformations that the explosive elements undergo. It is at this stage that the analysis is most vulnerable to termination as a result of excessively distorted elements. Manual manipulation of the time stepping is usually employed at this stage of the analysis. By showing the displacement with respect to time it can be seen that the profile during the stages of interaction as shown in Figure 4.2 compares with the theoretical

189 predictions as discussed by Wierzbicki and Nurick [9]. Wierzbicki and Nurick [9] categorised the plate deformation in two phases. During phase 1, the stage of transient deformation, it was predicted that there is a pronounced central bulge compared with the eventual profile at equilibrium, as clearly illustrated in Figure 4.2.

Figure 4.1: Expansion of explosive from time of detonation, t=0, to time of interaction, t=5,0. Units of time are ~ . From Figures 4.1 and 4.2 is observed that the plate-explosive interaction takes place from approximately t=3 las to t=28 lxs, i.e. over a time period of approximately 25 Ixs. The extent of the interaction can be seen not to extend beyond approximately the inner half of the plate. By the time the explosive extends beyond the inner half of the plate, the outer sections of the explosive are already experiencing an upward velocity as a result of the reflection from the plate.

190

Figure 4.2: Explosive-plate interaction from time, t=6,75, to time of separation. Units of time are ~ts.

Figure 4.3 shows the plate moving under its own inertia once the interaction with the explosive is completed. It is noted that the explosive is no longer moving after it has reflected off the plate (last four frames) and the interaction is completed. This is as a result of the artificial boundary conditions being imposed on the explosive. The 'freezing' of the explosive allows the plate to reach equilibrium without the analysis being terminated because the explosive elements have become excessively distorted.

191

Figure 4.3: Expansion of explosive from time of separation, to time of plate equilibrium. Units of time are kts.

Frame 1 within Figure 4.3 shows the plate deformation at t=40 ~ with a central bulge characteristic of phase 1 deformation. The phase 1 deformation is strongly contrasted by the deformation in the final frame showing the plate in equilibrium at 457 Its. The entire plate has deformed showing a distinctive 'three curve' profile as shown in the experimental results [1]. The 'three curves' relate to the final profile having curvature that changes from concave in the centre to convex and again concave at the boundary. This characteristic is especially notable in the larger impulses of the larger loading diameters. The outer concave section was a characteristic that was unable to be theoretically predicted using the mode approximation technique as used by Wierzbicki and Nurick [9].

192

4.2

Mid-point displacement results for the plates In order to gain an overall perspective of the mid-point displacement results, a plot of the experimental values plotted against the predicted values is shown in Figure 4.4. The data points are concentrated about the line, thus representing a favourable correlation between the experimental and predicted results. 30 E o ex

25

i~

2o

~

O

e~

~

1o

e~ o ~

5

~

0

, 0

5

10

15

20

25

30

Predicted Mid-Point Displacements (mm) O I B m m + 2 5 m m rt33 mm X4Omm l

Figure 4.4: Experimental mid-point displacements vs predicted mid-point displacements for 18mm, 25mm, 33mm and 40mm diameter load cases. 4.3

Plate profiles The profiles shown in Figures 4.5 to 4.8 represent an example of the results for each of the 18mm, 25mm, 33mm and 40mm load diameters respectively. The deformation profiles predicted by the model correspond well with the experimental profiles in all cases.

0.025

0.02 !" "~

~ 0.015

E ~, 0.01

0.005

0

0.01

0.02

0.03

0.04

0.05

Plate RmUm (m)

I--'--M~d . . . . V ~ n m t ! Figure 4.5" Comparison of plate profiles for loading diameter of 18mm and a 7,9 Ns impulse

193

0.016 0.014 ,~ 0.012 0.01 0.008 0.006 "~ 0.004 0.002 0.02

0.01

0.03

0.04

0.05

Plate Radius (m) Model . . . . Experiment ] i

Figure 4.6: Comparison of plate profiles for loading diameter of 25mm and a 5,2 Ns impulse 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

,

,

0.01

0.02

,

,;

,

0.03

,

0.04

0.05

Plate Radius (m)

I

Model. . . .

Experiment I

Figure 4.7: Comparison of plate profiles for loading diameter of 33mm and an 11,4 Ns impulse 0.025 ,~ 0 . 0 2 - " - . . ~ 0.015 g

""

"~''.~.,

O.Ol

i~ o.005 0

'

0

~

,

,

,

0.01

0.02

0.03

0.04

0.05

Plate Radius (m)

I

Model . . . .

Experiment I

Figure 4.8: Comparison of plate profiles for loading diameter of 40mm and an 8,9 Ns impulse

194 5

CONCLUDING REMARKS

While the predicted response of plates and beams subjected to blast loads has been widely reported with generally satisfactory results for large inelastic deformations, the understanding of the tearing mechanism is less satisfactorily predicted. This may be attributed to many factors such as, the knowledge of the structural material properties at these severely high strain rates or the interaction between the pressure wave and the structure. The latter is addressed in this paper, in which a model incorporating an equation of state for the explosive blast was included in the computation. This model predicted similar trends observed experimentally. These results included mid-point displacements and plate profiles. The results of the blast wave expansion provided an insight into the process of the pressure w a v e - structural interaction. These results highlight the advantage of the use of an equation of state in that loading of the structure need no longer be assumed but implemented through explosive-plate contact. ACKNOWLEDGEMENTS The authors gratefully acknowledge the secretarial support of Iris yon Bentheim. REFERENCES

[1] [2] [3] [4] [5] [6] [7] [s]

[9]

G.N. Nurick and A.M. Radford, Deformation and tearing of clamped circular plates subjected to localised central blast loads, Recent developments in computational and applied mechanics, CIMNE, 1997. Pp276-301. R. Bimha, Response of thin circular plates to blast loading, MSc Thesis, University of Cape Town, June 1996. M.E. Gelman, A numerical study of the response of blast loaded thin circular plates, with both clamped and integral boundary conditions, MSc Thesis, University of Cape Town, 1996. W.P. Grobbelaar, Modelling of fragmentation damage, BSc Thesis, University of Cape Town, November 1996. N. Jones, Structural Impact, Cambridge University Press, Cambridge, 1989. Hibbit, Karlson and Sorenson, INC. ABAQUS Example Problems Manual v5.7, 1996. F. Mostert, Personal consultation, 1998 Kamoulakos, V. Chen, E. Mestreau and R. Lohner, Finite Element Modelling of Fluid/Structure Interaction in Explosively Loaded Aircraft Fuselage Panels using PAMSHOCK / PAMFLOW Coupling, Conference on Spacecraft Structures, Materials and Mechanical Testing, March 1996. T. Wierzbicki and G.N. Nurick, Large deformation of thin plates under localised impulsive loading, Int. J. Impact Engng, Vol. 18, Nos 7-8, pp 899-918, 1996.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

195

An UNDEX Response Validation Methodology

James L. O'Daniel a, Theodor Krauthammer b, Kevin L. Koudela b, and Larry H. Strait b a. U.S. Army Engineer Research and Development Center, Structures Laboratory b. The Pennsylvania State University, USA

1. INTRODUCTION AND OBJECTIVES Underwater explosion (UNDEX) tests are costly and hazardous, generating pressure-time load histories that are not well quantified, response data that are difficult to acquire, and the tests may not be repeatable. Nevertheless, a validated approach to accurately predict the short duration dynamic response of a structure subjected to direct UNDEX pressure wave loading does not exist. Therefore, a more cost-effective method to validate the UNDEX response without using the "shock qualification" tests was sought. Krauthammer, et. alo (1996) suggested that precision impact testing can be used to produce peak loads, rise times, durations, and spatial distributions similar to those produced by explosions in air. For this study, the concept of equivalency between in-air explosive and impact loading was expanded to include UNDEX loading. It was postulated that the short duration dynamic response of structures subjected to UNDEX direct pressure wave loading can be reproduced by precision impact loading. Once equivalency is demonstrated, it is proposed that precision impact tests can be used to assess structural integrity of components subjected to direct pressure wave UNDEX loading. This investigation included two important aspects - precision tests and numerical simulations. These were combined to form the unique assessment methodology. Here, a generic UNDEX event is considered, and the concept could be applied to any far-field UNDEX problem. A more comprehensive coverage of this work can be found in O'Daniel (1998). The primary objective of this investigation was to explore the existence of a relationship between the short-duration, dynamic, structural response to UNDEX and to impact. Further, if such a relationship could be found, to develop a methodology to demonstrate an equivalency in structural response between impact and UNDEX testing. The impact tests and simulations were used to produce a response that envelopes the UNDEX response, matching important trends and characteristics. For this study, equivalency was assumed when structural strain time-histories produced by the UNDEX and impact Ioadings were equivalent. The second objective of this investigation involved the validation of a numerical code by precision impact tests. This was an essential part of the UNDEX-impact equivalency methodology. The

196

methodology was developed using an aluminum panel, and then it was demonstrated on a composite material panel to show that the developed procedures are independent of the structural material. Finally, the validated numerical code was used to predict structural responses in future UNDEX tests, and to demonstrate the effectiveness of this methodology by comparison with test data. 2. METHODOLOGY An initial simulation of the UNDEX configuration was developed to predict the short duration dynamic response that was then used to prescribe the parameters of a precision impact test that would produce an equivalent structural response. The most important part of this portion of the process was determining the UNDEX loading environment and, generally, the range of response of the structural component to that loading environment. Similitude equations were used to generate the pressure-time history using the parameters of the explosion. In parallel, the component to be subjected to the UNDEX loading was selected, and a finite element model of it generated. The approximated plane wave loading and the component model were used to produce the numerical UNDEX short duration dynamic structural responses, including strains, strain rates and accelerations. Selecting test parameters and simulating that test to check equivalency to the UNDEX response developed an equivalent structural response. The parameters of the impact test that could be varied were the impact velocity and the characteristics of the impactor, including geometry, mass, mass distribution, stiffness, and stiffness distribution. The iterative process of adjusting the parameters and simulating the new configuration was repeated until the precision impact simulation response and the UNDEX response were equivalent. Once the equivalence was established numerically between UNDEX and impact events, it had to be determined whether these simulations represented reality. A dedicated drop hammer was designed and built to accommodate the impact parameters derived from the numerical equivalency process. Precision impact tests were conducted and the measured responses were compared to numerical simulation results. If a correlation between impact test and simulation did not exist, it had to be determined whether the difference stemmed from inadequacies in the simulations or in the precision tests. Further, any new test and simulation configuration had to be analyzed to determine if the correlated impact response was still equivalent to the original UNDEX response. Once the precision impact test and the numerical simulation produced similar structural responses, the simulated UNDEX event had to be validated with a scaled UNDEX test. This established that the numerical code could simulate the real fluid-structure interaction. The process for the UNDEX simulation was the same as for the precision impact simulation. The UNDEX simulation was generated, followed by the UNDEX test, and their results were compared. The preliminary methodology was completed through the multiple comparisons between simulation and test results for the specific UNDEX and impact configurations. Its assessment was based on whether a satisfactory UNDEX event - precision impact test - numerical simulation equivalency has been established, or if further iterations were necessary. A picture of the impact-testing machine, including an aluminum test panel is shown in Figure 1. A catch mechanism ensured that the impactor was only allowed to strike

197

Figure 1: Test Structure and Aluminum Test Panel

Figure 2: Finite Element Mesh Used for The Structure

the entire face of the structure once. Tests were conducted at impact velocities of up to 4115 mm/s. The same composite test structure was used for the UNDEX tests. Blast and impact analyses share several common characteristics. Inertia effects must be considered for both types of problems. Wave propagation effects are needed to account for the interaction of stress waves between different materials and separate parts of the system. These effects lead to the use of explicit time integration to solve the problem. The explicit finite element code LS-DYNA (LSTC 1994) was validated for short duration dynamic loading by the impact study, and then the UNDEX tests were simulated using the same code. All the simulations - both impact and UNDEX contained three-dimensional solid finite elements and beam elements. The continuum elements were used to represent the test panels, steel support structures, impactors, and the fluid medium. The finite element mesh of the structure is shown in Figure 2. The geometry of the impactor was attained through the determination of the impact parameters; and through the observation of the impact test results. 3. RESULTS

The results of the impact simulations and tests compared well once some modifications had been made to the finite element model. A comparison between the aluminum impact test and simulation of the centerline longitudinal strains on the bottom of the panels is shown in Figure 3. The model captured the general behavior at this position, but over predicted the peak strain. A second hit between the centers of

198

15001

I ~ Test | [,...- Simulation]

_:~5OO

o~

o.~1

o.oo2 o.~3 \ l i ~ o 4 ~ ~ . ' s

.1000.0t |

~'~/-~ Time (sec)

Figure 3: Test vs. Simulation Aluminum Impact Centerline Strain

:e

:.:

/

::

-1500I

/

-2500 0.000

0.001

0.002

Time

...... 0.003 (see)

0.004

0.005

Figure4: Testvs. Simulation Composite UNDEX Centerline Strain

the impactor and target panel was due to the edges of the impactor still being in contact with the panel when the center of the panel rebounded. The strain was under predicted after the second hit, but the simulated strain paralleled the test data out to 5.0 msec. The actual pressure loading experienced by the test structure during the UNDEX test was very close to the simulated pressure loading calculated with the similitude equations. It was estimated from the dimensions of the pond and the acoustic wave speed in water that there would be approximately 2.5 msec of "free" time before reflections entered the test data. The centedine longitudinal strain responses of the test and simulation are compared in Figure 4. Much larger strains were seen initially in both simulations, and the correlation was not that good at this position. After 1.5 msec, both simulations deviate from the experimental results. This was sooner than the anticipated time of approximately 2.5 msec, when reflections were thought to enter the measured data. The correlation was much better away from the centerline, as evidenced by the comparison made at 177.8 mm (Figure 5). The longitudinal strains at the other offset positions reflected the same closer correlation when compared to the center position. The over prediction of strains was partially attributed to the inability of the urethane constitutive model in the composite to dissipate the high frequency energy components. The higher strains in the simulations were also partially due to the difference between the measured transient pressures and those used in the simulations. Both the differences in pressure magnitude, as well as the spatial distribution of pressure over the plate contributed to the discrepancy. The initial and final UNDEX-impact comparisons for the centerline longitudinal strain in the aluminum panel are shown in Figures 6 and 7. Initially, the peak positive strains are approximately equal between UNDEX and impact, and the initial period of

199

I

I

l--

=~[i

Test

1

I....... ~

/;

1"~

~

2600'

3000

=ooo

. ..,...".

i - - UN"DEX1

o

lO

"

-2000 Time (sec)

0.000

0.001

0,002 0.003 Time (sec)

0.004

0.005

Figure 5: Test vs. Simulation Composite Figure 6: Initial UNDEX-lmpact Strain UNDEX Strain @ 11.5" Offset Comparison - Aluminum Panel

4000t

,o~ =

I - - UNDEXI

~

~

/

,......... ,moac,

~~176176 I / ........~ 8.ot / .1"/~,/ '~..... ."

0

-1000

"'"'-

u/Pk~".~ " " " ~

Impact [

,

"l

0.003

"~0.'004"',' 0.005

~-2000 -3000 -4000 -5000

-2000 0.000

~ooo 2000

~ooo ~,1 ~ o

-o

:'.

4000

0.001

0.002 0.003 Time (sec)

0.004

0.005

Figure 7: Final UNDEX-lmpact Strain Comparison - Aluminum Panel

Time (sec)

Figure 8: Final UNDEX-Impact Strain Comparison - Composite Panel

200 response was also nearly equal. The test data exhibits a compressive behavior at 0.5 msec that was not reproduced by the simulation. In the final comparison, the UNDEX strain was much higher than the impact strain. The alterations made to the simulations in order to validate the response against test data worsened the equivalency. According to the methodology, another sequence of impact tests would be necessary with new impact parameters, but those tests were beyond the scope of this investigation. A comparison of the final UNDEX versus impact strain in the composite panel is shown in Figure 8. The peak positive strains compare well, but again the impact did not reproduce the initial compressive response seen in the UNDEX simulation. Additional work is needed for developing a representative constitutive model for both the rubber foam used in the impactor, as well as the rubber that is contained within the composite. 4. CONCLUSIONS The UNDEX-impact equivalency concept was shown to be feasible within the limited scope of this study. The foundational work performed here revealed that the equivalency methodology can reproduce some of the important characteristics of the short duration dynamic UNDEX response, but much more work is needed to determine the extent of the methodology's applicability. The methodology worked significantly better for the composite material. The final strain comparison between UNDEX and impact responses for the aluminum panel was not as good as expected, but this was mainly due to the modifications that were later made to both the UNDEX and impact simulations after the initial equivalency had been determined. The correlation between the precision impact tests results and the simulated results validated LS-DYNA3D for that impact event and range of response. Using the BlatzKo compressive foam for the bumper urethane material model provided the best results of the simple material models used. The correlation of the UNDEX simulations with the UNDEX test results was not as good as was present for the impact simulations, although the correlation was much better for the aluminum panel than for the composite panel. None of the simple material models used for the composite's embedded rubber layers generated a good correlation with the experimental results. A thorough visco-hyperelastic characterization of the urethane layers to more accurately model their behavior could be expected to improve the correlation. Additional work is required to refine the method and further develop its capabilities. REFERENCES Krauthammer, T., et. al., Precision Testing in Support of Computer Code Validation and Verification, Workshop Report, NDCS, May 1996. LSTC, LS-D YNA3D User's Manual 1994. O'Daniel, J., "An UNDEX-lmpact Equivalency Methodology", Ph.D. Thesis, Dept. Of Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

201

The Effects of Local Cavitation and Diffraction on the Underwater S h o c k Response of an Air-backed 2D Plate Structure with Large Deflections L.C. Hammond and C.J. Flockhart DSTO Aeronautical Maritime Research Laboratory, P.O. Box 4331, Melbourne 3001, Australia. The LS-DYNA/USA coupled finite element/boundary element codes are used for predicting the large deflection response of a 2D submerged plate structure subjected to a farfield underwater explosion. The response of the plate and surrounding water is investigated for a range of peak pressures of an incident exponentially decaying acoustic shock pulse. It is shown that the interaction of cavitation and diffraction phenomena result in significant reloading of the plate following reflection of the incident shock. 1 INTRODUCTION Numerical modelling procedures are currently being developed at DSTO-Melbourne to enable efficient simulation of far-field UNDEX (underwater explosion) shock-structure interaction effects on fully and partly submerged naval platforms. The methodology utilises the explicit non-linear finite element (FE) analysis code LS-DYNA [ 1] coupled with the boundary element (BE) code USA (Underwater Shock Analysis) [2,3]. The objective of these studies, in parallel with associated experiments, is to validate techniques for the prediction of hull damage and internal equipment response for RAN (Royal Australian Navy) vessels. Regardless of overall vessel size and hull geometry, numerical prediction of the vessels' response to far-field UNDEX shock relies on the ability of the numerical code to accurately model local details of the interaction of a high frequency short duration acoustic pulse with a compliant air-backed plated structure. In an effort to gain confidence in the validity of FE UNDEX simulation procedures prior to their application to full-scale vessels, the approach taken by DSTO has been to undertake detailed simulation and supporting experimentation on a range of small-scale simple air-backed plate structures [4,5]. This paper reports on the results of LS-DYNA/USA simulations of a simple submerged 2D plate structure subjected to UNDEX shock. The focus of this work has been to develop appropriate modelling procedures and to describe the important loading mechanisms that occur during shock-induced fluidstructure interaction. The aim of this study is to quantify the effects of local cavitation and diffraction phenomena on the loading and subsequent response of a 2D air-backed steel plate supported by a rigid baffle. The model dimensions and shock loading were chosen to correspond with an experimental study reported elsewhere [4]. The loading history and response of the plate are investigated for a range of incident shock pressures.

202 2 THE SIMULATION MODEL 2.1

The Finite Element Mesh

The LS-DYNA/USA model employed in this study (see Figure 1) is comprised of a 2D layer of plate and fluid finite elements (described below) surrounded by so-called DAA (Doubly Asymptotic Approximation) boundary elements [2,3]. The FE and the DAA BE computations are executed by the LS-DYNA and USA codes, respectively. A coupled computation for pressure and velocity at the interface between the FE fluid mesh and DAA boundary elements is achieved by means of a staggered solution procedure [2]. A far-field shock pulse, of magnitude and duration appropriate to the size and location of the explosive charge, is introduced through the DAA boundary elements. The 2D FE model (see Figure 1) consists of a l m deformable length of non-linear 4 node steel plate elements supported on each side by a l m length of rigid baffle plate elements. The effective boundary conditions of the deformable plate are thus built-in. A block of 8 node hexagonal solid fluid (water) elements extends outward from the plate to a distance of 1.5 m in the positive x-direction. A one-to-one correspondence of structure and fluid nodes is employed at the structure-fluid interface. Plane strain boundary conditions were applied to the FE model in the through thickness direction. The model dimensions were chosen to approximate a "slice" through an air-backed plate supported by a rigid box structure being used in a related experimental study [4]. The deformable target and rigid baffle plate elements were modelled using the LS-DYNA non-linear Hughes-Lui shell element formulation and Material-3. This material implements a bi-linear elasto-plastic constitutive law with a constant tangent hardening modulus [ 1]. The Material-3 parameters used for the plate elements are: Plate thickness: Density: Young' s Modulus: Poisson's Ratio: Yield Stress (static): Tangential Modulus:

6 mm 7850 kg/m 3 211 GPa 0.3 411 MPa 500 MPa

The fluid element mesh (see Figure 1) was divided into two parts. For the part directly adjacent the plate elements, an Arbitrary Lagrangian Eulerian (ALE) formulation was implemented. This was necessary in order to limit fluid element distortion arising from large deflections of the plate. An Euler formulation was used for the remainder of the fluid mesh. The entire fluid mesh was modelled using LS-DYNA Material-1, which is a linear fluid model with constant density and bulk modulus (and thus constant wave speed). Material-1 was chosen as it had previously been found [5] to be the most convenient material currently available in LS-DYNA for modelling far-field underwater explosions, since it incorporates the option of a simple, constant-pressure, cavitation model. This model prevents pressure from dropping below a user defined cavitation pressure, so that when the total pressure (ie shock overpressure plus hydrostatic pressure) in an element falls to the cavitation pressure, the element pressure is held constant at the cavitation pressure until further pressurisation occurs. A cavitation pressure of zero was implemented in this study since the cavitation pressure of water (approximately equal to the water vapour pressure) is known to be only a

203 few kilopascals over a considerable temperature range [6]. More sophisticated cavitation models have been proposed which take into the multi-phase nature of real cavitated flow by implementation of non-linear equations of state [7]. However the simple bi-linear constant pressure cavitation model available in LS-DYNA was considered sufficient for the purposes of the present analysis. The Material-1 fluid parameters used in this study are:

Figure 1 (a) The 2D FE/BE simulation model showing the deformable target plate, the adjoining fluid mesh and the DAA boundary elements, and (b) showing an enlarged view of the fluid region adjacent to the plate. 3 SIMULATION PROCEDURE 3.1

General

A series of UNDEX shock simulations were performed on the model shown in Figure 1. In all cases the plate and fluid geometry and material properties were held fixed with only the magnitude of the applied incident shock pulse varied. An exponentially decaying U N D E X shock pulse of the form, P(t) = P0ev~ was introduced through the DAA boundary where Po is the peak overpressure and 0 the decay constant. Simulations were performed for a range of peak pressures, P0, from 0 to 10 MPa, with the decay constant 0 = 0.1 ms held constant. The source of the pulse is assumed to be an explosive charge located at a very large distance along the target plate's normal, ie in the x-direction. The large stand-off distance of the charge ensured that curvature of the shock front was negligible, thus approximating plane-wave conditions incident upon the plate. The inclusion of hydrostatic pressure is an important consideration in UNDEX shockstructure interaction problems since it directly effects the onset of cavitation, particularly at

204 larger depths. Prior to the introduction of shock loads, the FE model was hydrostatically initialised to a pressure of 0.1 MPa corresponding to a depth of 10 m.

3.2. Hydrostatic Initialisation Procedure Due to the considerable time required to achieve hydrostatic equilibrium, it is appropriate to perform a separate hydrostatic initialisation simulation prior to performing the shock simulation. The hydrostatic initialisation run only requires the LS-DYNA code, since no DAA boundary is required. This run involves applying the hydrostatic pressure value, Phya, to the fluid and allowing the simulation to continue until all physical parameters (including total kinetic energy, displacements, strains and pressures) converge to constant equilibrium values across the entire model. The structural displacements, stresses and strains and the fluid element pressures (equal to Phya) are then used as initial conditions for the shock analysis run. For the present analysis where Phyd = 0.1 MPa, 250 ms of real time simulation was required for the initialisation run. The centre of the plate converged to a displacement of 15.35 mm.

4 RESULTS The same qualitative response behaviour was observed across the range of peak incident pressure values, Po, investigated in this paper. Figure 2 shows a chronological series of greyscale pressure contour plots which highlight the salient features observed in the early-time response for the case of Po = 3.0 MPa. The plane-wave incident shock pulse arrives at the plate at t = 1.0 ms. Figure 2a corresponds to a time (t = 1.121 ms) just after reflection of the incident shock from the plate. Early development of a cavitation zone is visible along the length of the plate behind the reflected shock front. At t = 1.734 ms (Figure 2b), the cavitation zone has fully developed into a semi-circular shaped region adjacent to the plate. Due the presence of the rigid baffle, a build up of diffracted pressure around the edges of the deforming plate is also evident at this time. The diffracted pressure tends to surround the cavitation zone, "linking-up" with pressure from the tail of the reflected wave. The diffracted pressure continues to propagate towards the centre of the plate (Figure 2c), progressively closing the cavitation zone. At approximately t = 3.294 ms (Figure 2d), a high pressure focusing of diffracted pressure occurs adjacent to the centre of the plate completely collapsing the cavitation region. The focused diffracted pressure then behaves in a manner similar to a point source, radiating a spherical wave and subsequently reloading the plate. Figure 2e (t = 3.569 ms) corresponds to a time just after reloading of the plate. A cavitation zone is evident in Figure 2e adjacent to the plate due to reflection of the reloading pulse from the plate. Subsequent closure of this cavitation zone is observed in at t = 3.845 ms (Figure 2f). At this time the spherical wave front originating from the diffraction focal point, propagates away from the plate. The pressure-time history for fluid element 2504, located adjacent to the centre of target plate (Figure 1b) is shown in Figure 3 for the case of Po = 3.0 MPa. This pressure history is a direct indication of the loading history on the plate. The principal features of the loading history are indicated in Figure 3 over the first 5 ms of response. These features can be correlated with the pressure contour plots of Figure 2. At tear = 1.12 ms (Figure 3) the incident shock pulse of magnitude Pi, is cut-off by the reflected wave from the plate. The

205 resulting primary cavitation (see Figure 2a-c) persists for a period, At~av = 2.09 ms, until the arrival of the cavitation closure pulse of magnitude Polos (see Figure 2d). Reflection of the closure pulse from the plate results in a brief secondary cavitation in front of the plate (Figure 2e) followed by a weak secondary closure pulse. Figure 3 indicates that significant shock loading of the plate effectively ceases after arrival of the primary cavitation closure pulse. The plate centre reached maximum displacement at tdmax "- 4.59 ms beyond which time the plate rapidly settles down to a damped fundamental mode, the frequency of which is apparent in the late-time pressure oscillations about the hydrostatic pressure (refer Figure 3 inset). Table 1 lists results obtained for the range of applied peak overpressures, P0, used in this study. From Table 1 it can be seen that the time corresponding to the onset of primary cavitation, t~v, is approximately equal to the time corresponding to maximum velocity, tvmax, at the centre of the plate. Furthermore, tmax and tcav are independent of Po. These results are consistent with simple Taylor plate theory [8], which predicts that for an infinite flat plate the time of maximum "kick-off" velocity is coincident with the time of cavitation onset. Taylor plate theory also states that for a given plate thickness and density, the latter time is dependent only on the decay constant, 0, of the incident shock pulse. Selected results from Table 1 are plotted in Figure 4 as a function of P0. The peak mid-plate velocity, Vmax,and displacement of the plate, dmax, are shown to increase linearly with increasing P0. The magnitude of the cavitation closure pulse, Pc~os, also increases approximately linearly with P0. The duration of cavitation, A~v, increases with increasing P0 to approximately P0 = 5 MPa, then tends to an asymptotic value of Attar ----3.7 ms. Table 1 Summary of simulation results. Po PI Pclo~ tr dmax tdmax V m a x

(MPa) (MPa) (MPa)

(ms)

tvmax

(mm) (ms) (ms"1) (ms)

1 0.69 0.59 1.97 0.98 2 1.26 1.17 2.64 2.13 3 1.84 1.54 3.28 3.35 4 2.40 3.00 3.78 4.61 5 3.00 4.26 4.10 5.97 6 3.55 5.37 4.36 7.29 7 4.17 6.09 4.49 8.39 8 4.73 6.87 4.68 9.63 9 5.25 7.89 4.77 10.74 10 5.79 8.73 4.78 11.79 e1:2504: element 2504 shown in Figure 1 Pi : peak incident pressure at element 2504 tclos : timecorresponding to Polos tdmax : time corresponding to dmax tvmax : time corresponding to Vm~ Attar : durationof primary cavitation at e1:2504

4.58 4.57 4.59 5.01 5.35 5.47 5.50 5.56 5.58 5.61 P0" Palos : dmax : Vmax: tear : fpl :

tc~

Atcav

fpl

(ms)

(ms)

(Hz)

0.84 1.12 1.19 0.74 31.4 1.53 1.12 1.19 1.45 31.4 2.21 1.12 1.19 2.09 31.4 2.90 1.12 1.19 2.59 31.6 3.58 1.12 1.19 2.91 31.3 4.31 1.12 1.19 3.17 31.1 5.00 1.12 1.19 3.30 30.8 5.67 1.12 1.19 3.49 29.7 6.35 1.12 1.19 3.58 28.7 7.01 1.12 1.19 3.59 27.9 peak overpressure of applied shock peakcavitation closure pressure at e1:2504 max.displacement at plate centre max.velocity at plate centre time of onset of primary cavitation at e!:2504 residualfrequency of plate vibration. ,,

,,

,,

206

Figure 2 Pressure-contour diagrams at successive times following the development and subsequent closure of the cavitation zone for P0 = 3.0 MPa. Deflections have been magnified by a factor of 50. Light and dark shading correspond to low and high pressures regions, respectively. The time shown for each plot is the time from the beginning of simulation (ie t = 0) when the shock pulse is introduced through the DAA boundary.

2.0 a.. = 2

2,0

incident shock pulse, Pi cavitation closure pulse, ~ ~ , , ~ "

1.5

cavitation CUt-Off

1.0~

I

(n

o. 0.5

/

o

.

.

.

.

.

n.s

0.0 *

secondary cavitation\

primary cavitation Atc,v

tcav

~ ,.6 i ,.o

.

.

.

time (ms)

.

.

.

.

.

.

.

.

"

.

.

.

.

.

.

.

.

llO ~.. (m)

.

.

.

.

.

IO

.

.

~00

secondary closure pulse t

.

.

.

l .

~ tdmax

Figure 3 Pressure-time history for fluid element 2504 located adjacent to the centre of the plate (see Figure 1) for Po = 3.0 MPa.

207

~

J'i

12-

ax

....-u)

4 >, ~176

E E

2'

~ o

"10

0

9

i

1

9

!

2

9

i

3

9

i

4

9

i

5

9

'1

6

9

!

7

9 i

8

'-

i

9

9

i

10

Po (MPa)

Figure 4 Graph of peak mid-plate displacement, dmax,peak mid-plate velocity, Vmax, cavitation duration, Ate,v, and peak cavitation closure pressure, Polos, against the applied peak pressure, P0, for the FE runs listed in Table 1. Least-squares lines of best fit are ovedayed for the displacement and velocity data. An asymptote has also been drawn at Attar = 3.7 ms. 5 CONCLUSIONS The response to far-field UNDEX shock of a simple 2D plate structure surrounded by a rigid baffle was investigated numerically over a range of shock pressures. The loading history experienced by the plate was shown to result from a complex coupled interaction of structure and fluid response. The simulation results indicate that cavitation closure and consequent reloading of the plate occur as a result of focusing of the diffracted pressure adjacent to the plate centre. The reloading pulse was shown to be of similar intensity to the incident pulse. 6 REFERENCES [ I] J.O. Hallquist, LS-DYNA User's Manual (Non-linear Dynamic Analysis of Solids in Three Dimensions), Livermore Software Technology Corporation, Report 1007, 1990. [2] J.A. DeRuntz, T.L. Geers and C.A. Felippa, The Underwater Shock Analysis Code (USA-Version 3), Defense Nuclear Agency, Washington, D.C., Report 5615F, 1980. [3] J.A. DeRuntz, The Underwater Shock Analysis Code and Its Applications, 1, Proc. of the 60 th Shock & Vibration Symposium, Portsmouth, Virginia, pp. 89-107, 1989. [4] L.C. Hammond, R. Grzebieta, Structural Response of Submerged Air-Backed Plates by Experimental and Numerical Analyses, accepted for pub. in J. Shock & Vibration, 1999. [5] L.C. Hammond, R. Grzebieta, The Requirement for Hydrostatic Initialisation in LSDYNA/USA Finite Element Models, accepted for pub. in J. Shock & Vibration, 1999. [6] H.E. Saunders, Hydrodynamics in Ship Design, Society of Naval Architects and Marine Engineers, 1964. [7] Sh.U. Galiev, Influence of Cavitation Upon Anomalous Behaviour of a Plate/Liquid/Underwater Explosion System Int. J. Impact Engng, 19, pp. 345-359, 1997. [8] G.I. Taylor, The Pressure and Impulse of Submarine Explosive Waves on Plates, Underwater Explosion Research, Vol. 1 - The Shock Wave, pp. 1155-1173, 1950.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

209

Ductile Failure of W e l d e d Connections to Corrugated Firewalls Subjected to Blast Loading Dr Luke A Louca, Imperial College, Department of Civil and Environmental Engineering, South Kensington, London, SW7 2BU, UK. Tel 0171 594 6039, Fax 0171 594 6042. Mr. Jesper Friis, Imperial College, Department of Civil and Environmental Engineering, South Kensington, London, SW7 2BU, UK. Tel 0171 594 6027, Fax 0171 594 6042.

This paper presents a number of numerical and experimental investigations of large scale tests and simulations which have established that the weld between the corrugated panel and its supporting frame is critical when assessing the maximum containment pressure of standard firewalls subjected to blast loading. Avoiding having to model the weld in detail, a fnite element technique based on limiting plastic strain has been proposed in order to predict the containment pressure of standard firewalls. Incorporating the presence of the surrounding structure in the model was found to be vital in order to obtain a realistic representation of the behaviour. A parametric study established safe containment pressures for a typical wall geometry over a broad range of load histories.

1. INTRODUCTION Structural design of corrugated firewalls under extreme loading is an important safety issue for the offshore industry in the UK where the Health and Safety Executive require safety cases for both new and exiting installations. This has placed requirements on the operators to consider the possible consequences of a hydrocarbon release in order to protect personnel and safety critical systems. Since the publication of the Cullen report 1on the Piper Alpha Tragedy in the North Sea during July 1988, a number of large scale tests have been performed on typical wall geometries to establish likely containment pressures and to assess current designs. However, due to the cost of conducting such tests, only a very limited study can be performed, and sensitivity studies must be based on numerical models validated against the experimental data. At the extreme end of the design, which is likely to involve large plastic deformations, weld tearing and possible contact with adjacent plant or structural components, interpretation of results from a large non-linear finite element analysis become complex. Simplifications, which are inevitable when modelling large complex structures, can also lead to misleading results, particularly at critical locations such as connection details.

210 At present there appears to be no universal approach to modelling failure for problems involving large strains and displacements under dynamic loading. Simple attempts at modelling failure of welded connections using a force based failure model controlling a contact surface between a corrugated panel and a flexible angle connection has been attempted by Louca et al2. The results gave a conservative estimate of the containment pressure but a good qualitative correlation with the failure mode. A node release algorithm to model rupture of plate structures has also been developed by Rudrapatna et al 3 using both a linear and quadratic failure criterion accounting for bending, tension and transverse shear. The influence of shear on a number of failure modes was highlighted with the numerical results showing good correlation with the small scale experiments conducted. Nurrick er al 4 used a simple strain based failure model to predict the onset of tearing of flat panels in small scale tests. Weld integrity assessment was also attempted by Plane et al 5 on a large scale corrugated panel using a simple strain based failure model. This provided a conservative estimate of the failure pressure of the wall. This paper presents a validation of a finite element model with two large scale tests on corrugated firewalls subjected to explosion loading in which weld failure occurred in one of the tests. The failure was modelled using a limiting equivalent plastic strain at the weld locations around the boundary of the wall. The limiting strain was established by a trial and error procedure such as to best represent the observed failure mode. The influence on simplifying the loading idealisation on the response is discussed and a failure envelope for an approximate containment pressure is presented.

2. E X P E R I M E N T A L INVF_.~TIGATION The tested walls, see layout in figure 1, measured approximately 3.5m square, and consisted of a 2.5mm thick corrugated steel panel supported by a frame constructed from 100x75x8mm angle brackets. The panel and the frame were joined by a continuous 3mm single-sided fillet weld. The angle brackets were fixed to the primary steel structure (test rig) by two lines of 5ram fillet welds. The panel was orientated with the corrugations running vertically, and the wider of the corrugation flanges furthest away from the explosion chamber. A diagonal 203x203 UC46 brace was attached to the primary steel structure with a clearance of 92mm to the surface of the panel. During testing, this gap closed and the brace bent about its minor axis. All structural members were manufactured from ordinary carbon steel. The cross brace was fabricated from Grade 50D steel, whereas the panel and the angle brackets both were fabricated from Grade 40C steel. The over-pressures generated during the tests by the burning natural gas/air mixture were recorded at various locations within the explosion chamber. The pressure transducers closest to the surface of the panel recorded a 387mbar peak over-pressure in the test designated R0888, and a 1600mbar peak in the test designated R0885. The recordings from these transducers, positioned 500mm in front of the midpoint of the panel, were believed to best describe the blast load received by the panels. The structural response was monitored in each of the tests by means of three displacement transducers. The central out of plane deflection of the diagonal cross member was recorded in both tests, but the location of the remaining transducers changed from one test to the other. The deflections recorded indicated that the applied pressure could be assumed

211 uniformly distributed over the surface of the panel.

3mm fillet 100x75x8RSA

.

.

.

.

.

.

.

.

L___ . . . . . .

A ~

+..:

............ Pnmory steel 5ram fillet71--3LL structure

i

m 1743 B L ,

.1743

.L203x203UC46

/

CORRUGATION DETAIL

.L.~

I

fi ' let

[ L / ~ m ~ et

+-+='Primorysteel in mm

Figure 1. Structural configuration of firewalls Post test inspection of R0888 showed significant plastic deformations, but the integrity of the firewall was preserved. The permanent deflection of the midpoint of the cross-brace was 30mm in the outwards direction. In contrast, the blast loading applied to R0885 caused rupture of the majority of the welding at the top and the bottom of the panel. Also, the lower connection between the cross brace and the primary steel work was severed. However, as only a few short failure lines developed in the vertical direction, the panel was not completely dislodged from the frame. The vertical failure lines developed in the vicinity of the corners as a combination of weld failure and plate tearing. The blast in R0885 caused the angle brackets transverse to the corrugations to undergo a maximum plastic rotation of approximately 45 o. In comparison the less severe load in test R0888 produced a maximum permanent rotation of about 5 o. The permanent deformations of the angles parallel with the corrugations were insignificant in both of the tests.

3. NUMERICAL ANALYSIS 3.1 Finite element model The dynamic response of the firewall was simulated using the ABAQUS/Explicit finite element program. The firewall was, as shown in figure 2, spatially idealised as an assembly of "S4R" shell elements, which are reduced integration elements. The brackets were assumed to be

212 rigidly connected along the first line of the 5mm fillet welds (see figure 1). The numerical simulation of the contact between the cross brace and the panel utilised a simplified algorithm, in which the relative sliding between the contacting bodies is assumed to be small. Both geometric and material non-linearities were included in the analysis. The progressive weld failure which occurred in test R0885 was numerically modelled by letting the panels outer elements represent the behaviour of the welding material. Without altering the shape of the stress strain curve quantified in section 3.2, the rupture strain of the boundary elements were reduced to either 8%, 10% or 12%.

Figure 2. Adopted meshing in the FE model

3.2 Material properties The plastic material behaviour was governed by the Von-Mises yield criteria combined with an isotropic hardening rule and an associated flow rule. Material failure was assumed to occu r immediately after the ultimate strain had been achieved. Table 1 lists the nominal material properties employed to define the true stress-strain behaviour of the Grade 43C and 50D steel respectively. Table 1 Nominal material properties .... Grade

43C

50D

Yield stress, fy, (MPa)

275

355

Range of yield plateau, e st, (%)

4.0

3.0

Ultimate strength, ft, (MPa)

430

500

30

25

Ultimate strain, e U' (%)

It is well known that the stress strain curve for steel is sensitive to the applied rate of straining. The maximum strain rate experienced in the models investigated was approximately 15 s 1, which occurred in the vicinity of the connections. In order to assess the significance of this, the empirical Cowper-Symonds overstress model was used in the numerical analyses for one of the tests.

213

3.3 Pressure-time curves Figure 3 illustrates the pressure-time curves recorded during the two tests. The traces have been reproduced by digitising graphs available from the test site. Since, the recorded curves were very complex, involving oscillations of varying frequencies, they have been simplified for the purpose of the numerical analysis. Two types of idealised pressure-time curves have been adopted in the present investigation. Type 1, represented by the curves ABCD in figure 3, incorporated an initial phase, AB, in which the pressure increases relatively slowly. This is followed by a phase, BC, characterised by a much higher rate of pressure increase. In both tests the pressure at the end of the initial phase approximately equal to one quarter of its peak value. The type 2 pressure history, the triangle B'CD in figure 3, ignored the initial phase. The triangular curves appear, at least in terms of curve fitting, to be a very crude representation of the recorded diagrams. ---450

~ .o "4-"~350 [-=~250

m

R0888 ldealised

Experim

!

dC ff -- /

-

~

50

o

-50

=

.

20

40

,It!

I~

-

0

A.C

.1400 i t.ff 1000

[

m 15o ~

"1800-.,~"

60 80 100 120 TIM E, T, ( m see )

140

~

600 "

I

/

I

Experim I

T!

'~i

,

,

c'7 (:, BJ;

a. 200 . , - ~ m ;> o .200 0 20

160

I .ITESTIR0885 tldealised

~,:

i D -~, " "..

40 60 80 100 TIM E, T, (msec)

120

Figure 3. Experimental and idealised pressure-time curves 3.4 Results and discussion Figure 4a shows a comparison between the deflections recorded in test R0888 and those predicted by the FE model both with and without rate dependence. Deflections were measured at the centre of the brace and at the midpoint of the panel quarter. It can be seen that the numerical model gives a good prediction with the peak displacements but significantly overestimates the residual when ignoring rate effects. Switching on the rate dependence produces considerable stiffening and underestimates the peak although the residual is well predicted. This trend in response has also been noted by Langseth et al 6 on impact loading of plates. The dynamic magnification of the panel (dynamic/static) deflection was 1.19. -300

m TEST: R0888'

-500

' I.

!~

Panel Brace No Rate : -"r Rate : --o--

"-'-200 ,~

_Experim

~.~w*~

,--.-250

. :

~

Analysis Experim__~ C Panel --4.,-Brace -..,-/

~ " -400

J~

J l

TEST: R0885

,-$

_g_

~, ,,

-300 Z O

<->

/

.-~

'=I

'g"%

-5o

,if

0

........ ~'~

0

20

40

60

':'p'

80

t

I

~I.o -200

/

~,~ ,,~ ,

""

I00 120 140 160 180 200 220

TIME, T , ( msec)

~ -100

Jf

,

0 0

10

20

30 40 50 TIME, T , ( m s e c )

60

70

80

Figure 4a. Influence of rate dependent material Figure 4b. Predicted load-deflection diagram behaviour on predicted load-deflection diagram when ignoring weld rupture

214 Figure 4b shows the displacement time history for R0885. The transducers used to measure the displacements were damaged during the test due to the exhaustion of their stroke, which appeared to occur prior to the initiation of weld rupture. Due to this, test traces were only available up to 28 ms. These are crudely shown in the figure as only a hard copy ofthe traces was available. Figure 5 shows the failure pattern obtained from the numerical model using an equivalent plastic strain of 10% at the outer elements of the panel connected to the angle connection. It was established that using a 10% rupture strain criteria caused the numerical model to mimic the experimentally observed weld failure very accurately.

Figure 5. Detachment of panel R0885 using 10% rupture strain

3.5 Influence of loading idealisations The v~,!idation of the models was carded out using load curves ABCD from figure 3, which is a rea~ able representation of the experimentally recorded load. A common idealisation used in sensiuvtty studies is to adopt a triangular load pulse represented by B'CD, which ignores the initial slow rise of the load pulse. Overall the triangular load pulse, B'CD has a more adverse effect on the response. The dynamic magnification on the maximum deflection was increased from 1.19 to 1.33 for 110888 and 1.08 to 1.22 for 110885. However, the dynamic magnification on the plastic strain was far more severe, particularly for R0885 where an increase from 1.29 to 2.17. For R0888 the corresponding increase was from 2.0 to 3.29 which has implications for structural integrity studies.

3.6 Estimate of containment pressure Based on a value of 10% as a realistic failure strain, a parametric study was conducted to establish the containment pressure for a range of rise times assuming a triangular load pulse with equal rise and fall times. Figure 6 shows a summary of the results with a resulting failure envelope. At low values of rise time which represents a steep rise in loading, the failure pressure is almost half the value at the other extreme of the curve which represents a static load condition (>80 ms). The graph also demonstrates that a single load parameter, such as the impulse, is insufficient when assessing the capacity of the firewall.

215 1600 1400

i

.,o

ered'ictior eSurvivc ! *Failed

1200 =.

/ .~.

.L_

~/

u.~lO00 800 -

/

, ,,~

600 400

m

200

10

20

30 40 50 60 70 RISE TIME, T s , ( m s e c )

80

90

100

Figure 6. Combinations of rise time and peak pressure initiating weld rupture

4. CONCLUSIONS Results from a large scale experimental study into the response and failure of corrugated firewalls subjected to a hydrocarbon explosion have been presented together with a numerical simulation. The study showed that the numerical models can be used to highlight important aspects of response, in particular to the sensitivity of the loading idealisation. The failure mode was predicted with some accuracy in a qualitative manner although more data is required to ensure the response can be quantified at failure. The use of a simple failure criteria based on an equivalent plastic strain value gave a good estimate of the containment pressure which has been shown to be sensitive to the rise time of the loading history.

REFERENCES 1. Cullen, Lord. The Public Inquiry into the Piper Alpha Disaster, HMSO, UK, 1990. 2. Louca, L. A., Harding, J. E. and White, G. Response of Comagated Panels to Blast Loading. Offshore Mechanics and Arctic Engineering,Fi0rence, June 1996. 3. Rudrapatna, N. S., Vaziri, R. & Olson, M. D. Deformation and Failure of Blast-Loaded Square Plates. Int J. Of Impact Engng, Vo122, No. 4, pp449-467, 1999. 4. Nurrick, G. N., Olson, M. D., Fagnan, R. F. & Levin, A. Deformation and Tearing of BlastLoaded Stiffened Square Plates. Int. J. Impact Engng., Vol 16, No. 2, pp273-291, 1995. 5. Plane, C. A., Bedrossian, A. N. and Gorf, P. K. FE Analysis and Full Scale Blast Tests of an Offshore Firewall Panel. Int. Conf. on Offshore Structural Design Against Extreme Loads. ERA, London, 1994. 6. Langseth, M, Hopperstad, O. S and Berstad, T. Impact Loading on Plates: Validation of Numerical Simulations by Testing. International Journal of Offshore and Polar Engineering. Vol. 9, No 1, March 1999.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

217

D e s i g n criteria for blast tolerant b u l k h e a d s I. Raymond a'b, M. Chowdhuryb, and D. Kellyb aThe Australian Maritime Engineering Cooperative Research Centre (AME CRC). bThe School of Mechanical and Manufacturing Engineering, The University of New South Wales (UNSW), Sydney, NSW 2052, Australia With increasing importance being placed on survivability of naval platforms, this paper looks at a structured approach to the development of design criteria to fulfil the operational requirements of naval transverse bulkheads by using finite element modelling and a J-integral analysis. Initially the operational requirements are established, followed by the explanation of the design criteria and their implementation to X-80 steel transverse bulkheads. 1.

INTRODUCTION

Survivability is becoming an increasingly important issue, and as stated in Chalmers (1993), "currently little information on the design of bulkheads to withstand internal blast effects" exists. Transverse bulkheads aid survivability by restricting the spread of the blast loads longitudinally throughout the vessel. Additionally, the transverse bulkheads are required to be at least a watertight boundary against flooding or fire in a post-explosion environment. The operational requirements are intended to ensure that if an explosion occurs inside a naval platform that the blast loads and related effects of flooding and fire are contained within the compartment that the explosion occurs in. These operational requirements have been developed into design criteria through considering safety factors and the accuracy of the finite element models. 2.

OPERATIONAL REQUIREMENTS

As discussed by Williams (1990), the determination of the customers' requirement for the product must be considered first. These are termed operational requirements. In this paper, the operational requirements will be based around the capabilities of a transverse bulkhead on deck 3 at 19.2 m from the bow on a vessel 118 m long moving at 30 knots. This transverse bulkhead will be considered as a generic worst-case naval transverse bulkhead for the operational requirements and subsequently used in the design criteria. 2.1.

Pre-air-blast loads

There are two static loads that must be considered in this section. These are a) A hydrostatic load, Pn, due to the transverse bulkhead being one side of a

218 liquid tank. The effect of sloshing will be included b) Structural loads, Ps, will cover the loads due to equipment above, and bending loads on the hull, and from such occurrences as dry-docking. The response of the transverse bulkhead to all of these loads should be in the elastic regime and no rupture is permitted. Additionally, buckling responses will be reviewed.

2.2.

Air-blast load

The air-blast load against the naval transverse bulkhead is assumed to be 150 kg of TNT equivalent explosive at 8 m. This load is comparable with the critical blast load considered by some western navies in the design of new vessels, Reese et. aL (1998) and OPNAV (1988). The response of the transverse bulkhead to this blast load will be critical in two situations. Firstly, no rupture is permitted within the transverse bulkhead structure. Secondly, a maximum permanent deformation of the transverse bulkhead will be set. This will be due to pipes penetrating through the bulkhead, equipment and walkways close to the bulkhead, and that a post-air-blast load has to be supported by the transverse bulkhead.

2.3.

Post-air-blast load

The post-air-blast load is flooding of the compartment after an explosion has occurred. The deformed transverse bulkhead is required to be able to support this load without extensive deflection and no rupture in the bulkhead structure. 3.

DESIGN CRITERIA

3.1.

Pre-air-blast load

The hydrostatic load due to a tank is solved by the method given in BV 104 (1982), which considers the effect of sloshing. In this case the value of the hydrostatic pressure is Pn = 90 kPa. The structural load comes from Chalmers (1993), where for the sides of the bulkhead it is 93.15 MPa and for the top it is 22.5 MPa. These methods have built in safety factors relevant to differences between the idealised bulkhead and a fabricated bulkhead. The hydrostatic pressure is applied as a pressure over the bulkhead plate area and the structural load is applied as a pressure field along the side and top edges of the bulkhead. The J-integral procedure will be used to confirm that no rupture occurs.

3.2.

Air-blast load

The air-blast load is equivalent to, by the Hopkinson scale method, 7 kg of Comp-B at 3 m in the Defence Science and Technology Organisation (DSTO) Bulkhead Test Rig (BTR). The blast data, from Turner (1999), consists of two separate position pressure profiles of the blast. The approximate positions of the pressure gauges relative to the centre of the bulkhead are 550 mm and 1150 mm in a radially outward direction. The blast pressure profiles, shown in Figure 1, are approximation of the raw data supplied. This approximation gives the

219 dominant feature of the blast load. The separation between the two initiation pressure values has been determined by the propagation speed of stress waves through X-80 and is discussed in Raymond et. a/.(1999).

Figure 1. Blast pressure history of 7 kg of Comp-B at 3 m in the BTR. These two blast pressure profiles are applied to the transverse bulkhead in the finite element model as two separate transient pressure loads. Due to the high strain rate experienced in an air-blast situation, material data is required. This material data can be obtained by undertaking Hopkinson bar tests, with the use of constitutive equations such as Johnson-Cook or Cowper-Symonds (Wang and Lok (1997)) to gain general constants that can be used in the finite element modelling. The maximum permanent deformation that can be accepted by a naval transverse bulkhead is set to 100 ram, Chalmers (1993). A safety factor is introduced, due to the modelling inaccuracy, which reduces this deformation by 20% to 80 mm, Turner (1999). Rupture of the transverse bulkheads is tested by the J-integral procedure.

3.3.

Post-air-blast load

The post-air-blast load that is being considered is flooding. BV 104 (1982) gives the flooding load to be 52 kPa. This is applied in the same way as the hydrostatic tank load. The deformed transverse bulkhead, due to the air-blast load, is required to maintain structural integrity due to the flooding load. The J-integral procedure will be used here to test for rupture. Structural redundancy within the vessel will absorb the structural load. As with the hydrostatic load the effect of sloshing has been covered in determining the flooding load and safety factors are built into this method.

220

3.4.

J-integral procedure

The J-integral procedure is used to determine if rupture has occurred or not due to any of the loads mentioned above. The use of the path independent J parameter for characterising materials toughness is well established by such work as Rice (1968) and Shih (1976). The parameter was designed to cater for elastic-plastic stress situations such as high toughness steel in naval transverse bulkheads. Therefore, a path independent J-integral application to determine if the naval transverse bulkhead has ruptured or not is an extension Of current practices. The J-integral cannot be applied to the whole transverse bulkhead arrangement at once. The solution is to apply the path independent J-integral to the appropriate faces of every finite element brick of interest that makes up the finite element model. This is possible as the shape of individual brick faces is simple and a solution can be obtained from EPRI (1981), Miannay (1998) and Tada et. al. (1973). These bricks have dimensions between 1 89mm by 20 mm by 20 mm to 3 mm by 20 mm by 20 mm. The transverse bulkhead will be modelled with three solid brick element through the thicknesses of the bulkhead material. The most likely position for a crack or tear to begin is in the weld. If this tear propagates through the thickness of the weld, rupture will eventuate. As shown in Figure 2 the minimal thickness of a weld, in a double weld situation, is 1.5 mm for a plate 4 mm thick. It is assumed that as long as the crack only exists in one of the two welds then the redundant strength in the entire structure will be great enough for the vessel to return to port for repairs. Bulkhead p l a t e - " - ' - Double side weld joint

2 rnn

Critical tear length, 1. Deck plate

Stiffener

~ ! / x~~~~l, ~Welded join~ _

"~

T r a n s v e r s ~

,

,

,

- -

Jv

Deck

Figure 2. Double-sided welding without fillets of the bulkhead plate the deck, additionally a standard transverse bulkhead is depicted. In the design criteria this crack length will be reduced for reasons of conservatism. The first reduction will be the crack length cannot be greater than the element thickness dimension. The second reduction is related to the inaccuracy in the modelling of the pathindependent J-integral value. This is due to the fact that the finite element processing programs, MSC/NASTRAN and LS/DYNA, will not be considering the cracks in their finite element models. A sub-program will calculate the J-integral at the completion of each load step from the node position and stress data. None of this crack information will be returned to the processing programs. As a conservative estimate of a safety factor the critical crack length will be set to 75% of the finite element thickness. This gives a safety factor of 89 to the permitted crack length. The technique for determining the J-integral value will be verified against empirical results in Hrovat and Hoffman (1999).

221

The J-integral will be calculated using the method for a single-edge crack plate in uniform tension applied to the faces of the brick element of interest in the thickness direction of the plate. Figure 3 shows the form of the single-edge crack plate under remote uniform tension. The J-integral can be evaluated by the following equation, from EPRI (1981)

J = fl (ae)--~7 + ~

(a~b,n

'

(1)

where ct, n are the Ramsberg-Osgood parameters; a o, e o are the flow stress and strain respectively; h l-(a/~b,n) is a function whose values are tabulated in EPRI(198 1).

Figure 3. Single-edge cracked plate under remote uniform tension, form EPRI (1981). This is one face of a brick element, where b is the thickness of the finite element. The J-integral will be calculated for both the parent and weld material. It is assumed that the initial crack length, a, will be 0.4 mm for the welded material and 0.01 mm for the parent plate material. At the completion of each load step the J-integral is calculated by the above method. This calculated J-integral value is related to a J-R curve to determine the crack extension value. This crack extension value is then used to produce the new a value, which is saved to the next time the J-integral value has to be calculated. If this new a value is greater than the critical crack length then failure is assumed and noted for review. Additionally, from the J-R curve a tearing modulus analysis will be performed to determine if the tearing is propagating stably or unstably. If the tearing is unstable then failure is assumed and the finite element evaluation is suspended. The decision of whether to base the analysis on plane stress or plane strain will be determined by the outcomes of the validation trials.

3.5.

Other relevant factors

In regards to X-80 steel plate, it is available in thickness from 4 mm up to 9 mm in intervals of 0.1mm. In the fabrication of the transverse bulkhead it is assumed that distortion is less than the thickness of the plate. This is in line with Ghose and Nappi (1994) findings.

222 Residual stress cannot be practically modelled, as discussed in Okumoto (1998). This is because X-80 has relatively high residual stress due to its' manufacturing process and there is limited data on the residual stresses that form during the fabrication of an actual structure. Hence, residual stress is addressed in the following manner 9 The effect of residual stress on the final deformation of the transverse bulkhead due to the air-blast load is a reason why the inaccuracy of the modelling is approximately 20%. 9 The safety factors built into the static loads cover effects of residual stress. 4. CONCLUSION This paper describes a structured approach to the development of design criteria for a blast tolerant naval transverse bulkhead. These design criteria will be used as the constraints in an optimisation procedure to develop optimised X-80 steel naval transverse bulkheads. ACKNOWLEDGMENTS The authors would like the thank Mr. Toman, Mr. Quigley, and Ms. Deeley, AME CRC, UNSW, DSTO, Tenix Defence Systems and BHP for all their continuing support. REFERENCES 1. BV 104-1 "Structural analysis of the ship's hull (strength specification)", July 1982. 2. Chalmers, D. W., Ministry of defence - design of ship's structures, HMSO, 1993. 3. EPRI, An engineering approach for elastic-plastic fracture analysis, General Electric company, NP- 1931, 1981. 4. Hrovat, R., Hoffman, M., AME CRC Internal Report-(in preparation), 1999. 5. Ghose, D. J. and N. S. Nappi, SSC-382, 1994. 6. Miannay, D. P. Fracture mechanics, Mechanical Engineering Series, Springer, 1998 7. Okumoto, Y., Journal of Ship Production, 14, No. 4, November 1998, pp. 277-286 8. OPNAV Instruction 9070.1, Chief of Naval Operations, Washington, 1988 9. Raymond, I., Chowdhury, M., Kelly, D., Design criteria for X-80 steel naval blast tolerant bulkheads report, AME CRC Internal report, 1999 10. Reese, R. M., Calvano, C. N., and Hopkins, T. M., Naval Engineers Journal, 100, January 1988, pp. 19-34. 11. Rice, J. R. A., Journal of Applied Mechanics, 35, June, pp. 379-386, 1968 12. Shih, C. F., and Hutchison, J. W., Journal of Engineering Materials and Technology, 98, October, pp. 289-295, 1976. 13. Tada, H., Paris, P. C., and Irwin, G. R., The stress analysis of cracks handbook, Del Research Corporation, Hellertown, Pennsylvania, 1973. 14. Turner, T., private communications, March 1999. 15. Wang, B., and Lok, T. S., 2 nd Asia-Pacific Conference on Shock and Impact Loads on Structures, 1997, pp. 569-576. 16. Williams, M., Navtec'90, RINA, 1990 17. Williams, J. G., Killmore, C. R., Barbaro, F. J., Meta, A., and Fletcher, L., B HP Internal Report, 1992

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta @ 2000 Elsevier Science Ltd. All rights reserved.

223

The ballistic impact of hybrid armour systems H. H. Billon Combatant Protection and Nutrition Branch Aeronautical and Maritime Research Laboratory PO Box 4331, Melbourne 3001, Victoria, Australia Research is conducted into the impact of armour systems by projectiles travelling at ballistic velocities. An attempt is made to relate the ballistic impact of woven fabric structures to the behaviour of impacted yarns. The effects of impact on structures consisting of hybrids of aramid, nylon and high-modulus polyethylene are described. The method of characteristics is investigated for its usefulness in providing an insight into the impact process.

1. I N T R O D U C T I O N There has been a number of studies into the impact on fabric structures by projectiles travelling at ballistic velocities [1-4]. Most of this work concentrates on ballistic impact on structures consisting either of a single fabric layer or of a collection of multiple layers of fabric where all the fabric layers consist of the same material. Less effort has been devoted to studying impact on hybrid structures comprising different types of fabrics [5,6]. Practical models for ballistic impact on multiple layer systems have been constructed using the insights obtained from studying ballistic impact on systems consisting of single yarns [7,8]. Therefore a thorough study of the ballistic impact of a two-dimensional hybrid system will provide insights into modelling the impact of ballistic projectiles on practical hybrid armour packs. The ultimate benefit of this work is the determination of the cost or protective advantages that may be obtained by constructing armours from combinations of different types of materials. 2. M A T E R I A L P R O P E R T Y D A T A Table 1 Material property data for ballistic yarns Material Density (g/cm3) 1.44 Aramid 1.14 Nylon 0.97 High Modulus Polyethylene

(m E)

Denier

1500 1187 1010

Young's Modulus

(GPa) 71

5.7 88

224 3.

A DESCRIPTION

OF THE PROBLEM

To provide a two-dimensional analogue for the interaction between two layers that occurs when a fabric armour pack is struck by a projectile travelling at ballistic velocities, the behaviour of two yarns struck by a projectile is investigated. In order to determine the applicability of the method of characteristics in problems of this type, a solution of the problem is attempted subject to the following assumptions and simplifications: (1) (2) (3) (4) (5) (6) (7)

The yarns possess constant mass per unit length values, namely pj and ,o2 Both yarns obey an elastic constitutive law in tension Neither yarn offers any resistance to bending Gravitational forces on the yarns may be neglected The yarns can only move vertically The deflections and slopes of both yarns are small A compressive elastic force exists between corresponding points on the yarns. The force will be zero when the yarns are separated (8) The yarns are subjected to initial tension values 7'i and/'2. It is assumed that horizontal components of the tension values remain constant during the course of the impact

4. E Q U A T I O N S

OF MOTION

During the impact process there are tensile forces on a yarn element as well as a force due to interaction between the two yarns. For convenience consider the first yarn which is assumed to lie below and in intimate contact with the second yarn. Using the assumptions in Section 3 and applying Newton's second law to the horizontal and vertical force components acting on a small yarn element on the lower (first) yarn it may be shown that:

~-f(x,t) +

a~w,= 10"~w,

(1)

where x is position, t time, w l the vertical displacement and the following identity has been used to describe the interactive force F (x,0 per unit length between the two yams:

f(x, t)=

lim ( F(x,t)) ~-,o~, 8 x

(2)

A similar equation holds for the upper yarn, the essential difference being that the force term has the opposite sign:

f(x,t) 82w2 1 02w2

_ _ _ _ +

T2

.

.

c3x 2

.

.

c2

Ot 2

(3)

225 In Equations (1) and (3) the definitions c 2 = _7"! and c22 = ~7"2 have been made. These are P!

P2

the squares of the wave speeds of the individual yams. The problem is to find solutions for Equations (1) and (3) subject to boundary and initial conditions appropriate to the impact of the two yarns by a ballistic projectile. In this paper the yarns are assumed to be initially at rest (except for the impact point) and to possess fixed extremities at the fight-hand boundary. The impact points of both yarns move at the same velocity as the impacting projectile. 5. S O L U T I O N

USING CHARACTERISTICS

Make the following definitions: OWI

0to l

u, = ~ " u2 = ~ ax' at

(4)

Using (4), Equation (1) may be converted into a quasi-linear first order system [9]:

- - - 1~ +au2 -c? at t~ I

&

Oua

ax

f -o 7,

~0u=20 Ox

(5)

(6)

By following an approach similar to that described in [9], it may be shown that:

d

u~) = cl f

(7)

tt - T

d-S u, +

=-c,~

Applying the same technique to Equation (3):

J

d(u

u4)

f

where u3=--~--,u4 = Ot

(9)

(10)

226 The essential difference between the above derivation and the approach taken in [9] is that, in this paper, the characteristic direction for Equations (7) and (8) is different from the direction for Equations (9) and (10). Two useful auxiliary relations are derived for the increments in ws and w2: &o, = Ow!,& + ~ d t Ox Ot

(11)

= u , & + u28t

&o2 = tgw2 & +-o ~ 2 ~ = u 3& + u48t

~x

(12)

&

The following form is used for the force term:

I:

(h-lw,-

(13)

where H(x) is the Heaviside step function. This force term is zero when corresponding points on the two yams are separated by a value equal to, or in excess of, h (initial yarn separation) and is a repulsive elastic force/length for separations smaller than h. E? is the net elastic modulus for the two yams. 6. R E S U L T S A N D D I S C U S S I O N The solution is obtained at the intersection of four characteristic lines with slopes +ci and +-c2. If Equations (7) - (13) are expressed in a finite difference form and applied to the boundary conditions then it can be shown that the yarns remain horizontal at the boundary until at least the time t = / J r . where Cta,~e is the larger of the two wave speeds and that a /"t

arge

point on either yarn with coordinate x will be motionless until a time t = x~_

. Equations

/ ' - 1 axlge

( 7 ) - (13) were used as the basis for a numerical technique to determine values of the dependent variables for a number of time steps. This technique uses linear interpolation to determine values of the dependent variables for subsequent time steps. The spatial increment 8x and the time step 8t are related to the wave speed c by the Courant-Friedrichs-Lewy criterion [10]" & < cSt

(14)

Numerical results are presented in Figures 1 - 3 for hybrid yarn systems struck by a ballistic projectile travelling at 100 m/s. The wave speeds for the individual yarns at a tension of 100 N are 775 m/s for aramid, 871 m/s for nylon and 944 m/s for HMPE. Figures 1 and 2 show that the wave speed values measured from the deformation profiles agree well with the wave speed values of the individual yarns and that the deformation profiles are in general agreement with previous experimental observations of yam impacts (see e.g. [11]). The

227 motion of the two yarns lends support to the hypothesis that a high wave speed material does not offer much constraint to the motion of a contiguous low wave speed material [5]. A different profile occurs when a low wave speed yarn is struck earlier than a high wave speed yarn as opposed to the reverse situation (Figure 3). The simulations were run with an increased time step of 2 Its. It was observed that the deformation profiles for Figures 1 and 2 did not change significantly. However, there was a significant change in the deformation profile for Figure 3. The most likely explanation is that there is instability in the solution for the case when the projectile strikes a higher wave speed material earlier than a lower wave speed material. A possible reason for this instability is the fact that there is a stronger interaction between the two materials in this case [5] and that the force term therefore creates a larger perturbation in Equations (1) and (3) away from the form of a wave equation, requiring a different stability criterion from Equation (14).

~-- 0.002 ~

~

0

0.01

0.02

x (m)

-0.05

-

ii

0

0.01

0.02

x (In)

Figure 1. Impact of a projectile with an aramid yarn (~) followed by a HMPE yarn (n) after 10 time steps. Time step 1 laS. Boundary at 0.19 rn.

0

0"002 0.001 ~I

-

,

00 2

0.03

x (m) Figure 3. Impact of a projectile with a HMPE (B) yarn followed by an aramid yarn ( . ) after 10 time steps. Time step 1 las. Boundary at 0.19 m.

Figure 2. Impact of a projectile with an aramid yarn ( . ) followed by a nylon yarn ( e ) after 10 time steps. Time step 1 las. Boundary at 0.17 m.

228 7. C O N C L U S I O N S Equations describing the impact of a projectile on two yarns in intimate contact have been derived subject to some simplifiying assumptions. The method of characteristics has been used to solve these equations and to derive some general features of the impact process. A numerical method was developed using the characteristic solutions as a basis and numerical results have been presented. These results indicate good general agreement with impact phenomena observed by other workers. Preliminary results indicate that the numerical method is stable for the case where the lower wave speed material is struck before the higher wave speed material. Instability can occur when a material possessing a higher wave speed is struck before a lower wave speed material and the stability of the numerical method has not been thoroughly determined. However, this paper demonstrates that the characteristic method is a promising approach for modelling the interaction between layers of dissimilar materials when they are struck by a ballistic projectile. REFERENCES 1. W.J. Taylor and J. R. Vinson, AIAA J., 28 (1990) 2098. 2. R. W. Dent and J. G. Donovan, Projectile Impact with Flexible A r m o r - An Improved Model, Technical Report Natick/TR-86/044L (Limited Release) (1986). 3. J. R. Vinson and J. A. Zukas, On the Ballistic Impact of Textile Body Armor, ASME Paper No. 75-APM-12 (1975). 4. B. Parga-Landa and F. Hernhndez-Olivares, Int. J. Impact Engng., 16 (1995) 455. 5. P.M. Cunniff, Textile Res. J., 62 (1992) 495. 6. H.H. Billon and D. J. Robinson, Modelling the Ballistic Properties of Fabric Armour, 3~a International Symposium on Impact Engineering, (1998) 511. 7. I. S. Chocron Benloulo, J. Rodriguez and V. S~inchez G~lvez, J. de Phys. IV, 7 (1997) 821. 8. B. Parga-Landa, Inst. Phys. Conf. Ser. No. 102: Session 11 (1989) 565. 9. G.B. Whitham, Linear and Non-Linear Waves, Wiley-Interscience, 1974. 10. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985. 11. J. C. Smith, C. A. Fenstermaker and P. J. Shouse, Textile Res. J. 33 (1963) 919.

Structural Failure and Plasticity (IMPLAST 2000)

Editors:X.L.Zhaoand R.H. Grzebieta 9 2000ElsevierScienceLtd. All rightsreserved.

229

Large-Scale Blast Analysis of Reinforced Concrete with Advanced Constitutive Models on High Performance Computers" Kent T. Danielson ~'2, Mark D. Adler, Stephen A. Akers 2, and Photios P. Papados 2 ~Mechanical Engineering and Army High Performance Computing Research Center Northwestern University 2145 Sheridan Road Evanston, IL 60208-3111 USA 2U.S. Army Engineer Research and Development Center, Waterways Experiment Station 3909 Halls Ferry Road, Attn: CEERD-SD-R Vicksburg, MS 39180-6199 USA

Abstract The analysis of concrete structures undergoing complex inelastic responses to loads, such as those resulting form explosive detonations, is a challenging mechanics problem. The task can also require significant computational resources. Recent collaborative efforts between researchers at Northwestern University and at the U.S. Army Engineer Research and Development Center (ERDC), Waterways Experiment Station have focused on these difficulties. A microplane constitutive model ~was developed for concrete that has been very successful in modeling such problems. The microplane model, however, is computationally intensive, which has excluded its use for many large-scale applications. Therefore, the microplane model has been implemented into a parallel finite element code developed by the authors 2 for execution on the DoD's high performance computing systems. Analyses of explosive detonations in a reinforced concrete wall were performed on as many as 512 processors of a Cray T3E-1200. The concrete, reinforcing steel, and C-4 explosive charge detonated in a cylindrical hole in the middle of the wall were modeled separately with hexahedral elements. A JWL equation of state and a programmed burn algorithm were used to model the explosive. The model consisted of approximately 1,000,000 elements and showed excellent scalability on the hundreds of processors. Analyses that would require over 1000 hours on a single processor were performed in only a few hours on the Cray T3E. The parallel performance demonstrates the ability to efficiently perform such analyses by the use of parallel computing.

' Approved for public release; distribution is unlimited.

230 I. INTRODUCTION In this paper, the microplane constitutive model (e.g. Reference 1) is placed into a parallel explicit dynamic finite element code, ParaAble 2, developed by the authors for threedimensional analysis of blast loading in reinforced concrete. The parallel development of the code has a similar structure to other parallel explicit dynamic codes 3'4. A SPMD paradigm is used with the code written in FORTRAN 90, and all interprocessor communication made with explicit Message Passing Interface (MPI) calls. The parallel procedure primarily consists of a mesh partitioning pre-analysis phase, a parallel analysis phase that includes explicit message passing among each partition on separate processors, and a post-analysis phase to gather separate parallel output files into a single coherent database.

2. EXPLICIT DYNAMIC FINITE ELEMENT ANALYSIS Using the principle of virtual work, the basic equations for dynamic equilibrium at time t are: Mqt = pt _ F t

(i)

where each dot refers to differentiation with respect to time, M is the mass matrix, q is the generalized displacement vector, P is the vector of applied loads, and F is the vector of internal resistance forces determined by inclusion of the interpolation functions into:

F~q

~(~:5EdV

=

(2) where 5 is the variational operator and, for geometric and materially nonlinearities, t~ and E are any work-conjugate pair of stress and strain measures associated with the reference configuration VR Eqn (1) is first used to evaluate the accelerations, /t'. A central difference scheme is then used for temporal integration of the velocities and displacements, i.e., At n+!

q

2

At n

=q

2 +

Atn+l qt+At n+! -_ qt + A t n . l q t + ~ 2

qt

(3)

(4)

231 where At with superscripts n+l and n refer to the current and previous time increments, respectively. The nodal mass is lumped so that the mass matrix, M, is diagonal. Therefore, the computations associated with eqns (1), (3), and (4) are primarily vector operations, and corresponding CPU usage is dominated by evaluation of the integrals of eqn (2).

3. PARALLEL CODE DEVELOPMENT For effective parallel computing, it is critical to balance the computational load among processors while minimizing interprocessor communication. Therefore, separate preprocessing software is used to partition any general unstructured mesh using the graph theory based software, METIS 5. The partitioning is made with regards to the calculations in eqn (2), since it involves more computational effort than the lumped mass equation solving. For METIS, elements are weighted according to their relative computational cost. Numerical investigations indicate that the microplane model is approximately fourteen times more expensive than typical elasto-plastic models. Partitions of nearly equal computational effort are produced while also generally minimizing the number of partition interfaces. The partitioning output provides a list of processor numbers for the elements, so that each is uniquely defined on a single processor. The diagonal nature of the mass matrix, M, permits the nodal equation of each degree of freedom to be solved independent of other degrees of freedom. To retain data locality, nodes are therefore redundantly defined on all processors with elements possessing these nodes. All loads, boundary conditions, material properties, constraints, etc., are only defined on the processors for which they apply. The entire preprocessing software can be reasonably executed for large models on a workstation. At each time increment, t, the basic parallel scheme first consists of creating a force vector (global for the partition) for the partitions on each processor from elemental contributions to pt and F t. Next, the forces belonging to redundant nodes are gathered into vectors and sent to the processors possessing duplicate definitions. The partial force vectors are then received from the other processors and added to the global force vector on the current processor. Finally, the critical time step on each processor is sent to all processors in order to determine the global value. At this point, other boundary conditions are accommodated in the tbrce vector, and the new accelerations, velocities, and displacements are determined by the relations in eqns (1), (3), and (4), respectively. Using the new configuration, the process is then started all over again for a new time increment. Communication of partition boundary nodal forces are overlapped with elemental computations at partition interiors.

4. NUMERICAL APPLICATION The explosive detonation in a reinforced concrete wall is depicted by the finite element model in Figure 1, which consists of 995,192 hexahedral elements and 1,030,89 nodes. The wall dimensions are 183cm x 183 cm x 30.5cm and a 94.97g C-4 charge is detonated in a 1 inch diameter cylindrical hole in the center of the wall. The event was experimentally staged at ERDC. Quarter symmetry was assumed for the calculation. The fully coupled explosive-structural analysis uses the microplane constitutive model for the

232 concrete, an elasto-plastic model for the reinforcing steel, and a JWL equation-of-state model for the C-4 explosive. Ignition of the explosive is treated by a programmed-bum algorithm. An example of the METIS partitioning for ParaAble is shown in Figure 1. Because of the large differences in computational effort among the different constitutive models, the microplane model elements were assigned fourteen times the vertex weighting of the other elements for METIS. The transient analysis was performed to 1 millisecond. The analyses were performed on a Cray T3E-1200 at ERDC Major Shared Resource Center. The scalability was excellent, as the analysis required about 8, 4, and 2 CPU hours on 128, 256, and 512 processors (PE's), respectively, of the CRAY T3E--an analysis that would take over 41 days on a single processor. Because of a communication-computation

Figure 1. Finite element model of blast load in reinforced concrete wall (995,192 hexahedral elements, 1,030,089 nodes).

233 overlapping algorithm in ParaAble, communication time for partition interface data was insignificant, thus achieving the near perfect levels of parallel efficiency. The damaged portion of the wall was predicted with a pressure dependent-effective inelastic strain damage model implemented in conjunction with the microplane model. The damaged portion is depicted in Figure 1 and compared favorably with the experimental observations.

5. CONCLUDING REMARKS Parallel computational capability was shown to be invaluable for large-scale application of a complex computational approach for blast loading in reinforced concrete. Analyses that would require over 1000 serial computing hours were performed in only a few hours on a large CRAY T3E platform. Although the methodology possesses some distinct benefits for complex modeling of nonlinear structural behavior, its computational expense may preclude its frequent use for large applications on serial computers. With the aid of high performance computing, however, the viability of the method has been greatly extended.

ACKNOWLEDGEMENTS The work is sponsored in part by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-003/contract number DAA-95-C-0008, the content of which does not necessarily reflect the position or policy of the government, and no official endorsement should be inferred. The work was also supported in part by a grant of computer time from the DoD HPC Center at U.S. Army Engineer Research and Development Center.

REFERENCES 1. Bazant, Z.P. et. al., "'Microplane Model for Concrete. 1" Stress-Strain Boundaries and l:inite Strain; II" Data Delocation and Verification",,/. Engng Mech. 122(3) 245-262 (1996). 2. Danielson, K. T. and Namburu, R. R. "Nonlinear Dynamic Finite Element Analysis on Parallel Computers using FORTRAN 90 and MPI", Advances in Engineering St~ware 29(36) 179-186 (1998). 3. Hoover, C.G., DeGroot, A.J., Maltby, J.D., and Procassini, R.J. ParaDyn - DYNA3D for massively parallel computers. Engineering, Research, Development and Technology FY94, l,awrcnce l, ivermore National Laboratory, UCRL 53868-94, 1995.

234 4. Plimpton, S., Attaway, S., Hendrickson, B., Swegle, J., Vaughan, C., and Gardner, D. Transient dynamics simulations: parallel algorithms for contact detection and smoothed particle hydrodynamics. Proceedings of SuperComputing 96, Pittsburgh, PA, 1996. 5. Karypis, G. and Kumar, V. A fast and high quality multilevel scheme for partitioning irregular graphs. Technical Report TR 95-035, Department of Computer Science, University of Minnesota, 1995.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

235

Fracture m e c h a n i s m of pre-split blasting A.V. Dyskin and A.N. Galybin Department of Civil and Resource Engineering University of Western Australia, Nedlands, WA, 6907, Australia

The paper investigates and models the mechanism of pre-split blasting, a technique used in excavations in hard rock and based on creating large splits by simultaneous blasting of thin, closely placed blastholes. The critical spacing between the blastholes is found separating the case of splitting from the case of bulk fracturing of the rock. The determination of the critical spacing is based on the comparison between the blasthole pressure required to start the preexisting cracks with the pressure required to bring the cracks to the length at which the interaction between them will make their propagation unstable (dynamically). If the initial cracks are relatively large such that the pressure of the crack start is less than the pressure of unstable growth, other cracks will appear fracturing the bulk of material. If the initial cracks are small enough, the starting pressure will be high, hence the cracks will propagate dynamically until the coalescence with the neighbours, which produces splitting. I. INTRODUCTION Pre-split blasting, a technique used in excavations in hard rock, is based on creating large splits by simultaneous blasting of usually thin, closely placed blastholes (e.g., Brady and Brown, 1995; Hock and Bray, 1997). Currently the parameters of this method are chosen based on experience rather than on an understanding of the fracture mechanics. A common misconception is that the splitting cracks are driven by gases penetrating the crack volume. This is obviously not the case, since the dynamic crack speed of one third of the elastic wave velocity in solids is an order of magnitude greater than the speed of sound in gases. The present paper attempts to investigate and model the mechanism of pre-split blasting. In particular, the critical spacing between the blastholes is determined which separated the case of splitting from the case of bulk fracturing of the material. Since the pre-splitting occurs through the formation of fractures connecting the boreholes it is natural to base the investigation of its mechanism on the consideration of crack growth form the pressurised holes. The consideration will be conducted only for the static case assuming that this will provide an initial approximation for the real dynamic mechanism of the splitting formation. This static modelling is important per se since it is relevant to fracturing using the non-blasting expanding materials.

236 2. MECHANISM OF FORMATION OF THE SPLITTING FRACTURE 2.1. Crack growth due to borehole pressurising Consider a chain of holes drilled parallel to each other along a line The holes will be assumed of equal radius, r and drilled at equal distance d Each hole is subjected to internal pressure p The influence of the in-situ stress is neglected In the case of a single hole, the pressurising will create cracks initiating from pre-existing flaws near the hole surface and growing from the surface in all directions, Figure 1 Therefore, in the absence of remote stress field the only mechanism creating the preferential direction of the crack growth and formation a single splitting fracture can be the interaction between the holes in the chain, Figure 2 It is also natural to assume that the formation of the splitting fracture will be caused by the coalescence of cracks growing toward each other from the adjacent holes The coalescence will probably occur in the form of overlapping due to the tendency of the tensile cracks to avoid each other ( e g , Melin, ! 983) The aim of the following consideration is to investigate the mechanism of preferential growth of the cracks that form the splitting fracture

...._ |

0

i

I

i

2r

~

-

9

i

,L

f,,

I

.

.

.

d

.

.

.

.

.

.

.

.

.

.

I

:'.

v i

i

Figure 1. Hole with initial flaws.

Figure 2. Formation of splitting fracture.

2.2. Crack interaction Consider the case when the distance between the holes is considerably greater than their radius, d >> r. In this case the cracks forming splitting fracture have to grow at the length much greater than the hole radius. Therefore in order to investigate the crack interaction, the hole with collinear cracks will be modelled in 2-D as a single crack opened by a pair of concentrated forces at its centre, Figure 3. The magnitude of these forces will be taken equal to the force per unit length of the hole created by the internal pressure on the upper/lower half of the hole

P= 2re

(1)

The influence of the gas flow into the crack will be neglected since the gas speed is an order of magnitude lower than the speed of crack propagation. The configuration for modelling the crack interaction is shown on Figure 4. It is a periodic array of equally inclined cracks. The consideration of inclined cracks in the chain is necessary since the cracks from the hole can in principle, be initiated at arbitrary angle.

237

I

i I I

! I

I

,

I

I

v

1

I

i

F

I

P

I

I

I

I

1

I I I I I .d

21

Figure 3. Model of a hole sprouting two cracks.

The solution of the elastic problem shown in Figure 4 is reduced to the following singular integral equation, Savruk (1981): 1

(2) -l where P is the magnitude of the concentrated force, 6(rl) is Dirac's delta function, g(~) is the density of the displacement discontinuity across the cracks normalized by the half crack length, 1:

g-(~'-)= 2(1 + v) d~ Uy K(r = - ~ Re exp(ict) cot

,2

--~-(exp(-kx) - exp(- 3iot)) cot rc~ exp(- i(x) 2

(3)

r t ~ exp(- iot) 2 sin 2 ~M; exp(- i~) 2

Here G is shear modulus, v is Poisson's ratio, d is distance between crack centers, ~,=21/dis the dimensionless crack length. The kernel of the integral equation is a singular kernel of the Hilbert type. It also contains a regular part of non-degenerate type, hence its solution cannot be obtained analytically. The Gauss-Chebyshev numerical integration pale for the singular operator was used to obtain the numerical solution (eg, Savruk, 1981) and, subsequently, the values of the stress intensity factors. In a particular case of collinear cracks (~=0 ~ they coincide with those provided for in Tada et al. (1985). For parallel cracks (ct=90 ~ the mode I stress intensity factor,/(i is in good agreement with an approximate analytical solution obtained in Savruk (1981). The results of the calculations for K~ are shown in Figure 5. It is seen that for shallow crack inclinations there are points of minimum of Kr A similar situation exists for the plots of the energy release rates. Therefore, regardless of the criterion of crack propagation the following conclusion can be made. Before the point of minimum of K~, the crack growth is stable, ie,

238 every step of the crack elongation requires an increase in the load. After the point of minimum, the growth becomes unstable (dynamic), ie, the crack elongation can be sustained even under decreasing load. Obviously, the existence of the regions of unstable crack growth underpins the mechanism of the splitting fracture formation.

Figure 4. Periodic array of inclined cracks.

K!

a=0~

/

a=30 ~ _

3

2

1 ct = 9 0 o

0

0

4~

o.5

1

"

1.5

- i~. . . . . . . . .

2

~-

21/d

Figure 5. Normalised mode I stress intensity factor vs. length for different crack inclinations. Another role the interaction plays is in straightening the crack trajectory. Indeed, regardless of the initial angle of crack inclinations, the mode II stress concentrations will try to reduce the deviation from the collinear arrangement. Subsequently, the growth of interacting cracks ends up in forming the splitting fracture connecting the hole centres, Figure 2. Therefore, the formation of the straight splitting fracture is a result of the interaction rather than the action of the in-situ compression directed parallel to the row of holes, as commonly believed (e.g., Brady and Brown, 1995). Figures 6 and 7 show respectively the crack length and the magnitude of Kz at the points of minimum for different crack inclinations. They show that the variations of the crack length and the magnitude of K/are not very high. Being added to the tendency of the cracks to arrange themselves collinearly, this implies that it is sufficient to conduct the analysis only for collinear cracks (or=0~

239 For collinear cracks the expression for the stress intensity factor (e.g., Tada et al., 1985) and the derived expressions for the crack length, l~r,and the stress intensity factor, Klnfin at the point of minimum have the form

K, pIdsin dl] -!/2 lcr d KI =

~

- -

'

-

-

4

~

rain 9

--"--

P

(d) -'/2

(4)

Using the conventional criterion of crack growth

Kr=Kt~

(5)

where Kz~ is the fracture toughness and expression (1), one finds the pressure, pf required to reach the unstable crack propagation and therefore the formation of the splitting fracture

K,r Pf = - ~ r ~/2

(6)

d 0.55 0.6 0.5 0.58

0.45 0.4

i

0~

10~

20~

30~

ct

Figure 6. Critical crack length versus crack inclination.

0.56 0~

10 ~

20 ~

30~

r

Figure 7. The minimum value of mode I stress intensity factor versus crack inclination.

3. CRITERION OF SPLITTING FRACTURE FORMATION When a row of holes is pressurised two mechanisms are in competition. One of them is the formation of splitting fracture; this requires pressure p~ Another mechanism is the growth of radial cracks from the hole contour, Figure 1. Let the pressure required to start the growth of a pre-existing surface flaw of length a0 be po.ack.Obviously, if pcrack
240 p~,o~k=pe

(7)

To find pcraa one can consider a single hole with a radial flaw (crack) of length a0<
(8)

The substitution of (8) into (5) gives the value for pcrack. Then the criterion (7) gives the critical value for the distance between the holes 7.13r 2 d~, ~ ~

(9)

~a o

When the distance between the holes, d<de,,the splitting fracture will be formed, otherwise the radial cracks will grow and break the bulk of the material. It is interesting that the critical distance is inversely proportional to the size of initial flaws. This implies that the smoother the hole surface the higher the critical distance.

4. CONCLUSIONS Critical spacing of the splitting fracture formation is determined by equating the blasthole pressure required to start the pre-existing cracks to the pressure required to drive the cracks to the length at which the interaction between cracks makes their propagation unstable, If the initial cracks are relatively large such that the pressure of the crack start is less than the pressure of unstable growth, other cracks will appear fracturing the bulk of material. If the initial cracks are small enough, the starting pressure will be high, hence the cracks will propagate dynamically until the coalescence with the neighbours, which produces splitting. The considered mechanism is insensitive to the in-situ stresses. REFERENCES

Brady, B.H.G. & ET. Brown, 1985. Rock Mechamcsfor Underground Mining. George Allen & Unwin. London, Boston, Sydney. Hoek, E and J.W. Bray, 1997. Rock Slope Engineering. E & FN Spon. London. Melin, S., 1983. Why do cracks avoid each other? Int. J. Fracture, 23, No. 1, 37-45. Savruk M. P., 1981. Two-Dimensional Problems of Elasticityfor Body with Cracks, Naukova Dumka, Kiev. Tada, C. Paris and G. R. Irwin, 1985. The Stress Analysis of Crack. Handbook. Paris Production Incorporated & Del Research Corporation. St.Louis, Missouri.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

241

Evaluation of energy absorption system for intense shock mitigation I.J.L.

Jaggi,

Rajesh Kumari, Harbans Lal, and V.S Sethi

Terminal Ballistics Research Laboratory, Sector-30, Chandigarh-160020, India

Abstract Inert materials having high order of porosity and low shock impedance characteristics act as good shock attenuators and energy absorbers. The shock absorbing capability of a material can be enhanced by creating air voids distributed over the material. In this paper we have studied the shock attenuation characteristics of alluvial soil and polypropylene. The shock attenuation efficiency of these systems has been discussed. The dry soil has shown 26% better attenuation characteristics as compared to water saturated soils. For polypropylene, it is found that a void of size 5 mm dia shows better shock mitigation. 1. INTRODUCTION The paper discusses the characteristics of various materials used as shock attenuators and the experimental methods for investigating these characteristics. The materials which are generally considered as shock attenuators have a high degree of porosity (e.g. alluvial soils). The study in shock compression and attenuation has been discussed in details by Rice et al~ The shock propagation and its attenuation in heterogeneous media (e.g. soil) is discussed in detail by Joseph Henrech <:). The experimental technique used fbr shock attenuation studies comprises direct and indirect measurement of shock pressure transmitted through these materials. In the indirect method, the pin oscillogravhic teclmique<3) has been employed for measuring transmitted shock pressure, whereas in the direct technique, the high pressure transducers are used. Shock attenuation capabilities of different materials viz. alluvial soil and polypropylene have been experimentally studied. 2. ENERGY ABSORFrlON PHENOMENON The intensity of the shock wave in a medium can be mitigated by creating low impedance barriers in the path of shock wave. These barriers are generally provided by creating voids in the medium. Alluvial soil is one such medium which acts as a good shock attenuator. Alluvial soil is a heterogeneous mixture of coarse sand, fine sand, silt, clay and water. The intense shock generated by detonation of high explosive attenuates very rapidly as it passes through alluvial soils. The cohesive soils have pores with dia of a few microns which can be continuous or discontinuous. Compositions of alluvial soil is shown as under:Course Sand Fine Sand Silt Clay

8.77% 49.42 % 22.33 % 19.48%

242 The energy absorption in soils is due to the deformation of pores and compression of air trapped inside these pores. Soils can be treated as three phase media i.e. mineral grains, water and air phase. The mass of each phase in unit volume is given by: P~o = 1 6 5 0 k g / m 3,1~o = 1 0 0 0 k ~ m 3 and p3o = 1.25kg/m 3

where subscript 10, 20 and 30 represent mineral grains, water and air phase, respectively.

To increase the shock absorption effectiveness, voids of known size have been artificially created in low density polypropylene. The internal structure of the voids in attenuating materials can be one or more of three types as shown in figure 1, namely soda straw, plates and orifices. In shock loading of such materials, some part of the energy is reflected at the interface of void and medium due to impedance mismatch. The quantity of reflected energy depends on the shock impedance and size of the medium. In addition, a part of the material loosened from side layers is filled in the pores thus producing frictional energy loses. The frictional energy loss is determined by the resistance of material flow within the matrix, vertically, horizontally or diagonally. Voids with plate type of internal structure were created in polypropylene samples. The effects of the size of voids on shock attenuation characteristics of materials were experimentally studied. 3. EXPERIMENT Instrumented trials were carried out to determine the induced shock pressures in various specimens of soil and polypropylene. The Set up for different experiments is listed below. 3.1 Shock attenuation pressure in soil samples The shock attenuation in the soil samples of diameter 75 mm and 10.5 mm thick with varying moisture content from 0 to 40 % by volume were studied The soil sample was sandwiched between tetryl pellet of 80 gm weight and an aluminium disk. The intense shock generated by detonating the tetryl pellet was induced in the soil sample. The detonation pressure of the tetryl explosive is 23 GPa. The shock velocity in soil and aluminium were measured using foil probes. The probes were fabricated by insulating two aluminium foils of thickness 0.015mm. A typical setup is shown in figure 2. These probes initially act as open switches. On arrival of shock wave, the probes become electrically short and generate Transistor -Transistor Logic (TTL) pulses. These TTL pulses were recorded by a digital storage oscilloscope. The following Hugoniot relation between pressure and shock velocity was used to evaluate the generated pressure in aluminium.

P~ -- Pal Ual(Ual-5.692)/1.108 Where Pa~=Pressure induced in aluminium in GPa pa~-density of aluminium in gm/cc Ua~=shock velocity in aluminium in mm/l~S

(1)

243

"'" i

h i

i

|11 i

i

i

i

i

I

| .

~ _

i

'

__

i l!! i i,

i

Soda straws

Plates

Orifices

Fig 1. Internal structure of artificial barrier

Foil probes

Electric detonator Aluminum plate Det holder Soil sample Tem-I peuet

Fig 2. Set up for soil experiments Induced pressure in soil was evaluated by using the following mismatch relation 4. Psoil = PaI(Zsoil+ Zal)/2 Zal (2) Where Psoii=pressure induced in soil in GPa Pal=pressure induced in aluminium in GPa Z~oi~= shock impedance of soil (i.e.product of shock velocity in soil and density of soil) Za~=shock impedance of aluminium (i.e. product of shock velocity in aluminium and density of aluminium)

3.2 Shock attenuation in polypropylene For shock attenuation experiments in polypropylene, plate type voids of were created in samples. In different trials, void size was varied from 3mm to 7mrn and the inter void distance kept at least equal to twice the diameter of the void. The voids of various diameter were chilled in discs of 5 mm thickness. A number of discs were joined together with adhesive to make a test specimen of desired size.

244

Fig 3. Testing assembly for shock attenuation in polypropylene

Fig 4. Typical oscillographic pressure profile record

These samples have been subjected to shock pressures of the order of 315 GPa. Such pressures were generated by attenuating the explosive pressure through soil samples of 20mm thickness. The transmitted shock pressures through these samples of polypropylene were monitored using a piezoelectric transducer PCB 109 A02. Line sketch of experimental setup is shown in figure 3. 4. RESULTS AND DISCUSSION Table 1 shows the experimental data acquired in trials with soil samples of thickness 10.5mm having different moisture content. Pressure attenuation in soils with higher porosity are higher than in soils having higher moisture content. This shows that shock energy dissipate at a faster rate with air voids. The efficiency of attenuation in dry soils with 40 % porosity is higher than the fully water saturated samples by 26%. Table 2 shows the attenuated pressure recorded in polypropylene of length of 62 mm having different void sizes. Variation in attenuated pressure and the size of void play a vital role. Initially with increase of void size, the attenuated pressure decrease but with void diameter more than 5mn~ the attenuated pressure starts increasing. In smaller size voids, the resistance is too low, the frictional loses are low and little energy conversion ocx~s. With bigger size voids the material loosen l~om side layers is able to create high resistance thus producing less frictional energy and more residual pressure. Table 1 Experimental data with soil sample Sr No.

Relative volume of Soil Water

Air

Density of sample (kg/m3)

Shock velocity in sample U soil(m/s)

Pressure generated Psoil(GPa)

1. 2. 3. 4. 5.

0.60 0.60 0.60 0.60 0.60

0.00 0.07 0.15 0.25 0.40

2180 1980 1880 1810 1650

3360 3530 3300 3218 2740

17.09 16.59 15.35 14.70 12.64

.

.

.

.

.

.

.

.

0.40 0.33 0.25 0.15 0.00 .

245 Table 2 Recorded attentmted pressures in polypropylene Sample thickness (mm)

Void diameter (mm)

Recorded pressure (MPa)

62 62 62 62 62

3.0 4.0 5.0 5.7 7.0

61.5 49.5 24.6 64.9 68.1

5. CONCLUSION From the preliminary results we found that low density polymers like polypropylene with voids can be used to reduce the transmitted shock energy and thus can have applications as shock attenuators. REFERENCES

1. M.H.Rice, g.J.Maqueen and J.H.Welsh,Compression of solids by strong shock wave, Solid state physics vol. I, Academic press, New York 2. JosephHenryeh, Explosive Dynamics, Elsvera publisher 3. V.S.Sethiand S.S.Sachdeva, Diagnostic technique for high speed events, 7th International symposium on structural failure and plasticity, IMPLAST 2000 4. Ya. B. Zeldevike, Physics of shock wave and high temperature hydro dynamic phenomenon, Vol ILAcademie press New York (1967)

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

247

Blast damage effects of an explosion of 5 ton TNT kept in a storage magazine Harbans Lal, RK Verma, MS Bola and VS Sethi Terminal Ballistics Research Laboratory, Sector 30 ,Chandigarh-160020, India The paper describes the air blast effect of a confined explosion of 5 ton TNT stored in a specially designed storage magazine of size 3.6 m X 4.5 m X 3.6m. Brick masonry targets of size 3m X 3m X 3m were constructed at various quantity distances ranging from 2.4 Q 1/3 to 22 Q 1/3 and in varying orientation around the test magazine. Q is the quantity of charge in kg of TNT. Extensive blast and ground shock instrumentation was deployed around the target structures to monitor the air blast and ground shock effects in the vicinity of these structure. The paper discusses the blast structure interaction and the determination of key blast parameters that result in damage. The acquired data was used for developing quantitative correlations for determining the safety distances for brick masonry buildings. I. INTRODUCTION The damage to framed structures under air shock and the effects of impulsive forces on structural members have been discussed by many investigators (1-3). The experimental methods to obtain loading on structures subjected to blast are described by Marriott, Norris & Kenny (4-6). The damage to different structures from high explosive bombs and the damage correlation with the yield, blast characteristics and the target parameters have been discussed by Held, (7). On the other hand lot of work has been done on improving the blast resistant features of the structures exposed to explosion hazards. New construction techniques and materials namely laced reinforced concrete (LRC), steel fibers reinforced concrete (SFRC) and geo-synthetie and composite materials are increasingly being used for the design of blast resistant sWactures. These techniques enhance the blast resistant features by increasing ductility, improving energy dispersal characteristics and by impeding the process of crack formation and spall. The new construction techniques for blast resistant design have resulted in the reduction-of safety distances and better utilization of land resources. The safety distances for various types of explosive storage and explosive processing buildings needs to be studied. For determining the realistic safety distances, an experiment was designed to simulate the accident of 5 ton of TNT in a specially designed explosive storage building: Simple equations are derived for determining the damaging effects of blast waves and ground shock in terms of quantity of charge and distance from the point of explosion. 2. DAMAGE EFFECTS The explosions produce damages to structures due to blast, ground shock and debris. The blast is the most important mode of transferring explosive energy to the stmcraral targets. It loads the entire structure produces large deformations in the structure. The strong ground motion associated with

248 near surface explosion results in vigorous shaking of structures and results in structural damages. On the other hand the debris results in the local damages to the structure. 2.1 Blast damage From the damage point of view the blast wave is fully described in terms of time of arrival, the peak over pressure, the duration of the positive phase and the positive phase impulse. The peak over pressure is a function of charge weight Q and the radial distance R in the form of scaled distance z= R/Q 1/3. The over pressure exceeding 0.1 bar results in the damages to civil structures in varying degrees. The damages to structure are correlated with scaled distances and the threshold for different categories of damages are given in Table 1. Ref (8) The table shows that a larger charges can cause the same level of damage as a smaller charge even though the associated pressure is less. Table 1 The scal.e distance for various type of d a ~ e Damage Damage Description category . . . . . . .

10 ton explosion

I t on explosion Scaled Peak overdistance pressure ( n ~ g 1/3) (kg/cm2)

Cat 'E' Cat 'D'

Cat 'C' Cat'B' Cat 'A'

Window panes broken (10~ Minor damage not necessitating the building to be uninhibited Structural damage requiring uninhibition for repair Damage severe enough to necessitate demolition Complet e collapse

Scaled distance

Peak overpressure (kg/cm2)

38.6 23.7

1.03 0.05

51.6 33.6

.017 .031

11.4-6.5

.13-.28

16.1-9.2

.16-.09

3.6

0.8

5.6

0.36

2.4

1.85 _

_

3.6 i

0.8 i

i

2.2 Ground shock damage. In the near region of the explosion, the ground particles experience unidirectional excursions and has the tendency to flow in the direction of propagation of shock. The shock wave attenuate very rapidly into strong ground motion characterised by oscillatory motion. The ground induced vibratory motion induce a dynamic response in the structures. Table 2 Type of damage to structure related with peak particle velocity (ppv) , Sr.No. Type of Damage .... ppv (mm/sec) 1. Appearance of first crack 50.0 2. Widening of existing cracks 90.0 3. Deep cracks 240.0 .... 4. , Falling of p!aster , . 282.0

249 If the wave length of the transmitting ground vibrations is comparable to the size of the structure, the entire structure tend to move as one. The building can be represented as mass spring system with single degree of freedom (sdf). The damage or the peak deformation in the structure can be related with the peak particle velocity (ppv). The minimum values of ground vibration at which damage is produced in brick structures are given in table 2 ref (9). 3. DESIGN OF EXPLOSIVE MAGAZINE AND TARGET STRUCTURES Two explosive mass storage magazines with storage capacity of 5 ton each were designed using LRC concepts and were located side by side with a clear separation o f 10m. The size of each storage magazine was 3.6m X 4.5m X 3.6m. Several target structures (each of size 3m x3m x 3m) having brick walls of thickness 23cm and RCC roof were constructed around the storage magazines at different distances and in different orientations. The distances of these target structures represented the safe distances for various types of activities. The photographs of the storage structures and the brick target structures are shown in figures 1 and 2 respectively. The layout of the target structures with the explosive storage structure is shown in figure 3.

3.1 Experiment details Simulated accidental explosion was conducted by detonating 5 tons of TNT crystalline powder of TNT stacked in 200 wooden boxes in one of the storage building. Piezoelectric crystal based pressure gauges were deployed in each of the target structure to record the incoming blast pressures. The velocity pick up were also fixed near each of the target structures to record the free field particle velocity.

Figure 1. Storage Magazine before trial. Wall to wall distance 10m

Figure 2. Brick Masonry walls with RCC roof target structure 3mx3mx3m

250 C] 137 m , ,

Earth covering Storage [Zl41m r J MS Door

'~1--

'Om []

El 62m

--~'

"'

"

82m

IZ! 137m 253m El 253m i~ 380m

[] 3mx3m Brick wall smletmr

Figure 3. Plan Layout of 5 Ton TNT trial 4. EXPERIMENTAL DATA Table 3 The blast data recorded at different target locations ~ elS r 9Distance(m) Scaled Peak overDurati no distance pressure on (m/K~; ~/~) (Kg/cm 2) (ms) 253(fr0nt) 14.8 0.07 83 137(front) 8.0 0.17 65 82(front) 4.8 1.02 55 62(front) 3.6 6.5 39 _ _ 137(rear) .... S.O. . . . ...... -

Impulse (Kg/cm2. ms)

Category of damage

2.7 5.03 16.5 47 -,

Cat'D' Cat'C' Cat'B" Cat'A" No damage i

Table 4 The record~ values Of particle velocity Sr Distance (m)

ii

_

_

ppv(mm/s)

no.

1 2 3 4 i,

62 82 137 253 iJ

,

75 44 19 8 i

i

,

i

|l|

|

5. DAMAGE DESCRIPTION After the explosion, the experimental storage magazine containing 5 tons of TNT was completely wiped out and a crater of apparent diameter of 16 m and apparent depth of 3.35 m was formed, whereas the adjoining storage magazine was safe and reusable. Figure 4 shows the view of the

251 survived magazine structure after the trial. The brick target buildings within the radial distance of 62 m were completely destroyed by the ground shock and the blast loadings. Figure 5 shows the collapsed target at 62 m. The targets within the radial distance of 82 m were destroyed beyond repair and the structural damage were mainly due to blast loadings. Figure 6 shows the view of a target at a distance of 82 m. The target at a distance of 137 m on the front side developed minor cracks and was hit by the debris on thefront wall resulting in the spall on the inner side whereas the target on the rear side at the same distance did not experience any damage due to blast and ground shock loadings. A big chunk of concrete piece fell near the target structure. Figure 7 shows the target structure at a distance of 137 m on the rear side. The target structures at 380 m suffered insignificant structural damage due to blast and ground shock but suffered repairable local damage due to debris hit. Broken pieces of reinforcement of size up to 30 cms were found up to a distance of 900 m from the point of explosions. The window panes, of the buildings situated up to a distance of 1000 m and directly exposed, were damaged.

Figure 6. Structure damaged Cat B at a distance of 82 m

Figure 7. Structure at a distance of 137 m after trial

252 6. SAFETY DISTANCES The blast damage data acquired in the trial is grouped in terms of sealed distances or quantity distances in Table 5. Table 5 Damage to storage magazine at various sealed-distances ............... Sealed distance Damage de~-ipJicm (Kg/m u3) -

0.5 0.7 3.6 4.8 8.0 14.8 22.0 60.0 >120.0 . t _

Crater Zone. No civil ~ e issafe LRC storage magazine with earth traversed in side-side configuration safe for HE contact. Cat 'A' damage to brick masonry target Cat 'B' damage to brick layout Cat 'C'. Structural damage due to blast and local damage due to debris hit. Cat 'D'. Minor structural damage and minor local damage due to debris hit. No structural damage, local superfluous damage due to debris hit Window pane cracking, superfluous damage due to blast and minor local damage due to debris hit Safe z o ~ with respect' to blast~ grotmd motion & debris: . . . . . . . . . . .

7. EFFECTS OF ORIENTATION OF TARGETS

The air blast produced in the contained explosion was highly directional in the near region. The blast overpressures produced in the direction of venting were considerably higher than those experienced in other directions. The brick targets located on the front side were exposed to higher levels of blast as compared to those at the same distance in other directions. The recorded blast overpressure at a distance of 62 m in the front direction was about 3 times higher than that expected from a blast of 5 ton explosive in the open. The target located at 137 m in the front direction experienced higher level of damage as compared to a similar target at 137 m in the rear direction. 8. CONCLUSION The earth covered storage magazine can be located in side-side configuration within quantity distance of 0.7 (m/Kgla). Higher quantity distances for safety are required for structure located in the direction of venting. Scaled distances for different types of damage are comparable with those reported in literature. The safety distance for structure and occupants should be based on all the three effects blast, ground shock & debris.

253 9. ACKNOWLEDGEMENT Authors are thankful to Dr AK Bhalla, Director CEES, Delhi for his valuable guidance in the conducting of the trial. Authors are grateful to the team members Shri IJL Jaggi, Mrs Rajesh Kumari, Shri Dhan Parkash, Shri SS Dhadwal, Shri Ramesh Chander, Shri PK Khosla and Mrs Nidhi Sood. Authors acknowledge with thanks the excellent secretarial support provided by Mrs Pankajavally. REFERENCES

1. J.F. Baker, EL.William, and D Lax, Civil Engineers in War, Vol 3 published by the Institute of Civil Engineer, London, (1948), Page 80 2. W.H. Thomas, Civil Engineers in War Vol. 3 published by the Institute of Civil Engineers, London (1948), Page 65 3. M.L Barman, Proceedings on the Symposium on Earthquake and Blast Effect on Structure, edited by CM Duke and M Feign, Los Angeles, California, June 1952,Page 233 4. M.L. Marriott, Proceedings of the Symposium on Earthquake and Blast Effects on Structure edited by C.M Duke & M Feign, Los Angeles California, June 1972,Page 74 5. G. N. Norris, R.J. Hanran, Molley Jr, Biggs J M etc. Structure Design for Dynamic Loads (Macron Co. Ltd 1959) 6. Kenny GF, Explosive Shocks in Air published by Macmillan in (1962) 7. M. Held, Blast Waves in Free Air Propellant, Explosive, Pyrotechnics 8,1-7 (1983) 8. "Blast Effects on Buildings" edited by GC Mays & PD Smith published by Thomas Telford Publication - (1995), Page 55 9. Professor P Roy, Characteristics of Ground Vibration and Structure Response to Surface and Underground Blasting, Indian Journal, Geotechnical & Geological Engineering (1998), 16, 151-156

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta @ 2000 Elsevier Science Ltd. All rights reserved.

255

Evaluation of damage and TNT equivalent of ammunition, explosive and pyrotechnics Pushpa Buri, MM Verma and Harbans Lal Terminal Ballistics Research Laboratory, Sector 30, Chandigarh-160020, India

A large explosion in general produces widespread destruction because of associated shock and blast waves. An explosion is a rapid release of energy in atmosphere and is characterised by the formation of shock wave subsequently resulting into a blast wave. The blast wave propagates in the media and interacts with structure produces large deformations. Methods to quantify the explosion of ammunition, high explosive and pyrotechnic and also the principles involved in determining their TNT equivalent with respect to blast are described. The experimental technique for determining the TNT equivalent of ammunition, pyrotechnic and explosives is discussed. The blast damage correlations as applicable to various types of targets are also discussed.

1. INTRODUCTION Sudden release of energy in atmosphere at a rate higher than the speed of sound results in the formation of a system of blast wave. The strength of blast waves depends on the amount and the rapidity of energy release. The blast waves attenuate in strength as it propagates outwards from its point of formation and at far off distances degenerate into sound waves. The blast interacts with structures and produces large deformations. The mechanism of blast structure interaction is described in reference (1,3). The blast effects of any type of explosion can be quantified in terms of equivalent TNT yield. The blast waves follow cube root sealing laws based on elastic forces. Safety distances for different types of activities and various categories of damage earl be defined in terms of sealed distance defined as the ratio of radial distance and cube root of the yield.

2. BLAST WAVES Blast waves are fully described by the time of arrival, peak-over pressure, positive phase duration, positive phase impulse and negative phase impulse. Over pressure is the peak over pressure in excess of ambient pressure. Positive duration is the time in which the over pressure decays to the ambient pressure. The positive phase impulse is the measure of the energy carried by the wave and is equal to the area under the blast curve. The negative phase is formed by the inertia of the media and has a longer duration than positive phase. For low yield conventional explosions, the damaging effects of negative phase are negligible. The propagation of blast waves is governed by the elastic forces of the media and therefore follows cube root scaling laws. That is the spatial and temporal dimensions of blast waves are scaled according to the cube root of the yield. Self-similar blast waves are produced at idemical

256 scaled distances i.e. peak over pressure, particle velocity, shock velocity and the temperature in the shock front remains same at the identical scaled distances. The arrival time, duration and impulse will be scaled according to the cube root of the yield. Peak over pressure, scaled impulse and scaled duration are functions of scaled distances (4,5).

3. TNT EQUIVALENT OF EXPLOSIVES AND AMMUNITIONS The blast effectiveness of any type of explosion is computed by the concept of equivalent weight. Explosion from any source be it chemical, mechanical or nuclear can be quantified in terms of equivalent explosive weight of standard TNT that generate blast of equal force at identical scaled distance. The equivalent weight for any ammunition, propellant or pyrotechnic is computed with respect to equal over pressure or equal impulse effects. For ammunitions, it depends on the shape of the charge, metal to charge ratio and the type of the explosive charge. The energy of explosive in ammunition is partitioned between the kinetic energy of fragments and blast depending on the metal to charge ratio and the type of explosive. The bare equivalent of an ammunition of cylindrical shape is given by

(1)

Weq = w . C / M + 2 a C/M+2

Where W is the weight of the explosive filling, C/M is the ratio of charge weight (C) to fragmenting mass (M) per unit length of the ammunition and a is the partition coefficient depending on the type of explosive and its value varies between 0.15<~<0.25. For Composition B the value of a is found as 0.15 and for aluminized explosive like DENTEX and TORPEX the value of a is found to be 0.25. The details of these High Explosive compositions are given in table 1. The average free air equivalent weights of some explosive types are given in table 2. Ref (6) Table 1 Co~os!~[ions of DE .NT.EX,: T 0 ~ E X a n d ~Composition B ................................................................................. SN Name .... Density ( ~ ) ..................... C0mposition Percentage ........... RDX/TNT/AL/WAX 1 DENTEX 1.75 (48.5/33.5/18/0.5) RDX/TNT/AL/WAX 2 TORPEX 1.76 (41/41/18/1.0) RDX/TNT 3 Composition B 1.6

_ 0/4o)

........

4. TNT EQUIVALENT OF PYROTECHNICS The pyrotechnics does not detonate under normal conditions of initiation and does not result in the formation of blast waves. But under conditions of confinements, these compositions have a finite probability of detonation. For safety considerations and hazard evaluation, the knowledge of TNT equivalent of these compositions is of paramount importance. The

257 Table 2 Average TNT equivalents SN Type of explosive 1 2 3 4 5 6 7 8

AFNO(ammonium Nitrate Fuel Air) Composition B Cyelotol HBX-1 HBX-3 HMX/TNT 70:30 TNT Lead Azide

~,IT Equivalent w.r.t Overpressure Impulse 0.82 0.07-7 1.11 0.98 1.13 1.09 1.17 1.16 1.14 0.97 1.20 1.11 1.0 1.0 0.34 -

Pressure range (Kg/cm2) 0.3 to 3 0.3-3.5 0.3-3.5 0.3-2 0.3-2 0.2-2 0.2-2

detonation of pyrotechnic depends on the degree of confinement, the weight and the type of pyrotechnic. A study was undertaken to determine the TNT equivalent of pyrotechnics.

4.1 Pyrotechnic compositions Magnesium based pyrotechnic compositions were chosen for determining their TNT equivalent and for determining their damage causing potential. The details of these compositions are given in table 3. Table 3 Pyrotechnic SR 703

SR 588

Contents Magnesium Powder Grade I- 42+1.5%,Boiled linseed oii[-"6~i5~ "................... Barium Nitrate Grade I - 15+1%, Potassium perchlorate - 25 +1%, Chlorinated rubber - 12 + 0.5%,Volatile matter - 0.2% maximum. Magnesium powder grade O - 55+_2.0%,Varnish Lithographic - 4_+0.5%,

4.2 Trial Setup Pyrotechnic composition weighing 50 kg was taken in a cylindrical box with length to diameter ratio of one. The pyrotechnic compositions were initiated with a detonation train consisting of tetryl pellet of dia 2" and electric detonator. Piezoelectric crystal based blast gages were deployed at distance ranging from 9 m to 15 m. The trials were repeated to acquire reliable data. Trials were also conducted with 50 kg TNT cylindrical charges having the same length to diameter ratio and the same height of burst. The blast pressures were also monitored in the trial with TNT charges at identical distances in the identical field conditions. A typical field setup for the trials is shown in figure 1 and a view of explosion of pyrotechnic composition captured by high speed video is shown figure in 2. The photograph clearly shows that the detonation of pyrotechnic composition is non-standard and part of the material remains undetonated and sprinkles around the point of explosion. A typical record of the blast waves at various distances in the trial with SR 588 composition pyrotechnic is shown in figure 3. The detonation of pyrotechnic composition resulted in the formation of blast waves of lower peak over pressures as compared to that formed in the case of TNT at the identical distances.

258 4.3 T N T equivalent

The blast data acquired in the trials with two pyrotechnic compositions are plotted on loglog scale in figures 4(a) and 4(b) respectively. The principle of equi-pressure scaling laws has been used for determining TNT equivalent. The weight of equivalent TNT charge is determined from equi-pressure scaling laws. The weight of TNT charge is determined from the graphs, which will produce the same overpressure at identical distances. TNT equivalent was determined with respect to 0.4 kg/cm2 peak overpressure. 4.4 Observations

The TNT equivalent for composition SR 703 and SR 588 for charge weights 50 kg varies from 0.161 to 0.233 and 0.083 to 0.127 Kg of TNT/Kg of pyrotechnic respectively. The trials were also conducted with 5 kg, 10 kg & 20 kg charge weights of these pyrotechnic compositions. There Were no consistency in the detonation pattern, however the maximum TNT equivalent for these weights were found to be less than that for charge weights of 50 kg. It can be safely assumed that the TNT equivalent of these compositions for charge weights less than 50 kg are smaller than 0.233 and 0.127 kg of TNT/Kg of pyrotechnics for SR 703 and SR 588 respectively. The TNT equivalency of pyro-composition varies from firing to

Figure 2. A view of explosion of pyrotechnic composition captured by high speed video

Figure 1. Trial set up

T

O.21

9.0m 10.5m 12.0m

0.14

0.14 Kg/cm2/div I div= lores

Time Figure 3. A typical record of blast waves for SR 588 composition

13.5m

259 firing and a range of equivalency has been determined for each composition. It also varies with the reference peak over pressure used for determining equivalency in the equi-pressure laws. The reason of variability lies in the non-standard detonation of pyrotechnics.

5. DAMAGE EVALUATION AND SAFETY DISTANCES For determining the safety distances for laying facilities, the extreme values of TNT equivalent should be taken. For SR 703 composition, the TNT equivalent should be taken as 0.233 Kg of TNT/Kg of pyrotechnics and for SR 588, the TNT equivalent should be taken as 0.127 kg of TNT/kg of pyrotechnics. The safety distances for compositions can be determined by taking the quantity of TNT as 0.233 Q for SR-703 and 0.127 Q for SR 588. Where Q is the quantity of pyrotechnic in Kg. The scaled distances for different type of damage because of pyrotechnics are reduced as cube root of the TNT equivalent. The comparison of safety distances for pyrotechnics and TNT for various types of damage are given in table 4. The safety distances for TNT are taken from (7).

t.,

1.0 0.9 0.8 0.7 0.6 0.5

SR-703~

~50Kg.

,'~ 0.9 1"0 f 0.8 0.7 0.6)" r/) 0.5 ' 0.4

TNT

0.4

~ 0.3 O ~ 0.2

0.3 O

0.2

0.1 -

56

9 A

9

~

it

, .

,

t

i

,,

i

!

t z

i !

il

78910 20 "" DISTANCE(m)

30

40

Figure 4 (a) Blast data of SR-703 compositions

\'\

\"

\ 1

! i

,

II

i

TN~

...... "'

|

ilX\ :i

l

50 K g ,

..... "''~" ~~.__,,.~.~i ...... "" "~""

!

9 ~

I

~ SR-588

0.1

: |

: s |.

56 78 910

20

- - -

30 40

-------~ll* DISTANCE(m) Figure 4 (b) Blast data of SR-588 compositions

260

Table 4 Sca!edfi!St~Cesforvgigu_s ~es~gfblast dm.ag~_ .......................................................................................... Sr. Structuraldamage to brick masonry buildings Scaled distance I ton of charge No. m/kg 1/3 TNT SR-703 SR-588 1. 2.

Window panes 10% broken 72 Damage calling for urgent repair but is not severe 23. enough to make building uninhibited. 7 3. Minor structural damage rendering the home 6.5 temporarily uninhibited 11. 4 4. Damage severe enough to necessitate demolition 3.6 ......5:..................comp!ete ~collapsoe~ _ 2:4 ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44.3 14.6

36.2 11.9

4-7

3.3-5.7

2.2 1.5

1.8 1.21

6. CONCLUSION The TNT equivalent of any type of ammunition, explosive and pyrotechnic can be determined experimentally and the safety distances in terms of equivalent TNT quantities can be estimated. The pyrotechnic composition Can be detonated under condition of confinement and their TNT equivalent various from trial to trial because of non standard detonation. For determination of safety distances, the highest value of TNT equivalent of pyrotechnic should be used. 7. ACKNOWLEDGEMENT The authors are thankful to the Director TBRL for his valuable guidance and encouragement for publication. The authors acknowledge with thanks the excellent secretarial assistance provided by Mrs Pankajavally. REFERENCES

1. Graham Kinney, Explosive Shocks in Air, Springer- Verlag Berlin Heidelberg, New York, Tokyo, 1985 2. W.E. Baker et.al., Explosion Hazards & Evaluation, Elsevier Scientific Publishing Company, Amsterdam, Oxford- New York, 1983 3. J. Henrych, Dynamics of Explosion, Amsterdam Elsevier Scientific Publishing, 1979 4. M. Held, "Blast Waves in free air", Propellants Explos. 8, 1-7,1983 5. M. Held, "TNT equivalent", Propellants Explos. 8, 158-167,1983 6. Conventional warhead system. Physics and engineering design. Chapter 6 Richard M Lloyd, Raytheon Systems company, Tewksberg, Massachusetts. Vol 179 Program in Astronautics and Aeronautics published by American Institute of Aeronautics & Astronomy, 1998 7. GC Mays and PD Smith, Blast Effects on Buildings, published by "Thomas Telford", !997.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

261

New approach to street architecture to reduce the effects of blast waves in urban environments. Ehab H. Mahmoud a and John. G. Hetheringtonb a e, b Engineering Systems Department, Cranfield University, Royal Military College of Science, Shrivenham, Swindon, Wiltshire, SN6 8LA, United Kingdom

"Protection against Terrorism" is an emerging factor which influences the urban planning process offering a new dimension to town planning. Although the risk to any individual structure of becoming the object of a terrorist attack is very small, analysis of terrorist attacks in the last decade indicates that this risk is greatest for facilities in city centres and public spaces. In this case the entire urban environment experiences the effects of the blast wave, which can cover miles of the surrounding areas. The blast protection process has to be a joint responsibility, in which several specialists have to take part. Each specialist has his own role and ideas in reducing the effect of the threat to an acceptable level. The Urban Planner may make a contribution to the mitigation of blast damage of structures by optimising town planning facets (streets, squares, land use, street architecture... etc.). These facets can play a significant role in blast wave management by the confinement, shielding, venting and dissipation through the spaces providing a measure of protection without compromising appearance and utility. This paper presents the findings from the phase of a programme of research which explores the protection offered by different street geometries against blast and the influence of space architecture, in urban environments, as assessed using the AUTOD'~q simulation package 2D & 3D V3.1.15 [Ref. 1].

1. BACKGROUND The Isle of Dogs is an area where planning procedures have been circumscribed; it is the flagship of market enterprise, business centre and free economics rule the day. In the 1980s many urban design studies were made to develop this area. All of them used urban planning parameters like unity, proportion, scale, harmony, symmetry, contrast, ...etc. On 9th February 1996, a vehicle bomb, carrying a charge equivalent to 200 Kgs of TNT, exploded in South Quay Street in the London Docklands. The effect of the blast exploded an area of 2 miles radius from the explosion. This case is one of many vehicle

262 bomb attacks, which have taken place in a city centre in the last decade. This means that there is emerging a new contemporary urban planning parameter being that of protection.

2. REFERNCE SCENARIO The geometry of South Quay street in the London Docklands, with some modifications, has been taken as a reference scenario with which all simulations will be compared. It is shown in figure 1. The AUTODYN simulation package V3.1.15 in 3D was used to evaluate the pressure and impulse values on the structures and the ConWep program [Ref.2] validated these values.

Figure 1. The Reference scenario (Layout of South Quay street, London Docklands.)

3. PARAMETER AND LIMITS OF SIMULATIONS The limits of the simulations were defined by groups of test simulations, which were analyzed carefully to define the most appropriate design limits, taking in consideration the following parameters: 1) Height of the structures. 3) Street shape. 2) Distance of the structure from the blast. 4) Damage category. 3.1. The height of the structures Due to the variation in the South Quay street buildings height, the analysis fixed the building height for all the simulations to represent typical potential target buildings. The assumed building characteristics were as follows: eight-storey (25m high), reinforced concrete structure system; the typical storey height was 3.00 m; and the first floor was 4.00 m high. An average of impulse and pressure values is presented for specific target points located on two sides, which form the street shape of the reference scenario [See figure 1]. 3.2. Distance of the structure from the blast The location of the blast was taken to be the location of the vehicle bomb at 10m standoff from the structure. It was selected to correspond to the Doeklands Bomb. [See figure 1].

263

3.3. Street shape Three major scenarios, as shown in figure 2 & 3, were investigated as candidate street forms to identify which can offer reductions in the effects of the blast. These scenarios are: 1. Straight street without sub-streets. [Closed sides]. 2. Curved street with one curve with perpendicular sub-streets. 3. S-shaped street with two curves with perpendicular sub-streets.

A. Straight street (Reference scenario).

B. Straight street (Closed).

264

Figure 3. Typical various street shapes in Autodyn simulations.

3.4. Damage category Damage was classified into five categories [Ref. 3 p. 13]. This classification depended on weight of charge and stand- off distance. The categories were as follows: l

Category A: Buildings completely demolished. 75% of the external brickwork demolished.

9 Category B: Buildings so damaged that they are beyond repair and must be demolished when opportunity arises. 50 - 75% of the external brickwork is destroyed. l

Category C: Buildings that are rendered uninhabitable with the roof and one or two external walls partially collapsed. Load bearing partitions would be severely damaged and would require replacement. 25% of the external walls demolished.

a

Category D: Minor structural damage though still sufficient to make the house temporarily uninhabitable with partitions and joinery being wrenched from fixings. Less than 25% of the external walls demolished.

9 Category E: Buildings requiring repairs, but remaining inhabitable. There would be damage to ceilings and tiling. More than 10% of glazing would be broken.

4. RESULTS All of the simulations were executed, analysed and tabulated. The output impulse and pressure were compared as follows:

4.1. The change in impulse and pressure values Figure 4 & 5 show the reduction, compared to the reference scenario, of average impulse and pressure values of specific target points located on structures, which form the street space.

265

The results indicate that: (a) Straight Street without any sub-streets (i.e. no venting), increases the impulse values by up to 71% and the pressure values by up to 6%, when compared with the reference scenario. (b) Curved street form with one curve and perpendicular sub-streets reduces the impulse values by up to 37% and the pressure values by up to 6.5 %, when compared with the reference scenario. (c) Curved street form with two curves (S-shape form) with perpendicular sub-streets reduces the impulse values by up to 55% and pressure values by up to 13 %, when compared with the reference scenario.

E

60

A

15

40

r

10

20

~

5

.e-

0

0 o~

c( 0 r

(~

9

-20

"'

-40

2 9 ,....o

* 9

"

."

3

",

4 9

",

........

,,

9,, , , .

r-

" ' "

-10

" ....

Target points

....

Target points

Straight street (Close street).

Curved street.

.... S - shape Street.

Figure 4. Change in impulse and pressure values on side A [See Fig. 1] compared with the initial case. 60

A .~ |

0~

40

~

20

.E

0

i

-20

2

o~ -40 t.-

-''

.r

4

(1)

,-

10 8 6

~

, ~."

4

,~

~

5"'.

6

Q"

.

-60

2 0 -2

".

-80

r

'

I "-2.-""

3

I

".

-6

4

.

-'"

-8

Target points

....

-"'"

Straight street (Close street).

Target points

Curved street.

, S - shape Street.

Figure 5. Change in impulse and pressure values on side B [See Fig. 1] compared with the initial case. 4.2. T h e c h a n g e in d a m a g e category:

Figure 6 presents the change in damage category for all of the street forms investigated. This classification indicates that the geometry of the street form can significantly affect the blast damage in urban environments.

266

Figure 6. The change in damage category by modifying street form. 5. CONCLUSIONS 1) The site selection process for specific buildings, which could be subjected to terrorist attack, should be taken in consideration not only as an architectural parameter of the buildings but also as a planning parameter for the surrounding urban environment. 2) The street geometry can have a significant reduction in the effects of the blast waves on the structure and the surrounding environment. 3) Modifying street geometry to reduce the effects of blast waves can be achieved without affecting the functionality of the structure and street space. 4) Street venting can offer significant reductions in the effects of the blast waves, which generate in the surrounding environment.

REFERENCES

1. Autodyn 3D V3.0.07, Century Dynamics Limited, 12 City Business Centre, Brighton Road, Horsham, England, 19.95. 2. ConWep, U.S. Army Engineer Waterways Experiment Station, 3909 Halls Ferry Road, Vicksburg, MS 39180, 1991. 3. R Merrifield, Simplified Calculations of Blast Induced Injures and Damage, Technology and Health Sciences Division, Specialist Inspector Reports, No. 37.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

267

Generation and measurement of high stresses and shock hugoniots S.S. Sachdeva, H. Lal, M.S. Bola and V.S. Sethi Terminal Ballistics Research Laboratory, Sector 30 Chandigarh - 160020, India Email: [email protected], mc. m High stresses of the order of hundreds of ldlobars and megabars are generated by detonating explosives and by impact method. Indirect methods are used to determine such a high stress by measuring shock and particle velocities.Polymethyl methacrylate (PMMA) gauge has been developed which can measure stresses between 100 - 350 kilobars while Lead Zirconate Titanate (PZT) gauges have been used to measure shock parameters in soil and water at low stresses. I. INTRODUCTION A disturbance of infinitesimal amplitude in any medium propagates with the speed of sound. In shock waves, the amplitude of the disturbance becomes very high and the forward gradient becomes discontinuous. When a material or an alloy is kept in contact with the explosive, the stress induced in the material depends upon the combination of explosive and the specimen as analysed by Graham et al [1]. Still higher stresses can be generated by an over driven detonation wave obtained in an implosion process. Collision of two detonation or shock waves at an oblique angle results in the formation of high-pressure stem. Stresses higher than one megabar can be obtained by the flying plate impact method. The advantage of this method is that the stresses of any amplitude can be generated by varying the charge to mass ratio and thus changing the impact velocity of the flyer plate. 2. GENERATION OF STRESS Two methods employed to generate different levels of stress are described below:

2.1. Contact geometry In this ease the material is kept in contact with the explosive and following detonation, a shock wave is generated in the material. The high stress induced due to shock loading is dependent on the material characteristics and the nature of explosive. Pin oscilliographie technique has been used to measure the shock and particle velocities and hence stresses in metals, alloys and inert materials. Experimental setup used to measure these parameters for metals and alloys is shown in fig. 1. Since it is not possible to measure the particle velocity easily and accurately in non-conducting materials, mismatch method [2] is employed to measure the induced stress. Only one parameter (shock velocity) is measured and stress is determined by using the mismatch equation ot = 2 oi ( ptU,t ) / ( piU,i + ptUst)

(1)

268 ,c" c b

9b

Al . . .

?

0.5

1.0

1.5 2.0

2.5

3.0

3.5

4.0

Upmm/~ Fig. 1 Experimental set up used to measure shock and particle velocities

Fig. 2 Graph between shock velocity vs particle velocity for aluminum, copper and rock sample.

Where oi is the incident stress, ot the transmitted stress, ptUsi is the shock impedance (the product of density and shock velocity of explosive or standard material through which shock is induced) and ptUa is the shock impedance of the specimen in which shock velocity is measured. Thus knowing the shock velocity in thin material specimen, the stress transmitted in Table -1 Shock parameters and stresses measured in various materials kept in contact with different

explosives. S. no

Explosive

Material (density)

(g/cm~) AI,

Pwrficle " S ~ vel. (G Pa)

Density of shocked

Volumetric strain

(mm/~s)

(mm/~s) 1.56 1.255 1.02

3i.5 24.3 18.8

3.431 3.29 3.191

0.2107 0.1758 0.150

7.44 7.14 6.8

..

m~ma~

(Vo-VYVo

2

TNT

3

Baratol

4 5 6

RDX/FNT 1 TNT Copper, Baratol po--8.93

5.15 4.91 4.53

0.82 0.582 0.4

37.7 25.5 16.2

10.62 10.131 9.975

0.159 0.1186 0.0883

7 8 9

RDX/TNT ~ Shale TNT J Rock, Baratol pc=2.105

4.565 4.125 3.695

2.14 1.73 1.43

20.6* 15.05" 11.15"

3.969 3.63 3.439

0.47 0.421 0.387

10

RDX/TNT

2.774

0.38

2.336

0.354

11

~

Shor vel.

po--2.712

Alluvium 5.72 2.17 21.38" soil, po--1.72 RDX/FNT Rubber 6.23 2.206 20.74* po--1.509 * Stresses determined by mismatch method throush ~ n u m

"

'

269 the specimen ot can be determined. When two parameters are known, all other Hugoniot parameters are determined by using jump conditions. Table 1 gives the experimental values of shock and particle velocities measured in aluminum and copper when they are shock loaded in contact with different explosives such as RDX/TNT (60:40), TNT and Baratol. Stresses induced and volumetric strains showing the compression of materials at different stress levels, are also shown in the table l. Shock parameters for shale rock samples of density 2.105g/cm 3 were determined from mismatch equation by using aluminum as standard base material through which the stress is induced. Equation of states for aluminum, copper and rock samples are determined by plotting shock velocity vs. particle velocity, which are straight lines as shown in fig. 2. Equations of state determined for these materials are as follows: Aluminum (po = 2.712g/cm3), Copper ( Po = 8.93g/cm 3 ), Rock ( 19o= 2.105g/cm 3 ),

Us = 5.692 + 1.102 Up Us = 3.94 + 1.489 Up Us = 2.809+0.748 Up

(2) (3) (4)

Stresses measured in inert materials such alluvium soil and rubber are also given in table 1.

2.2. Impact method In this method, a thin flyer plate of metal is kept in contact with the explosive. The plate captures the momentum on detonation of the explosive and gets accelerated to a maximum velocity in a short duration, depending upon the charge to mass (C/M) ratio of the explosive to the metal. Target is kept at a distance at 20 mm from the flyer plate and the probes for measurement of shock parameters are fixed on the other side of the target as shown in fig. 3. Different velocities of the flyer plate can be achieved by changing the C/M ratio and hence the desired stress can be generated in the target plate. The three materials studied in the contact geometry were also studied by shock loading with the impact method. The aluminum flyer plate was made to hit the aluminum target while the copper flyer plate was made to hit the copper target. In case of rock the stress was induced through the aluminum target being hit by aluminum flyer plate and the mismatch method was used to measure the resultant

Fig No. 3 Experimental setup to generate high stress by impact method

270 Table- 2 Shock parameters and stresses measured in aluminum, copper and shale rock samples generated bY impact method Stress ..... ~r S.No Flyer Target material ' Sh(~ck vei.' Particle Vel. generated strain plate (densit~y) in target in target (Ca'a) (Vo-V)No (g/cm'). (mm/pLS) (ram/its) 49.1 0.276 i .... Aluminum Aluminum 8.10 2.235 po=2.712 2.675 62.7 0.311 2 do do 8.64 2.938 71.15 0.329 3 do do 8.93 .

,,

4

Copper

5 6

do do

7

Aluminum

8

,,

do ,

J

,

.

..

Copper po=8.93 do do

7.045

2.15

134.8

0.3052

7.38 7.72

2.255 2.5

148.6 172.35

0.3055 0.3238

Rock po=2.105 do

4.947

3.28

34.2

0.663

3.81

45.4

0.673

,,

,,

5.66 ,

.i

ii

stress. The flyer plate velocity at different C/M ratios has been studied by Aziz et al [3], Kennedy [4] and Yadav et al [5]. It has been established that free surface velocity, which is twice the particle velocity induced in the target plate, is equal to the flyer plate velocity. Table 2 gives the shock velocity, particle velocity, stress generated and the volumetric strain produced in aluminum, copper and rock samples. Comparing the values of volumetric strain, which is a measure of compressibility of the material, we observe that aluminum is more compressible than copper while the rock sample is very soft and easily compressible. The relative values of volumetric strain for copper, aluminum and rock with RDX/TNT are 0.159, 0.2107 and 0.47 respectively. The density of aluminum increased from 2.712 g/em3 to 4.042 g/era3 under dynamic shock loading at a stress of 71.15 GPa. 3. MEASUREMENT OF SHOCK WAVE STRESSES High stresses generated by explosives in contact with solid materials or by the flyer plate impact method are generally measured indirectly. In direct methods, transducers are employed to determine the stresses of lower order. 3.1. Indirect method Measurement of intense shock stress is difficult by the use of transducers because the elastic limit of the materials is exceeded many times and thus plastic flow occurs behind the shock front. Therefore in this method shock and particle velocities are measured. A number of techniques are available to determine these shock parameters such as streak photography, pin oseilliography, flash radiography, optical interferrometrie technique etc. Knowing the shock and particle velocities and using jump conditions for the conservation of mass, momentum and energy, shock stress and other Hugoniot parameters can be determined from the following equations:

271

poU~ --p(u~ -Up)

(5)

a - ao = po u s Up

(6) (7)

E - E o = 1/2 (G + 6o) (Vo- V)

Where U.,, Up denote the shock velocity and particle velocity; po, ao and Eo denote the density, stress and energy before the compression, process respectively and the corresponding quantities without subscript are for the medium behind the shock front.

3.2. Direct Methods Direct measurement of shock stresses is generally carried out by two types of gauges. The first type of gauge is based on the principle of polarization and the piezoelectric effect, while the second type is based on piezo resistive and electromagnetic properties. Gauges of the first type have been developed and used in the present studies. Polymethyl methacrylate (PMMA) and Lead Zirconate Titanate (PZT) gauges have been developed for different ranges of stress. The phenomenon of dielectric polarization and depolarization occurs respectively on passage of shock wave. 3.2.1. PMMA Gauges It has been observed that PMMA is a polar type of material. When two faces of a PMMA disc are short circuited through a small resistance and the disc is shock loaded, current flows through the resistance. Initial amplitude of the voltage generated across the load resistance has been shown to be directly proportional to the incident stress, area of cross-section and inversely proportional to the thickness of the disc. To avoid side effects, the disc is divided in two equal areas by back electrodes. PMMA gauges have been successfully used to measure shock stresses between 100 to 350 kbar.

3.2.2. PZT Gauges The behavior of Piezoelectricity at low stresses up to 3 5 kbar studied by W.J. Halpin et al [6] showed a decrease in electrical depolarization with increasing stress. The response of PZT crystals at different stresses has been studied. To avoid ringing the free surface of the PZT is matched for its shock impedance. Only one single voltage pulse was obtained when the PZT disc was covered with a brass pellet, which also acted as a charge collector. 4. EXPERIMENTAL SETUP AND RESULTS The experimental setup used for studying the electrical polarization in PMMA is shown in fig. 4. A similar type of arrangement was used for PZT crystals. A plane shock wave is generated in the sensing element by detonating a plane wave generator. To achieve the different stress levels, the explosive pads of TNT, RDX/TNT and Baratol are used in contactwith PWG. The gauge is put either on the explosive surface or on the specimen in which the stress is to be measured. PZT disc faces were covered with the copper foil backed by a brass pellet of 10 mm thick on one side and moulded in araldite such that the other face of the disc is exposed to record the incident stress. PZT crystals have been tried to measure explosive pressures and stresses of low range generated at a distance from the point &burst in water and soil. The PMMA gauge was made of a 3 mm thick, 76 mm diameter disc and divided into two parts of equal areas separated by an air gap. The inner and outer halves were

272

Fig. 4 Expedmental setup to measure stress by PMMA gauge separately shorted through equal resistances to the aluminum base. In case of PMMA the value of the shorting resistance used was 100 f2 while in the case of PZT, it was kept low (of the order of 1 f2) to have a current mode operation. The choice of resistance value was dependent on the electrical response of the crystal materials at high stresses. PMMA needs high resistance because of its low electric~ response and high internal resistance as compared to PZT response. The voltage response of PMMA gauge at different stresses is shown in fig. 5. It has been observed that in the case of PZT, the current response at high stresses is not consistent and does not show a monotonic increase with stress. PZT gauges have been successfully used as stress detectors for measurement of shock velocity in liquids and non metals where pin contacter method fails. Fig. 6 shows the response of PZT gauge to stress transmitted through 45 mm soil when a charge of 45 gm tetryl was fired. The shift in the upper time base shows the arrival of the detonation wave at the surface of the charge, while a single pulse in the lower time base indicates the arrival of shock wave at the gauge. The shock velocity measured in this particular case is 1.52 mm/l~s. PZT gauges were also used to

Fig. 5 Voltage response of the PMMA gauge at different stresses

Fig.6 Response of the PZT gauge in soil at 45 mm from the point of burst

273 measure the shock velocity in water at a distance of 710 mm from the point of burst. 50 g. of spherical charges of compositions containing HMX and aluminum in various proportions were fired. Experimental results obtained using PZT crystals showed that the shock velocity decreased from 1.71 mm/~s for HMX/AI (82: 18) to 1.56 mm/~s for HMX/A1 (65:35) at a distance of 710 mm from the charge. 5. CONCLUSIONS Stresses upto 300 kbars can be generated by detonating high explosives. Higher stresses in megabars region can be produced by the flyer plate impact method. The impact method can also be used to get stresses of lower ranges. Indirect methods are used to determine higher order of stresses in which shock and particle velocities are measured. Equations of state for aluminum, copper and shale rock have been determined. Lower order stresses can be determined by direct methods using transducers. PMMA gauge has been developed and used successfully to measure stresses between 100 to 350 kbars (10 to 35 GPa).Use of PZT crystal as stress sensor is a good tool to measure the shock velocity and equation of state for nonmetals and liquids. REFERENCES

1. R.A. Graham and J.R. Asay, Measurement of Wave Profile in Shock Loaded Solids, High Temperatrures High Pressures, vol. 10, p. 355 - 390 (1978) 2. M.A. C o o l R.T. Keyes, W.O. Ursenbach, Measurement of Detonation, Shock and Impact Stresses, 3rd Symp. on Detonation p. 357- 385 (1960) 3. A.K. Aziz, H. Hurwitz and H.M. Sternberg, Phys. Of Fluids, Vol. 4 , no. 3, p. 380 (1961) 4. J.E. Kennedy, Explosive Output for Driving~,/Ietal Plate, Behaviour and Utilisation of Explosives in Engineering Design, Proc. 12 Annual Symp. Univ. of Mexico (!972) 5. H.S. Yadav, P.V. Kamat and S.G. Sundaram, Propellants Explosives Pyrotechnics 11, p. 1 6 - 22 (1986) 6. W.J. Halpin, O.E. Jones and R.A. Graham, Dynamics Behaviour of Materials, P. 208 218, ASTM Special Technical Bulletin, No. 336, Philadelphia, Pennsylvania. (1963)

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

275

Dynamic response of model reactor structure subjected to internal blast loads A.K. Sharma a, V.S. Sethi a and P. Chellapandi b, ITerminal Ballistics Research Laboratory, Ministry of Defence, Sector- 30 Chandigarh160020 India blndira Gandhi Centre for Atomic Research Kaipakkam-603102 India Sophisticated computer code FUSTIN has been developed for predicting the dynamic stresses/strains and large displacement elasto-plastic deformation of the PFBR structure. A well def'med experimental programme has been planned to validate this theoretical computer code through a series of experiments. The work. reported in this paper is confined to the design of suitable explosive charge assembly, construction of foundations, fabrication of scale down model vessels and the confmnation of desired expansion of vessels under dynamic blast loads. Special counter weight foundations provided with spherical holding down assembly have been constructed at the experimental site for supporting the test vessels. Experiments have been conducted by using deformable cylindrical vessels fabricated out of 1.25 mm thick SS-316 of diameter 370 mm and height 430 mm. Low density pentolite charges have been specially designed and developed to subject the scale down model vessels to desired blast loads through the water medium. The chemical explosive has also been characterised for simulation studies. Pressed explosive charges weighing 20 gms each have been centrally detonated in the water filled cylindrical vessels closed at both ends by rigid roof and bottom. The test vessels have withstood the blast loads and their expansion under dynamic loading has been found to be in agreement with the computed values.

I. I N T R O D U C T I O N It is important from a safety point of view that the structural integrity of the fast reactor system is ensured against impact force expended during the accident. The aim is to assess the impact of the shock that the reactor vessel receives in the sodium environment and the extent to which the radioactive sodium is escaped from the primary containment defined by the reactor vessel and its closure plugs. In order to assess this phenomenon, theoretical and experimental methods are available [1,2]. For theoretical investigations, sophisticated computer code FUSTIN has been specially developed. In the modelling of such complicated phenomena, a number of idealizations as well as boundary conditions and assumptions are made. It is therefore important to validate this code thoroughly on various aspects before using it for actual applications. Although, it is difficult to micro-scale the nuclear excursions,

276 but a conventional experimental method of assessing the integrity of structures lies with the use of a definite quantity of chemical explosive such as pentolite (50/50 mixture of TNT and PETN). Pentolite possesses a number of additional advantages over other explosives. In view of the ease with which it is detonated and its established replicability in small scale tests, pentolite has been chosen as the most satisfactory explosive to employ in containment programmes. The pentolite/TNT ratios of energy density and release of gas products per unit of reactant are 1.02 and 1.17 respectively [3]. As such the use of pentolite in containment studies is consistent with reasonable upper-bound criteria. Pentolite simulation is quite satisfactory where destructiveness and containment integrity are basic questions. A given quantity of pentolite has been exploded at the centre and dynamic stresses/strains are measured on the wall surfaces of the test vessels. The measured values are useful for validating the theoretical predictions of FUSTIN and also to assess the uniform expansion of test vessels. Further it is possible to understand more clearly the leakage of water through the top cover seals only by experiments. The presem paper describes the work for the fabrication of foundations at the experimental site, design of suitable explosive charge assembly, fabrication of deformable cylindrical vessels for simulation studies and the experimental trials for uniform loading of test vessels to observe their expected expansion under desirable detonation effects.

2. E X P E R I M E N T A L 2.1 Fabrication of Foundations

Special counter weight foundations provided with spherical holding down assembly have been constructed at the experimental site for carrying out the trials. Three foundations provided with M-30 bolts have been erected for supporting the test vessels. Since the experiments are to be conducted in three stages, there is an arrangement for carrying out all the three sets of experiments on the first foundation (see fig. 1). Whereas a provision has been

Figure 1" Special counter weight foundations constructed at the experimental site

277 made for conducting the second and third stage experiments on the second foundation. The third foundation (not shown in the figure) has been kept as standby for the trials. Bolts in circular array have been distributed on all the three foundations.

2.2 Development of low density pentolite explosive Compact low density pressed pentolite explosive (50/50 mixture of PETN and TNT) has been developed in the laboratory for undertaking the simulation experiments. PETN of particle size roughly 10 micron has been prepared by acetone filtration. Whereas TNT powder of almost same particle size has been obtained by machining the explosive with a sharp cutting tool. Before pressing, the two powders have been thoroughly mixed to form a homogeneous mixture. For the purpose of pressing of explosive, stainless steel die has been specially designed in the laboratory. Cylindrical pentolite charge has been pressed by using 2 ton pneumatic press procured from Ultra Engineering Corporation, Bombay. Pressing of the explosive has been carried out by maintaining a pressure of 2.5 kgJcm2. The dwell time has been kept 30 seconds. On the basis of p-t curve prediction of what will happen in the reactor core in an HCDA, cylindrical pentolite explosive of diameter and height 27.6 mm, weighing 20 grns having density 1.15 grns/cm 3 has been developed which can satisfy the conditions required for performing the scale down model experiments. 2.3 Fabrication of Vessels Deformable vessels of diameter 370 mm and height 430 mm have been fabricated out of 1.25 mm thick SS-316 rolled plate with a single weld. Welds represent a departure from the idealized vessel concept, but special precautions have been taken to ensure that the welds are capable of withstanding the same stress and deformation as the parent metal. The precautions such as the welds are continuous and are of full compaction, the weld material possessed essentially the same mechanical properties as the parent metal and the vessels are annealed to relieve any residual and thermal stresses resulting from the rolling and welding processes have been taken into consideration. By virtue of such precautions, welds of equal or perhaps better quality than those expected in actual reactor vessels have been ensured. 2.4 Explosive Characterisation Trials Several pellets of pentolite explosive of various densities have been pressed to observe the reliable detonation of this explosive in water without using any booster. Finally, an explosive pellet of density 1.15 gms/cm 3 has been prepared which is capable of producing desirable detonation effects in water on direct initiation with conventional detonator N0.33. Detonation characteristics of the pentolite explosive pellet have been determined by using high speed pin-oscillographic technique. Measurements have been undertaken by plane wave loading of the test pellet using probe geometry in conjunction with indigenous shock velocity recorders and specialised data acquisition systems. Underwater shock wave trials have been carried out by centrally detonating the explosive pellet in the shock pressure tank of 6.0 metres diameter and 6.0 metres depth. Weight of the pellet has been kept 20 gms in each trial. The distance between the point of explosion and point of measurement has been taken to be exactly equal to the radius of the deformable vessel to have an idea about the pressure loading produced in that range. Kistler pressure transducer model 6205 has been used in the reflected mode to record the pressure-time profile of underwater shock wave. The details of the trial setup have been reported elsewhere [4].

278

Figure 2. Experimental setup showing the vessel after the trial

2.5 Vessel Expansion Trials Vessel expansion trials have been carried out on the special counter weight foundation provided with M-30 bolts in circular array. The vessel has been fixed by means of metallic columns and closed by rigid top and bottom. The pentolite explosive pellet has been sealed in a water proof polythene jacket and lowered in the water filled deformable vessel through a plug provided at the centre of the top plate. The explosive pellet is detonated exactly at the centre of the vessel by using a conventional detonator No.33. The expansion of the vessel has been monitored by employing a FASTAX photographic camera. The camera has been operated at a framing speed of 2650 pictures per second and the total expansion has been observed in five frames. After that the vessel has shown a retreating trend. Fig.2 shows the photograph of the vessel after the trial.

3. D I S C U S S I O N Fracture analysis of SS-316 deformable vessel has been carried out by the computer software. The block diagram of the vessel showing mid point explosion of the cylindrical low density pentolite pellet weighing 20 gms has been depicted in fig. 3. Fig. 3 also includes the variation of hoop strain at mid point A in the vessel with respect to time. Fig. 4 shows the stress-strain curve for the vessel made of 1.25 mm thick SS-316 of density 7900 kg/m 3. The welded vessels have demonstrated excellent deformation properties and there is no case of premature rupture due to weld failure either in the weld material itself or in the heat affected zone adjacent to the weld. However, it is seen that in all cases, the vessels are capable of deforming without rupture to strains greater than l/3eu and with only few exceptions to strains greater than 1/2 eu. From these data it is postulated that the maximum strain of welded reactor vessels may be restricted to < 1/3 eu

279

.c_. 0

E

~0"~

-.

---- FUS'TIlq I~-0-

,

' ;~

'

~

,

i

9EX?ER~t"~ENT"~ O ' 5 ~ r "/

/

is-Z-

~Ti l'/l//hTl11)'llll/~Yb~'~;.~l --"I--

I v

El k. ,.,,., &'l 1:3_ 0 0 "1-

5"6i.5

-

/

/ "

0

-r ~ N~. S--_---_3"-_--_"T_------

"~

~~2)!i[,ir/......... i)1" ] i-

"'~,

9

I

"

"4

i ....

'~

...... 9 i

'

'8

i

"""

I

I

""

I'Z

i " " '

I"/'/.

t

I'~

"""'i'

9 "

I.~

7,.

Time (10-~ s) Figure 3. Block diagram of L.D. pentolite explosion in the deformable flexible tank.

IHICKNESS- 0.00125 M DENSIIY

=" 7900 K G I M 3 '=

600 559

0-3

_

7

L81 t~

352

"~ 2(;5

0

U 202 O-139

10.5

22.1A

3t.-s

Figure 4. Stress-strain curve for SS-316 deformable vessel.

E (~.)

280 The maximum deformation of the vessel has been found to be 65 mm. The expansion of the vessel has taken place at a maximum velocity of 24.9 metres/second. The experimental arrangement has worked perfectly well during the trials with no leakage of water from any point in the vessel including those from the top and bottom cover seals.

4. C O N C L U S I O N The aim of the work reported in this paper has been to assess the uniform loading of test vessels and their expected expansion under desirable detonation effects. The capability of test vessels to withstand impact loads and the performance evaluation of the support structures from design considerations has also been the part of studies. In conclusion, it may be stated that the explosive charge assembly has been designed to our satisfaction and the test vessels have shown desirable deformation characteristics. The measured parameters agree well with the computed values predicted through computer software FUSTIN.

5. A C K N O W L E D G E M E N T S The authors are thankful to all the team members for their help and co-operation in the successful conduct of experimental trials.

References 1. N.E. Hoskin and M.J. Lancefield,. Nucl. Eng. Des., 46 (1978) 17. 2. Y. Ando, S. Kondo, O. Kawaguchi, and T. Uchida, , Experimental analysis of vessel structures for fast reactor JOYO under shock loading, 2nd Int. Conf. on Structural Machanics in Reactor Technology, Berlin, Paper E 2/6, Sept. 1973. 3. R.H. Cole, Underwater Explosions, Princeton UP, Princeton, N.J., 1948. 4. A.K. Sharma, S.K. Shukla and D.S. Murty, Indian J. Pure & Appl. Phys., 31 (1985) 646.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

281

Spailation of Explosively Clad Metal Plates V.K. Sharma*, Vikas Srivastava, D.R. Kaushik Terminal Ballistics Research Laboratory Sector- 30, Chandigarh- 160 020, India. The scabbing of thick plates as a result of explosive loading has been studied by researchers world over. These scabs have high kinetic energy and posses high damage potential. This paper describes a method which reduces this damage potential through use of explosively clad metal plates. Such behaviour has been observed with mild steel-mild steel, mild steel-copper, mild steel-stainless steel clads. The results have been analyzed in the light of bond interface microstructure. 1. INTRODUCTION Spallation or scabbing of thick piates occurs near the free surface remote from the area to which an explosive load has been applied. This has been studied in details over the years [1,2] and is a result of interaction between the incident compressive wave and reflected tensile wave from the free surface, when the incident wave is decaying in nature. The phenomenon of scabbing/spalling has been of interest to the armament engineer and explosive metal working experts as it is frequently an undesirable feature. The normal methods of preventing spalling are i) Use of energy sinks on the reflecting surface, wherein the shock energy is partially absorbed in compaction of a loose powder or deformation of pipes placed in contact with the explosively loaded plate. ii) Use of momentum plates on the reflecting surface In these methods kinetic energy is transported away from the body being subjected to impulsive loading thus saving it from scab damage but the energy is required to be absorbed/trapped in another system to ensure total safety against such loadings. The authors were looking for a spall control system wherein no flying debris would come out of the system. For this purpose a study on the behaviour of clad plates subjected to explosive loading was undertaken. The cladding was achieved by explosive welding using inclined plate set up as described in [3] and samples had thin mild steel, Stainless Steel and Copper plates on 25 mm thick mild steel backing plate. These materials were chosen because their shock impedance values were approximately the same as the shock impedance of the mild steel plate on which the spalls were produced and hence could be used to evaluate the effect of the weld interface during shock propagation. 2.

EXPERIMENTAL 200 mm diameter, 28 mm thick mild steel samples were subjected to an attack of 100 g of plastic explosive ( p = 1.55 g/cc and VOD 6900 m/sec). For all experiments, the explosive was formed into a truncated cone geometry (fig 1) This produced scabs of about 300 g with an average diameter of 87 mm and average central thickness of 6.5 mm. The velocities were not measured but these scabs could puncture a 3 mm mild steel plate used as a witness plate at a short distance. These results served as a reference value for evaluating the response of clad plates. 25 mm thick and 200 mm diameter mild steel

282

MILD STEEL

MILD/STEEL

CLAD PLATE Fig. 1. Experimental set up for loading unclad and clad plate explosively

Fig. 2. M . S -

M.S Explosion weld interface 400 x

283 plates were clad with mild steel, SS-304 and commercial Copper using plastic explosive sheets. The weld joints were assessed for their weld integrity and interface characteristics using ultrasonic testing and microscopic examination. The joints were tested for separation strength (Tensile) on a Universal Testing Machine. Plate parameters and the bond characterization results of the combinations studied are presented in Table 1. The micro photographs of the bonds are shown in Figs. 2 & 3. 3. PHENOMENON DESCRIPTION The shock wave could pass through the backing plate to clad plate without any modification as the shock impedance of both metals were almost equal. However, the explosive welding introduces a bond interface, which affects the shock wave as follows.The analysis in the following paras is a special case and has been given as an example only. The actual position is however more complicated. The longitudinal components of velocities of elastic waves for cubic materials along the three crystallographic orientations <100>, <110>, <111> are presented in [4] in terms of elastic stiffness C ll, C12, C44and the material density : u = sqrt (Cll/P) (1) u = sqrt((Cll + Clz +2C44)/p)) (2) u = sqrt((Cll + 2C12 +2C44)/3p)) (3) The velocities of longitudinal plastic wave have. been obtained in [5] from the assumption that material can not resist shear stresses. He arrived at shock velocity values by taking into account only the density rise at 300 Kbars. For Copper Up<100> = U1 -- 3.99 * 105 cm/s (4) Up <110> = U2 = 5.22 * 105 cm/s (5) U p < l l l > = U 3 - 3.58"105cm/s (6) A normal metal for purpose of simplicity was assumed to have only three vertical orientations of grains. Thus the shock velocity is dependent on the fraction of each orientation present in the metal. Any change in any of the orientation fraction will lead to a change of shock velocity and ultimately the shock impedance. This results in a reflected and transmitted wave at the joint interface. The microstructural features of the bond zone have been studied in detail in [6]. The authors brought out the presence of highly deformed layer on both sides of the narrow zone with equiaxed randomly oriented ultra fine grains. This is seen in Figs 2 and 3. This deformed layer is highly textured and is a result of severe deformations as observed in extrusion of tubes of fee metals. Such deformations are known to give <111> and <100> textures which have lower shock velocities than other orientations [7].

4.

DISCUSSION OF RESULTS The results of explosive loading clad materials are summarised in Table 2. In all cases presently studied, scabs with lower diameters and masses were produced and were fully arrested by the ballooned clad plate (Fig. 4). This can be due to either (i) Presence of lower impedance crystal textures at the bond interface or (ii) Presence of relatively weak

284 Table 1. : Plate parameters and bond characterization of various metal combinations. Flyer Plate Material UTS Kg/mm 2 MS 50

Cu

25

Backer Plate Material, UTS Kg/mm 2 MS 50

MS

50

Bond Integrity

Clad Interface

No non weld

Wavy, no

regions

melting (Fig.2)

No non weld

Wavy, no

Bond Strength Kg/mm 2 55

23

melting (Fig.3)

regions ,,

SS-304

48

Table 2 :

S.No

1.

MS

50

No non weld

Wavy, no

regions

melting

50

Results of explosive loading of clad plates

Metal System

MS-28 mm

Explosive Loading Plastic Explosive g 100

Scab Details Dia Mass mm 87

g 303

Status Of Scab

Punctured a 3 mm MS witness plate

2.

MS(25 mm)-

100

57

141

Cu(3mm) Cu(3mm)-

Arrested, no effect on witness plate.

100

48

100

MS(22mm)-

Arrested, no effect on witness plate.

Cu(3mm) 4.

MS(25mm)-

100

61

163

MS(3mm) MS(25mm)SS 304(3mm)

Arrested, no effect on witness plate

100

57

150

Arrested, no effect on wimess plate

285

Fig. 4. Schematic of Clad Plate showing arrested scab

286 bond interface which allows the rear face of armour in the vicinity of detonation to separate and yet be attached to the backer plate on the rest of the area. The clad plate in the small separated zone absorbs momentum and is free to deform and absorb shock wave energy by plastic deformation. Since the clad interfaces of the plates used in the experiments are characterised by absence of melting and a bond strength close to the UTS value of flyer plate, separation of a small zone in our experiments is not likely. 5.

CONCLUSIONS The clad plates offer a viable method of arresting scabbed fragments by utilising the fact that different grain orientations have different shock impedance value. As the scab masses are reduced in such configurations, a better understanding of the deformations in cladding bond zone and the textures produced thereof can help in avoiding scabbing in many cases. The weak bond interfaces could also lead to similar observations. Further studies are required to assess their behaviour during explosive loading and for thorough understanding of phenomena involved. ACKNOWLEDGEMENTS The authors sincerely express their gratitude to Mr.V.S.Sethi, Director TBRL for his keen interest and kind permission to publish this work. REFERENCES 1. Rinehart J.S and Pearson J, Behaviour of metals under Impulsive Loads, American Society of Metals, Cleveland, 1954. 2. Meyers MA and Aimone CT , Dynamic Fracture (Spalling) of metals , Prog Material Science, 1983 3. Crossland B, Explosive Welding of Metals and its Applications, Claredon Press Oxford, 1982 4. Ghatak AK and Kothari LS, An introduction to Lattice Dynamics, Addison Wesley, 1972 5. Meyers MA, A "wavy wave" Model for the Shocking of Poly crystalline Metals, Proc. Fifth Int. Conference for high energy forming, Denver, Colorado, University of Denver, 1975 6. Hammerschmidt M and Kreye H, Microstructure and Bonding Mechanism in Explosive Welding in Shock Waves and High Strain rate Phenomena in metals, editors Marc A Meyers and Lawrence E Murr, Plenum Press, 1981 7. Polukhin P , Gorelik S and Vorontsov V, Physical Principles of Plastic Deformation, MIR Publishers, 1983

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

287

Stress and strain m a g n i f i c a t i o n effects in structural joints u n d e r shock loading. Gregory Szuladzinski Analytical Service Company, 5 Warwick St, Killara/Sydney 2071, Australia ABSTRACT It has been known for more than a century that structures with stress concentrations often fail prematurely when subjected to shock loading. This paper examines two cases of geometric discontinuities in presence of sudden loading associated with a stress wave traveling through the joint. Both static and dynamic responses are considered. Elastic as well as elastoplastic material properties are modeled for the purpose of identifying causes, which may help failure initiation. The examples discussed serve to illustrate why a rapid loading is more likely to cause failure than slow loading. 1. INTRODUCTION Although some structures and some steels in particular have been known to fail when they are subjected to rapid loading, the mechanism of that happening does not appear to be widely understood. Our main task will be to examine stress and strain fields under both static and dynamic loading near stress concentrations and highlight the differences between both. A discussion of material properties under dynamic loading creates a consistent picture of causes of failure. 2. STRESS AND STRAIN CONCENTRATION Consider a plate, or any other structural member subjected to a uniform stress S along an axis. A small hole, not necessarily circular, made in the plate causes a local discontinuity in the stress field. According to the theory of elasticity, the peak stress Sin, tangent to the edge, will appear at the edge of the hole provided there is no external loading along that edge. A geometric stress concentration factor Ks is defined as Sm ~-- Ks S

where the subscript indicates that the factor relates to stress. Similarly, we can write for strain em - " K e e, but in the elastic range the strain concentration factor is the same, i.e. Ks = IG. The above are conventional formulations of static elasticity. When the plate loading is a function of time, so is the peak stress along the edge of the hole. At some instant the absolute peak Sm is reached: S m - - ~,I~S

288 where S designates the largest applied stress and la, is the dynamic stress concentration factor. The value of this factor can vary broadly. In what follows, we limit ourselves to a step load in time. (The loading increases instantly to its peak value and holds there indefinitely.) The concept of a dynamic magnification is typically used when there are no stress concentration problems involved. This, of course, makes things more transparent. We know, for example that a simple, spring-mass oscillator experiences doubling of spring force and of deflection when the step load, instead of a static load is applied. (This can be demonstrated using work-energy principle.) We can therefore say that for a step load ~ = 2 in case of an oscillator. Many structural members, which are fLxed at one end and loaded at the other also have the same magnification factor. Consider an axial bar, fixed at one and pulled at the other. If the applied stress is S, then a stress pulse of this magnitude spreads along the bar and on rebounding from the fixed end it attains the magnitude 2S. Again, gs = 2, even though the physics behind it is quite different. Unfortunately, in case of discontinuities in the form of holes of various shapes, it is possible to make predictions only on a case-to-case basis. The complicating factor is that the stress pulse reflects from the edge of the hole, causing a complex wave interaction within the material. The peak stress is the result of such an interaction. To put things into a proper perspective one should note that after the stress pulse travels within the member for a sufficiently long time, the dynamic component gradually disappears and in the limit the static condition is reached. This means that during an event considered Its will be reached first and, after vibrations die down, Ks will be attained. Material nonlinearity, i.e. plasticity, is another complication. Once the plastic limit is exceeded for a ductile material, the magnitude of strain rather than stress becomes more meaningful. While plasticity reduces stress levels, compared with those calculated by elastic means, the plastic strains tend to grow rapidly in some situations. <.._-

,T

Y

i:

Fig. 1 Thin strip with a hole, subjected to tension S

3. A THIN STRIP WITH A HOLE The geometry of the long strip is shown in Fig. 1. The static stress concentration factor for this shape can be found in Peterson's [1] handbook as 4.33 (per Fig.86), with respect to the gross area stress. (Would be one-half that for the net section area across the hole.) The material selected was steel with E = 200 Gpa and v = 0.3. Also, a finite-element model of one-quarter of this joim (double symmetry) was prepared using Ansys code. There were 20 elements per one quarter of the edge of the hole. The solution gave nearly the same result, Ks = 4.354,

289 which will be used as a reference. (One should also notice that the symmetry boundary conditions imposed at the left and the bottom boundaries of the model are true with respect to the stress wave reflection.) The instantaneous stress distribution obtained from step load application of S = 100 MPa is shown in Fig.2. The peak transient stress at the edge was 1008 MPa, at point B in Fig. 1. The dynamic factor is = 1008/100 = 10.08 or 2.32x that of static concentration. For the study of the effects of plasticity the yield point of material was taken as 350 MPa and the ultimate strength of 430 MPa. The ultimate elongation was 20%. The assumption of bilinear plasticity gives the yield modulus of 403.5 MPa. Static load application resulted in a strain of 0.00244 at point B. Noting that the input strain of 0.0005 was applied at the edge, we have the static strain concentration of IG= 0.00244/0.0005 = 4.88 which is somewhat larger than the corresponding stress concentration calculated on elastic basis. Next, the edge load was applied as a step function in time and the material properties were treated as plastic. At this point the implicit code exhibited difficulties in converging the solution, therefore an explicit code (LS-DYNA) was employed. The calculation resulted in the peak strain of 0.01342. Relative to the input strain of 0.0005, we have the dynamic factor of = 0.01342/0.0005 = 26.84 which is 5.50x larger than static strain concentration. Also note that when we compare the dynamic strain magnification with the corresponding quantity calculated on elastic basis, the former is 26.84/10.08 = 2.66 x larger.

FIG.2 Dynamic response of a thin strip due to 100 Mpa tensile edge stress

290

~

RIO0.

S

300

.j

Fig.3. Plate with two rows of holes, subjected to tension.

4. PLATE WITH TWO ROWS OF HOLES The holes are staggered; their mutual position in a wide plate is shown in Fig.3. The static stress concentration factor for this configuration can be determined with the help of Ref.[2] as 3.525 (per Fig. 115). A finite-element model of this joint was solved statically and gave somewhat smaller result, Ks = 3.30, which will be used as a reference. There is a line of symmetry running horizontally through the center of each hole, which allows us to model only one long strip. (See "Model Width" in Fig.3) The instantaneous stress distribution obtained from step load application of S= 100 MPa is somewhat similar to that in Fig.2. The peak transient stress at the edge is 763.1 MPa to the left of point D and right of point C in Fig.3. The dynamic factor is therefore = 7.63 or 2.3 lx that of static concentration. The above Ks shows us that 100 MPa is not enough to induce localized plasticity under static conditions. Consequently, 150 MPa was applied at each end and the material properties

291 were modeled with the same plastic parameters as above. The strain in the undisturbed area near the end was 0.00075. The calculation gave peak strain of 0.002946, which set the concentration factor as IG = 0.002946/0.00075 = 3.928 Dynamic plasticity simulation was not carried out for this model, but one would expect a similar but somewhat smaller strain magnification ~ factor compared with the previous example. 5. THE CONCEPT OF RAPID vs SLOW LOADING As stated in the opening, the scope of this paper is limited to only one type of dynamic loading, namely that of a step load in time. This excludes most of vibratory type loading, intense as they might be, yet not as abrupt. If a resonance takes place during vibrations and greatly magnifies the effect of the applied load, that increased loading may, most of the time, be treated as static from design viewpoint. (Unless material sensitivity to strain rate is an issue, which is outside our scope.) Our purpose is to identify some events which result from stress/strain wave action and which take place in the interval of time much smaller than the natural period of a structural element. In other words we are speaking about a shock rather than of vibrations. The step loading is characterized by zero rise-time, which can't be attained in practice, but can only be approximated. A finite rise-time means a reduction of dynamic magnification. A study of the simplest example, namely a mass-spring oscillator shows that even if the risetime is as short as one-half of the natural period, Ix --- 1.9 can be expected, instead Ix = 2.0 for the instantaneous rise. (The actual figure depends on the shape of a load-time curve. Those curves which appear to result from physical events give larger ~ than a straight line. Details in Harris and Crede (1976), p.8.19). For real structural members the measure of rapidity of the load is the ratio of the rise-time to the natural period. Another factor to be kept in mind is a distance from where the load is applied to the critical spot of the member under consideration. A sudden application of the boundary load sends a stress wave traveling with sonic velocity along the member. Initially this wave has a vertical front; no stress before the wave front and a full boundary stress S behind it. During the travel the front becomes inclined and its edges curved. (This happens in reality as well as in computer simulation.) When the wave reaches the critical spot, the rise of load is much more gradual than due to the idealized step loading. A descriptive way of putting it is to say that a member softens the impact of boundary loading on its critical location. Still, one must be aware that the results reported here incorporate such "softening" due to distance. A strain rate, to which material is subjected, is often quoted as a measure of rapidity of the external load. For a uniaxial sam_pie subjected to stretching the increase in length divided by the original length is the strain and the latter divided by the time in which it took place is the strain rate. (These simplified definitions for small deflections and a steady process.) Dynamic properties of material are often quoted as a function of the strain rate, but there are marked ambiguities in these descriptions. Those result from the fact that the initial shock, to which the sample is subjected is an unsteady process with a broadly varying strain rate. The nature of these variations is machine-dependent. Unfortunately, most real processes also

292 have a varying strain rate, so presently used descriptions appear inadequate or ambiguous. 6. CONSEQUENCES FOR STRUCTURAL DESIGN Let us put ourselves in the shoes, so to say, of a structural designer for a moment. Suppose that a strip with a hole in Fig. 1 is to be subjected to an infrequem load application with such a magnitude that the end stress is to be 100 MPa. The first question is whether the middle section weakened by the hole is acceptable. As the net section area is one-half on the full section, the average applied stress is therefore 200 MPa at the weakest section. This is well below the yield point of 350 MPa, consequently there seems to be no problem. Without being specific, one knows that there will be a (net area) concentration factor of 2.0 or 3.0 (roughly) because of presence of the hole, but that will cause only a localized peak stress which will be smoothed out by plasticity. This reasoning quoted above is quite correct as long as shock loading is not involved. If it is, then the transient peaks of stress and especially of strain will be much larger than those resulting from static load application, as described previously in Section 2. It may still be acceptable for this particular example and many others, provided there is no reason to suspect a decrease in material ductility near the point of interest. Yet, such decrease may not always be ruled out. Even mild steel, otherwise tolerant of rapid loading, will fail at certain situations. This may happen after said steel had been subject to large strains during a formation process. The combination of properties resulting from cold working and the accompanying residual stress can often lead to failure under subsequem dynamic loads. Another example of loss of ductility in steel is brought about by the reduction in temperature below a transition point, which in mrn changes the nature of fracture from ductile to brittle. The material can withstand only a very limited strain and fails when the strain is exceeded. (Refer to Titanic and other marine examples earlier this century.) Material sensitivity is another important variable. The reasoning presented in the opening paragraph will hold for mild steel quite well, up to (but not including) the rates that can be induced by high explosives. True, the yield point will grow with the strain rate, but the ultimate elongation will decrease retaining the shock absorption capacity roughly unchanged. Yet, many steels do not behave as favorably and show a tendency towards brittle fracture at increased strain rates. A more common situation is that associated with welds, especially intermittent ones. The metal in a heat-affected zone has different properties from the parem material, typically with smaller ductility. Invariably, welds induce marked concentration factors, although magnitudes of those are strongly dependent on local geometry and finish. A combination of these two factors, i.e. increased stress/strain on one hand and a material more sensitive to overstraining on the other may cause cracks to appear under shock loading. And once a crack appears under a sudden load, it tends to continue without much resistance until it stops at a free edge or another discontinuity in the material volume. Even if a design team is competent in structural dynamics, the work is rarely carried out to an extent necessary to quantify the situation with regard to all structural details. The alternative to such a detailing is to make conservative estimates, which necessitate an increase in over-all weight of the member involved and often the entire structure.

293 7. SUMMARY AND CONCLUSIONS Most of structural members have notches and discontinuities, which act as stress raisers. Under shock loading those become more detrimental to crack-free performance of a structure than under a slow rate of load increase. In quantifying this effect we have used the following symbols: Ks = geometric stress concentration factor (static) IG = strain concentration factor (static). txs = dynamic stress concentration factor = dynamic strain concentration factor The first and the third of these are typically used to describe what happens in the elastic range. By definition, we must have ~ts > Ks because the static condition is the limit of the transient state imposed by the step load in time. Similarly, ~ > IG. Dynamic stressing is always higher than static. When we go beyond elastic modeling into plasticity (which is invariably reached near stress concentrations), the magnification factors for strain become much larger than in the elastic model. When this is coupled with the fact that material properties at the point of interest are often less favorable near the notch than those of the parent material, a local cracking, possibly followed by a general collapse, becomes more likely. Strain rate sensitivity is another contributing factor. While not critical for mild steel in the original condition, other steels may be quite sensitive to abrupt stressing. This again is conducive to failure. Acknowledgment: This writer is indebted to his metallurgist friend, Dr Michael Drew, and to Mr K. Mitchell of ABB for their valuable comments. REFERENCES 1. Peterson, R.E. (1974). Stress Concentration Factors. John Wiley & Sons, New York. 2. Harris C.M. and Crede C.E. (1976). Shock and Vibration Handbook. McGraw-Hill, New York.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

295

Numerical m e t h o d in u n d e r w a t e r shock simulations Hai Hoang Tran and John Marco

Aeronautical and Maritime Research Laboratory DSTO, P. 0. Box 4331 Melbourne 3001 Simulation of far-field underwater shock response of naval surface vessels using explicit finite element techniques has at present not achieved general agreement as to the modelling approach to be adopted. The questions that arise are whether the numerical model should include full geometry details of the explosive charge and the surrounding fluid or can simplified alternative models be used to generate the shock loads at the structure. Modelling full explosive and fluid details would substantially increase the global size (number of elements) of the model and therefore it would be an extremely expensive and time-consuming exercise to arrive at even a single solution for a particular scenario. The simplified technique requires the combined application of a calculated equivalent incident pressure wave and a much smaller size finite element fluid model, consequently, eliminating the use of additional fluid elements. To date there has been a lack of published information in the far-field underwater shock simulation area regarding an appropriate method that can be used to define a suitable configuration for the charge and fluid models. That is, the guidance on the global geometry, size and thickness of fluid elements required to transmit the detailed characteristics of the shock waves. Coarse meshes degrade the capability of fluid model in transmitting shock wave characteristics. In contrast high-density fluid mesh with very small element sizes would be extremely time consuming and very costly. To achieve a balance between cost and benefit in numerical simulation of a shock scenario, a suitable element size together with the utilisation of an appropriated fluid geometry shape is a better compromise. 1. INTRODUCTION Naval ship structures and on-board equipment when subjected to an underwater shock could be seriously damaged depending upon the level of shock (energy at the structure) received. An understanding of the shock response of the vessels would assist the RAN (Royal Australian Navy) to maintain and extend the operational service life of both the ship structure and equipment. Shock trials, sometimes called UNDEX (underwater explosion) tests, have been mandated for new platforms and have been widely applied in the defence maritime industry to assess the operational performance of both, the ship structure and the on-board equipment. Unfortunately, full-scale shock trials are expensive and only provide information for quite specific UNDEX scenarios. To thoroughly assess the shock response of a particular vessel, the use of numerous full-scale shock trials investigating various UNDEX scenarios would be required.

296 Because of concerns in saving resources and protecting the marine environment, the use of repeated physical tests have become increasingly infeasible. As a consequence, numerical simulation using the FEM (finite element method) is being increasingly applied in order to reduce reliance on full-scale UNDEX trials. When an underwater explosion occurs, a very large energy source from the explosive is discharged, which produces an extremely high-pressure pulse in the surrounding fluid. A shock wave, with a spherical wave front, will be formed and propagate radially outward at the speed of sound in the fluid medium. Due to the physical complexities of underwater shock phenomena, including the effects of bubble collapse, bulk and local cavitation, a detailed investigation would substantially increase the size of the numerical model. This paper will describe some of the current work being undertaken in investigating parameters relevant to the far field shock scenario. 2. NUMERICAL MODELLING The simulation parameters explored include the incident shock wave, boundary conditions, global fluid geometry and fluid element thickness and size.

2.1 Incident Shock Wave and Boundary Conditions To model a far-field ENDEX scenario using only a few metres in cross sectional thickness of surrounding fluid will significantly reduce the FE fluid model size. This would generally reduce the computing cost in constructing and analysing the numerical model. A pressuretime load curve, calculated at the external fluid face, could then be used to simulate the incident propagating shock wave. This incident shock wave could be simulated by a procalculated free-field pressure-time curve based upon the distance between the explosive source and point location on the external fluid surface where it acts. In order to account for the total shock load applied to the fluid, an approximation is used, whereby, the load curve is scaled to account for the 3D spatial orientation of all the fluid elements on the load surface face, see figure 1 and equation 1.

Figure 1" Technique for Load Application i

F~ Reference

dlstmnce

L i

||

..

Equation 1" Approximation for Scaling Load Curve S.F. ffi (R/Rx) a Where....~:.S.F: scaling factor R 9reference distance where S.F = 1, as shown in Fig 1 R~ :distance at which scale factor willbe calculated n 9index

297 To avoid the effect of any unwanted reflections caused by free surfaces around the simplified fluid model, an extra layer of fluid elements is required to be attached to those external surfaces of the fluid model, see figure 2. Non-reflecting boundary conditions can then be applied to these surfaces in order to simulate a semi-infinite fluid domain. This would save considerable computing time. Results from one dimensional fluid models have shown that, the thickness of elements used for the non-reflecting elements need to be less than or equal to the thickness of the fluid elements on the load surface side, where the shock waves will propagate. Thicker elements introduce more inertial effects to the incident shock wave as shown in figure 3, thereby, having a greater influence in modifying the amplitude of the initial shock load. Hence, the input shock energy would not be truly represented.

Figure 3- Effects of Different Thicknesses in the Non-Reflecting Boundary (NRB) Element ~...-.-..e....---.-r.-------..e--..----

1.0 (Origin)

(it NRB) ....... (10t NRB) .......... (,r~t NRB) .......

0.8 0.6 0.4 0.2 0.0 -0.2

0

1

2

3

4

nine(ms) 2.2 Fluid Element Size In order to determine the fluid element thickness which would transmit the detailed characteristics of shock waves through the fluid, several one-dimensional meshes with varying element thicknesses were examined, see figure 4. Element thicknesses examined are

298 T, 1.3*T, 2.0*T and 4*T, where T is a normalised thickness. Typical fluid element thickness needs to be about 2 - 5cm in order to transmit an underwater shock. Figure 4" One-Dimensional Fluid Model to Examine the Effect of Fluid Element Thickness on Transmitting of Shock waves A B

P ---:-~H I III I ililllllllllllillliilllltllllllllllll'filllH

elementthickness (T)

~

Figures 5a to 5d show the pressure time pulses at two locations A and B and how they are effected as a result of the fluid element thickness. Clearly, element thickness above 2*T produce results with considerable "numerical noise". In contrast, the numerical solution for small element thickness, < 2*T tend to be free of noise. Depending on the level of numerical noise, the characteristics of a shock pulse would not be truly represented from the appearance of excessive oscillations, which could excite or load the structure in an unanticipated manner. .

9

.

.

.

.

,

Figure 5a "Beme~ Size = 1T .

.

,

.

,

.

,

9

,

.

,

.

Rgure 5b: ~ ,

.

,

.

9

,

.

,

.

Size = 1.3T

,

.

,

.

,

.

,

.

,

.

,

.

i

-

1.0

~-~

i ---- {.~l~d) I . _

0.8

0.6

.

~

I

i-.. !

~

I .....

"..

. .....

9

.S

i

"..

.,, r

--

-O2

~.~ go g~ ,'.o ,'..~" ,'.o ' h

1

9

,

9

,

9

,

.

,

o.s

-0~. , ~ -0.5

3'.~ ~15 ;.o , ~

R g ~ 6c: Sement ~ 9

12

i

\

i . ~ ; -,

'.

,o

"..

r-

,

.

[:-=

.

i

1.4

9

__

12

~

1.0

i

9

1.0

l

1.5,,

-

w

-

2.0

,

9

2.5

J

9

3.0

|

9

3,5

4.0

4~5

,

9

,

9

,

9

,

-

,

9

~

9

,

~

i"

~

- -.

- ~nt

......

~

,

A

B

""

i I I!

:-.. i

. ......

0.4

..

~S

-

. . . 1.0 . . . 1.5 . . .2.0. . 2.5 . . . 3.0 . 0.5

Ln~ ('~) I

-

t

D_

'"

..

%.~ go .

~

.J

.._c_

i

0~5

Rgum 5d: Element ~'ze = 4T ,

Po~B

0.2

.

= 2T .

0.6

0.0

|

0.0

0.0

3.5

;~ " ,.~

.0.2

G5 9

9.

.............

9

=

0.0

.

'

0.5

.

'

1.0

.

~

1.5

-

'

ZO

.

.

.

'

.

2.5

.

,

.

~_~.____

I

3,0

,

i

~5

-

,

4.0

.

4.5

Extrapolating to 2D and 3D the same principle applies. Figures 6a and 6b show 2 meshes with different densities. Both have the same fluid element radial thickness, but different circumferential element lengths. In order to smoothly transmit the shock wave, it is necessary to have a uniform mesh with the element aspect ratio approximately l'l. The effect of aspect ratio is shown in Figure 7. As can be seen from the fringe plots, Figure 7a has a uniform and smooth wave front, whereas the contours in Figure 7b show the discontinuity in the shock

299 front. Hence, when this load reaches the structure it is not a true representation of the spherical pulse, and therefore will load the structure incorrectly.

300 2.3 Global Fluid Geometry To demonstrate the effect of the global fluid geometry, three fluid model shapes are examined. These models show the effect of geometry on shock propagation, see figure 8. In all cases, the same loading condition is used (ie. the same load pressure curve and its applied direction). Fringe plots of pressure contours at a number of time steps are also shown in figure 8. Clearly the concave shape, with the same curvature as the applied pressure wave, displays a shock front more uniform than the other two models, as the element mesh is biased towards the shock front. That means, the fluid element faces have orientations parallel and perpendicular to the direction of shock for a longer period of time than the other meshes and therefore more capable of transmitting the applied spherical pulse, and hence loading the structure would be therefore more accurate. (Note that: Since colours o f pressure contours can not be demonstrated in this paper, the wave fronts (ie. max. pressure with red colour) are shown as the dimensioning symbol).

3. CONCLUSION When modelling an UNDEX scenario an approximating solution to the shock wave transmission can be obtained by replacing the semi-infinite fluid domain with a limited fluid mesh and a non-reflecting boundary. The shock load can be approximated through the use of scaling the reference load curve. For models with large element aspect ratios, the global geometry and orientation of the fluid elements relative to the incoming shock wave can also influence the shock wave characteristics and hence these factors need to be taken into account for the specific IYNDEX scenario geometry being investigated. Due to the limitation in the current technology of computing power, the use of a high-density fluid mesh, would be extremely time consuming and very costly. In order to balance the cost and benefit ratio in creating fluid models, the use of a compromise in the size of fluid element is required. The thickness and aspect ratio of the fluid elements need to be determined for the specific scenario so as the initial characteristics of the shock pulse can be sufficiently transferred through the medium, and then onto the structure. REFERENCES 1. Ls-dyna3d User's Manuals, Version 940.2, 1998. 2. Swisdak, M., Explosion Effects and Properties Part II, Explosion Effects in Water Naval Surface Warfare Centre, NSWC/WOL/TR-76-116. ACKNOWLEDGEMENT The authors wish to thank Dr Craig Flockhart for his assistance during this work.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta @ 2000 Elsevier Science Ltd. All rights reserved.

301

R e i n f o r c e d m a s o n r y walls u n d e r blast loading C. Mayrhofer Fraunhofer-Institut ftir Kurzzeitdynamik, Ernst-Mach-Institut Am Klingelberg 1, 79588 Efringen-Kirchen, Germany

Masonry walls change brittleness and gain ductility with reinforcement in the bed joint. The dynamic force enhancement is more than a factor of 4 compared to unreinforced brickwork. Reinforced masonry walls can be applied under blast loading conditions.

1 INTRODUCTION Almost no information was available on the dynamic behaviour of masonry walls under blast loading conditions. The brittle behaviour of masonry under bending conditions can be improved to more ductile behaviour by reinforcing steel bars in the bed joint (figs. 1,2). Masonry walls, reinforced in the bed joint, were investigated in the shock tube. The effect of different brick materials (fig.3) and the amount of steel reinforcement on the ductility and load carrying capacity was statically tested. Different support conditions, single, fixed and arch effects (fig.4) was investigated. Yield line-theory was applied to determine the static behaviour. The dynamic response was computed on the basis of a single-degree-of-freedom system. Pressure-impulse diagrams are available.

Fig. 1" Possibilities of reinforcement by masonry m "m ~ir.m : i " II

I

I

II

IIII

IIIII

II

I

:

"Dj

:

""

III

I

I

Fig.2: Used reinforcement-coated Murfor

; li

II

j

" ~ ?

I

~

_

i"

302

2 EXPERIMENTS A total number of 700 masonry elements were tested. For the dynamic loading 10 model walls in scale 1:2 were used because the dimensions of the test facility. The shock-tube has a maximum diameter of 2.4 m (fig.5). The scaled models correspond to prototype dimensions of a wall-length * height = 3.8 * 2.4 m 2 and a wall depth of 11.5 cm to 49 cm. The air-driven shock-tube arrangement is about 50 m long. Typically load-time-functions are given in fig.6. The load duration changed from 15 ms to 38 ms. The static loading tests were done in different test facilities with partly full scaled or 1:2 scaled walls. Investigated was the influence of different brick material, different type of reinforce and amount of reinforcement and different support conditions (single, fixed, arch effects). The characteristic material strength of unreinforced brickwork - the bending tensile strength - was determined at 500 test beams.

lpressurechamberll - 17 m

expansion chambertm. = : 36 m

t i test object ,,

~ml

n

f

T

-~

M )

--,-!

!

:

.

.

.

.

.

.

.

.

.,r,'l.

Fig.5: Shock-tube

303

z.o

,to

P"Pd ""! Itl

_

......

'~

It=tSm 9

.S

.~

.

,

~

~

--

[

.iS

.~

.;S

.~

t-.-

* ;5

.,0 ms .~S

Z.5 9

bar

ZO 9

~

.I.S

p, i)d. e "till

P .111

-1t.5

~o

, =0

-11.5

:

.~

."

.,

I

I

.,~

.;,

.~I.

.~,I

I

.;~

.',I ,, .:~

Fig.6: Load-time function 3 STATIC RESULTS

3.1 Bending The type of brick material influenced the masonry behaviour strongly. Three different failure modes were observed. Failure can occur by exceeding of the brick material strength, the bond between the brick and the bed joint or by exceeding the tensile strength of the reinforcement. The problem was to get a high bond strength. That was reached by clay brick and sand-lime brick. Results are presented from clay brickwork. The ultimate load capacity under bending conditions can be calculated with the stress/strain relationship shown in fig.7. stmin bdckwoA section

':'""

de ~ 01

~176 ooo

~ ~Fe

~m~r f'orces

Emw

9

z = FeFe'~

-------.-.P Z

I.t = b-d"

Fig.7" Strain and equivalent stress diagram for design of reinforced masonry flexure member (pure bending)

304 The calculated bending moment must be limited for under-reinforced and strong reinforced brickwork. For under-reinforced brickwork (1s <0.03) the bending moment is dominated by the bending tensile strength of the masonry and for strong reinforcement ( 1s _>0.23) it is fixed by a maximum value of m=0.2 (fig.8). For failure criterion in the dynamic case a ductility factor 13" =XBdXE was defined. The definition of the elastic deflection xE and the maximum deflection XBr is shown in fig.9. Also shown in fig.9 is the description of the load-deflection curve. The accordance of this formula for the load-deflection curve with the experiments is pointed out in figs. 10 and 11. The result demonstrates the strong enhancement of the load capacity due to the reinforcement in reference to unreinforced brickwork (up to 4-times). The effect of the normalised reinforcement on the ductility factor is displayed in fig. 12.

" b.dZ.13R w

.IldW~

/

.3o //L ,..,~,.ik,,~ ] "'"0"

t 1 ""''"'

ddimim

"~-~:#,r

.20

~-I

,p~x,.x,,,)

%r

m"O,2

.....

.......

/ l'tr,-x .c, l

.00

e! I

.00 I

:

jl l

.tO

.~I

;

+l~c, "

'

'"

.30

O.OO/. < Ip.mm<0.03

XE

. . . . .

.411

.

I

I1=1.1-~

Fig. 9: Approximation of the static load capacity and ductility kN

N '2"p.

5G

Ps = IS, ~ " orcton (~" x 9c, l

t+O

7

50

I

X .r ) Ps " p''~ arcton l+t.'2.pc

i+O

P 30

P30

2O

* f + -

20

/J

10 0

XBr X '--'~--.

Fig.8" Normalized carrying capacity bending moment of reinforced brickwork with reinforcing in the bed joint kN 60

..'

c'dns

10 0

5

10 X - -

-

0 rum

Fig. 10: Static response of reinforced masonry elements with light weight concrete bricks

5 10 x . - - - - - ~ 111o1

Fig. 11" Static response of reinforced wall masonry wall elements with perforated clay bricks

305

T

'|?,

4.

p,x~,xt It/\\

3.,

%\\

t

1. oe , l l . ~

; q,, sso

N o.

,

.ol

.1~

.

,

-

: . . . . .

All

.~

.40

r.,p ~ Fig.

.50

---

12: Ductility of reinforced masonry specimen (beam simple supported)

3.2 Bending and axial force Panels that are confined within a frame due to floor, ceiling and adjacent walls are restraint because the motion in the plane of the wall panel is restricted. In this case an arching effect arises. The used restraint force model is given in fig.13. With this model the relationship between ultimate axial force and bending moment is determined in fig.14. Depending on the normalised reinforcement /~ the ultimate bending moment under consideration of the axial force can be calculated. l I I I

~I~I~-- N

********

§P

IoKli~ arch

/ / / l / .

maximum axial force

effect with reinforcement R

t

-

R

L

§ P

arch effect without reinforcement Fig. 13: Restraint force model

306

'~'ls 1"1)) 2

! V/[ i4d'/d;21~'('l'l)] 4

' -Z-i~'-p(-Z't'2". l'd' ld"il'*~'Z'li 1.o

~-..~-:.~...

R

b d" j3mw

'

01

o

j-.~ .,.....-""

....

......

................ ..... .:" o:.~ 0:5 ,,.-"""" "_,..,.."" ~ . y . ........."" .............,.....--' ........... .,....../...', .,...~ ......

T .00

Fig.

- c, l~;.pi

..~,.. ....

I .20

temile

I i -.40 .60 M m= b.d2~PRw ----,,-

-

9 .80

14: Combined normalised bending and axial force

4 DYNAMIC RESULTS To estimate of the dynamic load capacity the dynamic load factor ~, =Pd/Ps must be known. The dynamic load factor and the failure criterion were experimentally determined and compared to calculations on the of DOF-system. By introducing pu=~, *Ps the dynamic problem can be handled as a static problem. That is possible by using of the maximum value of Pd from the dynamic load carrying capacity diagram for example of fig.15 for the investigated load duration of t~=15 ms and t~=38 ms. For a normalised deflection and load duration the experimental results - where the wall panels failed - are shown in fig. 16. Also calculated lines for multiples of the elastic deflection 1/~, are shown in fig. 16. For comparison also an issue with an unreinforced wall is presented. With this diagram one can find out that reinforced walls are completely destroyed at a failure criterion of 10/~, to 13/~,. That means that reinforced masonry walls fail at a value 10 or 13 times of the elastic deflection. For unreinforced walls the value is 30/~,. Characteristic weapon effects from blastloading are peak pressure (p) and blastimpulse (I). Therefore its useful to have a p-I-diagram. That is carried through for a normalised pressure p =pa/p and a normalised impulse I =Pd * tl/pd * t2 (p=Xd,Br * C/2, t2=Xd,B~/r

Because p, I

m

only depend on x and t i fig. 17 can be developed by transformation of fig. 16. For a real existing wall a p-I-diagram is needed in a non normalised manner. Therefore the transformation points pd= p * c *XB~/2 and p * tl= 1 * c * xB~/~ must be determined to get the actual parameter c, XBr and r This is shown for a certain wall in fig. 18. Now it can be examined at which pressure and impulse that wall will be destroyed. Points above the boundary p-I-line indicate that the wall is destroyed. In the quasistatic region (long load duration) the maximum applied pressure is 60 kN/m 2.

307

I

I

~ot

o

*

~.~-.

. . . . .

,

.-'~"

-~'~"

~-10.

" "~! .

- - -

:

---

"

IIII

,h, ~=,.,,,.m ,

.

:~'

9

X . I,Z

/

/ I.

,

,,

, ,L,

,,,

I

"===~=~**

7 j~

,o

i

,'/

1, lot

0 Xr

~

:

:.t

Fig. 15: Dynamic response of reinforced masonry walls (Ix =0.3%) with clay bricks for different load duration . . . .

!. . . . .

Oa

1sll

.

Fig. 16: N0rmalised deflection vs normalised time with failure criterion

,___

otin

m.--.--?f.~, ~

.t I

...

/ Yl

i rNti

P')o'e't/t~--

_ _~

i~=

kNIm 2

0.3o/0 j

~'~

'--

3

-b -'%"

-'-- ~,a

/----

, _.,_. ~/~ ut,t

~L ~,- _ ~ r ""--~ -

I i

103

p~ -

102

/

qualm~

;~

60-k'r~ ~

~

/

~

~o~x 13/X

9 ~-0.3% II

#-C 101

J

n

~

t

, 0 t,

J

r ..~

, I 10 2

9, , , , . . 1

103

9 , ~, , , . , = 1

_. . 9 I 10/* k N . m s / m 2 10 5

I =Pd.t I -..,.--

Fig. 17: Normalized pressure-impulsediagram with failure criterion

ISO-damage curve of a reinforced masonry wall supported on two sides with the dimension of l*h*d=163* 101"12 c m 3

Fig.18:

308 5 RULES FOR THE DESIGN OF REINFORCED BRICKWORK In a first step the ultimate static load must be estimated. Required is the compressive strength of the brickwork. That can be determined by I]mwR = ~mw * ~ * 1.5 (t~coefficient for special type of brick) and I]mw= ko *l]s~ *~m~~ (ko = coefficient for number of masonry layers, 13s = compressive strength of bricks, 13m~= compressive strength of mortar). Using fie, the tensile strength of the reinforcement, and the amount of the reinforcement kt=Fe / b ' d , the normalised amount of the reinforcement # =Ix * t~ / ~mwR is known. For 0.03 < # <0.2 the D

bending moment is m= # (1-/z/2). The static pressure for a fixed support is Ps,Br = Mar * 12/L 2. With Mar= m * b * d 2 * I]mwR (b = width, d = effective depth) Ps,ar can be calculated. In a second step the spring constant c, the elastic deflection xE--ps,ar/c, the failure deflection of a reinforced brickwork xa,=13 * XE, the eigenfrequency and the load-/mass-factor must be determined for the design of the ISO-damage curve. More details to design reinforced brickwork can be found in r e p o r t / 1 / a n d for unreinforced brickwork in report/2/.

6 REFERENCE /1/Chr. Mayrhofer, Gemauerte W~inde mit Bewehrung unter Druckstoflbelastung, 1/93, ErnstMach-Institut, Efringen-Kirchen, Januar 1993 /2/Chr. Mayrhofer, Dynamisches Tragverhalten von Mauerwerk bei horizontaler Belastung, E 5/86, Ernst-Mach-Institut, Efringen-Kirchen, Juni 1986

Crashworthiness

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

311

Plastic collapse m e c h a n i s m s o f lifeguards for the class 465 E M U bogies E. C. Chirwa a, E. J. Searancke b, A. Hoe c and S. M. P. Wong c aAutomotive Engineering, Faculty of Technology, Bolton Institute, Deane Road, Bolton BL3 5AB, UK bBombardier Transportation, Horbury Junction Industrial Estate, Horbury, Wakefield WF4 5QH, UK eEQE International Ltd., EQE House, 500 Longbarn Boulevard, Birchwood, Warrington WA2 0XF, UK

Lifeguards are energy absorbing devices fitted to the leading edges of train bogies under driving cab ends. Their primary function is to deflect obstacles away from the wheel/rail interface and in the process absorb excessive impact energy through plastic deformation. Presented herein is an experimental investigation into the deformation mechanisms and failure phenomena of steel lifeguards for the Class 465 EMU (Electric Multiple Unit) train bogies. Their performance based on tight specification is evaluated under both quasi-static and impact loading. Under quasi-static loading, the lifeguards have shown to be capable of withstanding without permanent deformation to the bogie and its attachment to the axlebox, a load of up to 35 kN applied horizontally in the global longitudinal axis of the train towards the adjacent wheel. The deformation history is found to be that of a development of a plastic hinge at the neck or narrowest section, which proceeds by rotating the tip about the deformed section. Typical loads necessary for triggering the plastic failure are found to be between 40 kN and 58 kN, which rise to the maximum value of 60 ldq to 69 kN as the tip rotates around the single hinge mechanism. I. INTRODUCTION Derailment is one of the common accident scenarios accounting for over a third of the total number of railway crashes that occurred in the last 40 years. Reports published by HSMO (1, 2) and a statistical analysis carried out in (3) identifies the causes of many significant railway derailments in modem years as being one or more of the following: a) initially a collision occurs between two trains, then there is a de-coupling between the highly stressed rail vehicles and subsequently derailment follows. This scenario was for instance experienced in the Texas (USA) trains that derailed on 3 November 1997, the Clapham Junction on 12 December 1998 (UK), the Ladbroke Grove near Paddington station on 5 October 1999 (UK); b) a train negotiating a bend at high speed leaves the tracks. This was experienced in the Kurla (India) on 3 November 1997 and in the Sao Paulo (Brazil) derailment on 4 July 1998; c) a train having a fault on the undercarriage, that is say a broken wheel or a broken gear lever. This type of accident scenario was experienced in the intercity express Hamburg-bound

312 on 4 June 1998, 35 miles north of Hannover (Germany). There were 100 fatalities, making it the third worse rail accident in Europe this century; d) a train impacting a piece of maliciously placed rock, slab and natural landslide, snow, fallen rocks from mountains, plus animal and vehicle crossings. Such an accident scenario was experienced on 21 May 1995, when a group of teenagers placed a slab across the tracks and derailed a train near Missouri (USA). The Japanese accident on 28 November 1999 near Hokkorido was similar to that at Missouri. However in this case a block of concrete came loose in the under-pass ceiling causing a derailment. Analysing the statistics, many of the derailments are caused by obstacles lying in the trackworks. In Britain for instance such accident scenarios have accounted for 28% of the total railway crashes with 45% of occupant fatal injuries in the past 30 years (3). These high fatality figures have caused concern and European railway operators and manufacturers are searching for ways of reducing the number of derailments and passenger deaths by designing effective energy absorbing devices that deflect objects lying across the trackworks and in the process perform work through plastic deformation. One such device investigated herein is a lifeguard deflector that is fitted to the leading edge of the train bogie under the driving cab end. This paper presents the crashworthiness behaviour of lifeguards for the Class 465 EMU bogies under quasi-static and dynamic loadings. 2. EXPERIMENTAL DETAILS

2.1. Lifeguard/bogie interaction A bogie (7) consists mainly of a frame, suspensions, wheels, traction motors, brake hanger brackets and many more small parts including the lifeguards which are fixed to the radial arm axleboxes using M24 bolts according to Specialist Rail Products Specification No. SRP/PC/90/088E for Class 465 EMU (8).

2.2. Quasi-static testing of lifeguards The quasi-static testing of lifeguards was carried out at a speed of 1 x 105 ms "l in a DARTEC testing machine with a maximum capacity of 250 kN. Each lifeguard was bolted to a specially designed fixture with a rigidity that emulated the axleboxes on actual bogies. The fixture was then secured to the DARTEC anvil as shown in Figure 1. A 28 mm hole was drilled into the specimen end so that a chain ring could be slotted. This arrangement enables the specimen to be tested over a full range to a maximum stroke of up to over a 100 mm. An LVDT connected to the data logger and attached to the specimen, recorded the profile movement, X, of the leading comer of the lifeguard. A total of sixty lifeguards were tested under quasi-static loading.

2.3 Quasi-static test results Specimen No. LFl (neck width of 65 ram) Figure 2 illustrates a typical load versus crosshead displacement. The curve rises linearly up to a yield point of 50 kN with a displacement of 10 mm. The yielding is initially confined to the neck section just before the leading face of the lifeguard. At about 55 mrn deflection, a secondary change in the slope is observed at a load of 69 kN. At this point a second plastic hinge is noticed to be forming at the upper bolt end fixed to a rigid fixture. Further close observation shows slight loss in load, which is a result of the lifeguard separating from the

313 rigid body fixture near the upper bolt. Thereafter, despite gradual loss in load, the deformation continued for another 50 mm until the test was stopped at 66 kN. Figure 3 shows a typical displacement curve, X, of the LVDT against the crosshead displacement. The negative movement of the LVDT indicates an outward movement of the leading comer of the specimen. There is also a point of inflection on the curve at about 15 mm of the crosshead displacement, which is probably attributed to the first yielding of the specimen. A second point of inflection is observed at about 40 mm of the crosshead displacement, which is probably due to the second yielding of the specimen at the bolt joint end. The LVDT reached its maximum stroke of about 25 mm before exhausting its allowable extension. The specimen however, carded on deforming as stated above by moving outwards rather than longitudinally. At this juncture, it is appropriate to say that the excessive outward collapse of the lifeguard is also to a greater extent governed by the second yielding and the movement at the bolt joint, which exceeds the maximum allowable outward movement of 20 mm.

Specimen No. LF3 (neck width 50 mm) Figure 4 is the load versus crosshead displacement characteristic of the lifeguard with a neck width of 50 mm. The curve rises linearly up to a yield point of 41 kN at a displacement of 10 mm. The yielding is confined to the neck section just before the leading face. The curve reaches a maximum and almost steady load of 68 kN at about 78 mm. Figure 5 thereof, is the displacement, X, of the LVDT against the displacement of the crosshead. The negative and positive movements of the LVDT indicate an outward and inward movements of the leading comer of the lifeguard, respectively. At a point where the cross-head reaches 34 mm, the LVDT recorded a maximum value of- 10 mm, after which the leading edge of the specimen started to move inwards marked by the rising part of the curve in Figure 5. Specimen LF3 has met all the design specification, plus its deformation profile is confined and governed by a plastic hinge rotation about the neck. No deformations were observed around the bolted joint as the leading face plastically bent within the 20 mm maximum allowable distance.

2.4 Impact testing of lifeguards Impact testing of lifeguards was conducted in a drop hammer rig. The rig consists of a variable drop mass of up to 210 kg that can be released from a height of up to 9 m. A Laser Doppler Velocimeter was used to measure the velocity-time trace of the tup. The data from the Laser Doppler was then analysed using a computer software to obtain other information including force-time, displacement-time and thereafter force-displacement characteristics. Each trace was filtered at a cut-off frequency of 1000 Hz, which according to [9] is an adequate frequency for this type of specimen being tested. Based on the quasi-static parametric studies, a decision was made to impact only the 50 mm width lifeguard (specimen LF3).

2.5 Impact test results Specimen No. LF3 (neck width 50 mm) Specimen LF3 was impacted with a tup mass of 32.5 kg released from a height of 7.2 m resulting in an impact energy of 2.3 kJ. Since the neck had a smaller width of 50 mm, the resulting deformation was greater with more rotation about the neck as shown in Figure 6c.

314

.~.]Load ~ Crossheadarm I '1

L~ 1 7 6

\

X

I~ p Lvdt displacement= x : L2-LI

" ~ NX~'l~25mmPi n ~ 2 ~ ] ~___j ~3 ~.~ Lvtda f l e ~ ~/

Jig~

Lvtd

//////////////

Lvt'~ L-~ before ' Ll'

Figure 1. Lifeguard testing arrangements under quasi-static loading

10090 ! 8~TTT~T=TTI2~T~_~_T-I l T I~-T-~ o ii]! ........]il............... ~ -i ! --~i!-

r ~II~,

'

i

I . . . . {7-

9

l i ','

i

--F--~, 20

i

30

50

60

70

Displacement(ram)

80

90

.

.

.

8 4oi-- , - l ~ - - i

.

I

I

.

.

!

~

I00

0

5

]

'IV--! 0

!0

T

"] 20

i "

30

40

50

I]

70

1

.

~

;

.

.

!

~ ........... :.. . . . . '. ,

i

i

I0

'

. . . . .

,

......i.... ~-_:...=_2

15 20

25

30

35

i

'

40

..-,___................~**.~

45

50

55

60

CrossheadDisplacement(mm)

65

70

75

g

2 |

~

!

!

I

I

I

';

'--=--

i

3-

'

- ~

8

]

T

,,-T-- , - - - V

-~-

i.i *;.~ -T t * i, ;~ i.....

C~

' 60

"s

Figure 3.65 mm neck width

,

! ....

:

I,

9

!

-

i i i. i t.... !---~........ N ....

~1 ] J

20 ll

i

i i ~1 .... !

,

L 2 _ = L 40

,

.I

}

Figure 2.65 mm neck width

60

', ~1

t--HI

'

0 I0

!

,5

t

:

1----

20 ,411

r

g

70 40

:(----irj '"i

0 80

90

100

o

,o

20

3o

40

so

6o

~o

Displacemem(ram)

Lvdt distance (turn)

Figure 4. 50 mm neck width

Figure 5.50 mm neck width lifeguard

315 This depicts a typical failure mechanism. Some typical characteristics from the Laser Doppler Velocimeter are illustrated in Figure 7. Figure 7d shows that after a vertical displacement of about 5 mm, there is a variation in load oscillating at an average value of about 42 kN. The curve reaches a maximum force of 63 kN when rebound is observed after the maximum deflection. The mean load is therefore 47 kN. 2.6 Discussion of experimental test results The test of specimen LF1 lifeguard has shown that it is capable of sustaining statically applied loads in the order of 64 kN, but in the process excessively deflect downwards such that it makes contact with the beams representing the trackworks, hence fouling them. Two movements at the neck and at the bolt joint govern the deflection. It is the second movement at the bolt joint, which is of concern since it is this global deflection that allows excessive bending into the trackworks.

Figure 6. Collapse mechanism of a typical lifeguard under impact loading As with regards to LF3 lifeguard with a smaller neck width, the results were much better as the failure mechanism concentrated around the neck, which rotated with increase in load. The deflection was excessive, however well within the safe corridor of 20 mm that is necessary to avoid the fouling of the trackworks. Observations during these tests illustrated the tip rotation to be occurring along a predetermined arc type trajectory with linear translation well within the specified 100 mm. As a result, the lifeguard was unable to impair the safe operation of the vehicle or foul the trackwork or the running gear. Furthermore, under impact, a tup mass of 32.5 kg simulating an obstacle in the trackwork was released to make contact with the lower edge of the lifeguard at a velocity of 13.5 ms "i, hence an impact energy of 2.3 kJ. Test results show a well deformed lifeguard to have a linear displacement of between 23 m m - 50 mm in the longitudinal direction and its failure mechanism to be similar to that experienced under quasi-static loading. The collapse behaviour is believed to be mainly dictated by yielding and bending about the global lateral axis with a plastic hinge forming at the weakest point. 3. LIFEGUARD PLASTIC COLLAPSE MECHANISM The collapse mechanism of a typical lifeguard with 50 mm neck width is illustrated in Figure 6, while its load-displacement curve shown in Figure 8.

316 In order to understand the collapse behaviour, it is necessary to analyse Figures 6 and 8 simultaneously and describe what is happening at every stage of collapse. The deformation of the lifeguard has three distinct stages as shown in Figure 8. The mechanisms of collapse are namely yielding, rotation of the plastic hinge about the neck with the smallest width and the global collapse about the bolted joint. The first stage in Figure 8 shows a linear rise O-A in load. Close observation of the specimen shows an elastic behaviour that is terminated by yielding at the neck when point A is attained. Soon thereafter a plastic hinge is formed and rotates with subsequent increase in load. The rotation continues until the material strengthens due to the fact that the outer fibres on the compressive part of the neck becomes more stressed, while the plastic hinge of defined length equal to the neck width rotates about the neck centre of gravity. Point B in Figure 8 is attained after the plastic hinge rotation has fully rotated about 120 ~ when the leading edge locks to the main undeformed lifeguard body. Soon after, with little increase in load, the global rotation of the whole lifeguard follows about the root at the bolted joint. The global collapse mechanism happens usually just in front of the first bolt. Once the bending has occurred and the plastic hinge fully formed about the bolt joint, rotation continues until a complete collapse of the lifeguard is achieved. 4. CONCLUSIONS The parametric studies have clearly demonstrated that the modified lifeguards studied herein for Class 465 EMU bogies are more specification compliant than the earlier versions. The objective of this study has been to develop a new generation lifeguard that will function safely on a Class 465 EMU bogie and that will reduce the number of derailments. Both experimental work, using quasi-static and dynamic tests show results exhibiting typical deformation histories that are governed by yielding and bending of the plastic hinge at the neck or the narrowest section. Overall, the collapse mechanism energy absorption can be said to be carried out in three distinct works. That is the yielding, plastic hinge formation and rotation, and then global plastic hinge formation near the bolted joint. As a result the corresponding energy under the curve shows a 65% more energy absorption using the modified lifeguard shown in figure 6.

REFERENCES 1. The Department of Transport Railway Accident, "Report on collision that occurred o n 4 th March 1989 at Puley', HSMO Publication, London 1989. 2. Railway Safety, "Report on the safety record of railways in Great Britain during 1987", HSMO Publication, London 1987. 3. A W Ewans, "A statistical analysis of fatal collisions and derailments of passenger trains on British railways: 1967 - 1996"', Proc. Inst. Mech. Engrs. Vol. 211 Part F, pp. 73 - 86, 1997.

4. E C Chirwa, "Crashworthiness in railway vehicles", Proc. Of Railtech'94 IMechE conference, paper No. C47/13/002, 24 -26 May 1994. 5. A Scholes, JH Lewis, "Development of crashworthiness for railway vehicle structures", Proc. Inst. Mech. Engrs. Vol. 207 Part F, pp. 1 - 16, 1993. 6. J H Lewis, W G Rasaiah, A Scholes, "Validation of measures to improve rail vehicle crashworthiness", Proc. Inst. Mech. Engrs. Vol. 210 Part F, pp. 73 - 85, 1996.

317

7. B Webber, "Class 323 Electric Multiple Units (EMU)", Proc. Inst. Mech. Engrs. Vol. 213 Part F, pp. 4 9 - 62, 1999. 8. Specialist Rail Products (SRP) Technical Specification Project Reference 130/216/000, No. SRP/PC/90/088E, Issue 1, "Lifeguards for the Class 465 EMU bogie", 1990. 9.

R S Birch, N Jones, "Measurement o f impact loads using a Laser Doppler Velocimeter",

Proc. Inst. Mech. Engrs. Vol. 204, pp. 1 - 8, 1989. 14

60

12

'

10 ,--

6

o@

2

\X. I

!

l

1 ~

:

]

+,

:

'.~

:i!++!

I ' 5

--6 0

5

10

15

20

10

25

15

20

25

Time (ms)

Time (ms)

b)

a) 100

7(

80

6(

5(

I L I! I 1 I.

60

~.,i @ U.~

; +

i +

-4

,~,

,

+!

I

0 -20

/\I ~.,,! ~ / \ v r i "I,/I

3(

! ~__k~__.Lll,_i

+ I .....L___V~

ID

(

i

-!1

"V .

VI +

+r""

"

~

Y ""

~

[ .......

i

-21

-40 0

5

10

0

25

15

10

20

30

Time (ms)

Deflection (mm)

c)

d)

40

50

Figure 7. Typical 50 m m neck width lifeguard impact characteristics 100"

~-7"T~

90 -+-~+

..,

i

Rotation overall lifeguard

Rotation around

Neck section

60 . ~ ~ T ! / I .

20

- ~,. . . .

lO

i-- , ~ - ~

4-

': i ; . ' i ~ 0

!0

20

30

i 40

i 50

i i i ~'

ii

Li I

t/L_i__t

!1 Dfli _LJ 60

70

Displacement (mm)

80

90

100

Figure 8. Load-Displacement curve for the 50 mm neck lifeguard showing typical collapse regions

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

319

Application of multibody dynamics for simulating vehicle impacts on steel safety guardrails G. Sedlaceka; C. Kammela; U.J. GeBlera; D. Neuenhaus b a Institute of Steel Construction, RWTH Aachen, Mies-van-der-Rohe-StraBe 1, D-52056 Aachen, Germany b INIT GmbH, LennershofstraBe 160, D-44801 Bochum, Germany The new European Code EN 1317 [ 1], which is obligatory for members of the European Union since 1999, uses various criteria such as containment level, impact severity level and levels of working width to classify the severity of vehicle impact on safety barriers according to the required level of protection. One of the tasks of a recent study at the Institute of Steel Construction was to optimise the operation of existing road vehicle restraint systems and to categorise their performance for certification according to the severity classes as stated in EN 1317. Using the method of multibody dynamics a computer model of a typical steel guardrail system currently used in Germany was developed. Several impact scenarios were analysed in the time history domain, and numerical results were compared to experimental crash data with a good overall agreement. 1. DIN EN 1317 CODE REGULATIONS Various road restraint systems have been developed in nation-wide programs since the mid-fiRies to advance road safety in Germany. With the recent adoption of the European code EN 1317 as the German code DIN EN 1317 it was intended to harmonise vehicle impact testing procedures as well as standardise all different types of road restraint systems by classifying them into specific severity classes. The most relevant levels of protection of road vehicle restraint systems subjected to vehicle impact are as follows: 9 The safety barrier shall contain and redirect the vehicle without complete failure of the principal longitudinal elements of the system. The vehicle shall not override or penetrate the restraint system. 9 A restraint system should be capable of deflecting and absorbing the impact energy, and at the same time the maximum dynamic deflection should not exceed the specified level of working width. 9 Secondary collisions with traffic in adjacent lines should be avoided, therefore the vehicle should leave the restraint system at a flat angle after impact. EN 1317 defines a specific box-criterion for a safe redirection of the vehicle. 9 The vehicle shall remain upright during and after collision, although moderate rolling, pitching and yawing are acceptable. 9 Occupant ride down acceleration should not exceed a tolerable value. For evaluation purposes the Acceleration Severity Index (ASI) is defined by EN 1317. Also elements of the safety barrier shall not penetrate the passenger compartment of the vehicle.

320 9 No major part of the safety barrier shall become totally detached, being an undue hazard to other traffic or pedestrians. 2. MODE OF OPERATION OF SAFETY BARRIERS When considering the multiple tasks that have to be fulfilled by road vehicle restraint systems, it becomes obvious that it is not easy to find appropriate structural solutions which operate in the desired way and offer reasonably low maintenance. In general a safety barrier has to be capable of providing sufficient deflection to limit vehicle occupant impact severity while at the same time providing sufficient restraint to redirect the vehicle. These two main modes of operation are somewhat competitive, thus a compromise solution has to be found. Accordingly, the most promising barriers combine two system parts, where one part is used to redirect the vehicle via a deflecting element, whereas the second part is very stiff and restrains the vehicle from breaking through the barrier. Steel guardrails, operating like a cable, are commonly used for the pliable redirecting element taking advantage of the high ductility of steel in combination with the steel's ability to sustain high tensile loads. As a stiff restraining element either steel posts or concrete sections are suitable. To improve the interaction of the two parts additional deforming elements are often linked between them. Steel hollow sections are the most suitable components for these deforming elements, but they could also be constructed from rubber material. 3. APPLICATION OF THE METHOD OF MULTIBODY DYNAMICS 3.1 Multibody systems (MBS) To simulate the dynamic behaviour of structures with large nonlinear movements the method of multibody dynamics is preferable. To model a mechanical system as a MBS, it is necessary to assign properties such as elasticity, viscosity, friction, damping, inertia and force to discrete rigid body elements, springs and dampers, friction, impact and contact elements, which then are combined in a global system. The rigid bodies react according to external, internal, inertia and constraint forces.

3.2 The program MEPHISTO The multibody-system program MEPHISTO (Multibody systems with Elastic Plastic Hinges and changeable STructure Organisation) was specially developed for use in civil engineering applications at the Institute of Steel Construction. The program contains all the modules necessary to model the dynamic performance of members, connections and any mechanisms including contact. The Roberson and Wittenburg [10] algorithm is used for calculations. The equations of motion are based on Jourdain's principle of virtual power. Kinematics are described by generalised coordinates. For the numerical calculations in the time history domain the multi-step method by Adam-Bashforth-Moulton, which is a predictor-corrector-method for the solution of differential equations, is adopted. Detailed information on the features of MEPHISTO have been described by Neuenhaus [8], [9]. The program user can assign to each rigid body the required kinematical degrees of freedom. Time histories of accelerations, velocities, deformations, impact forces and internal forces can be specified for all MBS elements, and calculated values are available for a direct comparison with the measured values from full scale impact tests. 3.3 MBS-modelling for the numerical simulation of vehicle impact The study's objective at the Institute of Steel Construction was to apply the method of multibody dynamics to simulate vehicle impact on steel safety guardralls [6], [11], [12]. The

321 main focus of interest was to investigate the EDSP ("Einfache Distanzschutzplanke") steel guardrail system which is frequently used in Germany, see [2], [3]. Data of full-scale vehicle impact tests were used to calibrate the numerical simulation [4], [5], [7]. The test specifications are given in table 1. Table 1. Basic test specifications used for the study System and dimension

EDSP EDSP EDsP EDSP

Type and total ..... mass of vehicle [ton] on bridge cap " 104.0 m Rigid truck 10.0 with rammed posts 84.0 m Rigid truck 10i() with rammed posts i04.0 m Bus 12.7 with rammed posts 84.0 m Car l~b

Impact Impact speed angle [~ [km/h] 70.8 15 72.8 15 7210 20 102.2 20

Acceptance test specification to.EN 1317 [1] TB 42 TB42 TB 51 TB 11

Several impact scenarios were calculated in the time history domain for the four EDSP steel guardrail systems modelled as MBS including all properties of the guardrail-beam, the posts, the distance-spacers, the steel tension string going parallel to the guardrail-beam at the rear of the posts and all structural connections.

322

Plasticity, friction, contact, structural damping, soil conditions etc. were modelled. Internal force elements were defined using nonlinear load deformation and hysteresis characteristics. The resistance at the ultimate limit state was assumed for all structural elements with characteristic values. All necessary degrees of freedom (approx. 500) and their kinematical properties were also modelled. The MBS-models of the vehicles consisted of rigid bodies that were simply hinged together allowing all possible translational and rotational motions at the centre of gravity of the vehicle. Local deformation of the car body was assumed to be taken into account by a constant plasticity factor. Dynamics of the tyres and wheels were neglected. Due to the restricted amount of space, this paper can only report some of the results of the first test system as mentioned in table 1. The system is the 104 m long EDSP steel guardrail mounted on a concrete bridge cap with bolts in the middle section. It includes 12 m long terminals. This system represents the construction commonly used on bridge decks. The guardrail-beam is divided up into separate rigid bodies every 0.66 m in the longitudinal direction, i.e. half the post spacing, except for the terminals. That way local deformations could be sufficiently modelled. A drawing of a typical four m long section of the EDSP barrier taken from the German guidelines for road restraint systems, RPS 89 [2] and the subdivision of the MBS-model are illustrated in figure 1. The translational and rotational degrees of freedom in the defined hinges are marked for the different system axes (L~, Ly, Lz and Rx, Ry, Rz). All degrees of freedom are covered by accompanying longitudinal or rotational spring and damper elements. 4. NUMERICAL SIMULATION RESULTS Figure 2 illustrates via an isometric and plan view, the time history of the simulated vehicle impact with an EDSP mounted on a bridge cap. Figure 3 shows the vehicle accelerations, averaged over a moving time interval of 50 ms. The so-called Working Width or the maximum dynamic deflection at the relevant spot of the guardrail, respectively, is given in figure 4. This simulation indicates the barrier would successfully pass the test in a similar way as the full-scale experimental test. To compare the numerical and experimental results, the values of the ASI and of the maximum dynamic deflection are suitable. Taken from the simulation the ASI for the impact of the 10 tons-vehicle to the barrier is 0.22. But more important for categorising a barrier system are the maximum accelerations acting on a passenger car (1 ton): for simulating the impact scenario of the passenger car to the barrier the ASI has been determined to 0.77. According to real-world testing results the ASI-value was measured 0.72 in this case. Real-world testing results for the maximum deflection of an EDSP barrier system under impact of a 10 ton-vehicle varies slightly below 1.5 m. Some of the conclusions for a possible improvement of the safety barrier system due to the results of simulation are: 9 The bridge cap causes an undesired liiting of the vehicle during impact increasing the risk of the vehicle to override the barrier. 9 The bolted connections at the foot of the posts are designed to fail under impact loading. At the same time the bolted connections between guardrail-beam and posts are also designed to fail in order to prevent the guardrail-beam from being pushed down as a result of post rotation for the rammed post system. So there is a certain risk of posts becoming totally detached and thus becoming an undue hazard to other traffic or pedestrians. 9 Due to the post-soil-interaction, the rammed post safety barrier system has a lower stiffness than the same system with bolted posts and therefore reduces the risk of severe injuries to occupants during a crash significantly.

323

Figure 3. Vehicle accelerations, averaged over a moving time interval of 50 ms

Figure 4. Max. dynamic deflection of the guardrail at different distances from impact-point

324

5. CONCLUSIONS An accurate numerical simulation of different vehicle impact scenarios was achieved using the computer program MEPHISTO. Hence, a parameter study can easily be carried out of different types of vehicles, impact angles and velocities and for various safety barrier system properties. With a steadily growing and diversifying vehicle fleet on the streets the search for new and improved methods of protection using road vehicle restraint systems continues to be a lasting challenge for the future. The certification of such systems to uniform standards is an important step forward in evaluating any new developments. In the future, real-world fullscale impact tests will still be required as the final certification method for approval of a particular barrier system. However, during the barrier's development process the application of computer simulation methods as presented in this paper, will surely enable the prediction of their behaviour at a comparably low cost. 6. ACKNOWLEDGEMENTS The presented work was f'mancially supported by the Stitttmg Stahlanwendungsforschung, Essen, by order of the StudiengeseUschatt Stahlanwendung e.V., Dfisseldoff. The Institute of Steel Construction gratefully acknowledges this support. The kind support of the Bundesanstalt fiir StraSenwesen (BASt) through discussions and supply of data is gratefully appreciated. REFERENCES

[ 1] EN 1317 Road restraint systems- Part 1: Terminology and general criteria for test methods; Part 2: Performance classes, impact test acceptance criteria and test methods for safety barriers (1998) [2] RPS 89 - Richtlinien ~ r passive Schutzeinrichtangen an Stral3en, Ausgabe 1989, Forschungsgesellschaft fiir Straiten- und Verkehrswesen, German guideline (1989) [in German] [3] TL-SP 99 - Technische Lieferbedingungen flit Stahlschutzplanken an Bandesfemstral3en, Ausgabe 1999, German guideline (1999) [in German] [4] Ellmers, U.; Lukas, G.: Sonderantersuchung passiver Schutzeinrichtangen auf Brficken, Final report of project 94640, Bandesanstalt fiir StraBenwesen, Bergisch-Gladbach (1996) [in German] [5] Ellmers, U.; Schulte, W.: Aktuelle Anprallversuche an Schutzeinrichtangen nach europ~iischen Anforderangen, StraBenverkehrstechnik, No.4 (1997) [in German] [6] GeJ31er,U.J.: Untersuchungen zur Wirkungsweise yon Stahlschutzplanken bei Fahrzeuganprall, Diploma thesis, Institute of Steel Construction, RWTH Aachen (1999) [in German] [7] G6sswein, K.: Anfahrversuche an Schutzleitplanken mit SIGMA-Pfosten, StudiengeseUschafl fiir Stahlleitplanken (1977) [in German] [8] Neuenhaus, D.: Dynamik yon Mehrk6rpersystemen zur Beschreibang von Grenzzust~nden fiir Baukonstruktionen, Institute of Steel Construction, RWTH Aachen (1993) [in German] [9] Neuenhaus, D." A three-dimensional contact model to detect and represent contacts in multibodysystems, Innov. in Eng. for Seismic Regions, Civil-Comp Press, Edinburgh, pp. 103-108 (1997) [ 10] Roberson, R.E.; Wittenburg, J.: A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of a satellite attitude control, Proceedings of the 3rd IFAC Congress 1966, London, pp. 46D. 1-46D.9 (1968) [ 11] Sedlacek, G.; Neuenhaus, D.; Kammel, C.: Rechnerische Untersuchung fiber die Auswirkung von Fahrzeuganprall an Verkehrszeichenanlagen, Final Report FE 15.237 R93F, Federal Ministry of Transport (1996) [in German] [12] Sedlacek, G.; Kammel, C.; GeBler, U.J.: Anforderang an Stahlschutzplanken and deren Wirkangsweise bei Verwendang verschiedener Varianten and Altemativen and deren Weiterentwicklangen, Report P392, Studiengesellschafl Stahlanwendang e.V., Dfisseldorf (2000) [in German]

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

325

A large-deflection design technique for the collapse and roll-over analysis o f thin-walled tubular frames S.J. Cimpoeru a, N.W. Murray b and R.G. Grzebieta b aDSTO Aeronautical and Maritime Research Laboratory, P.O. Box 4331, Melbourne 3001, Australia. bDepartment of Civil Engineering, P.O. Box 60, Monash University 3800, Australia.

A large-deflection design technique is presented for the thin-walled tubular frames which form the superstructure and roll-over protection of passenger buses and other such vehicles. The load-displacement behaviour of a collapsing two-dimensional frame was modelled to large deflections by combining separate elastic and rigid-plastic analyses. Accurate modelling required accurate predictions of the frame's deformed geometry and insertion of the largerotation moment-rotation properties of thin-walled tubes into the rigid-plastic analysis. A unique pure bending rig was used to measure the moment-rotation properties and elastic stepby-step analysis was used to model the behaviour of the frame up to the point at which the frame formed a kinematic mechanism.

1. INTRODUCTION The upper body-work or superstructure of passenger buses is an example of a thin-walled tubular framework that requires an accurate large-deflection design technique. This is because this framework must protect the occupants of such vehicles in roll-over accidents and absorb the kinetic energy of the impact (1). While bus roll-over accidents are not common in Australia such tragedies still occur from time to time (2,3). In such events bus and vehicle frames generally fail in a collapse mode involving lateral sway of their side walls. Such frames will initially respond in elastic bending which is then followed by a loss of stiffness caused by plastic deformation in the various component members and joints of the structure. Plastic hinges will eventually form in the thin-walled members, and are best described as local plastic collapse mechanisms (4), because they involve large localised plastic deformations, with geometrical folding and significant changes in their cross-sectional profile. When a sufficient number of plastic hinges have formed in a frame to enable it to form a kinematic mechanism, the maximum load capacity is no longer sustained and structural collapse occurs with all further deformation being concentrated in these locations. Since the strength of these plastic hinges diminishes with deformation, a drooping load-displacement curve is exhibited beyond the peak load. The fall in the strength

326 of the frame causes the component members of the structure to unload with further deformation, as demonstrated by the recovery of the elastically stressed portions of the frame. Unfortunately the gross plastic deformation and geometric distortion that accompanies the formation of local plastic collapse mechanisms makes the analysis of collapsing frames particularly difficult. A reasonable approach is to conduct a static analysis, ignoring dynamic effects (5). This is because most bus roll-overs tend to occur in the lateral direction at low speeds (6,7). Inertial effects can be neglected because the deformotion is largely confined to the bus superstructure while most of the mass of the bus is concentrated close to the floor. In addition, at least for square and rectangular tubes, the geometry of plastic hinges formed dynamically is the same as those formed statically (8). Moreover, the dynamic enhancement of the material flow stress at large plastic strains can be accounted for, e.g. (9,10). The load-deflection behaviour of collapsing frames can be predicted to large deflections simply by superimposing separate elastic and rigid-plastic analyses as has been demonstrated in the past for many types of thin-walled sections (4). This paper applies such an approach. A complete load-deflection curve allows the kinetic energy absorption capability of the frame to be determined as well as possible intrusion of the frame into the occupant survival space (1). While a load-deflection curve will allow the ultimate strength of a frame to be determined, it also examines the potential of a frame to shed the applied load and collapse in a 'brittle' manner, i.e. collapse suddenly without significant energy absorption. The latter event can easily occur when a drooping load-deflection curve is exhibited (4). The modelling method described in this paper is aimed at providing a means that can be used to quickly evaluate whether potential vehicle designs have adequate roll-over protection at the early design stages (11). While a frame made from square thin-walled tubes is modelled in the present instance because many passenger buses are made from this material, the technique also has application for the design of any thin-walled structure that displays drooping load-deflection behaviour and requires a large-deflection design technique. When a bus rolls over, a roof comer usually contacts the ground first. For typical Australian buses, the line of action of the resulting load passes through the roof comer and is inclined at approximately 15 ~ from the horizontal datum. The horizontal load component is the most important component of the applied load in a lateral roll-over (5,6,12), being far more critical than a vertical crush load on the roof due to the vehicle self-weight (13). A collapse mode involving lateral sway is therefore analysed. As a first approximation, the prismatic rectangular shape of a three-dimensional passenger bus can be modelled as a combination of statically loaded two-dimensional frames (11). The energy absorption of each component frame can then be simply added to obtain the total energy absorption of a deforming bus. This assumes that the whole length of a bus contacts the ground at the same time and deforms evenly. Although this assumption is idealised, it is part of modelling procedures certified by ADR59/00 (1) and routinely practised (14,15,16). This approach has a number of practical advantages (17) and importantly should be conservative because longitudinal contributions to the strength of the overall bus frame are neglected.

327 2. EXPERIMENTAL DETERMINATION OF BUS FRAME LOAD-DEFLECTION BEHAVIOUR AND SECTION BENDING PROPERTIES

2.1. Frame Testing An idealised two-dimensional bus frame constructed from square thin-walled tubes was tested to confirm the validity of the model predictions (11). The experimental loaddisplacement response of the frame was obtained.by simply loading the frame by means of a hydraulic tension jack that pulled from an initial angle of 15 ~ from the horizontal. The applied load and the movement of its point of application were measured as deformation proceeded.

2.2. Large-Deflection Bending Properties of Square Thin-Walled Tubes The bending properties of the square thin-walled tubes from which the frame was constructed had to be characterised into the large-deflection range. This is difficult because of the large member rotations that the tubes must undergo as they form local plastic collapse mechanisms. While cantilever bending is commonly used to obtain collapse curves of thinwalled sections, this method of testing usually has a number of limitations (11,18). A unique pure bending rig was therefore constructed to stress a tubular specimen in a state of pure bending, and maintain a constant bending moment over the central portion of the specimen to large rotations beyond bending collapse. This rig and the details of the experiments are described elsewhere (11,18,19). One important feature of this bending rig was the minimisation of the development of tensile forces in the test specimens (19). Figure 1 is a typical moment-rotation curve that was obtained in pure bending for the square thin-walled tubes that were used to construct the tested frame. The main feature of the pure bending rig is that it allows the extent of the constant bending moment prior to collapse to be meaningfully and easily measured. It is also clear from Figure 1 that an enormous amount of energy can be absorbed by a member after the point of collapse. The key information from such a moment-rotation curve that is required for modelling purposes is: the collapse curve (obtained by curve fitting), the maximum load and importantly, the rotation at which collapse

'l

OCCURS.

A3 E Z I--

z

2

Ill

0

0

,

0

|

10

i

20

,

,

,

I

,

30

I

40

,

i

50

--

ROTATION (deg)

Figure 1. Typical pure bending moment-rotation curve for a cold-rolled 350 Grade 50x50x2 mm square steel tube.

328 3. MODELLING TO LARGE DEFLECTIONS BY SUPERIMPOSITION SEPARATE ELASTIC AND RIGID-PLASTIC ANALYSES

OF

3.1. Elastic Modelling Since elastic deflections are generally small compared to the relatively large deflections that occur during collapse, the elastic analysis need not be precise. In fact analytical methods have been used to derive simple, though adequate, elastic models for two-dimensional frames (11, 20). Finite element analysis can also be used to analyse more complex 3D frames. In the present instance, an elastic finite element program BEAM (Beam Element Analysis with Mechanisms) (21), which used a step-by-step analysis (22) of three-dimensional beam elements, was used to determine the load-deflection curve up to the point at which a sufficient number of plastic hinges developed to enable it to form a kinematicaUy moveable mechanism. The advantage of this type of analysis is that it allows a quick determination of the deflection of a vehicle frame up to this point and an estimate of its collapse load, in the process predicting the locations of the plastic hinges which form the frame collapse mechanism. This latter information readily allows a rigid-plastic modelling technique to be applied.

3.2. Rigid-Plastic Modelling Simple plastic analysis, which is commonly used to estimate the ultimate load-carrying capacity of steel structures (23), was extended into the large-deflection range to form the basis of a rigid-plastic modelling procedure (11). Such rigid-plastic modelling requires that the following assumptions be made: all deformation occurs at the locations of the plastic hinges; and the frame behaves as a kinematic mechanism that pivots about four plastic hinges that are assumed to be connected by rigid-members.

3.2.1. Deformed Geometry Once the initial plastic hinge locations are defined so too is the deformed geometry of the frame, i.e. the positions of the plastic hinges (in terms of their x and y co-ordinates) and their rotations can be defined for any horizontal displacement of the loading point, 6, by means of simple geometric relationships. These relationships can often be derived by inspection of the deformed geometries, e.g. Figure 2.

B~A-~

C

B

P

(a)

C

p

(b)

Figure 2. Initial and deformed frame geometries for the tested thin-walled frame, where (a) is the assumed and (b) the actual (refer Section 3.2.4.) deformed geometry, fi is the horizontal displacement of the loading point, P is the load and 0 and 0' are plastic hinge rotations.

329

3.2.2. Resisting Moment The resisting moment as a function of plastic hinge rotation (where the latter is obtained from the deformed geometry) can be obtained from experimental (11,18) or analytical (8,15) models of the moment-rotation curves of a number of different specimens. It is critical that these models predict or specify the rotation at which collapse occurs. 3.2.3. Equilibrium Equations Static equilibrium equations were derived by treating the members between the plastic hinges as rigid-bodies, ignoring shear effects (11). Figure 3 shows the free-body diagrams for the side-members of the frame, which can be used to derive an equilibrium equation for the applied load as a function of both the deformed geometry of the frame (Xl, yl and Y2) and the resisting moments at each plastic hinge. F1

~ Psin15 ~ ~ B~

F2

C

Yl

-Y2 A

Psin15 !~

1 1 =!

Figure 3. Free-body diagrams for the side members of the tested frame, where P is the applied load and F and M are internal forces and moments.

3.2.4. Modelling the Rigid-Plastic Load-Displacement Curve The calculation of the rigid-plastic load-displacement curve of the frame is achieved by incrementing the displacement of the loading point, 6, to obtain the plastic hinge positions and rotations. The latter allows the resisting moments to be calculated and these moments together with the deformed geometry of the frame (xl, yl and Y2) are simply inserted into the equilibrium equation for the applied load. This is an example of a heuristic solution technique, with the precise details of the local deformation at the plastic hinges not being required for a solution. An elastic model (11), the results from the elastic step-by-step analysis and a rigid-plastic model were superimposed to model the complete experimental load-displacement curve of the frame, Figure 4. The elastic analyses over-predict the stiffness of the frame because the joints within the frame and the fixed support were not perfectly rigid. The modelling of the loaddisplacement curve's elasto-plastic transition was improved by sketching a curve which intersected the point at which BEAM predicted the formation of a kinematic mechanism and the point of collapse as indicated by rigid-plastic theory. This is an extension of an established method of modelling the elasto-plastic region of collapsing thin-walled structures (4). Figure 4 shows that the load-displacement curve beyond collapse is faithfully modelled up to where the folds of the plastic hinges begin to interact with the fixed supports of the frame.

330 15

~,~----- elastic _.~.,'~_trigid.plastic

i ~ point of collapse

10

/0

I

I

I

I

I

100

200

300

400

500

Horizontal displacement, mm

Figure 4. A load-displacement curve for the tested frame. Elastic, step-by-step and rigidplastic analyses are superimposed and compared with experimental results. Open diamonds denote the points at which BEAM (21) predicts plastic hinge formations. While the rigid-plastic model assumed that plastic hinges would form at the comers of the frame, this was not found experimentally because the joints between individual members were specially strengthened (refer Figure 2(b). Despite this inaccuracy, Figure 4 shows that the agreement of the theoretical model with experiment was excellent. The current paper has concentrated on predicting load-displacement behaviour in the horizontal direction. While the vertical displacement of the loading point will also contribute to the energy absorption of the frame, this component is of negligible magnitude in the present instance because the frame only has a small residual strength when the vertical displacement of the loading point is large. In the more general case, however, the vertical component of energy absorption could be determined as well. Overall, the combined elastic and separate rigid-plastic approach was able to adequately define the elasto-plastic transition prior to collapse as well as model the load-displacement behaviour of the collapsing frame to large deflections. This vindicated the assumptions associated with the application of these techniques to such an event. The accuracy of the result depends on both the locations assumed for the plastic hinges and their moment rotation properties. Accurately determined moment-rotation curves will be beneficial and step-by-step analysis should ensure accurate predictions of both the plastic hinge locations and the modelling of the elasto-plastic region of the load displacement curve. 4. CONCLUSION The load-displacement behaviour of a collapsing thin-walled two-dimensional frame was modelled to large deflections by combining separate elastic and rigid-plastic analyses. This procedure could be used at the early stages of vehicle design so that potential vehicle structural designs can be quickly evaluated to ascertain whether legislated roll-over protection requirements are met. Accurate load-deflection curves require accurate predictions of the frame's deformed geometry and the large-rotation moment-rotation properties of its

331 component members, which can be directly inserted into the rigid-plastic analysis. Elastic step-by-step analysis will allow the plastic hinge locations and therefore the deformed geometry of a collapsing frame to be accurately predicted while moment-rotation properties can be predicted analytically or accurately determined to large rotations using a unique pure bending rig. Step-by-step analysis will also improve the accuracy of the prediction of the elasto-plastic region of the load-displacement curve. REFERENCES

1. 2. 3. 4.

Australian Design Rule 59/00, Federal Dept. of Transport, Australia. The Age Newspaper, Melbourne, Australia, 26 Sept. 1990. The Australian Newspaper, Sydney, Australia, 26 Sept. 1990. N.W. Murray, Introduction to the Theory of Thin-Walled Structures, Clarendon Press, Oxford, 1986. 5. D. Kecman, J.C. Miles, M.M. Sadeghi and G.H. Tidbury, Inst. Mech. Eng., 1977, C139/77, p. 67. 6. D. Kecman and G.H. Tidbury, in XIX FISITA Congress, Melbourne, 1982, paper 121. 7. H. Bruns, in The Bus '86, Inst. Mech. Eng., 1986, p. 13. 8. D. Kecman, in Structural Crashworthiness (eds N. Jones and T. Wierzbicki), London, 1983, p. 175. 9. P.S. Symonds, in Behaviour of Materials Under Dynamic Loading (ed N.J. Huffington, Jr), ASME, 1965, pp. 106-124. 10. S.J. Cimpoeru, in Dynamic Loading in Manufacturing and Service Conf., Melbourne, 911 Feb. 1993, IE(Aust.), pp. 17-23. 11. S.J. Cimpoeru, The Modelling of the Collapse During Roll-Over of Bus Frames Consisting of Thin-Walled Tubes, Ph.D. Thesis, Department of Civil Eng., Monash University, Australia, 1992. 12. G.A. Wardhill and D. Kecman, in XVIII FISITA Congress, Hamburg, 1980, Vol. 1, p. 281. 13. R.H. Grzebieta and P.H. Dayawansa, in Structural Crashworthiness and Property Damage Seminar, Monash University, Melbourne, 1987, p. 111. 14. J.C. Miles, Int. J. Mech. Sci., Vol. 18, 1976, pp. 399-405. 15. D. Kecman, Bending Collapse of Rectangular Section Tubes in Relation to the Bus Rollover Problem, Ph.D. Thesis, Cranfield Institute of Technology, Great Britain, 1979. 16. J.C. Brown, Int. J. Vehicle Design, Vol. 11, 1990, pp. 361-373. 17. J.C. Brown, in XXI Meeting of Bus and Coach Experts, Budapest, 1990. 18. S.J. Cimpoeru and N.M. Murray, Int. J. Mech. Sci., Vol. 35, 1993, pp. 247-256. 19. N.W. Murray and S.J. Cimpoeru, Int. J. Mech. Sci., Vol. 38, 1996, pp. 351-353. 20. S.J. Cimpoeru and N.W. Murray in Dynamic Loading in Manufacturing and Service Conf., Melbourne, 9-11 Feb. 1993, IE(Aust.), pp. 169-176. 21. S.J. Cimpoeru, BEAM: A Finite Element Program for the Collapse Analysis of Vehicle Structures, MRL Technical Report MRL-TR-94-6, 1994, Maribymong, Vic.: Materials Research Laboratory. 22. H.B. Harrison, Structural Analysis and Design, Some Microcomputer Applications, Part 1 and 2, Pergamon, Oxford, 1990, p. 125. 23. M.R. Home, The Plastic Theory of Structures, Thomas Nelson and Sons, 1971.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R,H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

333

A Method of E s t i m a t i n g Velocity in a C a r C r a s h Kenshi Fujiwara First Machinery Section, National Research Institute of Police Science Kashiwanoha 6-3,KASHIWA-CITY,277-0882,JAPAN

1. Introduction In. car crash accident investigation, estimating car's velocity is important. We are interested in two cars' crash accidents. When we can find skid marks of braking in the scene of an accident, we can estimate the velocities of the cars. But it is becoming difficult to find them because of a lot of cars with ABS. We propose a method of estimating velocities and directions of ears without information from skid marks of braking. In our method a conservation law of angular momentum is important as well as that of linear momentum and energy. Firstly we demonstrate that angular momentum is conserved in a real car crash experiment. Then we analyze it and propose our method.

2. Experiment We made a car crash experiment of side-impact collision. Fig.1 shows how they crashed and where they stopped. The large car B was going along the axis YY at right angles to the other small car A which was going along the axis XX at the same speed of sixty-five kilometers per hour. A large sedan type car B collided with a small compact car A on the left side. Then they moved separately and stopped. The cars loaded some accelerometer devices in the position of the center of gravity and other positions. In our study we were interested in a very short event of a collision. The collision starts at the initial time zero and ends at the final time z: ( t a u ) when the cars' bodies are the most deformed. In our case z" was about 80 milliseconds experimentally. RK~ ,d

,/.a,.;,>

20.5 m

: j'" rs.~ ".a, " , Z" e

.

Stoppingpoints

:..'" Sm

Crashing point

-Ib Fig.1

.."

9

...................... . . . . . . . . . . . . . . . . .

9

_

"

....

. ......... ' "

'

;

,,,~

~!

:-'~:;e\ i "~''r.... -~-

' ~9 "

,

,, i

i

..:"\

",b'\ ~ .,t \ ',,

a

"'

yy, u

tO.Sin 19.5u

The Side-Impact collision experiment

334

3. Analysis

of the experiment

Next we show the configuration of the cars at the crashing point in Fig.2. Ga is the center of gravity for the car A. Gb is for the car B. Now we imagine a point G which is the center of gravity for the two cars. In our analysis we use a special coordinates XX',YY' whose origin is denoted as G. The axis XX' or YY' has the same fixed direction as the ground coordinates system. In this coordinates system, both cars are moving with the same momentum in opposite direction toward the virtual barrier standing at the origin. We can show the trajectories of Ga and Gb on our special coordinates system.

,,

,.

.

XX' axis

xx ,,.,

A

'1-

.

.

,

C.O. ofcarb Gb

tml[nw-v-I

.........

> Ylf

.......

' YY'

C.G. r

a ~ ~

I-

lLm

, ,

'

,

Fig.2

axis

axis

of 8

i ;

i

i i

The Centers of Gravity for two cars at the beginning of the crash

Fig.3 is a trajectory of the center of gravity for each car. The left side is that of the car B, and the right side is that of the car A. Zero means a position at the initial time of collision and 7: at the final time. Each car has its velocity along the tangent of the trajectory. Both cars have 11"1 U . O

0.4 0.3

t--o

Rb

"

0.2 0.I

-1

-.5

1 "C V

-0.2

L

-0.3

"

R a ~ ~

1.

m

a

0.4 Fig.3

Moving traces of the center of gravity for each car

335 angular momenta around the origin in our experiment. We call it an orbital angular momentum. At the same time the car bodies rotate around their own center of gravity and we have to estimate the spin angular momenta. Summing up all of them, we can get the total angular momentum for the two cars. Fig.4 shows time dependence of total angular momentum for the two cars. The vertical axis indicates an angular momentum in units of ton.meter per second and the horizontal axis indicates time in units of miUisecond. La represents an orbital angular momentum for the striking 9 car A and Lb represents for the struck car B. They are decreasing with time. At about forty milliseconds they decrease rapidly and at eighty milliseconds they become constant. (Ia • co a + Ib • 09b) is the total spin angular momentum which is important in our study. It's derivation is represented later in this paper. It is increasing with time and becomes constant at about sixty milliseconds. We put four quantities all together and showed it as the total sum in this graph. It goes up and down with time and is not constant. But at the time z , it is the same amount as at the initial time zero. We think that the total angular momentum is presented in the collision as expected from angular momentum conservation.

30 . . . .

25

.

f

m.._....

J

~''~

-.,.. ~

J

Total sum

20 15

aX~a +mbX~b

!

10

La

5

--~____

~._.____. ,,

,,~.~

J'~ I~

2)

, , 43

60

Lb ]

8O

,

10o m (time)

-5

ton.d/s

Fig.4

Angular momentum of the two cars

Fig.5 shows spin angular momenta for both cars and their total sum amount. Spin angular momentum is a spin angular velocity multiplied by a moment of inertia. The spin angular velocities were derived from the data of the accelerometers in the cars. However estimation of moment of inertia about G for the car is difficult because the cars are changing their forms and positions. They are rotating around the center of gravity for the two cars. Then we approximated a moment of inertia as the product of M by the squared R (M • R z) where M is a mass of the car and R is a distance between the center of gravity for the car and the origin of the coordinates. R is changing with time while the cars are colliding. The top curve in Fig.5

336 12

10

Ia = M a x Ra ~ l'b= M b x Rb 2

8

I ilXma

J

Moment of Inertia

+lbX~b

/I axcoa

6 4

,f-

I bxcob 2

,

0

-T

.~

,

10o ms (eme) i

-2

ton.~ls

Fig.5

Spin angular momentum of two cars

indicates the total spin angular momentum for the two cars. In our study it is most important that we have approximated the moment of inertia as above. This approximation led us to the angular momentum conservation and by using conservation law, we can get a new-method estimating the car's velocity just before a collision.

4. Example of our method Now we are going to show an example of how to estimate the car's velocity before a collision. Fig.6 represents a frontal crash experiment. Both cars were the same sedan type and their masses were the same. t=O

Vax.V~

s i

MafMb

Initial of a Crash

i

i b

t----t"

Most

deformed

Bodies

Overlapping of the both configuration

w,~y

~oa. ,,.. Ra . = j [ _ ~ .

tffiO

x

tffi

............

I1 2..~:~,~:~.." 9

~'~!

.... ~.::..~:.~,_,-;r--. . . . . -. . .

9

. . . . . . . . . . . . .

. 1 ~ 1 ~ % . ~

" " ~

,

9 ,4~Q

_

_

L3--_:-..., yJ.

Fig.6

An example of an offset frontal collision

The left up figure shows the start of the collision at time zero. Usually we have skid marks of braking at the car accidents. We can then get information about the progressing direction of the car before a collision. But we do not always find skid marks of braking, especially for the car equipped with ABS. Then we have to estimate not only velocity but also its direction. The angular momentum conservation law is useful for estimating the direction.

337 Generally we have to estimate four components of velocities. Vax and Vay are two components of velocities for a struck car A. Vbx and Vby are the ones for a striking car B. They are described in the coordinates on the ground. We have to estimate the contact configuration for the cars at this time. The left down figure shows the final state of the collision when the time is z. First we have to estimate the contact configuration of the cars when the bodies were most deformed. Moreover we have to estimate numerically the velocities, the deformation energy of the car's body and the spin angular velocities. This is not easy in practice, but it is not impossible.. The right figure shows the left two diagrams superimposed. In these states of a collision, the mutual positions of the centers of gravity are changed. But the physical quantities of the total energy, the total angular moment and the total linear momentum are not changed. At the initial time of a collision, the total energy of the cars is expressed as a quadratic combination of relative velocity components as derived in Fig.7. At the final time of collision, total energy is composed of three numerical components. They are the kinetic energies, the bodies' deformation energy and the spin rotation energies. Then we get a quadratic equation of Vlx-V2x and Vly-V2y. On the other hand, the angular momentum is expressed as a linear combination of relative velocity components at the initial time of collision. At the final time of the collision, the angular momentum can be expressed numerically. Then we get a linear equation of relative velocity components. Moreover we have a linear momentum conservation law. It is expressed in terms of the ground coordinate system.

(1) Total energy

E = E'

E = (1/2) • Ma • Mb/(Ma+Mb){ (Vax-Vbx)2+(Vay-Vby) 2 } E' = Kinetic energy of relative motion for two cars +Deformation energy of bodies for two cars +Spinning rotating energy for two cars (2) Total Angular Momentum

L = L'

L = p • ( Vax -Vby ) + q • ( Vay - Vby ) p,q=numerical number L' = Orbital Angular Momentum around the C.G for two cars + Spinning Angular Momentum around the C.G.for two cars (3) Total Linear Momentum

Px = Px'

Py = Py'

Px = Ma • Vax + Mb • Vbx Py = Ma X Vay + Mb • Vby Px', Py' =Total Linear Momentum for two cars on the coordinate system on the ground Fig.7

Physical quantities of cars system at the beginning and the end of a collision

We can solve the equations graphically as in Fig.8. The vertical and the horizontal axes

338 indicate the relative velocity components in units of meters per second for the cars. In this figure a linear line represents an angular momentum conservation and a circular line represents an energy conservation. A point o[ the intersection is the solution, which gives us only a relative velocity for the cars. But we have another conservation law of a linear momentum in the ground coordinates system. Then we can get a complete solution for the four components of cars' velocities. They show that Vax = 10km/h, Vay = 0, Vbx = 104km/h and Vby = 5km/h. Experimentally their values were Vax=Vay=0, Vbx=100km/h and Vby=0.

Conservation of Angular momentum & Energy ( from the left graph ) V b y - V a y - - 1.5 V b x - V a x = 26.1

/~ Vbx- Vax ,,

Conservation of Linear momentum VI~,- V ~

~..

"~ ..... 30"t"~'(] ..... ~o'/"'!o'"'i'l~ .... -~0")'-~ (m/s)

Total Angular / Mommtum [

~ ~-30

.... Total J m ~

V b y + V ay -- 1.1 Vbx + V ax = 31.6

Total solution from the above equations Car A .... V ax---2.7 V ay= -0.2 Car B .... Vbx=28.8 V b y - 1.3

!

Fig.8

Solusion of the example

5. Conclusion Firstly we have shown the angular momentum conservation in the case of side-impact collision, assuming the moment of inertia as the product of M by R 2. M is a mass of the car and R is the distance to the center of mass for the two cars. Next we proposed a method of estimating velocities for two cars in a crash accident without knowing the velocity's directions for the two cars. The estimation in practice is difficult. But our method is helpful when we have no information about skid marks of braking. 6. Reference Ichiro Emori The engineering of the Automobile Accidents -The method of the reconstructionTokyo Gijutsu Shoin,1993 Hirotoshi Ishikawa Impact Model for Accident Reconstruction -Normal and Tangential Restitution Coefficients, SAE paper 930654 This experiment was carried out in the course '97 of NRIPS Training Center.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

339

D y n a m i c Characteristics of Bicycle Helmets S. K. Hui and T. X. Yu Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Polymer foam has been widely used in bicycle helmets as an efficient energyabsorption material. This study examines the energy-absorbing capacity and various failure modes of polymer foam laminated structure. The impact tests were conducted at a particularly designed drop weight test rig in accordance with relevant standards. Based on the experiment observations, a mechanics model is proposed which is capable of estimating the contact force and predicting the energy absorbed by the bicycle helmet. The predictions have shown good agreement with the experimental measurements. Alternative selection of materials is also suggested.

Keywords: bicycle helmet, impact tests, laminated structure, failure modes, mechanics model

1. INTRODUCTION In some studies, cycling is considered as an "unprotected" group in traffic [ 1] due to a higher frequency of head injuries than other road users [2]. Because of the serious consequence of head injury, the attention of using bicycle helmet is highly raised on head protection. The use of bicycle helmet has been cited and promoted as an effective means for reducing the head injury by 85% and brain injury by 88% [3]. In order to reduce human damage, it is necessary to minimize the impulse and the peak force transmitted to the head less than 15kN. In the present standards, the only option is a single main impact onto different anvils, within a maximum allowable acceleration. This raises the question of whether the current bicycle helmets have an optimum performance in their impact resistance and what kinds of failure they would eventually experience. The energy-absorbing capacity and various failure modes of the polymer foam laminated structure are investigated. Since the shape of the force-deformation curve is expected to vary at different impact points [4], the impact response of the bicycle helmet as a whole is also studied.

2. EXPERIMENTAL SET-UP

2.1 Impact Tests of Laminated Block The samples were laminated panels of dimensions 100x100x26 mm with PVC polymer skin and expended-polyurethane (EPU) foam core of density 96 kg/m 3. The tests

340 were conducted in Dynatup 8250 drop-weight tester with various impact energies to simulate various impact cases. From the requirement of the CPSC standard [5], the impact energy is 100J for flat striker and 60J for hemispherical and penetration strikers. The drop weight was a 100x 100 mm flat steel panel, a 100 mm diameter hemispherical steel striker or a 25 mm diameter penetration steel striker. Impact samples were firmly supported on a flat steel plate without additional constraint. The flat tup provided a uniform pressure on the samples (Fig. l a) and the total energy absorbing capacity of the laminated structure was measured. The hemispherical and penetration tups produced local damages to the samples (Figs. l b and l c), while the failure modes and penetrating characteristics were studied.

2.2 Structural Impact Tests of Bicycle Helmets The testing method was to attach a helmet to a headform and drop it in a guided freefall onto a flat steel anvil (Fig. 2). The headform was fitted with a PCB Piezotronics Model 350A04 shock accelerometer, which measured the acceleration applied to the headform during impact. The signal was transmitted to HP 54540A oscilloscope. All tests were performed as perpendicular to the test line of the helmet.

Fig. 5 Three different failure modes

341 3. EXPERIMENTAL RESULTS

3.1 Impact Response of Laminated Block Fig. 3 shows the transverse crushing behaviour of the foam-core laminated panels. An initial linear-elastic region ended at a peak load by the onset of plastic collapse. The crush load then remained approximately constant as the cells failed by a progressive folding mechanism and started to increase again once the core was fully crushed. This plateau with about seven or eight oscillations having amplitude of about 15% of its magnitude is manifested, similar to that observed by Porter [6]. The skin sheets played no significant role in the crushing process when the sample was smaller than the base plate. The results portrayed in Fig. 4 show four different energies' impact with a hemisphere striker, indicating typical dynamic force histories. Each exhibits an initial linear-elastic response followed by a drop from a peak load. This peak load coincided with the failure of skin. Samples then proceeded to collapse, with the skin bent into a concave profile. After this initial failure, crushing continued at low load. Progressive crushing with useful energy absorption did not generally occur since part of the load was carried by the thin skin rather than by the core material. The load history diagram terminated in a very high peak load on 100J and 150J samples. The final rise shows the over-loading of laminated structure. The skin recovered its shape, but the unloading path differed from the loading path. 3.2 Observation of Failure Modes of Laminated Block Polymer foam was compressed, while the central portion of the impinged skin area was bent. Further loading of shear force on skin made the skin cracking in a circular region. The striker sheared the skin together with the core underneath downward, while the foam fracture sometimes occurred. The failure modes in a laminated foam-core produced by a striker in impact area were quite complicated (Fig. 5). There are three common modes of failure: skin cracking, delamination and foam fracture. The physical variables governing the dynamic response and failure in the laminates under impact loading include the mass, shape and velocity of the striker, as well as the properties of the laminate. Therefore, it is necessary to develop a mechanics model to encompass all the variables necessary to predict the impact-induced dynamic deformation and damages. 3.3 Structural Impact Response of Bicycle Helmets Twenty-seven new helmets were tested. The load-deflection curved obtained in the tests demonstrated an initial elastic stage which ended at a high peak force, followed by a drop in force, implying a non-negligible rebound of the helmet and hence significant velocity change to the head. Evidently, the skin sheet plays a role of spreading the load widely so that although the impact occurred only in a small area of the skin, the crushable EPU foam of sufficient amount is engaged in absorbing energy.

342 4. A S I M P L E M E C H A N I C S M O D E L AND D I S C U S S I O N

4.1 A Simple Mechanics Model 4.1.1 Governing Equations The prediction of the mechanical behaviour of a bicycle helmet subjected to low velocity impact is a key to improve the efficiency, cost and safety margins of the primary parts of helmets. To simulate the impact situation depicted in Fig. 6, Fig. 7 sketches a one degree-of-freedom mass-spring model in which M~ denotes the mass of the head, M 2 the effective mass of the bicycle helmet, F~s the force acting on spring i and F~d the force acting on dushpot i. Springs 1 is elastic-plastic and spring 2 & 3 are elastic. This model accounts for the motion of the helmet, the contact behavior and the motion of the head. Thus, the governing equations are given by oo

Ml(w~- g) + (Fl s +/71 d ) = 0

(1)

M2 ( w 2 - g ) + F 2 ' - ( F ~ '~ +F~d) = 0 2

(2)

M2 (w3- g) + (F3 s + F3a ) - P2s = 0 (3) 2 The initial conditions of the motion at t = 0 for the head and the bicycle helmet are 9

9

9

oo

oo

w~ = w2 = w3 = Vo and wj = w2 F a=

{

oo

= w 3 -'~

g , while the damping forces are expressed as

r/,(w~-w2),

[(w'-w2)-wo]>O

0,

[(W 1

and F3a=

--'W2)--Wa]~_O

{ " T/3"W3'

W3 > 0

O,

W 3 ~--0

(4)

where w a is the physical gap between the headform and the comfort foam before impact.

4.1.2. Frictional Gap While the headform moves downward, friction exists between the comfort foam inside the helmet and the headform; that is, f =/~. N w h e r e / t is the coefficient of friction, and N is the normal force acting on the headform due to the compression of comfort foam before impact. The effective strain of the circumferential direction isE = (wt - w2 - wa ) / a , where a is the average thickness of the comfort foam and wo = 2mm is adopted in the model. The normal force applied on the headform can be formulated as N = Ac/.c rq, where A~ is the cross-sectional area of the comfort foam and crq is the membrane stress of comfort foam.

4.1.3 Spring 1 - Comfort Foam The comfort foams are polyurethane foams faced with a cloth layer. The uncompressed foam has thickness of 5 mm. The comfort foam can only be compressed as follows: E~E, Loading case: a q = Eqe. e [25r and if the strain

e

)~]

of the foam exceeds

0 < e _<E~r 9 9 e > •,r,( w~- w2 ) > 0 e~r =0.4,

(5)

the stress is multiplied

by,

exp[25(•-0.4 ~ ], to approximate the bottoming of the foam. Unloading case: trq = ix* . E~r (e* .

e),. e* >. e > e*

or* ,or , > Ec:t~r,(,.~-w2)
(6)

343

The normal force acting on the compression of the comfort foam is F c: = Acl -o'~:. In the vertical direction, the total force caused by deformation of the comfort foam reacted to the headform is F~s = (F~: + f ) s i n 0 where 0 is the inclined angle of the headform. From the model, the contact area is taken as constant and equivalent to a disc of radius 55mm. The experimental curve is fitted by F~ = k~(w~ - w 2 - w~ ) + r/~ (V~ - V2) and k~ = A~: .(1 + b t ) . E ~ / . s i n 0 .

4.1.4 Spring 2 - Global Deformation Due to the complex geometry of bicycle helmets, an approximate method is to perform a quasi-static compression test of the helmet with a loading plate from the top and a ring - like support under the helmet without any headform being present. This may result in a different stress distribution in the shell from those caused by a side or front impact. However, this simplified configuration was easy to be installed and tested. As a result, the elastic constant of spring 2 was determined experimentally as k2 = 700 kN/m. 4.1.5 Spring 3 - Yielded Foam Liner For an impact onto a fiat surface, the contact area of the liner is 2~Lx where x is the compressive deformation at the initial impact point and the helmet surface is spherical with radius R. Therefore, the liner is loaded by the force F as F = ~ 2Rx)~ >. = k 3x (7) with k 3 = 2Fta~a r . Hence, a perfectly-plastic foam with constant yield stress cr r results in force F3s as given by, Loading case:

F 3" = k3x 3

Unloading case:

F3" = F s ' "

w3 >_0 1+

t~yh

/~kX3

W3 < 0

(8) (9)

where h is the total thickness of the foam, E is the elastic modulus of the foam and & ' denotes the force immediately before unloading. If the liner deflection at the center of the contact area reduces during the impact the total force is assumed to reduce along a line of slope. 4.1.6 Discussion A numerical solution of the governing equation (Eqs. 1, 2 and 3) yields the dynamic response and contact force history in Fig. 8 and shows a good agreement with the experimental result from the whole helmet impact test shown in Fig. 9. The prediction of peak acceleration and pulse trend fairly agree with the experiments. Up to now, the model is only suitable to the flat anvil, while helmet is assumed to deform elastically. The model has incorporated the plasticity of polymer foam liner, as well as the inertia and viscosity of the helmet and headform. The distinct dynamic performances of various helmets under impact could be identified just according to their material properties and structural configurations. 4.2 Alternative Material Cellular textile composite may serve as an alternative material of liner inside bicycle helmets in view of the combination of good ventilation, high impact resistance and flexural stiffness. Some studies have been conducted in our lab, confirming the excellent energyabsorbing capacity of this kind of new material.

344

Fig. 8 The numerical solution for the model

Fig. 9 Experimental result from the whole helmet impact test

5. CONCLUSING REMARKS This paper presents an experimental study on bicycle helmets, identifying the failure modes on laminated panel. A simple mass-spring model is used, which is efficient but has limitations due to the simplifying assumptions on which it is based. With the help of the modelling simulations, it is easy to understand the physical meaning of impact pulse. The model can be further used to examine the influence of various components in the helmet structure so as to optimum its impact performance. REFERENCES [ 1] Eilert-Petersson, E. and Schelp, L., Accid. Anal. and Prev., 29,3,363-372, 1997. [2] National Center for Statistics and Analysis, Traffic Safety Facts 1993, Pedal Cyclists. National Center for Statistics and Analysis, Wastington, D.C., 1994. [3] Thompson, R. S., Riven, F. P., Thompson, D. C., New England J. of Med., 320, 361-367, 1989. [4] Mills, N. J., Br. J. Sp. Med., 24, 55-60, 1990. [5] CPSC Bicycle Helmet Standard, The Final Rule, Published in the Federal Resister, 1998. [6] Porter, J. H., SAE, paper no. 940877, 1994. [7] Gilchrist, A. and Mills, N. J., Int. J. of Impact Engng., 15, 201-218, 1994.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

345

Crash and High Velocity Impact Simulation Methodologies For Aircraft Structures C.M. Kindervater, A. Johnson, D. Kohlgrtiber, and M. Ltitzenburger German Aerospace Center (DLR), Institute of Structures and Design, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany Numerical crash and high velocity simulation studies of metallic and composite subcomponents of aircraft structures are presented which were performed within various European Commission (CEC) sponsored research projects. Both hybrid simulation techniques as well as Finite Element (FE) based crash codes were used. Several new composite material models were developed and validated against tests. Especially with the metallic structures, the simulations correlated well with data of impact tests which were performed within the programs.

I. INTRODUCTION One research project [ 1] intended to develop and validate non-linear dynamic analysis for commercial transport aircraft with the aim to generate a basis for the improvement of crashworthiness design. However, the crash impact conditions under which the components were tested and simulated went far beyond any currently applied crash safety standards for transport aircraft. Substantial damage in the structures should provide a data base to validate the numerical simulation tools. Examples of this project comprise hybrid and FE based simulation methodologies for a fuselage bay structure and a fuselage section of the A320. The research program CRASURV [2] intended to increase the knowledge of the crash behaviour of composite aircraft fuselages. Generic sub-floor sections representative for light aircraft and helicopters and sub-cabin belly structures of a commuter and an airliner type aircraft were designed in composites, built and dynanaically tested. The tests were simulated using modem explicit FE codes (LS-DYNA3D, PAM-CRASH [4], RADIOSS) with enhanced composite materials models. As an example a modular sub-floor section design concept and the application of PAM-CRASH to this concept will be demonstrated. In the HICAS project [3] a methodology was developed for simulating numerically the response of composite aircraft structures to high velocity impacts (HVI), such as bird strike or foreign object damage (FOD), e.g. on wing leading edges and engine fan blades. Procedures to measure the mechanical properties of composites under large strain and high strain rate loading were developed. New material constitutive laws and damage mechanics based failure

346 models for advanced UD- and fabric composites under HVI loading, typically up to 400 m/s, were developed and implemented into PAM-CRASH. The FE based methodology was validated by simulating the laboratory high strain-rate tests and HVI tests on idealised composite structures.

2. ALUMINIUM FUSELAGE SUB-STRUCTURES

2.1 Material and rivet properties used in FIE simulations Material tests with AI 2024 and AI 7075 alloys have not shown any sensitivity of the properties on the strain rate. Therefore, the Krupkowsky elastic-plastic material model with isotropic hardening, but without strain rate effects has been selected for the simulations with PAM-CRASH: = K ( e o + ep )n

(1)

In this formulation ff and e are true stress and true strain components, and K and n are material parameters. Elements are eliminated whose effective plastic strain reaches a value higher than a given limit strain. Rivets: All parts of the fuselage structure were modelled separately and connected with rivet elements. In PAM-CRASH rigid body links (rivets) between the nodes are allowed to separate upon violation of the following failure criterion:

( PN / PN* ) al + ( Ps/Ps * ) a2 < 1

(2)

The calculated normal and shear loads in the rivet PN and PS are related to the normal and shear failure loads PN* (8 kN) and PS* (5 kN), the exponents aland a 2 control the interaction between normal and shear failure.

2.2 Rear Fuselage Bay Structure - Component Dynamic Test 1 (D1) Structural details and FE models: The bay structure which was cut from the rear part of a A320 fuselage consisted of two half frames, 45 stringers, the skin and the entire cargo and passenger floor structure. The diameter was about 4000mm, the total length was 700mm. The final FE mesh consisted of 66440 nodes, 58884 shell elements and additional 3485 rivets. The trolley used in the test was represented by a moving rigid wall. Crash test: The structure was fixed at the passenger floor level and was loaded in the zdirection by a trolley having a mass of 1240 kg and an initial velocity of 8.12m/s. Trolley acceleration and three strain gage rosettes were measured. Two high speed cameras and one video camera were also used for the documentation of the crash sequence.

347

Correlation of FE simulations and crash test: Pre- and post-test simulations with different half and full models were performed. The simulation of a full model in Figure 1 shows the deformed structure 45ms after the impact. Plastic deformations started very close to the location where the skin failed in the test. Additional plasticity occurred just beside reinforcements around the intersection of the struts with the frames. In general, the correlation between the simulated deformations and loads and those found in the crash test was very good.

Figure 1. Deformed component D 1 at 45 ms

2.3 KRASH Simulation of a Fuselage Section Structural details and KRASH model: The structure represented a part of section 17 of the A320. Six seat rows, 14 dummies, and two overhead bins were installed. The KRASH [5] 3D-half model of the section consisted of 79 masses, 23 nodes, 30 springs, 136 beams and 42 plastic hinges. The linear beam properties and the structural mass distribution were determined from the NASTRAN file of the section. The non-linear properties of the springs and plastic hinges were generated from component crash test data and the respective PAM-CRASH simulations. Crash test: The section having a mass of 2330 kg was dropped with a z-velocity of 7 m/s on a concrete surface. The measurement channels comprised 48 at the dumnfies, 16 at the seats, 80 at the structure, and 36 at the overhead bins. Different views of the test were filmed with standard videos and high speed cameras. Correlation of KRASH simulations and crash test: More than 80 correlation of test results and KRASH simulations were performed. An overlay of the deformed structure and the KRASH post-test model is shown in Figure 2. The global deformation behaviour is represented very well and also accelerations, velocities and displacements at different locations were in good agreement with the test results.

348

Figure 2. Deformed structure and KRASH model at 100 ms

3. COMPOSITE AIRFRAME SUB-COMPONENTS 3.1 Modular Sub-floor Section Design Concept The design concept shown in Figure 3 comprises basically 4 modules which can be optimised separately with regard to EA performance and load carrying capability. In addition a fifth module representing the cabin floor panel could be considered. As an example, a module combination (box) which has been analysed and tested is shown in Figure 4. Module 1: Cruciform 1.) Simple Intersection 2.) HTP/HCP-Cruciform Modules 2/3: Beams

Module 4: Skin

1.) Plain webs 2.) Integrally stiffened web 3.) Sine-wave Beam 4.) Trapezoidal Beam 1.) Plain Skin

Figure 3. Modular construction of sub-floor aircraft sections 3.2 PAM-CRASH Applications to the Modular Sub-floor Section Concept

PAM-CRASH composite damage model: A homogeneous orthotropic elastic damaging material model was the most appropriate for fabric laminates, as this model is applicable to brittle materials whose properties are degraded by micro cracking. This type of material may be modelled in PAM-CRASH as a 'degenerate bi-phase' model in which the UD fibre phase is omitted, and the 'matrix' phase is assumed to be orthotropic. The assumed stress-strain relation in the model then has the general orthotropic form:

349

Figure 4. Sub-floor section design comprising HCP-cruciforms o = E E,

E = Eo[1-d(en)]

(3)

where o, E are the stress and strain tensors, E the stiffness matrix with initial values Eo, and d is a scalar damage parameter. This takes values 0 < d < 1 and is assumed to be a function of the second strain invariant En, or the effective shear strain. The composite fabric ply or larr~ate has orthotropic stiffness properties, but a single 'isotropic' damage function which degrades all the stiffness constants equally. The schematic fracturing damage function and corresponding stress-strain-curves are shown in Figure 5. fracturing damage function :

stress-strain diagram :

It

~(~)A /'-",

.

.

.

.

.

.

.

.

.

.

.

Figure 5. Schematic fracturing damage function and corresponding stress-strain curve

PAM-CRASH simulations of modular sub-floor sections: Different sub-floor concepts have been analysed using FE-meshes created in the modular sub-floor section model. Selected results, Figure 6, show quarter models of various box designs 8ms after the first impact. While the plain webs as well as the integrally stiffened webs in Box 2 tended to buckle and failed without absorbing much energy, the trapezoidal corrugated beams in Box 4 failed progressively and absorbed much more energy. In Box 6 with plain webs and a simple intersection, the webs just buckled and created a single fold and the simple intersection also failed without absorbing energy. Box 4 could absorb all the initial kinetic energy within 80 mm of deformation, Box 6 could not stop the added mass on top of the boxes until the simulation was stopped after 180 mm of deformation. At that point the downward velocity of the additional mass was still 6 m/s.

350

Figure 6. Comparison of different composite sub-floor section designs

4. HIGH VELOCITY IMPACT SIMULATIONS OF COMPOSITE AIRCRAFT STRUCTURES

4.1 Modelling of Composites under High Velocity Impact (HVI) Loading For composite materials dynamic failure behaviour is very complex, especially due to the possibility of both fibre dominated or matrix dominated failure modes, and the rate dependence of the polymer resin properties. Constitutive laws for orthotropic elastic materials with internal damage parameters are described in [6], and take the general form: e = S o

(4)

where tr and e are vectors of stress and strain and S the elastic compliance matrix. Using a strain equivalent damage mechanics formulation, the elastic compliance matrix S may then be written:

1/E,(1-d,) S =

- v!2 / E! 0

- v~2 / E~ 1/ E f f l 0

d2)

0 0

(5)

1 / G12(1 - d 12)

This general plane stress form for an orthotropic elastic material with damage has 3 scalar damage parameters d~, d2, d~2 and 4 'undamaged' elastic constants: the Young's moduli

351 in the principal orthotropy directions Et, E2, the in-plane shear modulus G~2, and the principal Poisson's ratio v~2 which is not degraded. The damage parameters have values 0 < di < 1 and represent modulus reductions under different loading conditions due to progressive damage in the material. Thus for unidirectional (UD) plies with fibres in the x~ direction, d~ is associated with damage or failure in the fibres, dz transverse to the fibres, and d~2 with in-plane shear failure. For fabric reinforcements then d2 is associated with the second fibre direction. In [6] conjugate forces Y~, Y2, Yl2 are introduced corresponding to driving mechanisms for the damage parameters: Yl = O112 / (2Et(1-dl)2),

Y2 = 13222/ (2E2(1-d2)2),

Y~2 = r

/ (2Gl2(1-dlz) 2)

(6)

and it is assumed that the damage evolution equations have the general form: dl= fl (Yt, Y/, Yt2),

d2 = f2 (YI, Y2, Yl2),

dl2 = fi2 (Yl, Y2, Yl2).

(7)

Multi-axial failure, or interaction between damage states can be included in the model depending on the complexity of the form assumed for the evolution functions fl, f2, f~2. These are determined from material test data.

4.2 Plate Impact Trial Simulation The basic features of the fabric ply damage mechanics model have been implemented into PAM-CRASH as Material Type 131 'Composite global ply model'. For validation, an impact case considered a 300x300 mm square 4mm thick glass fabric/epoxy plate placed on a 250x250 mm rigid frame. The plate was impacted at the centre by a 50 mm diameter rigid sphere having a mass of 21 kg and impact velocity of 3.13 m/s. Figure 7 shows the plate response at 4 ms at the point when the sphere begins to penetrate the plate due to fibre failures at the point of impact. The implemented fabric model distinguishes clearly between different failure modes in the structure. As examples contours of fibre strain and plastic shear strain are shown in Figure 7. This predicted behaviour is in line with observed failures in composite plates being tested.

5. CONCLUSIONS FE codes such as PAM-CRASH, although well accepted in the automotive industry, are just becoming established in the aircraft industry. Very detailed geometrical models with suitable materials models and property data are required for good structural failure predictions. FE simulation is being based only on a geometry model with appropriate materials constitutive laws. Any testing required is on materials specimens level. For composites under dynamic loads there are many possible failure modes such as crushing, fibre fracture, delamination, matrix shear, etc., and new materials models with associated test methods for measuring failure, generation of damage parameters and implementation of strain rate dependency are

352

Figure 7. GF/epoxy plate impact simulation after 4 ms - with contours of: (a) fibre strains (b) plastic shear strains (V0 = 3.13 m/s, M = 21 kg) currently active research areas. At present it is not practical to carry out detailed simulation for a complete aircraft. A hybrid code such as KRASH is well established in the aircraft industry and has been developed specifically for crashworthiness studies. Geometrical models are relatively simple as is demonstrated by the complete A320 section model. The designer can carry out in short time parameter variation studies. However, with hybrid codes skill is required in the idealisation of the geometry model, in determining mass distribution and spring stiffness characteristics so that essential structural features are included. Where spring properties are highly non-linear, as in crush elements, it is necessary to carry out crush tests on critical elements in order to characterise spring properties. Hybrid and FE crash codes can be coupled to a local/global approach where non-linear behaviour of representative structural elements is analysed with FE and used in a coarse hybrid model as macro-element, e.g. non-linear spring characteristic.

REFERENCES 1. IMT-2002 Crashworthiness for Commercial Aircraft, EU RTD Project, 1993-1995. 2. ,,CRASURV - Design for Crash Survivability", EU RTD-Project, 1996-1999 3. HICAS High Velocity Impact of Composite Aircraft Structures, CEC DG XII BRITEEURAM Project BE 96-4238 (1998) 4. PAM-CRASH, Engineering Systems International GmbH, D-65760 Eschborn. 5. DR.I/KRASH Version 9601 - Users Manual, Dynamic Response Inc., Sherman Oaks, California, USA, January 1996. 6. Ladeveze, E. Le Dantec, Damage modelling of the elementary ply for laminated composites, Composites Science and Technology, 43, 257-267 (1992).

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

353

Design for crash safety in mine shags G. J. Krige, W. van Sehalkwyk and M. M. Khan AATS, PO Box 61587, Marshalltown, 2107, South Africa Various recent accidents in South African mine shafts have focussed attention on aspects of crash safety in deep vertical mine shafts. The paper gives brief comment regarding the philosophy adopted for crash safety, and the holistie approach that is used. This includes the engineering design philosophy and the application of several impact resisting and energy absorbing devices, but also layout geometry, in-shaft signalling, and the use of specified procedures. The paper describes in detail some of the specific devices and measures used to prevent crash situations and alleviate their effects when they do occur. Several of these devices have been tested under close simulation of in-service conditions, and the paper gives typical results of these tests. Testing of other devices is impractical, so the paper describes the theoretical derivation of their behaviour. 1. INTRODUCTION On Christmas eve, 1989, the braking system on a man winder failed, and the cage bringing 105 men out of the mine collided with the crash beams in the headgear. There was loss of a number of lives and many others were paralysed with broken necks or backs. In 1994 a locomotive failed to stop at a shatt station and plunged into a shaft, falling about 100 m before colliding with an upcoming man cage. The rope was detached from the cage and the cage plunged to the bottom of the shaft, killing all 117 men aboard. Accidents such as these do occur, but they are not acceptable, and ways of preventing them, or reducing the severity of their effects when they do occur, must be sought in the drive towards greater safety in mines. A detailed consideration of crash safety requires not only the application of devices that will absorb the kinetic energy of heavy conveyances or vehicles moving at speed, but also a more holistic approach, if an appropriate result is to be achieved. In a technical paper, it is appropriate to concentrate comments on the engineering design philosophy and methods and the energy absorption devices themselves, but the paper would not be complete without some additional reference to the more general considerations. In different countries there are widely differing approaches to ensuring crash safety in shafts. The design practice in many countries is still rather simplistically to rely on the traditional use of high factors of safety. Experience with ultra deep shafts in South Africa has necessitated a much more realistic rational approach to the definition of loads and design methods, to achieve reliable shafts. In the application of safety devices, Canada requires the use of safety catches on man cages (Canadian regs, Roininen and GuUick, 1973), whereas in most other countries this is not required. This paper will focus on the technical aspects of current practice in South Africa

354 2. HOLISTIC PHILOSOPHY The fundamental approach when considering crash safety in mine shafts must be a holistic philosophy of understanding the entire chain of events leading to accidents. Amongst these are included a range of human issues, such as training, supervision, and the general safetyconsciousness of workers. Other factors in the chain of events are administrative, including the availability and use of proper equipment and procedures. The best engineered technical systems possible can fail if these contributory factors are overlooked. This paper, however, concentrates on some of the technical matters that have a contribution to accidents in terms of their prevention or mitigation of their severity. Again a holistic approach demands that a whole potential chain of events must be considered if the best technical solutions are to be achieved. Possible emergency situations must be identified, and the resulting loads need to be understood. Then the implications of reaching different structural limit states must be assessed, in order to define appropriate constraints for the design process. The manner in which this process has been followed is described below. 3. APPROPRIATE DESIGN PHILOSOPHY The holistic design methodology now adopted in South African shaft and conveyance design practice starts by ensuring the best possible definition of the static and dynamic loads that are actually applied in normal service and under emergency crash conditions. Current limit states design philosophy is then used to provide structures with an appropriate level of resistance. 3.1 Definition of Loads Throughout the 1980s and 1990s many research programmes have aimed at improving the current level of understanding of the loads actually applied in mine shaft design applications. Thomas (1990), Krige (199), Greenway (1990), Krige and Hofmeyr (1988), and AAC CTO (1997) describe the developments and give some of the results of this work. In all of these different investigations, the aim was similar- to improve safety and reliability by means of a better definition of the operational and emergency loads applied in mine shafts. A series of limit states design codes (SABS 0208) has also been prepared for the South African Bureau of Standards, to encapsulate the design philosophy and definition of loads resulting from many years of experience and from this research. 3.2 Limit States Design Approach These developments have led away from reliance on traditional factors of safety, to the defined level of reliability that is implicit in limit states design approaches. Traditionally, the member stress under static load only was compared with the material strength divided by a high factor of safety to ensure structural adequacy. When applied to the winding rope, which is effectively a very long tension member, and when applied to a limited range of shaft depths, this was perhaps an appropriate method to use. However, when applied to the full spectrum of structural members it ignored any dynamic effects that may have increased the static stresses, it ignored fatigue, it ignored the possibility that local or overall buckling may

355 have reduced the member strength, and it ignored any influence that deflections may have had. Thus, although this method was employed for many years, it gave a very uncertain indication of structural adequacy. The design approach now specified by all four parts of SABS 0208, and used by many engineering consultants to the South African mining industry, is the following: 9 Working loads are defined as accurately as possible. In numerous instances the codes require the use of rational methods to determine the loads, where it is intended that methods such as energy analysis or computational simulation should be used. 9 Emergency loads are defined on the basis of identifying possible crash or emergency situations. The maximum loads that can realistically be expected on shaft structures and conveyances in these situations are then assessed, and it is endeavoured to ensure the least severe consequences. Thus, for example, on fixed rope hoists the maximum load applied to any conveyance is the rope break load, because it is possible to break the rope, and there have been instances of the rope being broken. If the conveyance can resist the rope breaking load, then the maximum energy absorption of the system can also be mobilised, minimising the likelihood of the rope actually breaking. The mere application of a factor of safety did not achieve this desirable result, because in deep shafts it typically led to the conveyance failing without mobilising the full energy absorption capacity of the long rope. However, where friction hoists are employed, the rope will slip over the d_rum in most cases without breaking the rope. Rational simulation of circumstances that potentially lead to high loads typically indicates that the use of substantially lower loads is quite appropriate. 9 The normal limit states approaches, now almost universally applied in structural engineering, are then used for ultimate resistance, fatigue life, and deflection. It is interesting to note that this new design philosophy has resulted in slightly heavier structures in some instances, but in lighter structures in other instances. Heavier conveyance structures have caused some concern because hoist permits specify the conveyance maximum self weight, which may be exceeded under the new design requirements. More concern has been expressed where it has been possible to use lighter structures, because conservative users have been unwilling to accept what they perceive as reduced factors of safety. However, the new philosophy and procedures are becoming quite widely accepted. 4. PREVENTING SPECIFIC CRASH SITUATIONS Consideration of three crash situations is of primary importance to ensuring the highest possible level of safety in deep mine shafts. These are end of wind stopping, the conveyance snagging on something in the shaft, and vehicles inadvertently entering the shaft. Design philosophies and devices used to reduce the risks associated with these possibilities are described below. Figure 1 shows a vertical shaft with some of the safety devices used. 4.1 End of Wind Stopping Devices There are various reasons why a conveyance may continue to be hoisted beyond its intended stopping position, and result in an overwind accident. These include driver error, brake failure or other mechanical fault, or electrical signalling failure. Most of the effort that has been expended to reduce the effects of these incidents has concentrated on electronic devices to prevent the occurrence of overwinds.

356 End of wind stopping devices are a final effort to avoid a disaster once other safety devices have failed. Thompson (1973) made the statement that "spring-keps, jack-catches . . . . . are at best a last line of defence to mitigate the effects of an accident..", and Roininen and Gullick (1973) make reference to "last resort protection against overwind.." after describing various other safety devices intended to prevent accidents. The key here is the appropriate use of energy absorbing devices. Selda strips and various types of buffer systems are becoming more commonly used in South African mines. There are two main problems that arise in the application of these systems. The first is the length required for them to operate effectively. Because hoisting travel is vertical, it is costly to provide the space often associated with the most commonly employed road or rail crash energy absorption systems. The second problem is the mass of arresting beams required to span across the shaft. The arresting devices cannot easily be located in the path of the conveyances, so they must br placed alongside the hoisting compartment, with beams spanning the compartment to arrest the conveyance. Careful analysis of the dynamic performance of this system is necessary to ensure the safety of occupants of the conveyance under the high deceleration that results when the conveyance strikes these beams. End of wind stopping devices are usually only positioned in the headgear, but on some low speed hoisting installations, it has been considered pragmatic to use buffers at the bottom of the shaft as well.

4.2 Conveyance Snagging in the Shaft The possibility of any conveyance snagging on some obstruction in the shaft does exist. In a deep shaft, the guides may buckle due to compressive strain induced by increasing compressive strain in the rock surrounding the shaft. Or it is possible that a foreign body may lodge across the compartment in which the conveyance is nmning. The high inertia of the hoist motor and drum and the length of the rope may lead to extremely high forces being applied to the hoist rope before the hoist can be stopped. The primary consideration in this situation is that the hoist rope should not break. This is most likely to happen if the conveyance is descending when it snags. Extra rope is then paid out by the hoist, leading to many metres of loose rope above the conveyance. If the conveyance is then released, it will fall freely until the rope suddenly tightens again. Under these circumstances, the potential energy before the conveyance falls must be absorbed within the rope and the conveyance structure if the rope is to be prevented from breaking. The rope is a flexible structural member, with a large energy absorption capacity due to its length and coiled construction. On the other hand, the conveyance and its attachments to the rope are comparatively rather rigid. The energy absorption capacity of the rope can only be fully utilised if the rope does not break. The South African conveyance design code, SABS0208 : Part 3 (1999), thus requires that the conveyance must be capable of withstanding the full rope breaking force, unless it can be shown that the rope cannot break. Where friction hoists are used, it can often be shown that the rope cannot break, but that it will slip over the hoist drmn. When using this kind of hoist, it would thus be irrational to design for the rope break condition. Experience has shown that there are two important emergency conditions that may arise with snagging of friction hoists. These are that the conveyance may snag as described above, or that the balance rope below the conveyance may snag leading to high tension loads passing through the conveyance. Computer simulation of the entire hoist system is used to predict the maximum loads generated under these possible emergency conditions. This is may be as low as 50% to 70% of the rope breaking force.

357

4.3 Shaft Station Arresting Devices In the search for a comprehensive and holistic approach to shaft safety at shaft stations several different concepts have become frequent practice in deep South African mines. The first of these is procedures to ensure the proper functioning and use of locomotives in the near-shaft environment. The second mechanism commonly used to reduce the likely severity of an accident is the layout of the shaft stations. Trains carrying personnel or rock will usually not enter the shaft station area at all. Material and equipment cars will have to be shunted into the shaft station area to be loaded into the cage for transport up or down the mine, so other arrangements are required. Where trains travel at relatively high spe~d, they will typically not run straight into the shaft station, but will have to negotiate a sharp comer before entering the station. If a train is travelling too fast it will thus leave the rails and smash into the side wall rather than plunging down the shaft. Several shaft stations employ "tank traps", which are depressions in the concrete floor of the haulage way, spanned by a single track bridge, which is stored away from the tracks used by trains. Thus, a train travelling too fast, or that should not be in the shaft station area, will fall into the depression and stop against the end wall. The large size of these tank traps, and the possibility of harm being caused to personnel waiting in the shaft station area are two difficulties to be overcome when they are used. The final mechanism is arresting devices located close to the shaft in shaft station areas. Following the 1994 accident quoted above, it has become a legal requirement that all shaft entrances in shaft stations include an arresting device with shock absorption qualities. Previous legislation required the use of a device to prevent inadvertent access into the shaft, but it has now been recognised that many devices did not operate effectively because they had little or no energy absorption capacity, because even if they were quite substantial they were too brittle. Khan (1996) describes the testing and specification for the energy absorbing devices most commonly used at a particular mine. This device, commonly referred to as a "farm gate", is a simple steel beam across the rail tracks, supported about 200 mm above the tracks by two steel posts set into concrete foundations. The tests described by Khan involved driving loaded trains of known mass, at their operating speed measured by a radar device, into farm gates, and observing the behaviour of the gates. The energy absorption capacity of the farm gate is determined by the plastic moment capacity of the beam, and by the extent of deformation that can be permitted. Thekinetic energy of the trains is determined by their total mass, and by the speed at which they travel. These tests established that deformation of the farm gate beams accounts for between 69% and 75% of the total kinetic energy, and concluded that the remaining energy was absorbed in the train couplings and movement of the loose material carried. Other testing of arresting devices is reported by AAC CTO (1997). In these tests the devices included farm gates, as well as simple posts set in concrete foundations, and several proprietary devices. A study of the trains and trackless vehicles typically used in underground mining applications showed that it required an energy absorption capacity of up to 300 kJ to effectively stop trains, and up to 400 kJ for the trackless vehicles. The range of devices tested were rated in terms of their energy absorption capacity. In general, the forces implied by the plastic moment resistance of the sections used and equations based on simple conservation of energy, gave a good indication of the behaviour of the arresting devices. In both the above series of tests, the following conclusions were reached:

358 9 9 9

Energy absorption can be determined by the plastic moment of resistance of the sections used. The stopping devices must absorb at least 80 % of the kinetic energy. Foundations and other elements of the arresting system must be of adequate strength. Careful attention to detail is necessary, in order to ensure proper functioning of the system, and eliminate potentially dangerous derailments.

5. CONCLUSION Careful consideration has been given to investigating and defining the emergency crash situations that may be expected to arise while operating conveyances in deep mine shafts. This has led to the introduction of more rational design methodologies, and a more consistent level of safety and reliability than was previously provided by ensuring a prescribed factor of safety against static loads. REFERENCES 1. AAC CTO (1997) "Shaft Station Stopping Devices", Anglo American Corporation, Central Technical Office Report, Johannesburg. 2. Greenway M.E. (1990) "An Engineering Evaluation of the Limits to Hoisting from great Depth", eds Ross-Watt D.A.J. and Robinson P.D.K "International Deep Mining Conference : Technical Challenges in Deep Level Mining", S. A. Inst of Min and Metall., Johannesburg. 3. Khan M.M. (1996) "Safety Devices at Hartebeestfontein Gold Mine", Avmin internal report, Johannesburg. 4. Krige G.J. (1996) "The Design of Shaft Steelwork Towers at Reef Intersection in Deep Mines", The Struct. Eng., Vo174 no 19, pp 320-323, October. 5. Krige G.J. and Hofmeyr A.G.S. (1991) "Ore Pressures on a Skip Body", "Trends in Steel Structures for Mining and Building", SAISC in association with IABSE, Johannesburg, August, 1991, pp241-249. 6. Roininen L.J. and Gullick J.W. (1973) "Hoisting Plant at International Nickel", International Conference on Hoisting- Men, Materials, Minerals, S.A. Inst. Mech. Eng., Johannesburg. 7. SABS0208 :Part 1 (1995) "Design of Structures for the Mining Industry. Part 1 : Headgear and Collar Structures", South Bureau of Standards, Pretoria. 8. SABS0208 : Part 2 0995) "Design of Structures for the Mining Industry. Part 2 : Sinking Stages", South Bureau of Standards, Pretoria. 9. SABS0208 : Part 3 (1999) "Design of Structures for the Mining Industry. Part 3 : Conveyances and Counterweights", South Bureau of Standards, Pretoria. 10. SABS0208 : Part 4 (Draft) "Design of Structures for the Mining Industry. Part 4 : Shaft System Structures", South Bureau of Standards, Pretoria. 11. Thomas G.R. (1990) "Design Guidelines for the Dynamic Performance of Shaft Steelwork and Conveyances", COMRO User Guide no 21, Chamber of Mines Research Organisation, Johannesburg. 12. Thompson M.H. (1973) "Shaft Sinking and Equipping Techniques", International Conference on Hoisting- Men, Materials, Minerals, S.A. Inst. Mech. Eng., Johannesburg.

359

Figure 1 9Schematic Layout of Mine Shaft with Safety Devices

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

361

C o m p a r i s o n o f different car front structures u n d e r n o n a x i a l i m p a c t s M. Krtiger Institute of Mechanics, University of Hannover Appelstr. 11,30167 Hannover, Germany

The design of the bumper beam, the energy absorbers and the front part of the longitudinal members and their connection defines the crashworthiness of the front structure of each car. The structure has to absorb energy for axial impact, offset impact but also for nonaxial impact configurations, which are investigated in this paper. For nonaxial impacts the energy absorbers and their connections have a large influence on the kind of deformation. Often front structures are deformed by global bending at impacts with large impact angles. A change of the connection between the bumper beam and the energy absorber from a swivel joint to a bending resistant connection can increase the capacity for nonaxial impacts. Analytical descriptions of the front structure regarding the kind of connection show the influence of the design parameters of the front structure on the critical impact angle. At the critical impact angle the deformation process changes from axial deformation to global bending deformation. A general analytical description is useful to understand the behaviour of the front structure for nonaxial impacts. 1. INTRODUCTION The optimisation of the front structure for small impacts at a speed of 15 km/h (AZT Test) and for offset collisions often results in critical deformations at nonaxial impacts. In that case the front module deforms by global bending and dissipates less energy. Otte [ 1] shows, that 36 % of all front impacts have an impact angle greater than 15 ~. Front structures of cars have two different kinds of design at the connection between the bumper beam and the energy absorber or the longitudinal member: a bending resistant connection or a swivel joint. Here the bending resistant connection will be investigated. The swivel joint connection is described in [2]. The tested energy absorbers are cylindrical tubes, which are deformed by progressive buckling. The front structure model in Figure 1 a) describes the front structure at nonaxial impacts. In this model the bumper beam is rigid. The secondary absorber has two areas of deformation. One plastic hinge is at the connection to the car body, the second one is at the connection to the bumper beam. The primary absorber has an additional area of axial deformation.

362

Impact ~ mass

Bumper s

Axial deformationarea ~ ~

~ ~ ~ / a m

/Plastic hinge

--~

D t [_.._...__.~econdary kt// absorber

!

Primary ~ X absorber ~ ~ - - - - - 7 - ' " " ~ ~ Plastic hinge /

b)

_ l ~ . . ~ . , - . . . . . . . . . ~ ~ - - - - - ~ . . sq

i

B

9

a)

Car body

"i

-I

Figure 1" a) Front structure model, b) axial deformation and c) global bending deformation

Figure 1 b) shows the axial deformation of the front structure for small impact angles and c) the global bending collapse for large impact angles. The bending collapse usually dissipates less energy than the axial deformation. Therefore, it is important to know the impact angle ~r, at which the deformation process changes from axial deformation to bending collapse, in order to optimise the front structure. 2. ANALYTICAL DESCRIPTION OF FRONT STRUCTURES During the global bending deformation front structures show four plastic areas at the connections of the energy absorbers to the longitudinal members (car body) and to the bumper beam. It has been assumed that the bending moments in A, B, C, D are equal to a critical bending moment Mcr at the beginning of the bending deformation. The aim of this investigation is to find a relationship between act and the geometry of the structure. The free body diagram in Figure 2 shows the forces and moments. The impact force F,,~, follows from

F~= F~+F2 .

(1) cos~ The calculation of the critical plastic bending moment Mcr is described in [2]. In point B it yields

gBc r =GmtD2mCOS F2

,

(2)

_'-/

Mc

'

01

Mo."

C

D

01

02

2tTotD m

where am is the mean flow stress, t the thickness and Dm the mean diameter of the tube. The equilibrium of moments with respect to point B of the secondary absorber is M Dcr Q 2 J - M a~, , (3) with the free length I of the absorber.

Fl

MB X ~ F2

-~

Figure 2: Free body diagram of front structure

363 Because

MDcr and MBcr have the same value, equation (3) predicts 20.m tD2mCOS Q2cr = Mocr+ M scr = l

F2 2tYrotDm .

(4)

l

The axial force in the second absorber F2 follows from

(l'x + 2-l~) t an O~cr Fl~-~ Fz = - ~ . (5) 1+~ tana, 2a Impact angles a smaller than O~r still cause an axial deformation of the primary absorber. Therefore, the critical angle o~,. can be calculated using the axial deformation force of the absorber for the force Ft. At the ends of the primary absorber, points A and C, the bending moments are assumed to

I :/1

Mac r =Mcc r = CrmtDm2COS FI . 2CrmtD ~

(6)

The equilibrium of moments around the point A leads to 20-mtDm2cos

QI, = M Acr -I" Mccr l

F1

20"mtDm .

=

(7)

l

Now all forces are calculated and we can obtain an analytical expression for the critical impact angle

2/

El +COS2a.F2tO m ~a

2ormtO m cos 20"mtO.

Or, = arctan Q~c, + Q2c, = arctan

.

.

.

.

(8)

Fl + cos-20"mtDm F 2 1 l + 21,x) + Flal tYmtD2mI cos 20"mtDm

Fi + F2

If the impact angle is smaller than 30 ~ and the eccentricity of the impact load lex and the length I of the absorber are much smaller than the length of the bumper beam a, i.e. lda
20"mtDm .

~cr ~ a r c t a n ,

.

(9)

This equation shows the influence of the geometry and the axial deformation force/;'1. The calculation of F1 is shown in [3] or [4] for tapering. The same calculations for a structure with swivel joints at the connections to the bumper beam at point C and D, results in

CrmtD2m(l+co s 2tr,tDmF' ] act = arctan

.............

F,l

9

The calculation and the behaviour of this configuration is described in [2].

(10)

364

3. EXPERIMENTAL RESULTS The equipment for nonaxial impact tests is a drop hammer rig, Figure 3. It uses a bearing at the impact point to define the angle of the impact force by the angle of the impact mass, because in this configuration the friction force is neglectible. The energy absorbers are clamped at the connection to the stiff bumper beam and to the measuring devices of force and moment. The values of the moments are described in Figure 4. The measured moment MI (M2) is different to the moment at point A (B). After testing the results are filtered digitally. The tests are done using aluminium tubes 6060 (AIMgSi0.5 F22) with Din=50 mm and t=2 mm. The yield stress 6y=219 N/mm 2 and the tensile strength 6u=244 N/mm 2 results from tension tests. The mean plastic flow stress 6m is approximated by or.,,+ t r . . 2

365

1t:, . ot

--=- 40 .7 20

. 20.

_

0

.

0

.

.

|

.

10 20 30 40 50 60 70 S [mm]

~, 6o ~ 4o u,T 20 0

.

|

.

0

1

2

.

|

|

3 4 5 Sq [mini

|

|

6

7

o

:~-lj -2

3o

9a

b)

o.. 6o fo

~

~

~

J

s [ram]

-.9 ~'1

1~2

'

'

'

'

'

3

4

5

6

7

"~e)

-2

/"

t_J

c) . . . . . . . .

0 . . . . . . . . 0 10 20 30 40 50 60 70 s [mm]

Sq [ram]

1~1 0 , 0

, 1

, 2

', ' , , 3 4 5 Sq [ram]

, 7

'. 6

Figure 6: Force and momentsin dependence of the axial deformations and the lateral deformationsq of an absorbers with length l=125 mm at an impact under an angle of 10~ The front structure deforms for small impact angles mainly axially, Figure 5a). Figure 6 shows the corresponding force/71 and the moments M~ and M2 in dependence of the axial displacement s and the lateral displacement Sq of a tube with a free length/=125 mm testing an impact angle of 10 ~ The axial force in Figure 6a) and d) shows typical oscillations during progressive buckling. Because the moment M2 reaches the critical value only at the end of the deformation, the structure shows only small lateral deformations and the secondary absorber endures the loads without large deformations. The moment M~ is not significant. The same absorber is tested under an impact angle of 20 ~ see Figure 5b). Figure 7 shows force- and moment-characteristics of this test. The structure deforms at this test configuration by global bending with cracks at the top and the bottom of both absorbers near the clamps. After the first peak the axial force drops to zero, Figure 7a) and d). The moment M2 reaches the critical bending moment at the beginning of the lateral deformation, Figure 71). After a small bending deformation the absorber starts to crack and the moment drops quickly. When the second absorber starts to crack, the moment MI increases and the force FI decreases. But the first absorber is not able to carry the large axial force together with the bending moment and also starts to crack. Then the moment M~ also begins to fall. This two tests show the typical behaviour of progressive buckling and global bending deformation. 2 ~. E1

"~ 60 ! 40 L 20 0

b

)

~

==

"~s 0

10 20 30 40 50 60 70 S [mm]

-1

Vl

,0

s [mm]

6'0 7'0

60

0

|

|

0

.

.

,

.

|

.

10 20 30 40 50 60 70 S [mm]

E

~ 40

~g2(1 0

0

10

20 Sq

[cm]

.

.

.

30

40

50

~ o

.1~

,

-

10

2'0 304050 Sq tom]

.

0

'"

0

.

10

'

.

20 30 Sq [cm]

.

40

Figure 7: Force and momentsin dependence of the axial deformations and the lateral deformationSqof an absorbers with length l=125 mm at an impact under an angle of 20~

....

9

50

366

""

40 "! 1 30

I ~-Calculation I 9 buckling 9 mixture.

a)

I ~. 40 ] , I 930 J

t=

a 20

~10 t~

~r

lO

-

o

-E ~

.

50

.

75

.

.

1 O0

.

125

150

Free length I [mini

'

175

'

200

b)

9

-

-

50

75

100

125

-

-

-

150

Free l e n g t h ! [ram]

175

200

Figure 8: Critical impact angle for a structure with swivel joints a) or bending resistant connectionsb) to the bumper beam and deformation modesof impact tests The variation of the absorber length l and the impact angle (x results in a deformation mode diagram, Figure 8b). The lines in Figure 8 show the calculated critical impact angle ~ r using the calculated maximum axial force F~=F,,==70 kN. The maximum axial force is approximated by F.= = n" o',. t D=. This calculation gives a good agreement with the test results. Figure 8a) shows the same diagram for the structure with swivel joints at the connection to the bumper beam. The critical impact angle is here up to 15~ smaller than for the bending resistant connections. This demonstrates the big advantage of the bending resistant connection.

4. CONCLUSION Nonaxial impacts of car front structures with progressive buckling absorbers has been investigated. The calculated critical bending moment agrees well with the experimental test results. For the stability of the front structure, the maximum axial force is used in the analytical description. The critical impact angle is predicted, it agrees very well with experiments. The description can be used for the design of impact absorbers because it shows the influence of the geometry of the absorber on the capacity for angled impacts and helps to understand the deformation processes of front structures. The critical impact angle depends on the connection to the bumper beam. The bending resistant connection shows a much better behaviour for nonaxial impacts then the swivel joint connection and should be preferred. REFERENCES 1. D. Otte: Realit~itsbezug von Crashtestbedingungen zu den Situationen des realen Unfallgeschehens. Verkehrsunfall und Fahrzeugtechnik 29 (1991), No. 12, pp. 329-336. 2. M. Kr6ger and K. Popp: Nonaxial impacts on front structures of cars. 32th ISATA Advances in Automotive and Transportation Technology and Practice for the 21 st Century Automotive Ergonomics & Safety, Wien 1999, pp. 285-292. 3. W. Abramowicz and N. Jones: Dynamic progressive buckling of circular and square tubes. Int. J. Impact Engineering 4 (1986), pp. 243-270. 4. M. Kr6ger and K. Popp: Experimental and analytical investigations on the energy absorption by tapering. Int. Crashworthiness Conf. IJCRASH98, Dearborn, Michigan, USA 1998, pp. 499-508.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

367

I m p a c t attenuation o f frontal protection s y s t e m s in p a s s e n g e r vehicles Paul Bignell, David Thambiratnam and Frank Bullen School of Civil Engineering, Queensland University of Technology, Brisbane, Australia

Numerous types of Frontal Protection Systems (FPS) have been fitted to many different vehicle types for decades. The primary use of a FPS is to protect and provide mobility for the vehicle after minor frontal impacts. With the advent of modem vehicle safety systems, such as air bags and impact attenuating crumple zones, the design, manufacture and installation process of FPS have become more challenging and the process needs to be carefully considered. This paper highlights the research undertaken on the impact attenuation of FPS that will supplement a vehicle's existing safety systems.

1. INTRODUCTION The FPS has become an essential accessory for many vehicles for both city and country use. The FPS is mainly used to attenuate an impact load sustained by a vehicle during a collision, such as roadside obstacle and animal impacts, but is also used for the mounting of items such as winches, driving lights and aerials. In recent years the design of some of the FPS have changed from the earlier square edged cross-sections to a more rounded profile and finally to a more cosmetic accessory that provides little added contribution to vehicle strength and stiffness. With the increased safety issues of modem vehicles, the fitment of a FPS has given rise to concerns of possible detrimental effects on the safety characteristics of the vehicle. This has become more of an issue with vehicles fitted with air bags. The air bag is used as a protective cushion between the vehicle's occupants and the hard surfaces of the vehicle's interior, during the deceleration of a vehicle after an impact. Air bags are deployed when predefined values are exceeded. These values are stored as limiting curves and refereed to as the air bag triggering algorithm. The air bag triggering algorithms are generated from the vehicle's crash characteristics or impact response. Each vehicle has a different impact response dictated by parameters such as the vehicle size, weight and material properties. Thus the fitment of a stiffened member to the front of the vehicle could affect the crash characteristics of the vehicle, if it is not properly designed and mounted. Dependent on the added members, air bags could be deployed earlier or later than what would be appropriate.

368 2. INVESTIGATION FPS are made from tubular, plate and channel sections welded together to form a frame, which is then attached to the plates of the mounting system with bolts. The mounting system, whose design depends on both the FPS and vehicle type, is then bolted to the vehicle. The FPS is usually constructed from either steel or aluminium alloy. Figure 1 depicts vehicles fitted with FPS.

Figure 1. Vehicles fitted with frontal protection systems. Behaviour of the FPS is dependent upon the composite action of the FPS, impact zone and the incorporated mounting devices. Deflection, yielding and failure of the FPS must be such that air bags are neither deployed too early, too late, or not at all, while at the same time providing protection against animal and light impacts. Hence the design of FPS must satisfy dual design criteria, it must both be strong enough to offer protection during minor impacts, and remain flexible enough to avoid influencing the air bag deployment characteristics of the vehicle. This design criteria can be satisfied by limiting the energy absorbed by the FPS, ie the FPS must be strong enough to offer protection and yet sufficiently flexible to yield and deflect to set limits prior to the absorption of a designated amount of energy [ 1]. This concern initiated the study on impact attenuation and energy absorption of FPS.

3. TESTING Testing of FPS can be undertaken by the use of quasi-static and dynamic facilities, and by the finite element method. It is envisaged that all three testing facilities be used to better understand the action of a FPS under load, which will result in an enhanced design.

3.1 Quasi-Static Testing Quasi-static testing is conducted at Queensland University of Technology in a universal testing machine where the FPS is rigidly mounted to a pseudo chassis, to replicate the vehicle chassis. A load is gradually applied to the structure and displacements are recorded using an automatic data acquisition system. The load is typically applied to the FPS until the deflection of the FPS has reached a point where the FPS would come into contact with the vehicle. When the FPS has deflected this distance, any additional load will be transferred directly to the vehicle and the FPS becomes transparent to any further loads. A load deflection plot is then used to calculate the (quasistatic) energy absorbed up to this point and this is then compared with the kinetic energy of

369 the vehicle prior to impact. For this calculation the vehicle is assumed to be travelling at a speed of 25km/h at impact. The percentage of energy absorbed by the FPS before it becomes transparent can then be calculated. A static test is shown in figure 2 where yielding of the mounting plates has resulted in the large deflection. Figure 3 shows the load deflection response of a typical FPS. This particular FPS absorbs 1% of the energy of the vehicle prior to the FPS coming in contact with the vehicle.

Figure 3. Load deflection plot of frontal protection system under quasi-static load.

370

3.2 Dynamic Testing A pendulum rig is used for dynamic impact testing to gauge the maximum retardation, 'g' value, to which the FPS is subjected, as seen in figure 4. The FPS is mounted to a test rig and the height of the pendulum is adjusted to vary the speed of the pendulum impact. Maximum deceleration is recorded through the use of accelerometers, charge amplifiers and memory recorders. Deformation resulting from the impact is also recorded and data collected used to evaluate the effect that the FPS has on the crumple characteristics of the vehicle. The testing is conducted below the air bag deploy threshold for the vehicle.

Figure 4. Frontal protection system during dynamic testing.

3.3 Finite Element Analysis At the present stage of this research project, a number of FPS have been modelled using the finite element method and analysed for quasi-static and dynamic loads. The analytical results are compared with the experimental values to calibrate the computer model, which can then be used to study the influence of important parameters such as mounting plates and connections. The finite element analysis is carried out using the programs ABAQUS 5.8 [2] for quasi-static analysis and LS-DYNA 950 [3] for dynamic simulations. The FPS is modelled using beam and plate elements. It has been found that the complex mounting system of the FPS can be replaced by a single element with appropriate properties. The load is distributed across the FPS using a rigid contact surface that models the impacting surface of the experimental tests. Once the FPS is analysed with both quasi-static and dynamic loading, a parametric study of the model will be undertaken. Properties of the FPS will be changed, and results investigated. These properties will include, section thicknesses, geometry, impact velocities and mass of vehicle.

371 4. ANALYSIS Passenger vehicles manufactured after 1985 must comply with Australian Design Rule No. 69/00 [4]. This design rule provides crashworthiness requirements for passenger vehicles to minimise injury to vehicle occupants. During the test a vehicle is impacted into a fixed collision barrier at 48km/h. To meet the requirements, vehicle manufactures use progressive crumple zones and supplemental restraint systems (SRSs). The crtunple zones reduce the impact forces suffered by the vehicle occupants during a collision. It is the crumple zones of a vehicle that dictate the crash pulse for the vehicle. Two important parts of the crumple zone to consider when installing a FPS are the original bumper bar and the vehicle's crush cans, since these are the components most often replaced by the FPS. It is important that the crumple characteristics of the FPS offer similar characteristics to that of the components that are replaced. At speeds above 25km/h a properly designed FPS has little effect on the crush pulse of the vehicle, and as such air bag triggering. It is at less than the desired deploy speed that the fitment of a stiff FPS can result in premature air bag triggering, due to an increased deceleration. This highlights the importance of designing for low speed impacts, rather than high speed. The suitability of a FPS is determined by comparing the energy absorbed under static loading, and if necessary the retardation value against accepted standard "safe" values. At the desired energy level a FPS can be designed to fail either through the bar sections of the FPS or by failure of the mounting system [5,6]. The latter mode of failure can be activated by either yielding and/or buckling of the mounting plates or by shearing of the mounting bolts, before the FPS comes into contact with the vehicle. It has been shown from the numerous tests conducted that energy values up to approximately 8% have little or minor effect on the air bag triggering performance of most vehicles. Historically it has been found that these FPS under quasi-static loading have performed satisfactorily under dynamic testing. Values above approximately 8% may have some effect on the performance of the vehicle's air bag triggering performance, and thus quasi-static testing should be complimented and confirmed with dynamic testing [7,8]. Similarly FPS systems for passenger vehicles, which yield conservative retardation values less that 3.5g, have historically been found to be satisfactory [9].

5. CONCLUSIONS The introduction of a FPS to a vehicle can change the impact characteristics of the vehicle, which becomes more important when the vehicle is fitted with an air bag. When considering such vehicles it is not just the FPS that has to be considered, but the combination and composite action of the FPS, mounting system and the vehicle. With the appropriate design and testing, a FPS can be manufactured to supplement the original design of the vehicle. The FPS has to be designed to absorb a limited amount of the kinetic energy of the vehicle prior to impact. This will ensure that it offers protection during minor impacts, but does not interfere with air bag deployment characteristics of the vehicle during major impacts. During this project FPS have been modelled using non-linear finite element analysis and subjected experimental testing to study their elasto-plastic response and energy dissipation. The capacity to dissipate energy will be quantified as a percentage of the input energy. A set

372 of relationships will be established for predicting strength, stiffness, post yield behaviour, ductility and energy absorption based on material and section properties, configuration and other important parameters identified during the investigation. It is envisaged that this project will develop guidelines to improve the understanding and design of FPS.

REFERENCES 1. Bignell, P., Thambiratnam, D. and Bullen, F., (1999), "Energy Absorption In Structural Systems Under Random and Unpredictable Loads", Acta Polytechnica a Journal of Czech Technical University No. 5, Praha, Czech Republic 2. Hibbitt, Karlsson and Sorensen, Inc. (1998) "ABAQUS/Standard User's Manual, Volume I, II and III", Hibbitt, Karlsson and Sorensen, Inc., U.S.A 3. Livermore Software Technology Corporation, (1999) "LS-DYNA Keyword User's Manual, Nonlinear Dynamic Analysis of Structures", Livermore Software Technology Corporation, U.S.A 4. Federal Office of Road Safety, "Full Frontal Impact Occupant Protection", Australian Design Rule No. 69/00 5. Bullen, F., Thambiratnam, D., and Bugeja, M., (1994), "Modelling and Analysing a T-5 Bull Bar Subjected to Impact Loads", Report, School of Civil Engineering, Queensland University of Technology, Brisbane, Australia. 6. Bullen, F, Thambiratnam, D.P. and Bugeja, M. (1996), "Integration of Bull-Bars as Impact Attenuation Devices with Air Bags", Proceedings of 15th International Technical Conference on the Enhanced Safety of Vehicles", Melbourne, May. 7. Thambiratnam, D. and Bullen, F., (1999), "Evaluation of TJM T5 Steel Bar : Nissan Patrol GU", Report, School of Civil Engineering, Queensland University of Technology, Brisbane, Australia 8. Thambiratnam, D. and Bullen, F., (1998), "Evaluation of TJM T16 Alloy Bar : Toyota Hilux 11/97+", Report, School of Civil Engineering, Queensland University of Technology, Brisbane, Australia. 9. Taylor, L.R., (1998), "Developments In Vehicle Frontal Protection Bars", Report, TJM Products Pty. Ltd., Brisbane, Australia.

Tubular/Shell Structures

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

Unified T h e o r y Compression

for

Collapse

375

of T h i n

Rectangular

Tubes

under

C. W. Kim~, B. K. Han b, and C. H. Jeong r aDepartment of Mechanical and Electrical Engineering, Yonsei University, 134 Shinchon-dong, Sudaimun-gu, Seoul, 120-749, Korea bDepartment of Mechanical Engineerifig, Hong-ik University, Sangsu-dong 72-1, Mapo-gu, Seoul, 121-791, Korea CEMS Components Engineer, Elec 1. Technology Center, Korea Delphi Automotive Systems Corporation, P.O.Box 1 Dalsung, Taegu, 711-712, Korea In the present paper, the elasto-plastic buckling behavior of thin-walled rectangular tubes under compression is analyzed. The stress-strain relations of the panels of the tube are assumed nonlinear and the plastic buckling load of the panel is considered as the collapsing load of the tube. Applying the plasticity reduction factor, a single formula for the crushing strength of rectangular tubes in wide ranges of thickness to width ratio may be derived. The present theory is in good agreement with the experimental results for various thickness to width ratios and materials. 1. INTRODUCTION An accurate prediction of the crash response of a thin walled structure requires good understanding and modeling of its behaviour. Most components constituting a vehicle structure have sheet metal sections. The stresses in these components rarely reach the material yield strength before buckling occurs. The forestructure of cars mainly absorbs kinetic energy of collision in the case of frontal impact. Since the sheet metal section may be transformed into rectangular sections, the buckling and ultimate strength of thin-walled rectangular tubes subjected to compressive loads have been extensively studied w31. Mahmood and Paluszny t31 present comprehensive formulas for the collapse of rectangular tubes. The equations are formulated by using a semi-empirical approach for the purpose of developing methods of designing sheet metal columns for crash loading. For the maximum column strength the strength

376 parameter of Stowell [41 is utilized. Combining the elastic buckling stress into the strength parameter, a formula for the maximum strength of the tube is derived with the function of thickness to width ratio of the panel. The exponent n in the formula is determined by experiments and is found to be 0.43. However, this freed value of the exponent is not applicable in a wide range of thickness to width ratio and materials. In the present paper, the maximum strength of the tube is derived from the plastic buckling of the panels. The plastic buckling strength of the panel is determined precisely by the moduli of the material. The moduli are computed from the ideal stress-strain curves at specific stresses. Introducing the plasticity reduction factor proposed by Gerard Is], the formula may be extended to the elastic buckling of the panel. Thus the present crippling strength formula may be applied to a wide range of thickness to width ratio of the tube. The real stress-strain curves are idealized for nonlinear material of Ramberg and Osgood Is]. The present theory may be expressed as a single curve and experiments of previous studies are plotted.

2. T H E O R E T I C A L F O ~ T I O N

2.1. Plastic buckling of the plates When the thickness of a plate subjected to compressive stress is relatively large such that the elastic buckling stress exceeds the yield stress of the plate, then plastic buckling occurs. For a long plate the critical stress is given by Gerard [5] as follows:

712Es )2 a~,~.~)= 12(1- vp2) (-~ •

( 2 Et ) 1/2] v, +(1- v,2)-~--:-

(1)

where the Poisson's ratio vp is 0.5 for the incompressible plastic state, Et is tangent modulus and Es is secant modulus. By substituting the elastic counterpart for modulus and the Poisson's ratio, the buckling stress for the long compressed plate is given by a~t~)=

kx2E 2 12(1_v2)(-~1

(2)

In order to unify the buckling stress for all thickness to width ratios, the plasticity-reduction factor may be introduced such that -

c~p~tic) r cr( elastic)

(3)

377 By combining Eqs.(1) and (2), we find that the plasticity reduction factor for the long simply supported plate is r]= (1 - v 2) E,

( 1 - v 2) E [ 1 + 1 ( v 2 + ( X - v 2 ) - ~ )

Et

1/2

]

(4)

Thus the unified buckling stress for the simply supported plate is given by a ~ - 12( 1 - v2) 2.2. I n e l a s t i c

(5)

behavior

of materials

There are several expressions for stress-strain curves and tangent modulus for inelastic materials. Ramberg-Osgood E61 suggested the following relations: e =

eE,

s = ~

~1

0'1

(6)

where al is the secant yield strength as shown in Fig. 1. In Fig. 1 m l is a chosen constant between 0 and 1. Eq. (6) may be represented by a single curve for similar materials by fining the value of m l. Then the stress-strain relation may be represented as follows: e = s+ 1 - m______As" _l

(7)

ml

Using Eq. (7), a dimensionless stress-strain curve for a specific material may be obtained by the selection of appropriate values of m l and n. Since the tangent modulus Et is the tangent at the point of a specific stress on the stress-strain curve, the following expression is obtained: E Et

=

1+

When stress

n( 1 -

ml

m 1)

Sn_1

a is replaced by plastic buckling stress

(8)

a~r in Eq. (6), we may

obtain the following expression

Et E

_ -

1 1+

(9)

n(1-ml)(a~) -~1

The secant modulus is defined as the slope of the line connecting the origin to a point on the stress-strain curve. Thus the following relation may be obtained:

378 Es E--

1 1+ ( 1 -"m-l )i( dr -~1 c ar ) 1

(10)

Fig. 2 shows an example of the ratios of tangent modulus and secant modulus with the Young's modulus of the material such as ay = 354 MPa, o, = 394 MPa, a~ = 394 MPa, m~ = 0.2, and n = 25. 1.2

r.tJ r,t3

o'=F_~ ,,'"'"'"'

r/'j

0.8 ., 0.6 ~0.4

0.2

swdm Figure 1. Secant yield strength al.

~/E ..........

~/E

',~.

_

,

o 0

10

20

30

40

50

60

Figure 2. The ratios of tangent modulus and secant modulus with the Young's modulus of a mild steel plate.

2.3. Crushing of the rectangular tubes When a thin-walled rectangular tube is subjected to a compressive stress as shown in Fig. 3, the unloaded edges of the panel are approximately satisfied with the simply supported condition. In general, for a rectangular tube under compressive load, the wider panel reaches to buckling stress first. On the other hand the narrower panel does not reach yet, thus these panels restrict the buckling of the wider panel of the tube. Bleich [7] has presented a formula for the buckling coefficient of the rectangular tubes given by

K--(--~-)2+ p-Fq(~Ab)2

(11)

where p and q are the coefficients of restraining across the unloaded edges and A is a half-wave length. The buckling coefficient K varies between 4 and 7 depending on the section aspect ratio c/b of the tube. If we assume that the crippling strength of the tube is the same as the buckling stress of the wider panel, then the crippling strength of the tube is given as:

379

z2K~E

do.'--

2

(12)

The m a x i m u m crushing load is also given by Pn~ = 2ao,(b + c) t

(13)

2 5 0

......

.10-4

x

--" Present theory O Experimental Data for St~l Tube [8]

200

t,

i I

-~ ~

I

_

~" 150 , L --: =.====

a 50 z

/

--

/ .

b .

.

.

.

o

.

o.os

.

.

.

.

o.~

.

.

i,,,

o.~s

(fib) K~

0

Figure 3. A thin-walled rectangular tube subjected to uniform compressive force.

.

Figure 4. Nondimensional crippling strength vs. thickness to width ratio of rectangular tubes.

2.4 Unified theory of the crippling strength If we introduce new nondimensional variables for general plotting such as

a~, ~E

- C[

CR(~ )] 2

(14)

one can separate the nondimensional buckling stress from the material characteristics. Now Eq. (13) may be applicable to the materials which have the same mechanical properties in tension and compression. Fig. 4 shows the nondimensional crippling strength of rectangular tubes. In Fig. 4, the present theory shows good agreement with experiments.

380 3. CONCLUSION In the present paper the crippling strengths of thin-walled rectangular tubes are expressed with the analysis of plastic buckling. The stress-strain curves from the tensile tests are idealized into the nonlinear expression of Ramberg and Osgood. The elastic moduli of the material are computed from the idealized stress-strain curve. Applying the Gerard's plasticity reduction coefficient, the buckling stresses of the thin-walled rectangular tubes are expressed in single unified form. The restraining effects of the rectangular tubes are assumed to be elastic and the Bleich's buckling coefficient is applied. Through the present study the following results are obtained. (1) A new formula for the crushing of the rectangular tubes is presented. With a precise computation of the elastic moduli of the tube material the crippling strength of the tube is uniquely determined. (2) The present formula can be applied to a wide range of geometry and material properties of thin-walled rectangular tubes.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9.

C.L.Magee and P.H.Thornton, SAE Technical Paper No. 780434 (1978). P.H.Thornton, SAE Technical Paper No. 800372 (1980). H.F.Mahmood and A.Paluszny, SAE Technical Paper No. 811302 (1981). Z.E.Stowell, NACA Technical Note No. 2020 (1950). G.Gerard, McGraw-Hill Book Co., New York, 1962. W.Ramberg and W.R.Osgood, NACA TN 902, July 1943. F.Bleich, McGraw-Hill Book Co., New York, 1952. C.H.Jeong, MS Thesis, Yonsei University, 1999. B.K.Han and B.H.Park, KSAE Vol.6 No.2 228 234 (1998).

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

381

Stress-strain relationship for confined concrete in various shapes of concretefilled steel columns K. A. S. Susantha, Hanbin Ge, and Tsutomu Usami Dept. of Civil Engineering, Nagoya University, Nagoya, 464-8603, Japan. A method to predict complete stress-strain curve of concrete subjected to triaxial compressive stresses caused by axial load plus lateral pressure due to confinement action in various sectional shapes of concrete filled steel tubes is presented. FEM analysis procedure with the help of a steel-concrete interaction model is adopted to estimate the lateral pressure in box and octagonal shaped columns. Subsequently, an extensive parametric study is carried out to propose a relationship for the maximum average lateral pressure. The lateral pressure so calculated is correlated to confined concrete strength through a well-known empirical formula. A fiber analysis procedure is followed to calibrate the post peak region of the proposed model. The columns adopted for the calibration are composed of different steel and concrete material properties and various sectional shapes. Finally, the predicted concrete strength and post peak behavior are compared with test results and found to exhibit good agreement. I. INTRODUCTION Numerous experimental studies aiming to investigate the strength enhancement of concrete filled tubular steel (CFT) columns have been conducted in recent years. Beside the strength enhancement, ductility improvement and prevention or delaying of local buckling of steel tube are the other major advantages of CFTs in seismic areas. It is of this study's interest to examine the effect of sectional shape, geometric and material properties on the strength enhancement and post peak behavior of confined concrete in CFT columns. Three types of columns with circular, box and octagonal sections are examined. 2.

CHARACTERISTIC POINTS ON CONFINED CONCRETE STRESSSTRAIN CURVE

Referring to Figure 1 following characteristic points can be identified to define complete stress-strain curve for confined

i

....

............

!J

f,

Compressivestrain, 9 Figure 1. Stress-strain curves for confined and plain concrete

382 concrete. Here, f : , ~ - strength and corresponding strain of unconfined concrete; f:~, E c~ ffi strength and corresponding strain of confined concrete; f . , f . . -

ultimate strength and

corresponding strain of unconfined concrete; tt f : , ~ ~. = residual strength and ultimate strain of confined concrete at very high strain levels. The expression for the ascending part is taken as that suggested by Popovics [ 1] which was later modified by Mander et al. [2] and given by f~ = f:~

xr r-l+x

(1) r

s x=~

(2)

ecc r =

E~

(3)

(E~ - f : / e c ~ ) gc~ - e~ [1 + 5 ( f : ~ / f : - l ) ]

(4)

where, fc and e denote the longitudinal compressive stress and strain, respectively; E c stands for the tangent modulus of elasticity of concrete. 3. EVALUATION OF C O N F I N E M E N T 3 . 1 . C i r c u l a r section

In triaxial stress state the uniaxial compressive concrete strength can be given by

= g +my.

(5)

where, f ~ is the maximum radial pressure on concrete and m is an empirical coefficient that is assumed as 4.0 in this study. The maximum lateral pressure, f ~ , can be obtained by Eq. (6) proposed by Tang et al. [3]. 2t

fr~ = [~ (D- 2t-----~f '

(6)

Here, f y , t and D denote for the yield stress of steel, thickness and outer diameter of tube, respectively. In this method they consider the change of Poisson's ratio of concrete and steel upon loading. An empirical factor, [3, is defined as the difference between V, and Vs at the maximum strength point where, t~e and vs are Poisson's ratios of steel tube with and without filled-in concrete, respectively. The parameter v, is given by following expressions. Ive = 0.2312 + 0.3582 tf~ - 0 . 1 5 2 4 ( f ' l f y ) + 4 . 8 4 3 ~ ' ( f ' l f y ) - 9 . 1 6 9 ( f ' l f y ) I f = 0 . 8 8 1 x 1 0 -6 ( D I t ) 3 - 2 . 5 8 x 1 0 - 4 ( D I t ) 2 + 1 . 9 5 3 x 1 0 - 2 ( D I t ) + 0 . 4 0 1 1

z

(7) (8)

383 3.2. Steel-concrete interaction model for box and octagonal sections Steel In order to obtain the lateral pressure at //Concrete/ S~eeelmenb~tam.colum n "~. peak in box shaped columns, a concretesteel interaction model is used as shown in Figure 2(a). A similar type of model has been employed in the analysis of RC columns by Nishiyama et al. [4]. The V////II/IX\\\\%.'.,'II p, element confinement along the length of the r.4-- Center node column is assumed uniform and hence a I_. b ._1 I"1 unit length of the column is considered (a) (b) for the analysis. Here, the concrete is discretized into a number of segments Figure 2. Steel-concrete interaction model for box bounded by the lines joining the center columns point of model and mid points of adjacent steel beam elements as shown in Figure 2(b). Each of these segments is represented by an axial compressive bar element with an equal stiffness of corresponding triangular segments. Lateral steel is also represented by number of finite elements. Uniform lateral strain is assumed for all of the concrete bar elements. Consequently, for a pre-assumed lateral strain, corresponding displacements are computed and are applied incrementally at the each node of concrete bar elements at the center of the model. This leads to the expansion of outer steel cage since the steel elements are laterally pushed out by the concrete bar elements. At the end of each load increment, averaged lateral

/' Conc t ,ro.s

~/y///I]I\\\Nk~NN~_~

stress,

fr*,

and strain, I~r*, are computed. The maximum average lateral stress,

fr,,,*, is then

substituted for f~, in Eq. (5) to predict confined concrete strength. The same procedure can also be adopted in the case of octagonal shaped columns. 3.3. Material models for steel-concrete interaction model The material model for concrete truss elements is explained in Figure 3(a). This simplified model is based on the lateral stress-strain relationship of circular columns. In the case of box and octagonal shaped columns, the expressions proposed by Tang et al. (1996) will be used in a slightly modified form. Instead of diameter,/9, in Eq. (6), width of the section, b, and extreme dimension, D, should be substituted for box and f E' octagonal shaped columns, respectively. In Eq. (8), equivalent diameter, Des, which is obtained by equating the area of box or octagonal section to t2e e, e~ its equivalent circular shape has to be used for D. The falling (a) Concrete (b) Steel branch slope, k 2 , is calibrated Figure 3. Material behavior for steel-concrete interaction by using the test results on model reference [5]. Consequently, k~

f"

and k 2 are given by

I[. ..........

384

kI =

2t

(Deq - 2 t )

Es

k2=2744

(9)

:,JJ

-7637<0

(10)

Figure 3(b) shows the steel material model adopted in the interaction model. For steel material parameters, yield stress, f y , and the elastic modulus, E s , for each specimen are taken as reported in reference [5]. Other parameters as shown in Figure 3(b) are assumed as the same for a kind of mild steel SS400 [6].

3.4. Parametric study to determine maximum lateral pressure Based on the extensive parametric analyses, Eqs. (11) and (12) are proposed to estimate the averaged maximum lateral pressures on concrete in box and octagonal shaped columns, respectively. f # i.46

f * = - 6 . 5 R .,c

+ 0.12 f,l.o3

(11)

f,v 9= -35.0 R f" 1.3s + 0 . 2 2 f "l•2

(12)

f,

The geometry and material parameters are chosen in such a way that the local buckling could be ignored when columns are axially loaded. This is achieved by keeping a certain value for the width-to-thickness ratio parameter based on the relationship proposed by Ge at el. [7] for local buckling strength of plates in CFT columns.

4. DETERMINATION OF POST PEAK BEHAVIOR Since it would be a complicated process to determine post peak behavior purely in analytical manner, an approach based on the experimental results is adopted. Here, the test results reported in references [8-12] are utilized, local buckling is not expected in any of the selected columns since certain maximum width-to-thickness or radius-to-thickness ratio parameters ( R or R t ) are imposed as the selection criteria based on the buckling strength formulae reported in references [6, 7]. The definition of R and R, can be found dsewhere [6]. The columns are modeled using two beam elements representing concrete and steel having common end nodes. Load is applied incrementally in longitudinal direction as a prescribed displacement at the free end node while the other end is kept fixed. The stresses at integration points at each increment are recorded and, subsequently, total axial load against axial strain is plotted. Behavior of steel is assumed as explained in Figure 3(b). For concrete, the behavior up to peak is defined by Eq. (1) using f'c calculated through Eqs. (5), (11) and (12). It has to be emphasized that f " appeared in Eq. (5) is reduced by a factor, which is chosen as 0.85 in this study. This reduction is introduced to account for various uncertainties

385 associated with actual sized columns. Different linear stress-strain relations are assumed for post peak behavior. For every assumed slope, analyses are carried out and axial load and axial strain are computed. By comparing the predicted curves, which are having different concrete post-peak behavior, with the test curve, a slope that would lead to the best-fit curve can be decided. Similarly, ultimate strain is determined through reanalyzing the columns using the concrete model having proposed slope. This procedure is followed for all the selected cases. The mathematical expressions for falling branch slope, Z, and ultimate strain, ~`:,, are then determined through regression analyses and are summarized as follows.

(a) Circular section O

for

J

l.O•

f,)-600 .(f~'If,)-6000

Z-11.0• R , -

[(fy/283)~3~*[1.0• g`:, = 0.025

for for

s R, .(f'/fy)-600]

for

Rt.(f'/fy)0.006and fy ~283 MPa (13) R, .(f~'I f y)>O.OO6andf y >336 MPa R t .(f'l fy)> 0.006 and 283< fy <336 MPa (14)

(b) Box section

Z =23400R.(f'~/fy)-91.26>O e`:,- 14.5 [R 9( f ' l f y ) ~ - 2 . 4 R . (f~lfy)+"

(15) 0.166

(0.018 < e`:, -< 0.04)

(16)

(c) Octagonal section

R.(f~ -513>0 R.(f'/f,)+O.052<

(17)

Z - 2.85X104 8,:,=-0.566

0.035

In all the three cases the upper limit of ec, is enforced as ~c,< (~`:~ +f'`:`:lZ).

(18)

The

validity ranges of proposed equations are as follows: 10< f " <50 MPa, fy < 620 MPa, R < 0.85 and R, < 0.125. Three examples of predicted axial load-axial strain curves of circular, box and octagonal CFT columns using the above proposed stress-strain curves for the filled-in concrete are shown in Figure 4, together with corresponding experimental results [12]. It can be seen that the predicted curves agreed well with the experimental results.

386

~ , 2000 ]soo

_o 1000 "-~ soo .~

o 0

0.01 0.02 0.03 0.1)4

Axial strain

0

0.01 0.02 0.03 0.04

Axial strain

0

0.01 0.02 0.03 0.04

Axial strain

Figure 4. Comparisons between predicted and experimental axial load-axial strain curves

5. CONCLUSIONS A method to predict maximum lateral pressure and complete stress-strain curve for confined concrete in three types of columns were proposed. The highest confinement was found in circular sections and the least was in the box sections. As expected octagonal sections showed intermediate results. The validity of proposed method to evaluate confinement was confirmed through comparisons with test results. The relationships proposed for Z and I~c, are found to explain important characteristic of concrete-filled columns, including high ductility. It was found that the post peak behavior is dependent on several parameters such as width-to-thickness (or radius-to-thickness) ratio parameter and concrete and steel strengths. The predicted axial load-axial strain curves showed good agreement with the test curves. REFERENCES

1..S. Popovics, "A numerical approach to the complete stress-strain curves for concrete." Cement and Concr. Res., 3(5), 583-599, 1975. 2. J.B. Mander, J.N. Priestly, and R. Park, "Theoretical stress-strain model for confined concrete." J. Strcut. Engrg., ASCE, 114(8), 1804-1826, 1988. 3. J. Tang, S. Hino, I. Kuroda, and T. Ohta, "Modeling of stress-strain relationships for steel and concrete in concrete filled circular steel tubular columns." Steel Construction Engineering, JSSC, 3(11), 35-46, 1996, In Japanese. 4. M. Nishiyama, B.B. Assa, and F. Watanabe, "Prediction of stress-strain curve for confined concrete based on transverse steel-concrete interaction", Proc. of the Japan Concrete Institute, 19(2), 543-548, 1997. 5. H. Watanabe, T. Sakimoto, K. Senba, and S. Onishi "A simplified analysis for the ultimate strength and behavior of concrete-filled steel tubular structures." Proc. of the 5th international colloquium on stability and ductility of steel structures, Japan, 893-900, 1997. 6. T. Usami, and H.B. Ge, "Cyclic behavior of thin-walled steel structures-numerical analysis." Thin-walled structures, 32(1/3), 41-80, 1998. 7. H.B. Ge, and T. Usami, "Strength analysis of concrete-filled thin walled steel box columns" J. Construct. Steel Res., 30, 259-281, 1994.

387 8. A. Mukai, K. Yoshioka, I. Nishiyama, and S. Morino, "Structural behavior of concrete filled steel tubular columns under axial compressive load." Abstracts of the Annual Convention of the Architectural Institute of Japan, 735-740, 1995, in Japanese. 9. S.P. Schneider, "Axially loaded concrete-filled steel tubes." J. of Struct. Div., ASCE, 124(10), 1125-1138, 1998. 10. K. Tanaka, Y. Kanoh, M. Teraoka, and A. Sasaki, "Axial compressive behavior of composite short columns using high strength concrete." Proc. of the Japan Concrete Institute, 12(2), 83-88, 1990. 11. M. Tomii, K. Sakino, and Y. Xiao, "Triaxial compressive behavior of concrete confined in circular steel tube." Trans. of the Japan Concrete Institute, No. 280, 369-376, 1988. 12. M. Tomii, K. Yoshimura, and Y. Morishita, "Experimental studies on concrete filled steel tubular stub columns under concentric loading." International Colloquium on Stability of Structures under Static and Dynamic Loads, Washington, D. C., 718-741, 1977.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

389

Experimental behaviour o f internally-pressurized cone-cylinder intersections Y. Zhao and J. G. Teng Department of Civil and Structural Engineering The Hong Kong Polytechnic University, Hong Kong, P. R. China Cone-cylinder intersections are used in many shell structures including tanks, silos, pressure vessels and piping, and internal pressurization is often an important loading condition. For the intersection of the large end of a cone and a cylinder, internal pressurization causes large circumferential compressive stresses in the intersection. These stresses can lead to failure of the intersection by either axisymmetric collapse or non-symmetric buckling. Several thorough theoretical studies have examined the buckling and collapse strengths of internally-pressurized cone-cylinder intersections, leading to simple design approximations. However, very limited experimental work has been carried out, due to the great difficulties associated with testing these thin-shell intersections at model scale. This paper describes a recent experimental study on these intersections. A sophisticated experi_rnental facility is briefly presented first, followed by a summary of the experimental results and observations. 1. INTRODUCTION Cone-cylinder intersections are used in many shell structures including tanks, silos, pressure vessels and piping, and internal pressurization is often an important loading condition. For the intersection of the large end of a cone and a cylinder, internal pressurization causes large circumferential compressive stresses in the intersection. These stresses can lead to failure of the intersection by either axisymmetric collapse involving excessive inward axisymmetric deformations or non-symmetric buckling featuring periodical waves around the circumference. The theoretical collapse and buckling behaviour and strength of uniform thickness conecylinder intersections under internal pressure have been thoroughly investigated by Teng (1994, 1995, 1996) who also developed simple design approximations. Non-symmetric buckling rather than axisymmetric collapse is likely to govern the strength of these intersections when they are relatively thin (Teng 1995). A recent buckling failure of a real cone-cylinder intersection was reported and analysed by Jones (1994) and further studied in detail by Teng and Zhao (2000). A review of other relevant studies can be found in Teng (2000). Only a small number of buckling experiments on internally-pressurized cone-cylinder intersections have been conducted (Gabriel 1996; Teng and Zhao 2000). The three tests conducted by Gabriel (1996) had large geometric defects which were unrepresentative of full scale structures, leading to the development of localised buckles and substantial discrepancies between the test results and finite element results based on the perfect geometry. The buckling load from a recent test on a relatively more perfect specimen was found to be close to the finite element bifurcation load of the perfect geometry (Teng and Zhao 2000). This test was conducted as a trial before commencing a large experimental program on both steel silo transition junctions

390 and intemally-pressurized cone-cylinder intersections. This paper reports briefly the results of the part of this experimental program on cone-cylinder intersections. A sophisticated experimental facility is briefly presented first, followed by a summary of the experimental results and observations. Further details of the experimental results are given elsewhere (Zhao and Teng 2000). 2. MODEL FABRICATION The method of rolling thin steel sheets followed by seam welding was adopted in the study and careful fabrication techniques were employed to produce good quality model intersections. The models were of one meter in diameter and made of 1 mm and 2 mm thick thin steel sheets. A long cylinder was used to ensure that the boundary effects at the far end of the cylinder did not affect the behaviour of the intersection. The steel sheets were cut using a plasma electric cutting installation and then rolled into desired shapes using a bending machine with three rolls. Welding along the meridional seams and at the cone-to-cylinder joint was carried out using a TRANSTIG 16Pi Pulsed TIG (Tungsten Inert Gas) welding machine. Pulsed TIG welding reduced heat input and consequently distortions due to excessive heat buildup. In addition, a former was built to keep the flexible shell components in position during welding. Fig. 1 shows the former supporting a cone-cylinder intersection during welding. 3. EXPERIMENTAL SET-UP An overall view of the experimental set-up is shown in Fig. 2. This set-up was built as a multi-purpose test rig for shell buckling experiments with special attention to the testing of steel silo transition junctions (Teng et al. 2000). For the present experiments on cone-cylinder intersections under internal pressure, the loading frame was not used. Instead, loading was applied by filled water pressurized using a hydraulic pump. As buckling of shells may be very sensitive to geometric imperfections, precise surveys of initial geometric imperfections are an essential step in high quality experiments. A laser displacement meter supported on a measurement frame was employed for precise measurement of initial imperfections and deformations at selected load levels. The chosen laser displacement meter was able to provide high-speed non-contact measurements of high accuracy. Stepping motors were used to drive the laser displacement meter up and down and to rotate the measurement frame around the specimen. The measurement system is similar to that described by Berry et al. (1996). Further details of the experimental set-up are given elsewhere (Teng et al. 2000).

Figure 1. Model intersection during welding

Figure 2. Overall view of experimental set-up

391 4. EXPERIMENTAL RESULTS

4.1. Model Geometries and Material Properties Altogether, four tests were planned on internally-pressurized cone-cylinder intersections, with one of them stiffened at the cone-to-cylinder joint by an annular plate (Zhao and Teng, 2000). This latter test has not been completed, so only the three tests on unstiffened cone-cylinder intersections are discussed in this paper. Nominal dimensions of the test models are given in Table 1, where R is the radius of the cylinder middle surface, tcon~ and to,tinder are the thicknesses of the cone and the cylinder respectively, and c~ is the apex half angle of the cone. Models 1 and 3 had a uniform thickness while model 2 had a cone half as thick as the cylinder. All models had a cone apex half angle of 40 ~ Table 2 details the material properties of the steel from tensile tests according to BS18 (British Standard Institution 1987). As the coupons were cut from steel sheets before rolling and welding, the effects of rolling and welding are not represented in these material properties. Such effects have only a small influence on the overall structural behaviour (Zhao and Teng 2000) and are not further discussed here. Table 1 Nominal geometries of test models

Model 1 Model 2 Model 3

R (mm)

tco.e (mm)

500 500 500

1 1 2

to,linder a (mm) (degree)

1 2 2

40 40 40

Table 2 Material properties of steel sheets Nominal thickness (mm)

Actual Thickness (mm)

Young's Modulus (MPa)

Yield stress (MPa)

1 2

0.950 1.966

2.04 x 105 1.99 x 105

253 165

4.2. Geometric Imperfections Before each test, a careful three-dimensional survey was conducted on the specimen to obtain the initial imperfect shape. The whole cylinder was surveyed, but only the portion of the cone near the cone-to-cylinder joint (22 from the joint, where ~ is the linear elastic bending half wavelength) was surveyed because of the limited measurement range of the laser displacement meter. This limitation is not a significant drawback for the present tests, as the critical zone of the cone was included in the measurement. As the measured initial shapes were to be included in subsequent finite element analyses using ABAQUS (Zhao and Teng 2000), the measuring grids were determined by required finite element meshes. The measurements were compared with those taken on a precisely machined steel cylindrical shell with an outer diameter of 1000 mm using the same measuring grid to determine the geometric imperfections. The maximum norrnal geometric deviations from the perfect surface in the vicinity of the joint (within 22 from the joint) are about 3 mm in the 1 mm thick cones and 2 mm in the 2 mm thick cone, and 2 mm in the cylinders. Such imperfections are expected to be similar to those in real intersections, although there have been no measurements of imperfections on real cone-cylinder intersections. 4.3. Failure Behaviour and Strength As mentioned before, the model cone-cylinder intersections were filled with water which was then pressurized using a hydraulic pump. The applied pressure was recorded by a pressure transducer. A number of strain gauges were installed on the cone around a circumference at a meridional distance of 20 mm from the joint. Fig. 3 shows the pressure-strain curves of all three model intersections. The circumferential positions of the strain gauges are omitted here as their

392 readings are used here only to illustrate the divergence of readings as buckling displacements develop. Such divergence provides a useful mean to determine the buckling pressure. In the initial stage of loading, the strains were similar and approximately proportional to the pressure (Fig. 3), indicating linear and dominantly axisymmetric behaviour. As the pressure reached a certain value, the strains at different locations started to diverge fi'om each other and the relationship between strain and pressure became obviously nonlinear (Fig. 3). The divergence of the strain readings is an indication of the growth of non-symmetric buckling deformations. With further increases of the pressure, roughly uniform short-wave buckles centred around the circumferential weld could be observed clearly. The buckles continued to grow with the applied pressure until rupture and/or severe leaking occurred at one or more of the welded joints, leading to the release of pressure and signifying the end of the test. One of the models (Model 1) failed by the rupture of the circumferential weld. Model 1 after failure is shown in Fig. 4a, while Fig. 4b displays the deformed shape of the same model plotted from survey data of the laser displacement meter. Only half of the grid points were used in Fig. 4b to avoid congestion. The circumferential waves are obvious in the plot (Fig. 4b) and can also be seen in the photograph (Fig. 4a).

Figure 4. Postbuckled model intersection 1

393 Obviously, the postbuckling behaviour of these intersections is stable. Consequently, the buckling load of them cannot be determined in a straightforward manner. The buckling pressure can be taken as the pressure at which the strain readings started to diverge, although this does not allow a precise definition of the buckling pressure as the strain readings differed from each other right from the beginning of loading due to the presence of initial imperfections. The nonlinear bifurcation loads based on the perfect geometry are also shown in Fig. 3. These results were obtained using the NEPAS program for nonlinear bifurcation analysis of shells of revolution (Teng and Rotter 1989). It can be seen that these bifurcation loads correspond well to the pressures at which strains started to diverge. It may therefore be concluded that due to the stable postbuckling behaviour, cone-cylinder intersections show only limited sensitivity to initial imperfections and that the bifurcation loads, on which Teng's (1995, 1996) design approximations are based, are a good measure of the buckling strength of real imperfect intersections. The NEPAS analyses also showed that the change in slope in the pressure-strain curves is due to the spread of yielding near the cone-to-cylinder joint, and that all three models buckled aiier yielding. During the test, two circumferences were scanned at each load level to record the development of deformations. One circumference was on the cone, and the other on the cylinder. The positions of the monitored circumferences were chosen to coincide with nodal lines of the finite element meshes. Fig. 5 shows the results measured from the monitored circumference on the cone of Model 1, which was at a vertical distance of 6 mm from the cone-to-cylinder joint. The pressure-radial displacement curves of several points within a selected half-wave (Fig. 5) are shown in Fig. 6. The bifurcation load based on the perfect geometry is also shown. The pressure-displacement curves show the same phenomenon of divergence in the postbuckling range as that observed from the pressure-strain curves, although the divergence of displacements is less severe than that of the strains. In all three tests, the formation of buckles was not associated with a drop in the load carrying capacity. This stable postbuckling behaviour does not diminish the importance of the buckling phenomenon, as the postbuckling growth of short-wave buckles can lead to large strains near the circumferential weld and consequent rupture failure. The buckling load is thus a good and conservative strength measure. In cases where the prediction of the weld rapture load is required for a safe design, as in the design of a fragile cone-to-cylinder joint in tanks (Yoshida 1999), the growth of postbuckling deformations and strains should be carefully studied and a failure criterion developed. The present experiments should be useful for future research on this important issue.

]

Selectedhalf-wave

3

~

~_~

-~-~"

~I 02

-6

0.1

~"~

~.

'-- 0

-

~l~ v

'~/"

I

30

60

Model1 I

I

, I

.... ~ 0 . 0

90 120 150 180 210 240 270 300 330 360 Circumferential Angle (degree) Figure 5. Deformation growth on the cone

z_

,

'

!

1 2 3 4 0 Inward Displacement (mm) Figure 6. Pressuredisplacement curves

394 5. CONCLUSIONS This paper has described a recent experimental study on the buckling of internally-pressurized cone-cylinder intersections. A sophisticated experimental facility was briefly presented first, followed by a summary of the experimental results and observations. These experiments showed that the postbuckling behaviour of the intersections is stable but the postbuckling growth of deformations and associated strains can cause rupture failure at the weld. Therefore, the buckling load should be taken as the ultimate strength of the structure. It has also been shown that this buckling load can be closely approximated by the nonlinear bifurcation load of the perfect geometry. Consequently, simple design equations previously developed based on this bifurcation load can provide a good estimate of the buckling strength of real intersections. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. PolyU 66/96E) and The Hong Kong Polytechnic University. The authors also wish to thank the technical staff of the Heavy Structures Laboratory of the Department for their support. REFERENCES

British Standards Institution (1987). British Standard Method for Tensile Testing of Metals (Including Aerospace Materials). BSI, London. Berry, P. A., Bridge, R. Q. and Rotter, J. M. (1996). "Imperfection measurement of cylinders using automated scanning with a laser displacement meter." Strain, 32, 3-7. Gabriel, B. (1996). Behaviour and Strength of Plate-End and Cone-End Pressure Vessels. MEngSc Thesis, Dept. of Civil and System Engineering, James Cook University, Australia. Jones, D. R. H. (1994). "Buckling failures of pressurized vessels - two case studies." Engineering Failure Analysis, 1,155-167. Teng, J. G. (1994). "Cone-cylinder intersection under internal pressure: axisymmetric failure." Journal of Engineering Mechanics, ASCE, 120(9), 1896-1912. Teng, J. G. (1995). "Cone-cylinder intersection under internal pressure: non-symmetric buckling." Journal of Engineering Mechanics, ASCE, 121 (12), 1298-1305. Teng, J. G. (1996). "Elastic buckling of cone-cylinder intersection under localized circumferential compression." Engineering Structures, 18(1), 41-48. Teng, J. G. (2000). "Intersections in steel shell structures." Progress in Structural Engineering and Materials, accepted for publication. Teng, J. G. and Rotter, J. M. (1989). "Nonsymmetric bifurcation of geometrically nonlinear elastic-plastic axisymmetric shells subject to combined loads including torsion." Computers and Structures, 32, 453-477. Teng, J. G. and Zhao, Y. (2000). "On the buckling failure of a pressure vessel with a conical end." Engineering Failure Analysis, accepted for publication. Teng, J. G., Zhao, Y. and Lam, L. (2000). "Techniques for buckling experiments on steel silo transition junctions." to be published. Yoshida, S. (1999). "Bifurcation buckling of aboveground oil storage tanks under internal pressure." Advance in Steel Structures, S. L. Chan and J. G. Teng, eds., Elsevier, 671-678. Zhao, Y. and Teng, J. G. (2000). "Buckling experiments on cone-cylinder intersections under internal pressure." in preparation.

Structural Failure and Plasticity (IMPLdST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

395

FEM Analysis of Buckling of Thin-Walled Tubes under Dynamic Loading B. Wang a and G. Lu b

"Department of Mechanical Engineering, Brunel University, UK bSchool of Engineering and Science, Swi_nburne University of Technology, Australia

Extensive studies on axial crushing of thin-wall robes have been c ~ e d out due to their wide applications for crashworthiness. Most of the studies have been conducted under static and low speed loading conditions to predict the deforming mechanism, resulting in typical progressive buckling and dynamic plastic buckling, from which the energy absorption capacity of the stmctm~ component can be predicted. However, two features associated with this problem are not fully understood, ie. the inertia and strain rate effects. It is understood that when the loading speed is relatively high, these two factors cannot be neglected and will lead to dynamic plastic buckling in the format of wrinkles and folds. However, the current study shows that different deforming mechanisms will occur when the impact speed is severely high. In particular, mushrooming, or thickening of the tube wall may occur and this will increase the resistance to buckling, significantly altering its deformation mechanism_. In this study, we examined the effect of inertia and strain rate by simulating an axial crush of thin-wall tubes through finite element modelling. Instead of crushing a tube by a moving mass, we simulate a moving tube section striking against a flat rigid surface, causing axial crush solely by the inertia force associated with its own mass. It is clearly shown that the combination of these two factors significantly alters the deforming mechanism in terms of velocity. 1. INTRODUCTION An understanding of axial crushing of thin-walled tubes under static loading has been well established over the last two decades. The nature of the progressive buckling process under the static mode leads to a stable deforming response of the structure in which folds generally occur at one end of the tube, though these folds may at)pear in different forms, such as axisymmetric, concertina and mixed mode [1]. Many industrial applications have been developed to use its energy absorption capacity. Generally speaking, due to the low speed of deformation, the inertia and strain rate effects are not substantial and can be neglected. Alexander [2] was the first to provide a rigid, perfectly plastic model of the progressive buckling process and since then the model has been improved in many ways. When a tube is subjected to a severe dynamic axial load, the response would be a dynamic plastic buckling which may become quite different from a progressive one. Wrinkling will occur along a quite large portion of the sample and folding may also happen if the load is severe enough. Tests and modelling have been conducted on cylindrical shells of finite length struck by a moving mass. Both inertia and strain rate have been considered. A comprehensive review can be found in Jones' book [3] on both dynamic progressive and plastic buckling.

396 The problem studied in the present investigation differs from the above in that the tube section is not struck or pressed at one end with the other end resting against a hard surface. In this study, the tube is shot at high speed against a hard surface so the driving force for deformation comes completely from the inertia force generated by the tube with deceleration occurring at the moment when the front end of the tube hits the rigid sm~ace. The difference in the loading condition is substantial in the above two cases. Assuming that the whole tube sample is subjected to a uniform deceleration when stopped, the magnitude of the axial force applied in the stopped tube would be linear rather than uniform, having a zero value at the free end of the tube to a maximum value at the stnTAng end (the tube head). The high value of the compressive force may increase the wall thickness at the tube head causing it to mushroom, thus, increasing the resistance to buckling in that portion. As a consequence, folds may occur only in the portion a distance from the stnTdng end. Practical importance of such phenomenon may arise in crashes of high-speed train coaches and sudden thrust of rocket engines. The inertia and strain rate are two key issues in the study. We are interested in how theses two factors influence the structural response, such as the location of the first fold, final wall thickness of the tube, magnitude of the impact force at contact and total energy absorption, etc., in terms of tube dimensions and the striking velocity. The complexity of the problem prevents restdting a theoretical modelling at this stage, thus a finite element modelling approach was adopted. Only numerical simulation results are reported in this paper while the experimental investigations will be discussed elsewhere. 2. PROBLEM DESCRIPTION

The tube segment studied has a length of 129.6mm, an outer diameter of 6.4ram, and three different wall thickness to diameter ratios (t/D) at 1/16, 3/32 and 1/8, marked as tube 1 to 3, respectively. This t ~ range represents relatively thick tubes where diamond-patterned deformation with a circumferential wave would not g ~ y occur [4]. The material is mild steel with the Young's modulus of 206.8Gpa and a Poisson's ratio of 0.3. Cowper-Symonds relationship was assumed for the material for dynamic deformation: ~=D

-1

,

cr0 >or 0

\~0 where D=40.4/s and q=5 [5] and the tested static stress-strain curve is given in Fig. 1. Only axisymmetric deformation is assumed. A commercial code ABAQUS - Explicit was used and a two-dimensional model was constructed using 2691 four-node quadric axisymmetric elements. A frictional coefficient of 0.1 was assigned between the tube and the hard flat surface and 0.25 between the contact of the tube itself (folds), respectively. All nodes in the tube were assigned with an initial velocity before impact ranging from 200m/s to 600m/s. 3. SIMULATION RESULTS Generally, the nominal pattern of deformation can be divided into three categories: folds for thin tubes at low speeds; mushrooming and folds at medium speeds for all tubes, and mushrooming and wrinkles only for thick tubes at high speeds. The initial wall thickness has

397 a strong influence on the response. Fig. 2 shows the sequence of deformation for tube 1 with an impact velocity of 300m/s. The dynamic buckling is progressive, starting from the striking end, and thickening of wall can be seen. Fig. 3 shows tube 2 deforming under the same velocity. Mushrooming at the ~ g end is evident as the wall end becomes thicker than the undeformed portion. The mushrooming also significantly alters the fold formation as the first complete fold does not occur at the very end. The increased wall thickness enhances the resistance to buckling, thus shifting the first fold to a distance from the tube end where the thickening effect diminishes. This is clearly a different phenomenon from the progressive buckling and plastic buckling reported previously. When the original wall thickness increases ~r, mushrooming becomes predominant and the final deformation displays mushrooming and wrinkling with no complete folds, as shown in Fig. 4 for tube 3 at 300m/s.

600 5OO A W

m400

g'3oo 2oo W

100 0

|

0

....

)

0.2

'

0.4

'|

)

0.6

0.8

'

'

'"".

1

strain

Fig. 1 Static stress strain relationship

ii

ii ui

i

ii

IIIIII

I

-

II

I

II III

Fig. 2 Deformation of tube 1 at 300m/s. Frame interval 0.0275ms.

Fig. 3 Deformation for tube 2 at 300m/s Frame interval 0.055ms.

Fig. 4 Deformation for tube 3 at 300m/s Frame interval 0.055ms.

The influence of the striking velocity shows a similar trend with the deformation mechanism evolving fi'om folding at the striking end to mushrooming and folds at a distance through to excessive mushrooming as the velocity increases. The effects are also illustrated in Figs. 5 to 7, showing the increase in the wall thickness at the first wrinkle, the position of the

398 first wrinkle and the total length reduction for tube 2 at various impact velocities. Though the total length reduction appears to be approximately linear in terms of smT~g velocity, there appears to be a trend in the change in wall thickening and the first wrinkle position at the velocity of 400m/s. Further increase in velocity seems to produce less effect, particularly for wall thickness increase, indicating that the higher kinetic energy is mainly dissipated by more fold formation, rather than mushrooming. Figs. 8 and 9 show the history of the impact force-at the ~ g point and the energy dissipation in terms of the striking velocity for tube 2. Interestingly, the figures indicate that the duration of the impact event generally lasts for 0.2 ms irrespective of

350 300 250 200 150 100 50 1

'

250

9

350

,,

450

1

550

650

S~k~vdodty(~)

Fig. 5 Percentage of wall thickness increase at the first wrinkle for tube 2 30 28 26

f.

~'24 ~z2

J

J

J

.~N 18

the striking velocity. ,J

4. DISCUSSIONS AND CONCLUSIONS The axisymmetric FE model demonstrates the effect ot mushrooming in the axial crushing process of tubes with a relatively thick wall. It shows that various modes oi deformation will emerge and they can be significantly different from those under a static or low speed loading conditio~ Generally, three patterns of deformation may be expected: dynamic progressive folding for relatively thin tubes under a low impact speed; end mushrooming with folds formed at a distance from the st~'king end for all tubes at medium speeds, and mushrooming and wrinkling for thick mbcs at high speeds.

14 12 10

!

-

350

25O

9

"

450

,

'

550

'i

650

S ~ Vek~y(mls) Fig. 6 Position of the first wrinkle from the 8o smTAng end for tube 2 70 60 50 40 '

250

9

350

I

f

9

450

550

650

VeJoc~ (m/s)

Fig. 7 Percentage of total length reduction For tube 2

399 A preliminary experimemal study using a high pressure gas gun confirms the mushrooming scenario at the striking end of a tube section for the range of dimensions discussed in this paper. However the tests also show fractures emerging in the outer surface of folds in thick tube samples at l'figher impact speeds. Apparently these cracks form as a result of the excessive tensile stress due to large deformation. The current FE model does not have the capacity to simulate material failure under dynamic loading conditions.

Fig. 8 History of impact force at the striking end vs velocity

Fig. 9 History of energy dissipation vs. velocity

Though the strain rate effect has been fully considered through the adoption of the Cowper-Symonds relationship, corresponding experimental results indicate that dynamic material failure would be the next task to be tackled. REFERENCES

I. Andrews, K. R. F., England, G. L. And Ghani, E., Classification of the axial collapse of cylindrical tubes under quasi-static loading, Int. J. Mech. Sci. 25 (1983) 687. 2. Alexander, J. M., An approximate analysis of the collapse of thin cylindrical columns, Quart. J. Mcch & Appl. Math. 13 (1960) 10. 3. Jones, J., Structural Impact, Cambridge Univ. Press, Cambridge, 1989. 4. Tvergaad, V., On the transition from a diamond mode to an axisymmctric mode of collapse in cylindrical shells, Int. J. Solids Structures, 19 (1983) 845. 5. Cowper, G. R., and Symonds, P. S., Strain hardening and strain-rate effects in the impact loading of cantilever beams, Brown University Division of Applied Mathematics Report No. 28, 1957.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

401

Axial crushing of aluminium columns with aluminium f o a m filler A.G. Hanssen, M. Langseth and O.S. Hopperstad Structural Impact Laboratory (SIMLab), Department of Structural Engineering, The Norwegian University of Science and Technology (NTNU) N-7491 Trondheim, Norway

An overview of previous experimental results leading to a compact design formula for determination of the average crush force of circular and square foam filled extrusions is presented. The design formula is applied in order to assess the effect of cross section geometry on the energy absorption capacity of foam filled columns.

I. INTRODUCTION For new materials and designs to be considered for structural applications, as in various automotive components, additional requirements beyond that of the main function are usually posed. Aluminium foam is initially attractive because of its low weight and highly efficient energy absorption, but also benefits from promising sound absorption/insulation properties, non-combustibility, efficient recycling and high stiffness to weight ratio. Combined with new cost effective manufacturing routes, this makes aluminium foam a candidate for the next generation of automotive energy absorption systems. This paper considers the axial energy absorbing properties of square and circular aluminium extrusions filled with aluminium foam, Figure 1, and is a limited overview of the experimental database generated by Hanssen et al [1-4], comprising more than 300 quasistatic and dynamic tests.

Figure 1. Test specimen geometry

Figure 2. Typical material behaviour

402 The components presented in Figure 1 may easily be implemented in bumper systems and accurate design formulas for prediction of the energy absorption (average crush force) is therefore of great advantage. A brief summary of experimental details, material characteristics and visual observations will be given in the following (Section 2), before presenting a design formula (Section 3) which again is applied to compare the energy absorption capacity of square vs. circular foam filled columns (Section 4).

2. EXPERIMENTAL DATABASE The bottom end of the components tested in [2,3] was clamped during testing, see Figure 1. Furthermore, the ratio between the effective component length and cross section width/diameter was approximately equal to 3 for all tests. A trigger was applied in the top in order to initiate the folding during dynamic loading conditions. However, only the results from the quasi-static tests will be considered herein. In order to evaluate possible design formulas for the given components, uniaxial material tests were carried out for both foam and extrusions, Figure 2 (engineering values). All extrusions investigated were of the aluminium alloy AA6060 in a variety of tempers. The choice of temper generally determines the strain hardening as well as the strength, here quantified by the stress at 0.2% plastic strain Cro2 and the ultimate stress cry. For later use in design formulas, the characteristic stress tr0 of the extrusion material is defined as the average value of tr02 and cry. Cubic specimens of aluminium foam in the density range from 0.1 to 0.5 g/cm 3 were tested in compression. In Figure 2, the plateau stress crf of the foam is defined as the average stress at 50% strain (absorbed energy at 50% deformation divided by corresponding deformation). The final deformed shape in axial crushing of some square and circular components is presented in Figure 3. Briefly, the foam filler was found to have significant effect on the

Figure 3. Deformation behaviour as function of foam filler density

403 deformation behaviour, causing the square extrusions to develop more lobes [2] whereas a critical foam filler density caused the circular extrusions to change their deformation behaviour from diamond to concertina mode [3]. For detailed descriptions, see [2,3].

3. DESIGN FORMULA Based on the results from Refs [ 1-3], the following design formula was found to represent the average crush force F,,vg of both square and circular foam filled extrusions with satisfactory accuracy F,,,,#= F/,~ + cr/ A/ + Ci 4O'oCr/ A o .

(1)

Three terms are present in the above equation. The first part 1) is simply the average crush force of the corresponding non-filled extrusion F~ whereas the second part 2) constitutes the uniaxial resistance of the foam filler given by the product of foam plateau stress crf and foam core cross sectional area ,4:. As observed experimentally, the average crush force of foam filled extrusions always exceeded that of the sum of 1) and 2). This increase in capacity is referred to as an interaction effect and is represented by the third term of Equation 1. Here, Ci is a dimensionless interaction constant, whereas Ao is the cross sectional area of the extrusion. The properties of the extrusion material are represented by the characteristic stress or0. In order for the design formula in Equation 1 to be robust and generally valid, the expression for the interaction effect should satisfy some trivial requirements beyond that of correlating well with experimental data in the investigated parameter range. These requirements are basically that the expression for the interaction effect should vanish (evaluate to zero) when Equation 1 is applied to either a non-filled extrusion (o"t = 0) or a single foam cube (tr0 = 0). As seen, Equation 1 obeys these requirements. For the design formula to be complete, an expression is needed for the average crush force of non-filled extrusions. For non-filled square extrusions exhibiting the asymmetric deformation mode (definition after [5]) as well as circular extrusions obtaining diamond modes, the average crush force is represented by [5] F~

= Coq~2/3~oA o .

(2)

Co is a constant dependent upon cross section geometry. Moreover, Co will in practice also be dependent upon the definition of the characteristic stress tr0. The solidity ratio (relative density) of the cross section tp is defined as the ratio between the solid extrusion cross sectional area Ao and the area Ac enclosed by the centre lines of the cross section walls. Since rather thin walled tubes are considered herein, Ac is approximately equal to the foam cross section area Aj; hence tp = ,40 / A f . Based on the above discussion, a complete description of

the average crush force of foam filled extrusions can now be written as F,,~ t3 1 a f ty f cr o Ao = C o tP 2 + _tp tr o + C i ~] Cro .

(3)

404 A summary of the parameters involved when using Equation 3 for square and circular cross-sectioned columns is given in Table 1. Here b represents the outer width of the square extrusions, whereas d is the outer diameter of the circular ones. The wall thickness of the tubes is given by h. The corresponding correlation plot for Equation 3 vs. 180 experiments is shown in Figure 4. For most tests, the accuracy of the proposed design formula is within an error margin of +10%. The parameter range for the tests was given by 0
A1

,

Circular

( b - 2h) 2

~(d - 2h) 214

Ao

4 h ( b - h)

rib(d-h)

(p

4 h ( b - h ) I ( b - 2h) 2

4h(d - h)/(d - 2h) "

Co

1.30 1.39 (see Figure 5)

2.15 0.88 (see Figure 5)

Ci

200-

Square

.I

Experiments (kN)

I

I

~

1.60

<>1~ ~ =t

t

I

(-)

150 -

1.20

1oo -

,

I

,,,

I

Cj

r-y

S

0.80

o~

Square, Ret H]

50

Square, Ref [2]

0.40

Circular, Ref [3]

0

0

I

50

I

100

Equation 3 ' I

150

'

(kN)

200

Figure 4. Model correlation, average force

0.00

- ~

Square

-

--~--

Circular

i Axial deformation

0

'

I

2o

'

I

4o

'

t

60

---r

(%) ....

8o

Figure 5. Interaction constant development

The values for the interaction constant Ci given in Table 1 are valid for a deformation of 50%. However, it was found [2,3] that Ci is an increasing function of deformation, see Figure 5. A plausible explanation to this is that the foam has been assumed to have a perfect plastic behaviour represented by the plateau stress GI, whereas in reality the foam shows some strain hardening.

405 4. SQUARE VS. C I R C U L A R FOAM FILLED COLUMNS There are several ways to compare the energy absorption characteristics of square vs. circular foam filled extrusions. However, the comparison should strictly focus on the effect of geometric cross section shape, and thus not compare extrusions of different material properties containing foam of different strength. For this reason, the best comparison is probably done when the solid cross section area Ao and the cross section of the foam filler Af are equal for both square and circular components. As seen from Section 3, this implies that the comparison should be carried out at constant solidity ratios q~. A great benefit of this approach is that both cross section types will have the same weight, assuming that the density of the foam filler is equal. Furthermore, if Ao and At are equal for both cross section types, it is easily shown that both the diameter and wall thickness of the circular extrusion is increased by a factor of 2 / ~ (=1.13) compared to the wall thickness and width of the corresponding square extrusion. Hence, the dimensions of the two cross section types to be compared are also roughly the same. Figure 6 shows how the total average force (Equation 3) are divided between non-filled extrusion, foam and interaction effect as function ofcr:/cro, given that the solidity ratio r is kept constant at O. 10. When increasing the strength of the foam filler, the relative contribution to the average force of the extrusion naturally decreases whereas the contribution of the foam filler increases. However, it is interesting to note that the interaction effect appears to have a maximum value for a given value of the stress ratio or: /cro. Increasing the strength of the foam filler beyond this Value does not lead to any further relative increase in the interaction effect. Moreover, whereas the contribution from the non-filled extrusion is significantly higher for the circular compared to the square cross section, the opposite is true for the interaction effect. Hence, the interaction effect is more pronounced for square extrusions than for circular ones. The average force ratio for square vs. circular columns can be studied in Figure 7 for a variety of solidity ratios. When q~= 0, no extrusion material is present and there are naturally no difference in average force of a circular vs. a square foam block. For non-filled extrusions it is seen that the average force ratio is approximately 0.6, concurring with [5], showing that a 100

,

1

,

.1

~

I .

%

(%)

\

I yExtrusion

80-

\\

. ,

I,.

Square Circular

~ Q

o,o

,oao

I

Square vg Favg

J

I

,,~

I

,

I

i

_,

TM

1.00

~

,

1.26

..

o?2S/.-~-~f---

.[3 2 26:" " -;5

40 ~

20

" 0.75

__~V,~#~ '''-''~" Interactio~ ~,',,

o-f-.

0,00

_~

~= 0.20

Gf / Go ,

i

0.02

'

i

0.04

'

i

0.06

'

i .....

0.08

O'f / O" o

0.50

'

0.10

Figure 6. Contributions to total average force

'

0.00

l

0.02

"

'"i

0.04

'

I

0.06

'

I

0.08

0.10

Figure 7. Force ratio, square vs. circular

406 circular cross section is superior to the square shape. However, introducing foam filler diminishes the difference in average force between the two cross section geometries. This is most rapidly occurring for thin walled extrusions (low value of tp). Above a certain value ofcrt/or o, square foam filled columns even slightly exceed the performance of the circular ones. Eventually, as crs I tro increases, the average force ratio between the two cross section types will approach 1.

5. CONCLUSIONS The average, axial crush force of both square and circular extrusions filled with aluminium foam has been assessed. The design formula applied for this comparison has been validated against an extensive experimental database presented in previous publications. It was found that the average crush force could accurately be represented by three parts, being 1) the average crush force of the corresponding non-filled extrusion, 2) the uniaxial resistance of the foam core and finally 3) an interaction effect originating from the changed deformation modes of the extrusion. An interesting observation is that the relative contribution of the interaction effect on the total average force appears to have a maximum value for a given strength of the foam filler. According to the presented design formula, the foam filler induced interaction effect on circular foam filled extrusions is significantly lower than that of square extrusions. On the other hand, the crushing resistance of non-filled circular extrusions is higher than for square extrusions. For this reason, circular foam filled extrusions still have a higher capacity for energy absorption than square extrusions of the same weight. However, this difference decreases as the foam filler strength increases, eventually leading to a more or less break-even in performance.

A c k n o w l e d g e m e n t - This project was made possible by deliverance of aluminium foam and economic support from Hydro Aluminium.

REFERENCES 1. 2.

3.

4.

5.

A.G. Hanssen, M. Langseth and O.S. Hopperstad: Static Crushing of Square Aluminium Extrusions with Aluminium Foam Filler. Int. J. Mech. Sci., Vol 41/8 pp. 967-993 (1999) A.G. Hanssen, M. Langseth and O.S. Hopperstad: Static and dynamic crushing of square aluminium extrusions with aluminium foam filler. Submitted to the Int. J. Impact Engng. (1999) A.G. Hanssen, M. Langseth and O.S. Hopperstad: Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler. Submitted to the Int. J. Impact Engng. (1999) A.G. Hanssen, M. Langseth and O.S. Hopperstad: Optimum design for energy absorption of square aluminium extrusions with aluminium foam filler. Submitted to the Int. J. Mech. Sci. (1999) N. Jones: Structural Impact. Cambridge University Press (1989)

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta @ 2000 Elsevier Science Ltd. All rights reserved.

407

Failure M e c h a n i s m and Behavior o f Thin-walled Reinforced Concrete Barrels under Lateral Loading Mahmoud A. Issaa, Mohsen A. Issab and Robert H. Bryant b aT.Y. Lin International Bascor 5960 North Milwaukee Avenue Chicago, IL 60646, USA bDepartment of Civil and Materials Engineering The University of Illinois at Chicago 2095 ERF, 842 West Taylor Street Chicago, IL 60607, USA

ABSTRACT This paper presents the failure mechanism and behavior of short thin-walled RC tubular cylinders subjected to lateral loading. Five cylindrical specimens were cast and tested. The prototype barrels were 5.4 m in diameter, 35.6 cm thick, and fixed to a massive concrete footing at the bottom and to a 76.2 cm thick slab at the top. These dimensions were scaled down accordingly to 1/10. Height-to-diameter ratios, the percentage of steel reinforcement, and the wall opening were chosen as study parameters. The structural evaluation was helpful in determining the bending and shear behavior of closed RC tubular walls, as well as the behavior of the specimen with the wall opening. The short specimen sustained more load to failure. The longer specimens deflected more in terms of their ultimate load with the exception of the specimen with the least amount of reinforcement. Shear cracking started at approximately the same locations for the closed specimens, i.e., a shear crack at an angle of approximately 45 ~ however, this behavior was somewhat different for the specimen with an open wall as the first shear crack appeared at the comers of the opening. In comparing the results obtained from the testing and nonlinear finite element analysis (NFEA), the NFEA predicted satisfactory results. 1. INTRODUCTION As a consequence of significant research efforts since the early part of the century, the necessary design equations and provisions for flexure and flexure combined with axial load in the American Concrete Institute (ACI) Building Code Requirements are well developed. Unfortunately, the current shear in combination with bending design provisions have not reached that level of understanding, hence they are still based on a wide variety of empirical information.

408 Inclined cracking in structural concrete beams is generally considered to be caused by either web-shear stresses or flexure-shear stresses. Under loading, web-shear cracks are diagonal cracks that tend to form in the web of the member near the centroid level of the cross-section in untracked concrete. Before cracking occurs in the concrete the shear stresses in the web of a structural member give rise to compressive stresses in one principal direction accompanied by diagonal tensile stresses in the second principal direction. When diagonal cracks form, the ability of the concrete to transmit principal tensile stresses is seriously diminished, and unless appropriate reinforcement is provided, sudden failure may occur. The objective of shear design is to avoid premature brittle shear failures, and to ensure that member shear capacity can ensure ductile flexural failure. Large reinforced concrete thin-walled structures are widely used in modem concrete construction. These structural members can exist independently as water tanks, silos, etc., or they can be part of a structural building, e.g., generator barrels in submerged, hydropower generator buildings. The structural behavior considered here is that of a relatively short deep beam, whose behavior is somewhat similar to that of a shear wall [ 1]. 2. BACKGROUND Research efforts on shear were intensified after the collapse of the Wilkins Air Force Depot warehouse in Shelby, Ohio in 1955. Many researchers have conducted experimental programs pertaining to the development of basic concepts involved in specifications dealing with reinforced concrete subjected to shear in the United States and worldwide. Cardenas et al. [2] carried out an investigation at the structural development section of the Portland Cement Association to evaluate Section 11.16 (special design provisions for walls) of the ACI 318-71 Building Code [3]. In this investigation, thirteen large rectangular shear wall specimens were tested under static combinations of axial load, bending, and shear. Veechio and Collins [4] tested 17 panels to determine the softening coefficient and to get a better understanding of the compression response of reinforced concrete. Hsu and Mo [5] derived a truss model theory to predict the strength and behavior of low-rise shear walls. This theory was based on equilibriurn and compatibility conditions and the constitutive laws of sottened concrete. By comparing the theoretical predictions with the results of 24 tested shear walls, the theory was found to be applicable throughout the loading history. Mau and Hsu [6] established a sottened truss model theory to analyze frame wall panels subjected to shear and vertical loading. Their theoretical predictions agreed very well with the experimental data of 14 test specimens. Adebar and Collins [7] tested 9 large-scale reinforced concrete offshore structure wall elements subjected to combined membrane shear and transverse shear. They concluded that the three-dimensional modified compression-field theory accurately predicted the influence of membrane shear on transverse shear and that it is a more appropriate shear design method for complex concrete structures. Bathe et al. [8] developed nonlinear finite element modeling procedures for thin circular reinforced concrete walls, forming the tubular structure, in order to predict the stress level and stress distribution in the entire structure. In addition, they were designed to predict the ultimate load as well as the load-deformation behavior throughout the post cracking loading history of the member. Bathe reported more description of the modeling and analysis technique in a book [9].

409 3. MATERIAL, SPECIMEN GEOMETRY AND TESTING PROCEDURE Concrete composed of water, cement (Type I), and aggregate ratios of 0.65:1:4 by weight were used. The sand to gravel ratio was 4:1. The maximum aggregate size was 4.75 mm (0.187 in.). Two sets of standard 50 x 100 mm and 75 x 150 mm cylinders were prepared for each model from the same batch used in casting each barrel. Strain gauges were mounted in longitudinal and transverse directions on the surface at mid-height of the cylinders to obtain modulus of elasticity, Ec (29 GPa), and the Poisson's ratio, v (0.16). The obtained compressive strength of concrete ranged between 35.8 and 40.7 MPa. A total of five specimens were cast and tested. The first four specimens, which were designated as MC-1, MC-2, MC-3, and MC-4, are closed sections. The first specimen (MC-1) with a height-to-diameter ratio of 1.0, was reinforced with a 1.022 percentage of steel. The second specimen (MC-2) had the same height-to-diameter ratio but was reinforced with a 0.62 percentage of reinforcement. The third specimen (MC-3) was the same as the first one but with an additional nine 4.45 mm bars in the tension side. The fourth specimen (MC-4) had a height-to-depth ratio of 0.5 and was reinforced in the same manner as the third specimen. The fifth specimen (MO-l) was identical to MC-1 but with an opening of 30 ~ at a fight angle to the direction of the applied load. This specimen was reinforced as specimen MC-1 was, with an addition of four 4.5 mm diameter bars added above and below the hole in the wall, and four 4.5 mm diameter bars added on each side of the hole. The reinforcement for the upper and lower slabs was made of 6.4 mm diameter wires for longitudinal and transverse bars, and of 3.4 mm diameter wires for the stirrups. A typical reinforcement assembly for a specimen is shown in Figure 1. The test setup is shown in Figure 2. A servo hydraulic testing machine with a capacity of 225 kN was used. To fix the specimen to the machine bed a very stiff frame was designed and manufactured. This frame was fixed to the machine bed and test specimens were tied down to the frame. Based on the preliminary finite element analyses, 11 to 12 points were selected on each of the specimen wall to measure strains on the concrete surfaces and on the reinforcing steel. At each point, two three-dimensional rosette strain gages were mounted on the inner and outer faces of the concrete wall. In addition, four electrical resistance strain gages were mounted to the reinforcing bars in the vertical and horizontal directions on the inner and outer wire meshes. Two linear variable displacement transducers (LVDTs) were mounted at the two lower ends of the specimen to measure deflections and end rotations. In addition, two LVDTs were mounted at the same height (76 mm below the top slab) on the tension and compression sides of the specimen to measure deflections and to detect the degree of ovaling. Another LVDT was mounted at 76 mm height on the tension side of the specimen. Two crack mouth opening displacements were mounted on the specimen; one in the longitudinal direction at mid-height on the tension side, the other in the transverse direction at mid-height at the centerline of the cylinder. 4. FINITE ELEMENT ANALYSIS The finite element package ALGOR [10] was used to perform the analysis presented in this paper. In order to predict the stress level and stress distribution at different heights across the thickness of the wall, three-dimensional isoparametric eight-

410 noded brick elements were used to model the concrete wall and slab. Six-noded brick elements were used in the critical areas at the crack fronts. The steel reinforcement is modeled by three-dimensional truss elements. Results obtained from nonlinear finite element analysis were compared with experimental results.

Figure 1. Reinforcement caging

Figure 2. Testing Setup

5. RESULTS AND DISCUSSIONS In all specimens the first crack was observed near the fixed end of the specimen. In MC-1 specimen the first crack was observed at a load of 37.4 kN and this crack propagated at an angle of 42~ The ultimate load was reached at 115 kN. In MC-2 specimen, the first crack was observed at a load of 20.5 kN and this crack propagated at an inclination of 70~ The ultimate load was reached at 73 kN. In MC-3 specimen, the first crack was observed at a load of 68.5 kN and this crack propagated at an angle of 57 ~ The ultimate load was reached at 141 kN. In MC-4 specimen, the f'n-st crack was observed at a load of 90 kN and this crack propagated at an inclination angle of 60~ The ultimate load was reached at 210 kN. In MO-1 specimen, the fast crack was observed at a load of 44 kN and this crack propagated at an angle of 60~ The ultimate load was reached at 133 kN. In all specimens final failure was accompanied by steel necking and tension reinforcement breakage. Front side cracking pattern for specimen MO-1 is shown in Figure 3. As expected, the short specimen sustained more load until failure (See Figure 4), while the specimen with minimum reinforcement sustained the least amount of load. As a result, cracking progressed to a further extent in the specimens with a larger percentage of steel, i.e., more shear and flexure cracks developed. Longer specimens deflected more in terms of the ultimate load. The additional steel in MC-3 specimen provided more strength as compared to the MC-1 specimen. The smaller first cracking and ultimate loads reached in MC-2 are attributed to the fact that this specimen was reinforced with only the minimum amount of steel. Furthermore, more ovaling was observed in the longer specimens with the exception of the specimen with minimum reinforcement. The analytical and experimental loaddeflection curves, ovaling effect curves, moment-curvature curves, cracking loads, ultimate applied loads and the corresponding moment, and steel stresses at ultimate capacity for each specimen were compared. Deflection of model I (MC-1) at ultimate loading is shown in Figure 5. The results from the finite element analysis for each model were analyzed in

411 evaluating the behavior of such complex structures. The load-deflection curves for the finite element analysis reasonably predicted the corresponding experimental behavior of the specimens up to the ultimate load (ascending part), while due to software limitation we were not able to predict the descending part (See Figure 6). After ultimate, gradual loss of the tensile reinforcement at the cracks, micro and bridging cracks direction and orientation were hard to predict and model. The FEA predicted the ovaling diameter effect with the same accuracy as of the load-deflection relationship. The overall nonlinear finite element analysis prediction was satisfactory.

Figure 5. Deflection of model I (MC-1) at ultimate loading (NFEA)

Figure 6. Comparison of experimental and NFEM results for specimen MC-3

412 6. CONCLUSIONS The focus of this study was to investigate the structural behavior of circular type thinwalled reinforced concrete structures subjected to lateral loading. Two types of specimens were tested, closed and open, with two different height-to-diameter ratios, under lateral shear applied at the top of the cylinder. Based on the experimental observations and nonliniear finite element analysis, the following conclusions can be drawn: 1. The reduction in height-to-diameter ratio reduced the moment due to the applied flexural load and increased the ultimate shear capacity of the specimen. 2. The longer specimens deflected more in terms of the ultimate load with the exception of the specimen with the least amount of reinforcement. 3. The specimen with the least amount of reinforcement sustained the least amount of load. 4. More ovaling was observed in the longer specimens with the exception of the specimen with the least amount of reinforcement. 5. The load-deflection curves for the finite element analysis reasonably predicted the corresponding experimental behavior of the specimens.

REFERENCES [1]

[2] [3] [4]

[5] [6] [7]

[8] [9] [10]

Issa M.A., Structural behavior of thin-walled reinforced concrete cylinders under lateral loading. Ph.D. Thesis, Department of Civil and Materials Engineering, University of Illinois at Chicago, 1997. Cardenas A.E., Hanson J.M., Corley W.G. and Hognestad E., Design provisions for shear walls. ACI Journal, Proceedings, 70,3 (1973):221-230. ACI building code requirements for reinforced concrete (ACI 318-71) and commentary (ACI 318R-71). American Concrete Institute, Detroit, 1971. Vecchio F. and Collins M.P., Stress-strain characteristic of reinforced concrete in pure shear. IABSE Colloquium, Advanced Mechanics of Reinforced Concrete, Delft, Final Report, International Association of Bridge and Structural Engineering, Zurich, Switzerland (1981):221-225. Hsu T.T.C. and Mo Y.L., Sot~ening of concrete in low-rise shear walls. ACI Journal, 82,6 (1985):883-889. Mau S.T. and Hsu T.T.C., Shear behavior of reinforced concrete framed wall panels with vertical Loads, ACI Structural Journal (1987):228-234. Adebar P. and Collins M.P., Shear Design of Concrete Offshore Structures. ACI Structural Journal, 91,3 (1994):324-335. Bathe K.J., Walczak J., Welch A. and Mistry N., Nonlinear analysis of concrete structures. Computers and Structures, Vol. 32, No. 3/4, pp. 563-590, 1989. Bathe K.J., Finite Element Procedures. Prentice-Hall, Inc., 1996. ALGOR Manual, Accupak Reference Manual, Nonlinear Stress and Vibration Analysis, 1996.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

413

Crushing behavior o f composite domes and conical shells under axial compression N.K.Gupta a, R.Velmurugan b and M S Palanichamyb Department of Applied Mechanics, Indian Institute of Technology, Delhi, New Delhi - 110 016, India. b Department of Civil Engineering, Mepco Schlenk Engg. College, Sivakasi - 626005, India. Composite domes and conical shells made of short, randomly oriented glass fibre mat with polyester resin were subjected to axial compression in an Instron machine both in empty and foam filled conditions. The domes were of different sizes and their generator was a parabola. The conical shells had different cone angles. The deformation modes and energy absorbing characteristics of the shells were studied, and their load compression curves presented. The effect of foam filling on the response of the domes and the conical shells is discussed. Analytical expressions were formulated to find the average crush load and the crush length of the conical shells and domes. The results thus obtained were compared with experimental results and these match well. 1. INTRODUCTION Composite thin walled cylindrical, spherical and conical shells and domes have been investigated [1-7] in the past and their collapse modes and energy absorbing characteristics have been studied for applications to design for crashworthiness of vehicles. The crushing behavior of FRP round and square tubes (glass, graphite and kevlar tubes with epoxy resin) for different lay-ups and sizes has been studied by Thornton & Edwards [3] and Hull [4]. Farley and Jones [5] in their experiments on the kevlar/epoxy tubes with different ply orientations at different crushing speeds observed that the energy absorption capacity increases with speed for all the cases considered. Experimental and theoretical investigations on shells of conical, spherical and dome shapes, are very few. Price and Hull [6] carried out experiments on truncated conical shells made of short and randomly oriented glass fibre mat with polyester resin for different cone angles and made a comparative study with the collapse mechanisms of composite tubes made of same material. Gupta and Velmurugan [7,8] carried out experimental and theoretical studies to find the average crush load and the average crush length for the cylindrical and conical shells and Mamalis et.al [9,10] carried out analysis to find the mean collapse load. In the present study, composite domes of different sizes and conical shells of different cone angles, all made of short, randomly oriented glass fibre mat with polyester resin were subjected to axial compression in an Instron machine. The mode of deformation and the energy absorbing characteristics of the shells were studied. The shells were filled in-situ with polyurethane foam and its effect on their mode of deformation and on the energy absorbing capacity is discussed.

414 An approximate analysis is carried out by considering various failure modes, observed in the crushing process of the shells, and results thus obtained were compared with experiments and these match well. 2. EXPERIMENTS Composite domes of different sizes and conical shells of different cone angles were made of short randomly oriented glass fibre mat with polyester resin and fabricated by hand lay up method. The base of shells was circular and was machined perpendicular to the central axis of each shell. The domes employed in the present tests had a parabolic profile, given by H = C R 2, where H is the height and R is the radius of the shell. Values of C for different shells were 1.2, 0.8, 0.6, 0.52 and 0.2. The semi cone angles (~) of different series of cones varied from 11 ~ to 45 ~ The dimensions of domes and cones are given in Tables 1 and 2. These shells were subjected to axial compression in an Instron machine at a cross head speed of 2mm/min. Experimental observations showed that, in shells with C = 1.2, the progressive crushing initiated immediately after a small amount of bending from the narrow edge, and it continued over the entire length. The crush zone remained circular in shape. In the shell of base diameter (D) = 103ram, thickness (t) = 2.7ram and C = 0.8, the crushing mode was same as in the case of the shell with C = 1.2. As the value of C is decreased, the formation of the crush zone was delayed. The fibre layers bending inside were more than that bending outside the shell radius. In a shell of D = 105.5mm, t = 2.5mm and C = 0.6, as the loading progressed the fibres kept on bending inside the shell radius. There was no bending of fibres outside the shell radius and no crush zone was formed. Also the shape of the cross section was changed from circular to elliptical due to the shell geometry. Shells of C = 0.52 with D = 102 ram, t = 2.6 mm and D = 98mm, t = 2.4mm were also subjected axial compression. In the former, as the deformation progressed the cross section changed from round to triangle and in the later case it changed to square. After a deformation of 60mm a vertical crack started from the bottom end and it soon moved upwards, which made the shell inefficient to absorb the energy. The mean collapse load for domes was calculated to the crush zone for which the height from the vertex was HI and the corresponding profile angle was ~l. The shape of the shell of D = 104ram, t = 3mm and C = 0.2 was almost like a hemi spherical. In this case, height and radius of the shell were almost same and the loadcompression curve was smooth with no oscillations. In this case, the bending of the fibres inside the wall radius continued and no crush zone was formed. After a bending deformation of 25mm a vertical crack started forming from the bottom, and there was sudden drop in the load-compression curve for the above specimen. On removal of the load, the bent fibres came almost back to their original position. Some dome specimens were filled in-situ with polyurethane foam and from the loadcompression curves (see, Fig. 1) it is seen that the presence of foam increases the load carrying capacity and delays the formation of vertical crack for smaller values of C. Fig. 2 shows the load-compression curves of the empty and foam filled conical shells. In conical shells there was a sharp rise in the load-compression curves at the initial stage of deformation unlike in domes. The mean load and the crush length calculated from experiments are given in Table 2 along with the smaller diameter (d) and thickness (t) of the conical shell. Experimental observations in conical shells show that as the cone angle is increased beyond 18 ~ their progressive crushing is followed by vertical crack formation.

415

Tablel Average crush load in a crush cycle for GFRP domes under axial compression. S.No

D

t mm

C

H1

mm

{~1 deg.

P~v kN Ex.

Pay kN Th.

1

104

2.3

1.2

22

5.55

34

32.28

2

103

2.7

0.8

25

6.38

38

40.01

3

106.5

2.5

0.6

28

6.96

23

*

4

102.2

3.1

.52

-

-

-

*

5

108

3.0

0.2

-

-

-

*

* - No progressive crushing

Figure 2 - L o a d compression curves o f empty and foam filled composite conical shells

Figure 1 - Load compression curves o f empty and foam filled composite d o m e s

Table 2 Average crush load and crush length of the GFRP conical shell in a crush cycle. S.No

d mm

t mm

~ deg.

h mm

P kN Th.

Ex.

Th.

Ex.

1

12.35

3.15

9.5

24.6

25.5

3.8

5.22

2

14.99

2.9

14

24.8

21.0

4.1

4.63

3

30.97

2.5

11

35.0

42.0

5.5

7.00

4.0

11

68.8

70.0

7.35

10.27

4

34.63

416 3. ANALYSIS The analysis is carried out by considering the work involved in the mechanisms of the crushing processes, i.e., bending of fibre layers, circumferential strain in the material and matrix cracking in the vertical direction. The work required to bend the fibres inside the shell radius, see [8], is computed as the total energy due to bending in the case of each shell. The expressions in each case of domes and cones respectively are, W b = ~1R n(t213n + tz20n)O0

(1)

Wb = n_r 0 (t2~ + t20)( R + hsint~)

(2)

R and Rn are the radius of the conical shells and domes respectively, c~0 is the ultimate strength of the composite laminate in uni-axial tension. 13i, 0i are the fibre angles bending inside and outside the shell radius and ~ is the semi cone angle, tl and t2 are the thickness of the fibre layers bending inside and outside the shell radius. The expressions for total work due to hoop strain in domes and conical shells are respectively written as, n

2

(3)

Wh = rCaoY.X [t, O-sin q~i)+ t2(sin ~i -1)] l

where x is the length of each segment of the dome, see Figure 3(b). Wh = rccr0h2[tl (sin(~ + [3) - sin~) + t2(sin ~ - sin(~ -0))] (4) The work required for propagation of the central crack, which moves lengthwise in a single crush cycle, is given by (5)

W m : ~o m ~"~X[tl 0 - s i n Oi)+ tz(sin Oi-1)Ri] i=l

h

Wm = ~ m c~

0

=2nCm(tl(1-sin~) + t 2 ( s i n ~ - l )

h

R+-sin6 2

)

(6)

O'm is the shear strength of the polyester resin and is equal to 26 Mpa [8] and em is the

corresponding shear strain. Therefore the total strain energy, which is the sum of the work required to bend the fibres, to strain the material in the circumferential (hoop) direction and to delaminate the shell wall, is given by,

Wr =W b+Wh+Wm

(7)

417

Figure 3 - Idealised models of conical shell and dome. The work done by the applied load is Wp = P~5

(8)

where P is the applied load and ~5 is the reduction in length of the shell in a cycle. The expression for 5 and is given by 5 d = h[cos~n - 1 ] ;

5 c = h[cos~b-1]

(9)

Equating the total strain energy to the work done by the applied load in a cycle the expressions for the crush length and average crush load in a cycle are obtained. Thus the crush length for dome and conical shell are

l

croRtd

1" l

,

hd = 2 ~ (6 0 + Om sin {~)td**

9 h =

'

c

)'

cr~

(10)

2(cr 0 + crm sin ~)tc**

The computed values of crush length are compared with the experimental values in the case of cones in Table 2. The corresponding average crush loads are Pd =~-

.5 ts0(Rn)t d 9 + 2

(o 0x2 +2tsm(Ri)x

d

;

Pc =-2 0.5 o0(R + hsin~)t c + o0h 2 + 2Om R + - s i n ~ h tc** 2

(11)

418 Here t d ** = (tl(1-sin~i)+t2(sin~i-1)) ; t e ** = (tl(1-sin(h)+t2(sin~-l))

(12)

and td* = tl2~n + t220n ;tc* = tl2~+ t220

(13)

The values of 13and 0 are given by 1I 1I 1I 1I (14) ~n :'2"--(~n'0n :'2+(~n "~:-~-I~,O=-~+~ The calculated average crush load in a cycle of the domes and conical shells are compared with experiments, in Tables 1 and 2, and these match well.

4. CONCLUSIONS Composite domes of different sizes were subjected to axial compression. Different deformation modes were identified and the energy absorbing capacity of the shells was studied. Domes of higher values of C deformed in progressive crushing, whereas those with smaller values of C deformed in progressive bending followed by a progressive crushing mode. The presence of foam increases the energy absorbing capacity of the domes and delays the formation of vertical crack in domes with smaller values of C. Experiments on conical shells show that, as the cone angle is increased, the crushing mode changes from progressive crushing mode to progressive crushing followed by vertical cracking. Analytical expressions were formulated to find the average collapse load of the conical shells and domes. The results showed good agreement.with the experiments. ACKNOWLEDGEMENT

Financial support for carrying out this work was received from Aeronautical Research Development Board. REFERENCES

1.

A A Ezra and R J Fay, In Dynamic Response of Structures, G Herrmann and N Perrone (eds.), ISBN, (1971) 225. 2. W Johnson and S R Reid, Metallic Energy Dissipating Systems, Mech. Engg. Publications Ltd, London (1978). 3. P H Thornton an P J Edwards, J. of Composite Materials, 16, (1982) 521. 4. D Hull, In: Structural Crashworthiness, N Jones and T Weirzbicki (eds.), Butterworths, London, (1983) 118. 5. G L Farley and R M Jones, Journal of Composite Materials, 25, (1991) 1314. 6. J N Price and D Hull, Composite Science and Technology, 28, (1987) 211. 7. N K Gupta, R Velmurugan and S K Gupta, J. of Composite Materials, 31, (1997) 1262. 8. N K Gupta and R Velmurugan, J. of Composite Materials, 33, (1999) 567. 9. Mamalis, A.G., Manolakos, D.E., Demosthenous, G.A. and Ioannidis, M.B., Thin 9Walled Structures, 24, (1996) 335. 10. A G Mamalis, D E Manolakos, G A Demosthenous and M B Loannidis, Int. J. of Impact Engg., 19, (1997) 477.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

419

The Influence o f Residual Stresses in the Vicinity o f Circumferential WeldInduced Imperfections on the Buckling o f Silos and Tanks M. Pircher and R. Q. Bridge School of Civic Engineering and Environment, University of Western Sydney, Nepean, PO Box 10, 2747 Kingswood, NSW, Australia

The load carrying behaviour of cylindrical thin-walled shell structures under axial load is strongly dependent on imperfections invariably caused by manufacturing processes. Axisymmetric imperfections have been recognised to result in particularly severe reductions in strength. Imperfections in the vicinity of circumferential welds in steel silos and tanks fall into this category and therefore deserve special attention. Finite element models were used to analyse imperfect cylindrical shells and special care was taken to model the weld-induced circumferential imperfection. The geometry was calibrated against data gained from measuring such imperfections on existing silos and residual stresses were taken into account. Interaction between neighbouring weld imperfections and the role of the strake height in this interaction was investigated. Residual stresses were found to increase the buckling load. The extent of this increase proved to be dependent on various parameters such as weld geometry, weld depth and strake length. The buckling and post-buckling behaviour of imperfect structures either with or without residual stresses was studied in close detail. This study enabled the mechanism leading to the strengthening effect of the residual stresses to be clearly identified. 1. INTRODUCTION The stability of circular cylindrical shells has long been known to be highly sensitive to deviations from the assumed design parameters. Until the 1970s, axisymmetric imperfections had been dismissed as less important from an engineering point of view as they were thought to be unrepresentative of real structures (Koiter [1 ], Arbocz [2]). This idea had its origin in the aerospace industry where measurements of fabricated cylinders showed that the axisymmetric component of these specimens was actually small (Arbocz and Babcock [3], Singer et al. [4]). However, when circular cylindrical shell structures such as silos or tanks are built, axisymetric imperfections do occur and have been found to be of an extremely detrimental nature. Techniques commonly used when erecting silos or tanks explain the presence of this type of imperfection. Rolled steel plates are usually formed into a series of individual strakes and joined together by circumferential welds (Figure 1). At each circumferential joint a depression occurs. These deformations are partly caused by the rolling process, and partly by

420 the shrinkage of the heated area in the vicinity of the weld. In addition to these geometric deviations, residual stresses of considerable magnitude can be found near the weld.

Rolling

Welding

Figure 1. Erection of a Circular Silo or Tank The shape of these depressions has been shown to have great influence on the buckling resistance under axial load (Pireher & Bridge [5]) and a conservative shape function which is based on measurements (Rotter [6]) was chosen for the work for this paper. On the other hand, residual stresses due to welding of thin plates and shells have been measured (Dwight & Moxham [7]) and again simplified formulations have been utilised in analytical derivations (Ravn-Jensen & Tvergaard [8]). Three papers are known to have looked at circumferential welds in circular cylindrical shell structures including the effects of residual stresses (Bornscheuer et al. [9], Rotter [6], Pircher & Bridge [10]). These papers reach different conclusions. Bornscheuer et al. report a decrease of the buckling strength of their model by up to 10% while Rotter [8] concludes that "Circumferential residual stresses in the welded joint, developed by shrinkage of the weld, appear to increase the buckling strength...". The investigation by Pircher & Bridge [ 10] confirmed Rotter's findings. This paper explains the role of residual stresses for axially compressed cylinders using a procedure by Esslinger [ 11 ] and confirms the findings by Rotter [6]. 2. BUCKLING MECHANISM OF THIN-WALLED CYLINDERS 2.1. Buckling of Structural Members Classical buckling theory works well to predict the stability of perfect structural members such as flat plates, columns and cylinders. Taking all the usual simplifications of classical theory into account the following can be observed: all three of these structural members show a linear response until a critical load is reached (see solid line in Figure 2). However, following the load-displacement path beyond this point, the differences between the three members become apparent:

Idealised columns cannot develop transverse stresses to restrain out-of-plane displacements. The response ager buckling is therefore undefined. Imperfections cause increased lateral displacements and the maximum load is reached asymptotically. The load capacity of fiat plates increases even after the critical load has been reached as transverse tensile stresses work against the growth of out-of-plane displacements and provide

421 for a stable post-buckling behaviour. Again, the post-buckling response of the perfect plate is approached asymptotically by an imperfect plate, smoothing the kink in the load-deflection curve.

The post-buckling response of a perfect cylinder is significantly different. A steep descent in the stress-strain characteristic can be observed after the critical load has been reached signifying a highly unstable post-buckling behaviour. Imperfections reduce the classical buckling strength dramatically as the narrow gap between the pre-buckling and the postbuckling segment of the response curve is bridged at a much lower load level. This bridging is made possible by the fact that unstable states of post-buckling equilibrium exist for cylinders with loads smaller than the theoretical buckling load. Increasing the load during the pre-buckling phase also increases existing small imperfections until such unstable equilibrium becomes possible and snap-through occurs.

A

I- "--Perfect Member / - - Geometricallyimperfect ~ ~~ 1. / Member ~ FlatPlate

|

/~'~!

i Snap " thr~

05

i

1.0

2.0

Member strain I classicalbuckling strain Figure 2. Post-buckling behaviour of elastic fiat plates, columns and cylinders These unstable states of equilibrium in the initial post-buckling of the cylindrical shells are the key to understanding the imperfection sensitivity of these structures in general and explaining the strengthening effect of weld-induced residual stresses in particular. Esslinger [11] offers an elegant model based on Dormel's [12] equations explaining the imperfection sensitivity of thin-walled cylinders, and pointing out the major difference between the buckling behaviour of columns, plates and cylinders. Esslinger's paper [11 ] clearly shows the importance of circumferential membrane stresses induced by axial load and their stabilising function before buckling. However, as soon as the bifurcation load is reached and buckling deformations occur, this effect is reversed and circumferential forces become detrimental, thus leading to a post-buckling minimum below the bifurcation load. 2.2. Numerical Results for Circumferential Weld Imperfection

Finite element analyses have shown that weld-induced residual stresses generally increase the buckling strength of thin-walled circular cylindrical shell structures in comparison to geometrically equal models that neglect these stresses [5, 6, 10]. Circumferential membrane

422 stresses are by far the most influential component of the stress field induced during the welding process and the development of these stresses along one meridian of a cylinder is shown in Figure 3.

,.- 1.2 I o "~r 1 i x ! w e l dI .

1 I induced stresses m 0.8 V' I I-"0 0.6 " - - ~ Stresses at buckling o ~ 9 0.4 ill I 1

0.2

~

o

"~ .0.4 I .0.6 ..~ oE

0

~

\ Stresses at buckling " (initially stress free) 25

50

75

100

xlt

Figure 3. Circumferential Membrane stresses As axial load is applied on the initially stress-free model, circumferential compressive stresses develop near the centre of the weld thus creating an area of two-dimensional compression. When the applied axial load approaches bifurcation load, infinitesimally small buckles start to form which subsequently cause buckling of the structure. Circumferential compressive stresses accelerate the development of these pre-buckling deformations as is shown in Figure 4 (a). In a welded cylinder circumferential residual stresses reach yield-level at the centre of the weld and even at the point of buckling under axial stress this area is still under considerable tension (Figure 3). Compressive residual stresses further away from the weld typically range from 0.2 to 0.4 of the yield stress and are further increased by axial compression. The stabilising effect of the tension stresses near the weld is illustrated in Figure 4 (b). With the increase in axial load the tension stresses decrease and the compressive stresses further away from the weld are further increased until a point is reached where the structure buckles. Depending on various parameters such as wall-thickness or shape of the imperfection a gain in buckling strength of up to 10% compared to the initially stress-free models was observed in numerical parameter studies performed over the last few years [5, 6, 10, 13].

423 <

(a) compressive circumferential stresses

rcumferential tension stresses

Figure 4. Effect of circumferential residual stresses 3. CONCLUSIONS During the construction of silos and tanks, circumferential imperfections are commonly introduced into the structure by various manufacturing processes including welding. Due to thermal shrinkage, residual stresses develop in combination with the geometric imperfections. These residual stresses have been found to increase the buckling strength of these structures and a gain of up to 10% in buckling strength compared to models with purely geometric imperfections has been reported. This paper offers an explanation of this rather unexpected behaviour. REFERENCES

1. Koiter, W. T. The Effect of Axisymmetric Imperfections on the Buckling of Cylindrical Shells under Axial Compression, Proc. Koninklijke Nederlandse Akademie van Wetenschappen, 66(B), pp. 265-279, 1963 2. Arbocz, J. The Effect of Initial Imperfections on Shell Stability, Thin Shell Structures, Y.C. Fung and E.E, Sechler, Eds., Prentice Hall, Englewood Cliffs, N.J., pp. 205-246, 1974 3. Arbocz, J. & Babcock, C.D. The Effect of General Imperfections on the Buckling of Cylindrical Shells, J. Appl. Mech., v.36, pp. 28-38, 1969 4. Singer, J., Muggeridge, D.B. & Babcock, C.D. Buckling of Imperfect Stiffened Cylindrical Shells under Axial Compression, A/AA JNL., v.9, pp. 68-75, 1971 5. Pircher, M. and Bridge, R.Q. The Influence of Weld-Lnduced Localised Imperfections on the Buckling of Cylindrical Thin-Walled Shells, Proc. 16~ Australasian Conference on the Mechanics of Structures and Materials, Sydney, 1999 6. Rotter, J.M. Buckling and Collapse in Internally Pressurised Axially Compressed Silo Cylinders with Measured Axisymmetric Imperfections: Imperfections, Residual Stresses and Local Collapse, Proc. Imperfections in Metal Silos Workshop, Lyon, France, 1996, pp. 119-139 7. Dwight, J.B. & Moxham, K.E. Welded steel plates in compression, Struct. Engineer, v. 47, pp 49-66, 1969

424 8. Ravn-Jensen, K. & Tvergaard, V. Effect of Residual Stresses on Plastic Buckling of Cylindrical Shell Structures, lnt. J. Solids Struct., v. 26, n. 9-10, pp. 993-1004 9. Bomscheuer, F.W., Hafner & L., Ramm, E. Zur Stabilit~it eines Kreiszylinders mit einer Rundschweissnaht unter Axialbelastung, Der Stahlbau, v. 52, pp. 313-318, 1983 10. Pircher M., Bridge R. The Influence of Weld-Induced Residual Stresses on the Buckling of Cylindrical Thin-Walled Shells, Proc. Thin Walled Structures Conf., Singapore, 1998, pp 671-678 11. Esslinger M. Eine Eklarung des Beulmechanismus von dfirmwandigen Kreiszylinderschalen, Der Stahlbau, vl 2, pp. 366-371, 1967 12. Donnell L.H. A new theory for the buckling of thin cylinders under axial compression and bending, Trans ASME, vol. 56, 1934, pp 795-806 13. Pircher M., Bridge R. Q. The Influence of Circumferential Weld-induced Imperfections on the Post-Buckling of Silos and Tanks, Proc. Advances in Steel Sructures 1CASS 99, Hong Kong, 1999

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

425

I m p r o v e d Marshall strut element to predict the ultimate strength o f braced tubular steel offshore structures K. Srirengana and P.W. Marshall b aStruetural Engineer, Zenteeh, Inc. 8582 Katy Freeway, Suite 205, Houston, TX 77024, USA bMoonshine Hill Proprietary 5100 San Felipe #107 E, Houston, TX 77056, USA

The Marshall B-strut element with 2 nd generation improvements has been implemented in the StruCAD*3D* to predict the ultimate strength of braced tubular steel offshore structures. The Marshall B-strut is a phenomenologieal model to predict the inelastic behavior including compressive buckling, post-buckling, tensile straightening and tensile yield of a brace when subjected to extreme monotonic and cyclic loads. Based on Sherman's extensive series of tests, improvements made to the original Marshall strut model include the addition of compressive yield plateau prior to the post-buckling regime in the action-deformation envelope and the degradation of the ultimate compressive capacity due to cyclic loading. T o evaluate the performance of the Marshall B-strut element, psuedo-static pushover analysis of an X-braced frame subjected to cyclic overload was considered. The predicted overall frame response matched well with the experimental results. The decay in the load-deflection curves with inelastic cycles and the progressive failure sequence of the braces was conservatively predicted with respect to the experimental results.

I. INTRODUCTION Pushover analysis is the recommended practice [1 ] by the American Petroleum Institute (API RP 2A) to predict the ultimate strength of braced tubular steel offshore structures. Traditionally in static pushover analysis, member buckling is predicted from nonlinear geometry considerations and member yielding is predicted using plasticity models [2]. However, this traditional approach does not account for local buckling, reduction in the member capacity and the member failure. This leads to the over-prediction of the stiffness and the strength of the member especially for cyclic loading. Phenomenological models which are primarily based on statistical correlation of the experimental data; are representative of the overall behavior of the member and hence give a closer and conservative prediction.

' - 3-D Structural Analysis and Design Sottware of Zentech, Inc.

426 Several phenomological models [3-5] based on the experimental works [5-8] have been proposed to predict the strut behavior. The Marshall strut [3,4] is a phenomenological model to predict the strut behavior and its original version is referred to as the Marshall A-strut. Improvements to the Marshall A-strut have been made based on the tubular beam-column tests conducted by Sherman [6-8]. The tests involve four strut series of slenderness ratios 20 and 50, compactness ratios ranging from 35 through 95 and ductility ratios ranging from 2 through 8. The improved version of the strut model is referred to as the Marshall B-strut. The Marshall B-strut has been implemented in the StruCAD*3D. The goal of this paper is to present and evaluate the performance of the Marshall B-strut.

2. MARSHALL B-STRUT ELEMENT The Marshall B-strut element or simply the B-strut is essentially a one dimensional inelastic truss element. The inelastic axial response of the B-strut is governed by the actiondeformation envelope (or the backbone curve) shown in Figure l(a). The backbone curve is comprised of nine regimes namely linear tensile AG, tensile yielding GG', linear compressive AB, compressive yielding BC, first post-buckling CD, second post-buckling DE, zerocapacity 'beyond E', first tensile straightening EF and second tensile straightening FG. The direction of the loading and the unloading paths along the backbone curve and inside it is indicated in Figure l(a). Figure l(b) shows the maximum stretch and flexural energies that can be accumulated by traversing along the backbone curve. The backbone curve for the B-strut can be defined by nine parameters including the modulus of elasticity E, strain hardening modulus Eh, tensile yield stress Oy, ultimate axial load Pu, yield plateau length Ap, post-buckling slopes of the bilinear load decay CI and C2, stress at the slope change in compressive load decay Cn and the stress at the slope change in tensile straightening Ob. These parameters are functions of the material stress-strain curve, strut diameter D, thickness t, length L and the amount of applied lateral load Q. Only the parameters that are associated with the improvements to the A-strut are discussed herein. Ax;ol Load

~ Axial Load

~

i

//i

/," i

-"~.~ #

#

#

-

K:,--:tT.

l

C B1 B

~

,

J

Axial Deform~ion

"Pn 'Pu

(a) Loading and unloading paths

stretch energy

'ormatlon

energy

(b) Accumulated stretch and flexural energies

Figure 1 The backbone curve for the Marshall B-strut shows the loading and unloading paths and the accumulated stretch and flexural energies.

427

2.1 Ultimate axial load Pu The ultimate axial load Pu is a function of the lateral load Q and the interaction [6] between Pu and Q is given by 1 Q 1-Pu/PE Qp

=cos(2 Pu) Pcr

where PE is the Euler buckling load for a fixed-fixed column

Qp is the lateral load corresponding to a mechanism failure for a zero axial load condition Per is the critical column load for a centrally loaded member and is given by the AISI linear column equation [8].

2.2 Compressive yield plateau The compressive yield plateau represents the pronounced compressive shortening at the ultimate capacity for short thick-walled tubular braces, and is beneficial to load sharing and ductility in the overall structure. From the test data [8], the axial displacement Ap corresponding to the end of the yield plateau is given by

Ap/Au =[833/(D/t)l'7](1.5-O.02 L/D) A p / A u =1.8(1.5-0.02 L/D)

forD~t>35

forDIt <35

where Au is the axial displacement corresponding to the ultimate axial load. Due to the unavailability of data for D/t < 35, it is assumed that the length of the yield plateau is not affected by local buckling and thus is reflective of only the general column instability.

2.3 Reduction in compressive capacity The reduction in the compressive capacity with successive strut cycles in the B-strut is another major improvement. A strut cycle is complete when the axial response reaches the tensile yield after it has reached the compressive yield plateau or the first post-buckling regime of the backbone curve. The reduction is applied only to the ultimate capacity and at worst the ultimate capacity is reduced to the load P, (See Figure l(a)). Based on test data [8], the peak compressive capacity Pi at the completion of the i th strut cycle is given by

Pi = Pu R2 (/-1) >--Pn where R 2 = (1.2 - 0.005 L / D) //2 0.2

_~ 1.0

for D / t <_50

R 2 = [(1.2 - 0.005 L / D) //20.2 ](1.3 - 0.006 D / t) _<1.0 for D / t > 50 and ~t is the ductility (which is the ratio of the peak displacement to the displacement at Pu).

428 2.4 Strut Failure

An energy based failure criterion as given below is used to predict the failure of the B-strut.

E* =[(Es/Ecs)2 +(El IEcf)2] where Es and Ef are the accumulated stretch and flexural energies respectively

Ecs and Ecf are the critical stretch and the critical flexural energies respectively At strut failure, the stiffriess and the capacity of the strut is reduced to essentially zero over a few subsequent load steps. Due to the lack of test data on the accumulated stretch energy, the equation for the critical stretch energy in the A-strut algorithm [3] is used herein. The critical flexural energy is indicative of the failures due to local buckling, low cycle fatigue and the rupture at the plastic hinges. From the test data [8], the critical flexural energy Ecf is approximated as Ecf = E m (1.2 + C/,/da ) where F-,mis the monotonic flexural energy and is given by E m =(12.5-0.1D/t)(1.3-O.O15 L / D ) Pc2r L/(2r,.DtE) C = 60(0.8 +O.O1L/ D) f o r D ~ t < 5 0 C =(llO-D/t)(O.8+O.O1L/D) f o r D / t > 50 and ~a is the average ductility per strut cycle. In contrast to the A-strut algorithm, the critical flexural energy in the B-strut algorithm varies with strut cycles since it is a function of the average ductility per strut cycle.

3. TEST CASE AND EVALUATION Psuedo-static pushover analysis of the X-frame [5] was considered to evaluate the performance of the Marshall B-strut in predicting the reduction in the strut capacity with strut cycles. The X-braces of the lower and the upper bays of the frame were modeled with the Marshall B-strut elements. All the other members were modeled with elastic beam elements. For the full evaluation of the structural collapse mechanisms, inelastic portal beam-column modeling of the legs would be required. A cyclic displacement profile [5] was imposed at the top right corner of the frame. Geometrically non-linear analysis was performed. Figure 2 shows the maximum frame load versus push-pull cycles for the X-frame. The StruCAD*3D predictions match well with the experimental results [5] and the INTRA predictions [9] which also uses an enhanced version of the Marshall A-strut model. Figure 3 shows the variation of the maximum load with push cycles for the lower and the upper braces. The ultimate capacity (which is the maximum axial load at the end of the 2"d

429

0

.

.

40 20

~

.

.

.

.

.

.

.

.

.

.

.

f

f

.d

l i

"

~ -40

| - ,

~--...~.,

- 6 0

i

0

1

_

. . . . . _.

!

!

i'

!

!

2

3

4

5

6

'

!

,~ '

"

w .

i

7

!

8

i

|

!

!

!

!

9 10 11 12 13 14 15 16

push and pull cycles Figure 2 Comparison of the StruCAD*3D predictions of the overall response of the X-frame with the experimental results and the INTRA predictions. load cycle) for both the upper and lower braces as predicted by the StruCAD*3D is more conservative than the INTRA predictions. This is due to the difference in the expressions used for the ultimate axial load calculations. However, the general pattern of the reduction in the compressive capacity for the upper braces matches well with the INTRA predictions. For the lower braces, the INTRA predictions do not show any reduction in the ultimate capacity during load cycles 10 through 15, even though the magnitude of the push and pull cycles are large enough to cycle the braces into the post-buckling regime. However, both the analyses predict the failure of the lower braces at about the 16th load cycle. Even though the experiments [5] show severe local buckling and tearing of the upper braces by the end of the 9t~ load cycle, the reduction in the measured ultimate capacity in Figure 3(a) is not reflective of that. The comparisons of the total energy dissipation with push and pull cycles for the analyses and the experiment are shown in Figure 4. For both the upper and the lower braces, the StruCAD*3D predictions are close to the INTRA predictions. Both the analytical predictions for the upper braces are considerably lower than the experimental results. However, those for the lower braces are much closer to the experimental results. o~

0

r,#'J

I::a,,

9m -10

~_, -30 r . r

~ -35

-10

,,m -20 t'll o

-15

-30

0

. . . .

I 2 3 4 5 6 7 8 9 1011 12 13 141516

push ~ ~ (a) upper brace

s.o..D/ /

'

-4o 5o

~ -6o 0

\'t -'t\

-7

experimen~.~ --

0

!

!

'!

!

1 2 3 4

!

|

|

5 6 7

|

l

! ......

I|

|

|

!

m

8 9 10111213141516

push cycles (b) lower brace

Figure 3 Comparison of the StruCAD*3D predictions of the maximum axial load for the lower and the upper braces with the experimental results and the INTRA predictions.

430

/ /

250 "7' 200

..~

1"

1

o 1t~)

Ii

..........

& 20o

~

.....

!50

expe

o

"N

50

"-"

9

50

0

0

0

0

1 2

3

4

5 6

7

8

9 10 !1 12 13 14 15 16

.,-

I

-

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16

push cycles

push cycles

a) upper braces

(b) lower braces

Figure 4 Comparison of the StruCAD*3D predictions of the total energy dissipation for the lower and the upper braces with the experimental results and the INTRA predictions. 4. CONCLUSIONS The prediction of the stiffness, the strength and the energy dissipation by the Marshall B-strut for strut behavior is conservative with respect to the experimental results. 5. ACKNOWLEDGEMENTS The support of the president Maini, R., and the executive vice-president Guntur, R., of Zenteeh, Inc. for the publication of this paper is gratefully acknowledged. REFERENCES 1. American Petroleum Institute, Recommended Practice for Planning, Designing, and Constructing Fixed Offshore Platforms, 20th ed., July 1993. 2. Chen, P. F. and Powell, G.H., Generalized Plastic Hinge Concepts for 3-D Beam-Column Elements, Report No. UCB/EERC-82/20, UC, Berkeley, California, November 1982. 3. Marshall, P.W., Gates, W.E., and Anagnostopoulus, S., Inelastic Dynamic Analysis of Tubular Offshore Structures, OTC 2908, Houston, Texas, May 1977. 4. Marshall, P.W., Design Considerations for Offshore Structures Having Nonlinear Response to Earthquakes, ASCE Preprint 3302, October 1978. 5. Zayas, V.A., Mahin, S.A., Popov, E.P., Cyclic Inelastic Behavior of Steel Offshore Structures, Report No. UCB/EERC-80/27, UC, Berkeley, California, August 1980. 6. Sherman, D.R., Ultimate Capacity of Tubular Members, Report CE-15, University of Wisconsin, Report to Shell Oil Company, August 1975. 7. Sherman, D. R., Post Local Buckling behavior of Tubular Strut Type Beam-Columns: An Experimental Study, Univ. Wisc., Milwaukee, Report to Shell Oil Company, June 1980. 8. Sherman, D.R., Interpretive Discussion of Tubular Beam-Column Test Data, University of Wisconsin, Milwaukee, Report to Shell Oil Company, 1980. 9. ISEC, Inc., INTRA Enhancements and Analytical Correlation of API Test Frames, Final Report to Shell Oil Company, June 1981.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

431

The aseismatic behaviour o f high strength concrete filled steel tube Zhan Wang a and Yonghui Zhen b aShantou University Daxue Road, Shantou City, Guangdong Province, P.R.China bHarbin University of Civil Engineering and Architecture Haihe Road, Harbin City, Heilongjiang Province, P.R.China

In this paper, the force-displacement hysteretic loops of high strength concrete filled steel tubular members under compression and bending are calculated using finite element method with a steel constitutive model which is suitable for multiaxial cyclic loading and for concrete a modified bounding surface model for multiaxial cyclic compression. Six new tests were carried out on steel tubes filled with concrete with a cube strength of 77N/mm 2. The theoretical lateral force-displacement hysteretic loops are compared with these tests and tests by other experimenters and the results are discussed.

I.INTRODUCTION The use of high strength concrete (HSC) in structures is growing above ever increasing rate. The weakness of HSC is its brittleness. Its failure, especially under complex stress states, is controlled by this brittleness ,and its reliability ,when used in structures, is lowered. High strength concrete filled steel taabe (HCFST) have high strength and high ductility; this is the best way of the applying HSC to practice. On the basis of the research of the HCFST forcedisplacement hysteretic loop under compression and bending we can model the hysteretic loop and thus analyze the HCFST elasto-plastic earthquake reaction by using a shear model for the building. The behavior of HCFST member under compression and bending, the aseismatic behavior and the determination of the force-displacement hysteretic loop model require theoretical calculation of force-displacement hysteretic loop. This paper makes use of finite element method to obtain the hysteretic behavior of HCFST under compression and bending and also describes six tests that were carried out. This research not only has an important practical value on HCFST aseismatic design, but also indicates the need for further research, rut21

2.THE CALCULATION OF FORCE-DISPLACEMENT HYSTERETIC LOOP 2.1. Calculating hypothesises and test method HCFST under compression and bending is belonged to three dimension problem. Three dimension finite element should be used to resolve it. In this paper author resort to three dimension twenty nodes iso-parametric element which has so high precision in each element that it has being widely used on resolving three dimension problems. There are some hypothesises on HCFST analysis by using finite element method, t31 a) Plane cross section normal to the deformed member axis

432 b) c)

Constitutive relationship of steel is two linear random strengthen model. (see Figure 1) Constitutive relationship of core concrete is modified bounding surface model. (see Figure 2) static water

O',fo~fy ~

f~

0"3

pressure

~=~

Critical surface

Projection of the critical surface on the ~ plane Figure I Figure 2 There are two loading ways in calculation. One is loading with force, which is used for vertical load. The other is loading with displacement, which is used for horizontal load. It mainly accords to the following. a) If loading with force, we will have great trouble near peak value and find no ways to calculate decent part. b) With respect to horizontal section hypothesis, we know the resultant force of imposed load and that section is still plane, but we don't know how the imposed forces distributed. In this case, lateral load P and axial load N are not able to turned into equivalent joint force on member joint. It is hardly carried out even loading with force in elastic part on this question. 2.2. Calculating method The target of discussed in this paper can be regarded as a part of a frame column between inflection point and fixed end with lateral deflection, which stands for real work conditions of the column. The member under compression and bending is only discussed here, that is, a constant axial force is applied on a cantilever column first, lateral force is increased continuously later. In this case, we research the relationship between lateral force and lateral displacement. There are two methods on calculating force-displacement hysteretic loop, i.e model column and data analysis method. As there is bigger error in model column method, data analysis is used here.

3.Test 3.1. General features of test In order to provide further research on the behavior of HCFST under compression and bending and check the accuracy of theory, six experiments producing a force-displacement hysteretic loop were carried out. The columns were circular of two diameters and two wall thickness and two lengths(see Table 1).Two axial loads(200 KN and 300 KN) were applied to pairs of similar columns.. Loading sensors, displacement sensors and strain gauges were connected through the Isolated Measurement Pods (IMP) to the computer. Test data was gathered automatically. The interval time was 1500 milliseconds with continuous gathering

433 control. The P -~5 hysteretic loops are drawn from the test data obtained.

Table I

features of test specimens loaded cyclically

Number of specimen

D x t x L (mm)

f y ( N / ram)

f,~ (Nlmm)

~:['1

N(kN)

Z1-20 Z1-30

108x4.5 x 1250 108x4.5 x 1250

312.4 312.4

77.1 77.1

1.07 1.07

200 300

Z2-20

114x6.0 x 1250

319.3

77.1

1.04

200

Z2-30

114x6.0 x 1250

319.3

77.1

1.04

300

Z3-20 Z3-30

114• • 1450 114x6.0 x 1450

319.3 31.9.3

77.1 77.1

1.36 1.36

200 300

Note[*]: +

= ~r~fck t'l"

, r~ ---radius of the concrete core

3.2. Experiment result Figure 3 is the comparison between experiment (on the left) and theoretical result (on the right). The two are basically identical. The difference is mainly because of the following, a) The stiffness of theoretical curve is higher because only bending deflection is considered on theoretical calculation. b) The stiffness of test curve is lower because of crack in the concrete and frictional force between each link of loading equipment. In addition to this, experiment results of concrete filled seamless steel tube from others [4ItS]are gathered here. Figure 4 and Figure 5 show the comparisons between experiment (on the left) and theoretical result (on the right). The parameters of the member in figure 4 were: 133 x 5.1 x 1260 mm seamless steel tube, the yield strength of steel(fy) was 347.7 N / m m 2 , the cube strength of concrete(fou) was 70.2 N/ram2 . The axial loads were 385 KN . The parameters of the member in Figure 5 were: O 108 x 5 x 1100 mm, the yield strength of steel (fy) was 327.8 N~ mm2 , the cube strength of concrete(ft,) was 33.8 N~ mm2 . The axial loads were 200 KN and 270 KN respectively. 51]

~:3e

...., 9 3D [1. 2D

10 O

1O fl -ln

JS

.10 i

-21] -3] -lO

,,~

-,;,

,;

'--,;,

,;',

b

(,.,,,)

.o

..'1o

(a)Specimen Z 1-20

(ram)

434

~-~

~

x~'tO

~"30

N

tO

'~

0

o lO

-at)

,,

///~,

//~;

-4O

~

io ;o |

~(mm )

(ram)

(b)Specimen Z1-30

c,~. 40

,

-4D

-40

,

~

,,

O

20

r

|

el) W ~( mill )

~

...-

~ - ~ " : & " - o ~

~

to

so

(c)Specimen Z2-20

0

-I)

~ " -

6[mm

~ -

:& . . . .

~ .... i .... ~ ....

6 (men)

)

(d)Specimen Z2-30 r~

2o

~20

lo

Z -2O .~0 40 -6O /',( m : )

(e)Specimen Z3-20

8 (m~

435

//

/,,'./"

,

.~o ~

.,or

~'~-

.ao

~,~

....................................

-80 ~

40

-~O

[]

~n

lO

80

80

~,~,

r6.'////

,~-7

1130

8 (i..i) (f)Specimen Z3-30 Figure 3. The comparison between theoretical calculation and the test result (aim)

r

,,-,,- B O , -

u "

~',o!I! .,olJ

I' IJJmrr~ I ~/l'~/b',,l~,/v' I

z

'

ol ! I : _l_1

I ', I ; I

",,I i- -!- ~~.,~A

~

-2ol !--'Ir i _,,,I , t ' ~ ~ " I ..ml~-li +I 2 I ..ml I I I I I . . . .I. . . I1 I'-i. . . . . . . -U:i-IO

-GO-~0-20

0

~0

I I ! I

~0 a O 8 0 t 0 [ ~.; ( m m *]

(ram)

Specimen HSC3-2 Figure 4. The comparison between theoretical calculation and the test result in reference 4

tO I1. 2 0 0

,n.

,"

, " / . / J

-m

+,

~,

0

,.I

/ .~/, , , /

" //

"i"_,,,J

;,

;,

+~

"~ .3,

,

1

/I/XJ

,m-+0

.

+,

~

fit.M)

I

3.

,~

5

(ram)

.

(a)riysteretic loop

. . . . .

...... I

~- ~*t

:

....% - ' 1

/

,/

,//

; /,"/S../ 4 0 1

-

-

.

,

~

.

t

-

-

_ n

.

t" m m )

.

~ (mull)

]0

(b)Hysteretic loop Figure 5. The comparison between theoretical calculation and the test result in reference 5

436 4.THE FORCE-DISPLACEMENT HYSTERETIC LOOP CHARACTERISTIC We can find some characteristics of force-displacement hysteretic loop of high strength concrete filled steel tubular members from the theoretical analysis and experiment research result. a) The shape of hysteretic loop is closed to that of steel member under the condition of without local buckle. And it is also analogous to the loop of general concrete filled steel tube member. b) No matter how the parameters change, the hysteretic loop has great plumpness and no pinched or reduced phenomenon appear.

5.CONCLUSION The calculation method in this paper has its new characteristics on how to select constitutive relationship and construct the model of finite element. On the basis of theoretical analysis and experiment study, the characteristic of force-displacement hysteretic loop of HCFST under compression and bending are discussed. From above, the following should be further studied: a) The basic property of polygon HCFST should be studied by making use of programme in this paper. b) The property of the member of eccentric compression should be studied by making use of calculating method in this paper. c) The lateral force resisting property of short column of HCFST should be studied considering of shear deflection.

REFERENCE 1.S. Zhong, Concrete filled steel tubular structures. Heilongjiang science and technology press, 1994. 2.L. Han, Mechanics of concrete filled steel tubular. Dalian science and engineering university press, 1996. 3.Y. Zhen, The hysteretic behavior studies of high strength concrete filled steel tububular members subjected to compression and bending. Master thesis of HUAE, 1998. 4.W. Yah, Theoretical analysis and experimental research for the hysteretic behaviors of high strength concrete filled steel tubular beam-columns. Master thesis of HUAE, 1998. 5.Y. Tu, The hystersis behavior studies of concrete filled steel tubular membersw subjected to compression and bending. Doctor thesis of HUAE, 1994.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

437

Stub-column failure test o f welded b o x steel section under axial compressive loading Y. C. Zhang~ J.J. Zhang~ W.Y. Zhang ~ D.S. Lib aBox 703, Harbin University of Civil Engineering and Architecture, 202 Haihe Rd. Harbin, 150090, P. R. China bHarbin Boiler Company Ltd. 17 Daqing Rd. Harbin, 150040, P. R. China Several stub-column failure tests of Welded Box Steel Section (WBS) under axial compressive loading were conducted, in order to study the new failure type of WBS caused by the extra tension in the comer weld connection due to the post-buckling of plate assembly, which was revealed by the recent engineering accidents. The agreement between the nonlinear analyses and experimental results is seen to be satisfactory. The influences of the width thickness ratio, size of fillet weld and the form of the box cross section to the weld-failure ductility ratio of WBS were observed by numerical study based on the nonlinear analyses. The formula of determining the minimum size of fillet weld according to the ductility demanded in the static loading is proposed.

1. INTRODUCTION The welded built-up section members are widely used in the steel structures. Up till now, the connection welds of the welded built-up steel section colurrms have been designed to resist the shear forces which are caused by the external forces or develop in the global buckled columns. But the engineering accidents recently revealed that a new failure type of the Welded Box Section (WBS) caused by the extra tension in the comer fillet welds due to the post-buckling of plate assembly was observed. During the 1985 Mexico City earthquake, one of three identical 22-storey steel flame buildings of the Pino Suarez complex collapsed, when another one was close to collapse and the other had severe structural damages [1]. These flames were constructed of fillet welded box columns and specially fabricated open-Web girders. The column section sizes are as follows: the overall dimensions are 600x500mm; the thickness of plates varies from 7.9mm to 31.Smm. Several locally buckled columns have been observed in the frame close to collapse. Although some columns located on the fourth storey were compact sections, they also inelastically buckled locally near the end of the columns. Buckling of these plates results in

438 the failure of comer fillet welds, and the column plates were no longer connected to each other. Another example is the collapse of two welded box steel piers caused by the 1995 Kobe Earthquake [2], when local buckling occurred in many similar piers. The steel piers cracked at comer welds, and then collapsed a few hours after the earthquake in a manner rather like a peeling banana. The overall sizes of the box piers are about 2700x2700mm, and the reduced width-thickness ratios of the stiffened plates are all about 20. A WBS column of a boiler-supporting frame in China cracked at a comer weld, when the whole construction almost finished in the September 27, 1996. The crack abruptly propagated to 1.4m long along the connected weld in the end of column, after a sharp sound. Local buckling of plates near the crack was observed. The overall sizes of the box column was 680x600mm, the thickness of the plates was 20ram. It can be seen as the new failure type of WBS in the static loading case. Although there are many cyclic loading tests and studies of WBS column or pier [3~5], only few papers stress the importance of the behaviour of the comer welds. The purpose of this paper is to try to introduce the importance of comer welds for the stability and ductility of WBS columns, using the stub-column failure tests and the nonlinear analysis results. The influence of the width-thickness ratio, size of fillet weld and the form of the box cross section (square or rectangular) to the weld-failure ductility ratio of WBS was also observed.

2. OUTLINE OF EXPERIMENTS 2.1. Test specimen In order to investigate the inelastic behaviour of the WBS columns, 4 compact section specimens were designed and fabricated according to the dimension limits of the stub-column test procedure [6]. The dimensions of the specimens are shown in Figure 1 and Table 1, where L is the length of the specimens; hI is the leg size of the comer fillet welds; S and R express the square and rectangular box section respectively. The reason for using small-sized specimens is that the maximum compression force is limited by the loading capacity of the machine. The steel employed is Q235 which has an average measured yield stress fy, Young's

Table 1 Dimensions of the WBS specimens Specimen b Name I/lm S-1 200 S-2 200 R-1 200 R-2 200

D

t

Innl

mill

200 200 100 100

10 10 10 10 .

b/t 20 20 20 20

L

h~

rnlTl

mill

600 600 400 400

4.5 4.5 3.2 3.4

439

-T

1

F

!

n

/

I

6 .-X

I

/////////////////

I

I.

b

)/

1-specimen; 2-dial gauges; 3-base; 4-testing machine cross head; 5-electrical gauges. Figure 2. Test setup

l

1-1 Figure 1. Configuration of specimen

modulus E and Poisson's ratio ~ of 275N/mm 2, 2.2 lx105N/mm2 and 0.299 respectively[7]. 2.2. Test setup and procedure The test setup is shown in Figure 2. Mechanical dial gauges and electrical resistance gauges were used for alignment and testing. In order to measure the strain of the comer welds, the electrical gauges perpendicular each other were also installed in the throat face of the fillet welds along the length of the specimens. After the alignment using the electrical gauges at the four plates of the specimen, the stage axial compressive loading was gradually applied until the failure of comer weld occurred. 2.3. Test results All tests failed in the inelastic range. Severe local buckling of the plates and crack 1.6

"

1.4 1.2

9

9

9 mlm

mm

m

1.0 9

0.8 ~"

0.6

1

Experimental rcsu result! Analytical result

0.4 E=2.21x105N/mm2 !.t=0.299 fy=275N/mm 2

0.2

0.01 0

,

,

I

2

i

I

4

t

S/ey

I

6

,

,I

|

8

Figure 3. Stress-strain curve for S-1

i

10

440

_

1.2

1"01 ~

0.8

_

.JI

9

n m

result - - 9- - Experimental _

-

0.6

- - Z Analyticalresult

0.4 0.2 t 00

0

2

4

6

S/Sy

8

10

Figure 4. Stress-strain curve for S-2

1.4

-

1.2

liB

1.0

~

m

m i

Experimental result Analytical result

0.8 0.6 0.4 0.2~ I 0.0= 0

E=2.21x1~N/mm2 ~ 0 . 2 9 9 ~275N/mm2 2

4

6

8

8/8y

Figure 5. Stress-strain curve for R-I

1.4 9

1.2

~

mmm

9

9

9

m

1.0

Experimental result Analytical result

0.8 0.6 0.4

E=2.21x10SNImrn2 P=0.299 fy=275NImm 2

0.2 2

4

8/gy

6

8

Figure 6. Stress-strain curve for R-2

10

441

0.54 0-36I ~

8t

o

'

~

-~ -0.1

-0.36~L..

-----=--1700KN -= 1850KN --v---1900KN .~ 1950KN . ,', , 20p0K~l 600 700

5oo

Column 200x200x10x600 hf=4.5mm

Figure 7. The distribution of direct stress at welds for S-1

propagation in the comer welds was observed. The results of the stress-strain relationship of each specimen are shown in Figure 3-~Figure 6, and the direct stress distribution at welds before failure along the specimen length is shown in Figure 7. The axial compressive loading given in Figure 7 varies from 1700KN to 2000KN. The additional tension stress in the welds was found and it became noticeable with the increased loading.

3. WELD-FAILURE DUCTILITY RATIO OF WBS The modified nonlinear (including material and geometrical non-linearity) fmJte element analyses method of plates and shells [7,8] was used to analyze the whole deforming path of the test specimen. The initial imperfection and dimensions of welding were considered in the modeling Figure 3 to Figure 6 indicate the agreement between the analyses and the test is satisfactory. Then the study for the influence of the width-thickness ratio, size of fillet weld and the form of the box cross section to the weld-failure ductility ratio of WBS under axial loading was conducted by this method. From the statistical analysis, a formula for the weld-failure ductility ratio of WBS was obtained as, 6m~x / 6y = 9.026 + 0.626h I - 0.137t - 0.0556b / t

(1)

where e , ~ is the maximum strain when the crack of the comer weld occurs; Ey is the yield strain of the steel; b/t is the width-thickness ratio for the wide plate of rectangular box section. It is indicated that the form of the box section has very small influence on the behaviour in the inelastic range. The minimum leg size of the fillet weld for the certainly demanded weld-failure ductility ratio of WBS can be derived from equation (2):

442 hI > 1.618,~x/e.y + 0.22t +O.09b/t-14.56

(2)

4. CONCLUSIONS Experimental and analytical research has been carded out to study the behaviour of WBS under axial compressive loading. The influence of dimensional parameters on the weld-failure ductility ratio was also investigated. The cyclical behaviour of stress on the weld connection of WBS also needs to be studied in future.

REFERENCES

1. J.F.Ger, F.Y. Cheng and L.W.Lu, Collapse Behaviour ofPino Suarez Building During 1985 Mexico City Earthquake, J. Struct. Div., ASCE, 119(3) (1993), pp.852-870 2. K.Horikawa and Y.Sakino, Damages to Steel Structures Caused by the 1995 Kobe Earthquake, Struct. Engrg. International, Vol.6, No.3 (1996), pp.181-182 3. S.Kumar and T.Usami, An Evolutionary-Degrading Hysteretic Model for Thin-Walled Steel Structures, Engrg. Struct. Vol.18 No.7 (1996), pp.504-514 4. H.Tajima, H.Hanno and H.Kosaka, Experimental Studies on Steel Rectangular Piers under Cyclic Loading, Proceedings of the 5th International Colloquium on Stability and Ductility of Steel Structures, Nagoya, Japan, July 29-31, Vol.1 (1997), pp.205-212 5. H.Otsuka, et al, Failure Mechanism of Steel Rectangular Piers under Cyclic Loading, Proceedings of the 5th International Colloquium on Stability and Ductility of Steel Structures, Nagoya, Japan, July 29-31, Vol.1 (1997), pp.213-220 6. B.G.Johnston, Guide to Stability Design Criteria for Metal Structures, John Wiley & Sons, 1976 7. J.J.Zhang, Comer Weld Behaviour of Welded Box Steel Columns under Axial Compressive Loading, Master dissertation (in Chinese), Harbin Univ. of Civil Engrg. and Arch. May, 1998 8. Y.C.Zhang and Y.T.Dong, Finite Element Method for Nonlinear Local Buckling of Plates and Sections under Monotonic or Cyclic Loading, Building for the 21st Century, Gold Coast, Australia, 25-27, July, Vol.3 (1995), pp.2099-2104

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

443

S t r e n g t h and ductility o f c o n c r e t e filled double skin square h o l l o w sections Xiao-Ling Zhao and Raphael Grzebieta Department of Civil Engineering, Monash University, Clayton, VIC 3168, Australia

Abstract This paper describes a series of compression and bending tests carded out on concrete filled double skin tubes (CFDST). Theoretical models developed to predict the ultimate strength of CFDST stub columns and beams are also described. Both outer and inner tubes were cold-formed C450 (450 MPa nominal yield stress) square hollow sections (SHS). Four different section sizes were chosen for the outer tubes with a width-to-thickness ratio ranging from 16.7 to 25.0. One section size was chosen for the inner tube which had a widthto-thickness ratio of 20. It was found that there is an increase in ductility for CFDST both in compression and bending when compared to empty single skin tubes. It was also shown there was good agreement between theoretical and experimental results.

1. INTRODUCTION The concept of "double skin" composite construction was originally devised for use in submerged tube tunnels [ 1]. It is believed that it also has a potential in nuclear containment, liquid and gas retaining structures and blast resistant shelters [2, 3]. Concrete filled double skin tubes (CFDST) consist of two concentric steel cylinders or shells with the annulus between them filled with concrete. This form of construction can be applied to sea-bed vessels, in the legs of offshore platforms in deep water, to large diameter columns and to structures subjected to ice loading [4, 5]. Recently CFDST have been used as high-rise bridge piers in Japan [6] to reduce the structure weight while still maintaining a large energy absorption capacity against earthquake loading. A research project on "Tubular Structures under High Amplitude Dynamic Loading" is currently running at Monash University. Some results on void-filled tubes subjected to bending and compression tests were reported elsewhere [7, 8]. Part of the project is to study the behaviour of CFDST in collaboration with Osaka City University. This paper reports a series of stub column compression tests and beam bending tests carried out on CFDST. Both outer and inner tubes were cold-formed SHS with a nominal yield stress of 450 MPa. An increased ductility was Obtained using the concept of double skin tubes. Formulae were derived to predict the ultimate strength of CFDST in compression and in bending. 2. MATERIAL PROPERTIES

The SHS sections were supplied by BHP Steel - Structural Pipeline and Products. They are in-line galvanised tubes (also called Duragal SHS) manufactured using a cold forming

44

process [9]. The average values of measured dimensions and material properties for the SHS used in this project are shown in Table 1, where a cross-section ID number (S1 to $5) is given. The tensile coupons were extracted from the flat surfaces both opposite and adjacent to the weld seam (for O'yf yield stress and t~,f ultimate strength) and comers (for frye and flue) of each tube size. The tensile coupon tests were performed according to AS 1391 [ 10]. The 0.2% proof stress was adopted as the yield stress. The compressive strength of the concrete was determined using concrete cylinders with a diameter of 100 mm and a height of 200 mm. The concrete used for the CFDST stub column tests was cured for 28 days. A compressive strength of 58.7 MPa was obtained. The concrete used for the CFDST beam tests was cured for 6 months due to the delay of the testing program. A compressive strength of 71.3 MPa was achieved. Table 1 Measured section dimensions and material properties Section D B t (~yf (~uf ID No. (mm) (mm) (mm) (MPa) (MPa) S1 99.74 99.74 5.97 485 532 $2 100.49 100.49 4.01 445 546 $3 100.18 100.18 2.94 464 545 $4 100.46 100.46 2.06 453 539 $5 50.00 50.00 2.44 477 542 Mean 465 541 COV 0.035 0.010

aye (MPa) 559 521 594 568 591 567 0.052

t3'uc (MPa) 615 647 656 618 652 638 0.031

3. STUB C O L U M N TESTS Eight stub column tests were carried out in a 5000 kN capacity Amsler machine. The specimens were labelled as shown in Table 2 where the last letter refers to the first sample (A) and the repeated sample (B). The length of the specimen was about 375 mm which was determined according to AS4600 [ 11 ]. The ends of the stub columns were milled fiat before testing so that they could be properly seated on the rigid end platens of the testing machine. Shortening of the column was measured by using two Linear Variable Displacement Transducers placed between the two parallel end platens and measuring each platen's movement relative to the other. The maximum test load (Ptest) for each specimen is listed in Table 2.

Figure I Failure modes of CFDST stub columns

445 The failure modes of both outer and inner tubes are shown in Figure 1. It can be seen that the outer tube behaves the same way (i.e. forming an outward folding mechanism) as a concrete filled tube [12], whereas the inner tube behaves the same way (i.e. forming both outward and inward folding mechanisms) as an empty compact tube [ 13]. A typical load-deflection curve is shown in Figure 2 (a). Table 2 Results Specimen Label CSIS5A CS 1S5B CS2S5A CS2S5B CS3S5A CS3S5B CS4S5A CS4S5B Mean COV

of stub column tests Ptest Pth~o~y Ptheory]Ptest (kN) (kN) . . . . . . 1545 1538 0.995 1614 1538 0.953 1194 1201 1.006 1210 1201 0.993 1027 1060 1.032 1060 1060 1.000 820 811 0.989 839 811 0.967 0.992 0.0242

eyui ratio (DST/OutTube) 0.970 0.932 1.194 1.222 2.125 1.818 2.455 2.727

Energy Ratio (DST/OuterTube) 1.27 1.32 1.78 1.90 2.71 2.79 4.15 4.37

4. PLASTIC BENDING TESTS Five beam tests were carried out in the same Amsler machine that was used for the stub column tests. The specimens were labelled as shown in Table 3. A four point bending rig was used to apply the moment similar to that described in [ 13, 14]. Both strain gauges and spring pots were used to measure the *bending curvature. Typical dimensionless moment-versuscurvature graphs are shown in Figure 2 (b) where Mp-outer is the plastic moment capacity of the outer tube and kp-outcr is the curvature corresponding to Mp-oater. It can be seen that the results from the strain gauges are almost the same as those from the spring pots. The maximum test moment (Mtcst) is listed in Table 3. The failure modes of both outer and inner tubes are shown in Figure 3. No tensile fracture was observed on the tension flange. Similar to CFDST in compression, it can be concluded that the outer tube behaves the same way as a concrete filled tube [7], where as the inner tube behaves the same way as an empty compact tube [13]. 1,5

1200

~1000

~'~ o

~

eoo

,

600 400

i]

200

~ ..........

o

[ 0

I0

.i'~L-

..,~

5

:~ui

10

15

20

i

.i

0

,,

0

Axial Shortening (e) in mm

(a)

1

|o5

,:, ....

"

2

4

6

8

--~-- Curvaturu

on string pots Curvatures based on strain gauges

. . . . Estlmated~e for R ,!

............

10 12 14 16

k/kp_outer

(b)

Figure 2 Typical experimental curves (a) compression (CS2S5A), (b) bending (BS 1S5A)

446 Table 3 Results of beam tests Specimen Mtest Mtheory Label (kNm), .(kNm) BS1S5A 49.93 47.11 BS 1S5B 51.06 47.13 BS2S5A 36.34 37.69 BS3S5A 30.04 30.44 BS4S5A 22.05 23.64 Mean COV

Mtheory/itest 0.943 0.923 1.037 1.013 1.072 0.998 0.0631

Rotation Capacity (R) >10.4 >9.4 >9.0 >5.0 >4.7

Estimated R values 15 18 13 12 11

Figure 3 Failure modes of CFDST beams 5. S T R E N G T H AND DUCTILITY

5.1 CFDST in compression The ultimate strength (Ptheory) of CFDST can be estimated using the sum of the section capacities of the Concrete, the outer steel tube and the inner steel tube, i.e. Ptheory = econcrete + Pouter + Pinner in which,

(1)

Pconcre,e = 0.85" fc' Aconcre,e (the reduction factor 0.85 is defined in AS3600 [ 15]) 2 Pouter = Pcorner+ Pfla, =Oy~ ./t. (r2x,o- rin,o)+4.Oyfo .beo .to einner = ecorner -I- eflat = O'yci" ~" (re2xti- ri2nti)+4"O'yfi "(hi-2" rexti)" t i

A ...... ,e=

(b-2.to)2-4.(r,2o-~.r,~,o)-

b~-4.(ri2

_-~-.r~,i)

where beo = b o - 2. rex,~

if ~ _<~y = 40 specified in AS4100 [ 161

~ey beo = (b o - 2" r~x,o). )---~-

if ;~ > ~y = 40 specified in AS4100 [ 16]

b ~e = (

o - 2. rex,o)" . / ~ o as defined in AS4100 [ 16] to ~/250

447 rexto and rinto are external and internal corner radii of the outer tube, rexti and rinti are external

and internal corner radii of the inner tube, to and ti are wall thickness of the outer and inner tube respectively, ~ is the plate slenderness defined in [ 16] and ~y is the yield slenderness limit specified in [ 16]. It should be noted that there is no reduction due to local buckling in calculating Pi..er since the inner tube (SHS 50x50x2.5) is a fully effective section [ 16]. The predicted ultimate strength (Ptheory) is compared with the experimental value (Ptest) in Table 2 A mean of 0.992 and a COV (coefficient of variation) of 0.0242 are obtained. It seems that Equation (1) gives a good prediction of the ultimate strength of CFDST in compression. The yield deflection on the falling load-deflection curve (eyu0, as defined in Figure 2 (a), for CFDST is compared in Table 2 with those for empty outer tubes tested elsewhere [ 17]. The same comparison is shown in Table 2 for energy absorption calculated using the area under the load-deflection curve when axial shortening is up to 15 mm. It can be seen that there is a significant increase in ductility and energy absorption especially for more slender sections. 5.2 CFDST in bending

Based on the experimental observation that the outer tube behaves like a concrete-filled tube and the inner tube behaves like an empty tube, the ultimate moment capacity (itheory) of CFDST can be estimated using the sum of the section capacity of an inner tube and that of an outer tube filled with concrete. That is: M,h,ory = M inner"~" M outer-with-concrete (2) in which, M inner -" Sinner " (~ ufi where Sinner is the plastic section modulus of an inner tube and Gun is the ultimate tensile strength of the fiat faces for the inner tube. bo

-r r',--

-1--I

C~fo(compression) 9 ~I ~

*

bo - 2to I~

'

1

f =1 I~_

~

fc

N

Neutral Axis (b) Figed Concrete

, (a) Outer Tube

~fo (tension)

Figure 4 Model for ultimate moment capacity of DST beams The determination of Mout~r-with-concreteis similar to that described in [7] for void-filled SHS beams. A full plastic stress distribution in the outer steel tube is assumed, as shown in Figure 4 (a), where Gufo is the ultimate tensile strength of the outer steel section. The stress distribution of the concrete is shown in Figure 4 (b) where inner tube corners are replaced by right angle areas for simplicity and there is a void bi x di in the centre of theconcrete fill. The

448 term di is the depth of the inner tube. The position of the neutral axis can be found by enforcing equilibrium (Equation (3)) of the forces in the tension and compression zones as shown in Figure 4. Thus 3

3

4

Y_.To,- Y, Co, + i=i

i=l

(3) i=]

where Toi is the tensile force in the outer steel tube, Col is the compressive force in the outer steel tube and Cci is the compressive force in the concrete. The neutral axis position can be approximated as: fc d -2.t o -d i bi (do - 2 " t o ) + ( ~ f o ) - t o .[0.215-0.25.( o to )" (~'o)] dn =

1 fc b -2.t o-b i 2 +~. (~fo). ( o to ) fc (d o - 2. to) + (~----) 9to .[0.215]

do

"-" ufo

d. =

1 fc bo_2.to 2 + ~- (~-~fo). ( t-----~)

do - 2 - t o -d~ if d > n

2- to

2

~ d i

if d n < 2

The ultimate moment Mouter-with-concreteis the summation of moments caused by all the forces, i.e. 3

3

4

Mo~,~_,,i,_co,~e = ~z_.,T~"droi+ ~__,C,~ "dcoi + ~z__,Ca -dca i=l

i=i

(4)

i=i

The predicted ultimate moment capacity (Mtheory) is compared with the test value (Mtest) in Table 3. Good agreement was obtained with a mean of 0.998 and a COV of 0.0631. Rotation capacity, defined as R = k~p_outer- 1.0 when M]Mp_.outer falls to 1.0, for each specimen was determined from the dimensionless moment versus curvature curves. An example is shown in Figure 2(b) for specimen BS 1S5A. An estimated line is drawn in order to get an intersection point with M/Mp_outer = 1.0 line. The estimated R values are listed in Table 3, which range from 11 to 18. It is interesting to note that the rotation capacity for an empty C450 SHS with the same range of width-to-thickness ratio (16.7 to 25) varies from 3 to 6 [ 13]. Smaller R values (ranging from 1 to 4) were obtained for empty RHS (rectangular hollow sections) with the same range of bo/to due to the interaction between web and flange [ 14]. It seems that the rotation capacity of CFDST is about 3 times as that of empty SHS for width-to-thickness ratios ranging from 16.7 to 25. 6. C O N C L U S I O N S 1. Increased ductility and energy absorption have been observed for CFDST subjected to compression. 2. Based on the limited test results, the rotation capacity for CFDST has been found to be about 3 times as that for empty outer tubes for bo/to ratios of 17 to 25. 3. The theoretical models developed to predict the ultimate capacity of CFDST in compression and in bending show good agreement with experimental values.

449 ACKNOWLEDGMENTS The authors are grateful to the Australian Research Council for financial support. Thanks are given to BHP Structural and Pipeline Products for providing the steel tubes. Thanks are also given to Roger Doulis, Ban Chuan Lim and Maggie Leung for performing the tests. REFERENCES 1. Tomlinson, M, Chapman, M., Wright, H.D., Tomlinson, A. and Jefferson, A. (1989), Shell composite construction for shallow draft immersed tube tunnels, ICE Int. Conf. on Immersed Tube Tunnel Techniques, Manchester, UK, April 2. Wright, H., Oduyemi, T. and Evans H.R. (1991a), The experimental behaviour of double skin composite elements, Journal of Constructional Steel Research, 19, 91-110 3. Wright, H., Oduyemi, T. and Evans H.R. (1991b), The design of double skin composite elements, Journal of Constructional Steel Research, 19, 111-132 4. Goode, C. D. (1988), Composite construction to resist external pressure, Speciality Conference on Concrete Filled Steel Tubular Structures, 46-52 5. Shakir-Khalil, H. (1991), Composite columns of double-skinned shells, Journal of Constructional Steel Research, 19, 133-152 6. Sugimoto, M., Yokota, S., Sonoda, K. and Yagishita, F. (1997), A basic consideration on double skin tube-concrete composite columns, Osaka City University and Monash University Joints Seminar on Composite Tubular Structures, Osaka City University, Osaka, July 7. Zhao, X.L. and Grzebieta, R.H. (1999), Void-filled SHS beams subjected to large deformation cyclic bending, Journal of Structural Engineering, ASCE, 125(9), 1020-1027 8. Zhao, X.L., Grzebieta, R.H., Wong, P. and Lee, C. (1999), Void-filled RHS sections subjected to cyclic axial tension and compression, Second International Conference on Advances in Steel Structures, Hong Kong, December, 429-436 9. Zhao, X.L. and Mahendran, M. (1998), Recent innovations in cold-formed tubular sections, Journal of Constructional Steel Research, 46 (1-3) 10. SAA ( 1991), Methods for tensile testing of metals, Australian Standard AS 1391, Sydney 11. SAA (1996), Cold-formed steel structures, Australian Standard AS4600, Sydney 12. Ge, H.B. and Usami, T. (1996), Cyclic tests of concrete filled steel box columns, J. Struct. Engrg., ASCE, 122(10), 1169-1177 13. Zhao, X.L. and Hancock, G.J. (1991), Tests to determine plate slenderness limits for coldformed RHS of grade C450, Steel Construction, AISC, Sydney, 25(4), 2-16 14. Wilkinson, T. and Hancock, G.J. (1998), Tests to examine compact web slenderness of cold-formed RHS, J. Structural Engineering, ASCE, 124(10), 1166-1176 15. SAA (1994), Concrete Structures, Australian Standard AS3600, Sydney 16. SAA (1998), Steel Structures, Australian Standard AS4100, Sydney 17. Lim, B.C. (1999), Void-filled double skin box stub columns, final year project report, Department of Civil Engineering, Monash University, Clayton

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

451

The quasi-static piercing o f square tubes G. Lu and J. Zhang School of Engineering and Science, Swinburne University of Technology, Hawthorn, VIC. 3122, AUSTRALIA

This paper investigates the energy absorption properties of square tubes pierced by pointed punches under quasi-static conditions. In a series of tests, pyramidal punches of square section and conically headed cylindrical punches were pierced slowly into square steel tubes having 40x40mm outside dimensions. Wall thicknesses 1.6 and 2.5mm were tested, and the length of tubes was varied. Some typical loads were plotted against deflection, and a number of interesting conclusions were drawn from the test.

1. INTRODUCTION The use of readily available metal structural products as energy absorbing devices has gained increased attention over the last two decades. The effective design of these devices for use in pressure vessels and automobiles where damage may cause catastrophic results is important. Study of tubes as energy absorbing devices has been extensive. Most studies have focused on axial and lateral loading of round tubes under both static and dynamic conditions. Axial collapse of cylindrical tubes under dynamic loading has been extensively investigated by Grzebieta [1 ]. Thomas et al [2] investigated the behaviour of thin walled circular tubes under transverse loading where the load was applied at a central cross-section of the tube with a wedge shaped indenter. All of these researches found that energy absorption is a function of the physical dimensions of the tube and the material used. By using this information, tubes could be designed to absorb energy in a controllable manner. Johnson et al [3] described the process of piercing using a punch penetrating a thick or thin metal plate. Johnson et al [4] fin~er studied the quasi-static piercing of cylindrical tubes under different loading conditions, and provided a relationship between the different loading conditions and the energy each individual test piece was able to absorb. They found that the type of failure for a test piece with a length less than 2.5 times that of the diameter was structural collapse and a test piece with a length greater than 2.5 times that of the diameter was localised penetration. The investigation undertaken for this project involved piercing square tubes under similar testing conditions to Johnson et al [4]. Load applied versus the deflection of the punch will be recorded and analysed. Deformation pattern of the square tube will also be examined as the load is applied.

452

Figure 2 Typical nominal stress strain curves for steel 2. APPARATUS AND SPECIMENS The tests were performed using a Shimadzu Universal testing machine with a 250kN capacity. Traces of load against punch movement were recorded automatically. A punch speed of 5mm/min was used for all the tests. We used five pyramidal punches of square or circular cross section with 12.7mm sides or diameter (see Figure 1 and Table 1). The punches were made of high quality tool steel (ASSAB grade XW-1). After machining, the punches were case hardened. All tests were conducted with the punch normal to the top surface of the specimen. The sides of the punch were either parallel to, or at 45 ~ to the sides of the tube. The specimens tested were commercially available square hollow section, grade C350 Steel to AS1163-1991. The size of the tube section was 40x40mm and the wall thicknesses 1.6ram and 2.5mm. The length of tube tested was varied. Tensile samples were taken for both wall thicknesses, average yield stress was 350MPa (see Figure 2).

3. TEST RESULTS: TYPICAL CASES 3.1. Test 1 : A W l 6 0 7 In this test, we found that penetration of the specimens was the primary form of energy absorption without structural collapse (see Figure 3). Five distinctive stages were observed (as labelled in Figure 4). Load initially increased in stage 1 with local plastic deformation of the

453 Table 1 Test specification Test 9 AWl 601 AW2502 AWl 603 AW2504 AWl 605 AW 1606 AW 1607 AW 1608 AW 1609 AW2510 AW2511 AW2512 AW2513 AW2514 AWl 615 AW2516 AWl 617 AW2518 AWl 619 AW2520

Thickness (ram) 1.6 2.5 1.6 2.5 1.6 1.6 1.6 1.6 1.6 2.5 2.5 2.5 2.5 2.5 1.6 2.5 1.6 2.5 1.6 2.5

Dimensions Tube Length (ax a) (ram) (L) (ram) 40X40 40X40 40X40 40X40 40X40 40X40 40X40 40X40 40X40 40X40 40X40 40X40 40• 40• 40X40 40X40 40X40 40X40 40X40 40X40

340 340 75 75 100 100 100 100 100 100 100 100 100 100 40 40 58 58 75 75

Punch Type Pyramidal Pyramidal Pyramidal Pyramidal Conical Conical Pyramidal Pyramidal Pyramidal Conical Conical Pyramidal Pyramidal Pyramidal Pyramidal Pyramidal Pyramidal Pyramidal Pyramidal Pyramidal

Semi-Angle (degree) 30 30 30 30 30 15 30 22.5 15 30 15 30 22.5 15 30 30 30 30 30 30

Orientation (degree) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 45

specimen sidewall. A plateau was reached in stage 2 when the punch broke through to the sidewall. In stage 3, the load steadily increased again due to cracks propagating from the comers of the punch and petalling of the sidewall between these cracks. As the petals expanded the contact area enlarged, increasing the friction forces acting on the punch. The maximum load was attained when the shank of the punch entered the deformation zone where there was maximum contact area. The load then rapidly decreased in stage 4 as there was a reduction of contact area. Finally in stage 5, there was only contact between the sides of the shank mad the sidewall petals and the load exhibited another force plateau.

3.2. Test 2 : A W l 6 1 5 In this test, structural collapse of the tube occurred (see figure 5), there were three different stages evident (A, B and C) (figure 6). In stage A, there was a steady increase in load as the sidewall of the tube plastically deformed but was not penetrated. In stage B, there was further plastic deformation until there was full contact between the inclined face of the punch and the sidewall. In stage C, plastic deformation continued and the sides of the punch shank began to contact the deformed tube sidewall. With increasing contact area the required load increased.

454

~ 4-

C

~3 2 1

~

'

lb

'

io

'

3'0

'

40

Displacement (ram) Figure 6

Load-displacement curve for test AWl 615: L/a=/. O, t=l. 6m, a = 3 0 o

455 "-'14

~

12

~

,.a

8

///,..

~6 5

AW2502

,

,"/

2

AWl 60~_ ..,/~"~'k a,. /

~

160,1

1

15 20 25 Displacement (mm)

00

I

00'

'

5

10

Figure 7 Effect of wall thickness

,

I

5

,

10

I

I

15

20

AWl 608

I

I

,

25 30 35 Displacement (ram)

Figure 8 Effects of semi-angle

~,7 AWl603

~

AWl617

o 5

4

4

3

3 2

2

1

1

00

~~-~

5

'

j

'

1;

'

1'5

' 2s0 ' 25 Displacement (ram)

Figure 9 Effect of tube length on mode of failure

00

/

.~

/

./

// // i

,

,..~.

I

5

/,,

\

A~W1605 AWl607 '\

\

\\ \\

i

I

1

,i

I

15

i

I

,

20 25 Displacement (ram)

Figure 10 Effect of indentor geometry

4. OBSERVATIONS 4.1. Effect of wall thickness on punch load The two curves shown in Figure 7 exhibit the five stages of tube penetration. For the 2.5mm wall thickness tube maximum load was more than twice that for the 1.6mm thick tube. 4.2. Effect of pyramidal punch semi-angle on punch load From Figure 8 it can be seen that as the semi-angle of the pyramidal punch increased, the maximum load also increased. This was largely due to the rate of plastic deformation required for petal formation. 4.3. Effect of tube length on mode of failure From our series of tests we found that mode of failure was mainly determined by the length of the tube. For tubes of lengths 100 and 75mm with thickness 1.6mm (L/a=2.5,1.87), the mode of deformation was by penetration (see Figure 9) with localised plastic deformation. For a tube of length 58mm (L/a=l.45), after the tip had penetrated the tube wall, structural deformation occurred around the point of penetration. The mode of failure varied between penetration and structural collapse, whichever path required the least energy. When the tube

456 length was reduced to 40mm collapse.

(L/a= 1.0) for test AWl615, the tube failed by structural

4.4. Effect of punch geometry on punch load Figure 10 shows that the pyramidal punch had a larger maximum load when compared to a conical punch with the same semi-angle. The shape of the load-deflection graph was similar for the two punches, only the maximum load varied.

5. CONCLUSIONS From this investigation the following conclusions can be drawn. Depending on the length of the tube, two different modes of failure were observed: penetration of sidewall and structural collapse of the tube. As expected, the maximum load attained (and energy absorbed) was greater for the thicker walled tube. The predominant failure mode was the one that required the least energy to achieve. Increasing the semi-angle of either punch resulted in a higher maximum load. The maximum load occurred at a smaller deflection for the penetration mode of failure. The point at which the maximum load was attained was a function of the geometry of the punch. This result was consistent for both conical and pyramidal punches. Once the length of the tube increased beyond a critical point, the tube length had no effect on the maximum load or mode of failure. It was found in our tests, that for lengths greater than 75mm (L/a=l.87), structural collapse occurred rather than penetration for both wall thicknesses. Further study is required to clarify this critical length. The pyramidal punches attained higher maximum loads than the conical punches with the same semi-angle. Further tests need to be conducted to explore this relationship more thoroughly.

ACKNOWLEDGEMENT The authors wish to thank I. Duguid and P. Kozina for their help with experiments. The work is financially supported by the Australian Research Council.

REFERENCES 1. R. Grzebieta, An alternative method for determining the behaviour of round stocky tubes subjected to an axial crush load. Thin Walled Structure, 9 (1990) 61. 2. S. Thomas, S. Reid, W. Johnson, Large deformations of thin walled circular tubes under transverse loading- 1. Int. J. Mech. Sci., 18 (1976) 325. 3. W. Johnson, A. Mamalis, The perforation of circular plates with four sided pyramidalheaded square-section punches. Int. J. Mech. Sci., 20 (1978) 866. 4. W. Johnson, S. Ghosh, A. Mamalis, T. Reddy, S. Reid, (1979). The quasi-static piercing of cylindrical tubes or shells. Inter. J. Mech. Sci., 22 (1979) 9.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

457

The splitting o f square tubes G. Lu a , T.X. Yub and X. Huang a aSchool of Engineering and Science, Swinburne University of Technology, Hawthorn, VIC. 3122, AUSTRALIA bDepartment of Mechanical Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Hong Kong

This paper presents an investigation into the energy absorbing behaviour of splitting square metal tubes axially. Tubes (50mm square) with a thickness of either 2.5 or 1.6mm were pushed slowly against rigid pyramid shaped dies, which had various semi-angles. The results of typical cases are given in the form of the load versus displacement relationship. The effects of material, thickness, semi-angle of dies, and lubrication are studied. Theoretical considerations are presented.

I. INTRODUCTION Energy can be absorbed by deforming structural members under conditions of lateral loading, axial loading and by piercing. The ability to absorb energy is important in many structures such as automobiles where damage can occur. The energy is dissipated by various failure mechanisms such as plastic deformation and fracture or tearing. Over the last two decades much experimental and analytical work has been undertaken to investigate these failure mechanisms. These studies are important in better understanding energy absorption in order to reduce the damage caused during collision of structures. Johnson et al [1] studied buckling of plastic thin-walled tubes under axial compression. They presented an inextensional collapse mode of the tube. Reddy [2] investigated another type of impact energy absorber, that is, tube inversion. He divided both external and internal inversion into two phases, i.e. transient and steady state phases. His analysis included the effects of bending, stretching and compression of the tube. Tube splitting tests have been conducted by Stronge et al [3]. The method used in their experiments was that a square tube was pressed axially against a die or a flat plate. They identified several energy dissipation mechanisms: fracture energy associated with tube splitting; plastic deformation associated with the development of the curl and friction work as the tube interacted with the die/plate. In this experiment plastic deformation energy and frictional energy could be estimated reasonably well, but it was difficult to determine the amount of energy dissipated in the tearing of four comers. Another experimental study in splitting square robes was conducted by Lu et al [4]. This study focussed mainly on the tearing energy associated with tube tearing. The values of the coefficient R (the tearing

458 energy per unit tom area) for both mild steel and aluminum square tubes were successfully derived. These values, however, are only true for problems with similar stress states to the specimens being used. The objective of the current study is to further investigate the energy absorption involved in tube-splitting process under quasi-static conditions. Here, different dies are used from those by Stronge et al [3]. The energy was dissipated in three mechanisms: bending, tearing and frictional. The effect of several factors on these three energy components will be discussed.

2. EXPERIMENTAL SET-UP AND SPECIMENS All the experiments were performed in a Shimadzu Universal testing machine. The experimental set-up is shown in Fig. 1. The cross-head of the testing machine forced the tube slowly down against the pyramidal die at the bottom at a speed of 3mm/min. Traces of load against the cross-head movement were recorded automatically. Pyramidal dies of five different semi-angles (a (see Figure 5 for the definition of a)) were selected: 30 ~, 45 ~, 60 ~, 75 ~ and 90 ~ respectively. All dies were made from mild steel and heat-treated to increase their surface hardness. All the specimens tested were commercially available square hollow section tube. The external dimensions were 50mmxS0mm and the length 300mm. There were two thicknesses: 1.6mm and 2.5mm. And two different materials were selected: aluminium and mild steel. The type of specimen and the coding used in the experiment are listed in Table 1. In order to prevent buckling, initial cuts were made in the end of the four comers. The code "Normal" in column 6 means that no lubricant was applied and the test was stoped as soon as one or more of the four leading edges of the tube touched the walls of the tube. Mechanical properties of all the specimens used were obtained from standard coupon tests and typical stress-strain curves are shown in Fig. 2. Average values of yield stress (or 0.2% proof stress in the case of aluminium) q., ultimate stress ~, and fracture strain e~ are as follows. For mild steel tubes 2.5mm thick, ~=387.1 MN/m 2, o;,,=451.6 MN/m 2, e~.= 0.20, and mild steel tubes with a thickness of 1.6mm, ~ =252.8 MN/m 2, ~,=323.6 MN/m 2, e~ =0.20. For aluminium tubes with a thickness of 2.5ram, ~. =157.6 MN/m 2, ~,=183.5 MN/m 2, =0.10.

3. TEST RESULTS AND ANALYSIS A total of 10 specimens were tested: seven mild steel and three aluminium tubes. During the test, the applied load, P

Fig. 1 Experiment set-up

459

II/ " i oorl/ 2so

\

l'5~ . " ' " - ' " ~ ' " " - " I g,e ,d'

"

"~

loo~//

mild steel for t=2.5 mm mild steel for t = l . 6 mm --- - - aluminium for t =2.5 mm

50 0

i

....

,

'

0

I

'

5

I

10

t

t

t

20

15

8n-nin (%) Fig. 2.

Nominal stress strain curves for mild steel and aluminium

Table 1 Summary of test data Test

Material

t

(mm) 1A 2A 4A 1B 2B 3B 4B 5B 4E 4LB

AL AL AL MS MS MS MS MS MS MS

2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 1.6 1.6

Semi-angle Initial cut (a ~ (ram) 30 45 75 30 45 60 75 90 75 75

30 0 30 30 2 30 30 12 2 2

Condition

Pave.

(kN) Normal Normal Normal Normal Normal Normal Normal Normal Normal Lubricated

10 13 17 16 20 27 32 90 18 17

Rave.

Pcalc.

(mm)

(kN) (mm)

33.93 21.19 8.76 54.61 24.94 15.29 8.76 6.78 6.82 6.35

17.84 18.74 22.90 21.90 24.21 28.73 40.03 51.58 15.44 9.33

Realc.

10.92 9.95 8.52 14.06 13.02 10.72 9.32 6.49 5.50 5.07

and crosshead displacement, I, were recorded and plotted. Typical curves are shown in Fig. 3 for Test 4A, Test 4B and Test 4E. Typical specimens after test are shown in Fig. 4. When the test started, the force increased with the crosshead movement. The initial saw-cuts at the four comers caused the tube to divide into four plates, the first steady state was reached with an almost constant force. Further movement of the crosshead resulted in a sharp increase of the load until it reached the peak, which corresponded to initiation of four fractures at the comers. After that, the load decreases rapidly and the fracture propagates along the tube by ductile tearing. Curling of the tube began because the end of the tube consists of four plates that are free to bend outward in response to the applied moment. This moment results from the normal force and the friction on the contact surface between the curls and the die. The

460

~

1)

.

.

.

.

.

.

,,OW

=ga.

Test4B ~ tr,~T~

~ //~

t4E

2kN T/est4A

te

O L 0

,

I 10

,

I 29

,

I 3O

,

I 40

~

I SO

40

Displacementof Crosshead1(ram) Fig. 3.

Fig. 4.

Typical force-displacement traces. (For specimens 4A and 4B, the initial plateau of low values corresponds to the pre-cut of the four comers)

Typical specimen after test: (a) Test 4A (b) Test 4E P,V

Fig. 5. Forces on split tube curling against the die of tx semi-angle

461 load in this stage took on the second plateau and the curling continued with an almost constant radius. The curve for Test 4E has the same pattern as that of Test 4A and Test 4B except that the first plateau disappears because of the shorter pre=cut length in this specimen. This typical pattern of having the second plateau and an almost constant curling radius were observed in all tests. There are three primary sources of energy dissipation: plastic deformation, fracture propagation and friction. So the rate of work done by the axial force I~"e = PV is expressed as follows, l~ -- Wp § l~f § l~v

(1)

Plastic deformation occurs in a circumferential band from point A to point B (Fig. 5). Forces on the curling end of the tube are concentrated on the line of contact with the dies, and the largest moment for each side, M s, occurs at an intermediate point in the curl, B. In a strain hardening material the moment at the interface of AB increases from M0 to MB, where M 0 = trybt2/4 is the full plastic bending moment. Hence, the rate of plastic deformation energy for four sides will be G = 2(Mo + MB)V/ R.

(2)

The tearing energy would mainly depend on the torn area and the wall thickness (t). So the rate of energy dissipation is expressed by 1V1 = 4 P~V t .

(3)

The tearing energy per unit tom area (Re) can be obtained according to Reference [4]. For mild steel R~ = 8.8cr,r for aluminium P~ = 37.2cr~6I . Friction is an important energy dissipation mechanism in tube splitting. Careful experiments give a frictional coefficient/z of about 0.49 without lubrication, mad about 0.10 with lubrication. The rate of energy lost to friction is: lir'F = 4NI~V.

(4)

there N = P / 4 ( s i n a + / t c o s a ) is normal force for each side. From the above analysis, the force required to split and curl the tube by a die of semi-. angle (a) is P =4[(M 0 + M B ) I 2 R + Rct]l[1-ul(sina + / l e o s a ) ] .

(5)

From our experiments, we found that the curl curvature (1/R) is nearly constant and can be calculated by geometry and equilibrium considerations. If the material is assumed as linear strain-hardening, it will have a flexural rigidity per unit width, D p, associated with a strainhardening modulus Ep. From Fig. 5, it can be shown that

462 4bDp

R:

(6)

[(~/1 +/t 2 - I~)R - /J t / 2]P /(sina + / ~ c o s a ) - 4M0 where Dp = Ept 3/12. Calculated and measured values for P and R are shown inn Table 1, colmmas 7 to I 0. There are large discrepancies between the experiments and calculations. However, the equations (5, 6) may be used to explain qualitatively the effect of several factors. When splitting square tubes, firstly load increases as the semi-angle of the die increases. The radius of curling decreases with increasing the semi-angle, which leads to more dissipation of plastic deformation energy. Wall thickness naturally has a great effect on the applied load. Clearly, thicker wall tubes require more energy for both fracture and plastic deformation. Material properties (Cry) and lubrication directly affect load as shown in equations (5, 6).

4. CONCLUSION Square tube splitting is an efficient energy absorbing system for a crashworthy system. There are three components combined to make up an energy dissipating system: tearing energy, plastic deformation energy and friction energy. Equations have been presented for each of these three energy components. Several factors affect these three energy components, such as the semi-angle of the die, material properties, tube thickness and friction. This study has shed an insight into splitting square tubes. More research is needed in order to thoroughly investigate the mechanics involved.

ACKNOWLEDGEMENT The authors wish to thank the Australian Research Council for the financial support to undertake this work.

REFERENCES

1. W. Johnson, P. Soden and S. AI-Hassani, Inextersional collapse of thin-walled tubes under axial compression. Journal of Strain Analysis, 12 (1977) 317. 2. T. Reddy, Tube I n v e r s i o n - An experiment in plasticity. International Journal of Mechanical Engineering Education, 17 (1988) 277. 3. W. Stronge,T. X. Yu and W. Johnson, Long stroke energy dissipation in splitting tubes. International Journal of Mechanical Science, 25 (1983)637. 4. G. Lu, L. Ong, B. Wang and H. Ng, An experimental study on tearing energy in splitting square metal tubes. International Journal of Mechanical Science, 36 (1994) 1087.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

463

S t r e n g t h a n d ductility o f concrete-filled circular c o m p a c t steel tubes u n d e r large d e f o r m a t i o n pure b e n d i n g M. Elchalakani, X. L Zhao and R. H. Grzebieta Department of Civil Engineering, Monash University, Victoria 3800, Australia Current design codes and standards provide little information on the flextural behaviour of circular concrete filled steel tubes (CFT) as there have been little experimental studies. This paper presents an experimental investigation of the flexural behaviour of circular CFT subjected to large deformation pure bending. Whilst the tubes tested are compact, local buckling still occurs at very large rotations for hollow tubes. This may cause catastrophic failure during a low cycle fatigue event such as an earthquake. The paper compares the behaviour of empty and void-filled, cold-formed circular hollow sections under pure bending for tubes where d/t_<40. The confinement of concrete and void-filling of the steel tube prevents local buckling for very large rotations and thus enhances strength, ductility and energy absorption of the composite section. A simplified formula is provided to determine the ultimate flexural capacity of CFT. 1. INTRODUCTION Composite members consisting of circular steel tubes filled with concrete are extensively used in structures involving very large applied moments, particularly in zones of high seismicity. Composite circular concrete filled tubes (CFT) have been used increasingly as columns and beam-columns in braced and unbraced frame structures [ 1]. Void-filling is an efficient way to delay premature local buckling and to enhance ductility of structures built with cold-formed hollow sections (CHS). The CFT columns are cost effective in building structures compared to conventional reinforced concrete ones [2]. In spite of the bulk literature written over the last 4 decades on CFT, little was devoted to the large deformation flexural behaviour of these members. In particular, few experimental studies focused on the difference in behaviour between circular hollow and CFT. Kilpatrick and Rangan [3] carried out tests on non-compact circular tubes (d/t=35, 43) filled with high strength concrete (fc=80 MPa) under 3-point and 4-point bending. They concluded that bond between concrete and steel has little effect on the strength of CFT under flexure. They also found that concrete filling increases the strength by approximately 40% over the hollow steel tube. This increase was significantly higher and of the order of 93% for CFT (d/t=92) with fc=73 MPa under 4-point bending [4]. Limited studies on the deformation capacity of CFT were carded out by Toshiyuki et al [5]. Nethercot et al [6] studied analytically the required rotation and moment redistribution of composite frames. They developed empirical equations, based on their numerical results, which link these two parameters together. Studies by Eltawil et al [7] on Cb-'r beam-columns showed considerable differences in the bending capacity of CFT between the available design rules. These differences among others were quantified in a recent international seminar [8].

464 Early research at Monash University on square hollow sections (SHS) beams subjected to large deformation cyclic bending showed that a plastic mechanism may form in the flange of SHS and the residual strength rapidly reduces after a few cycles [9]. This is true even for compact sections that have adequate rotation capacity under static bending. Recent research on void-filled SHS beams subjected to large deformation cyclic bending demonstrated the significant increase in ductility of these members [10]. Similar phenomena are expected for CHS. Therefore, it is necessary to investigate empty and void-filled CHS subjected to large deformation pure bending for a single half cycle test before extending the study to multiple cycle tests. This paper examines the strength, ductility and energy absorption of CFT constructed from cold-formed compact steel tubes with d/t_<40 filled with normal concrete subjected to very large bending rotations. It also compares the failure modes of hollow and CFT members under pure bending. The failure modes of unfilled tubes under pure bending are discussed in Elchalakani et al [ 11 ]. The inelastic rotations of CFT are given in the current paper, which are necessary for seismic analysis and design. The differences between the design rules in the ultimate bending capacity of CFT are quantified. A simplified formula is derived to determine the ultimate flexural capacity of c v r under pure bending. 2. PURE B E N D I N G T E S T RESULTS The procedure of pure bending tests, description of the bending rig, and ancillary tests to determine material properties are all given in Elchalakani et al [11]. The ends of the specimens were left uncapped to allow slippage to occur. This was believed to be the worst case in regards to loss of composite action. A total of 8 concrete cylinders (200x100 mm) were tested in a 170 kN capacity Amsler universal testing machine to AS 1012.9 [ 12] on the day the pure bending experiments were carried out. The average unconfined compressive strength of the concrete was 23.4 MPa. Table 1 shows the key mechanical properties of the specimens. Table 1 Pure bendin[ test results Specimen Size d/t Yield Maximum Ultimate (Mu)finea Strength Rotation Moment No. Dxt fy 0max M, (M.)unfilled (mm) (MPa) ( D e g . ) (kN.m) cBCI 101.83x 2.53 40.3 365 38.2 11.33 1.29 CBC2 88.64 x 2.79 31.8 432 69.2 10.86 1.36 CBC3 76.32 x 2.45 31.2 415 66.6 6.92 1.37 CBC4 89.26 x 3.35 26.6 412 30.5 10.47 1.06 CBC5 60.65 x 2.44 24.9 433 71.0 3.78 1.23 CBC6 76.19 x 3.24 23.5 456 70.9 9.87 1.30 CBC7 60.67 x 3.01 20.2 408 66.6 4.75 1.14 CBC8 33.66 x 1.98 17.0 442 63.6 0.90 1.12 CBC9 33.78 x 2.63 12.8 460 60.8 1.17 1.03 !. Energydetermined up to Op--30~ bending rotation. 2. Energydetermined up to %=600 bending rotation.

(0max)filled ( 0 max ) unfilled

2.27 6.87 7.94 2.29 2.29 5.44 2.42 1.76 1.12

E filled E unfilled

(to Omax) (to Op) 4.34 1.62' 9.36 1.742 9.31 1.872 2.36 1.11i 2.51 1.602 1.80 1.222 1.90 1.212 1.52 1.222 , 1.09 1.012

2.1 Strength Strain gauge measurements indicated no sudden increase in strains, and no sudden drop were observed in the moment-curvature response. These observations suggest that no slippage

465 occurred during testing of CFT specimens. This emphasises the significance of the binding action arising from longitudinal curvature. Figure 1 shows the typical behaviour of deformed unfilled and filled specimens with the same d/t ratio after testing in the pure bending test rig. It can be seen that the CFT specimens have better performance at overload than the hollow tubes, which suffered local buckling. In general, a kink formed in the hollow tubes after considerable inelastic rotation. Unlike the hollow tube, the CFT did not exhibit any form of buckling, plastic ripples or a single local buckle, or even a tensile fracture at the same rotation. These later failure modes are found in CFT beams and beam-columns constructed from slender tubes [4]. The numerical values of the ultimate moment (Mu) obtained in the test for the CFT are listed in Table 1. Figure 2 shows a typical sample response of the normalised bending moment (M/M0t) versus the normalised curvature (~cot) for the filled and unfilled tubes. Note •vt is the plastic curvature (Kpt=Mpt/EI) and Mpt is the plastic moment capacity. In general, the CFT specimens showed strain hardening during the full course of experiment, and did not exhibit any form of local buckling, unlike the hollow ones.

Figure 1 Large deformations in unfilled and filled CHS 88.9x2.6 at 600 angle of rotation The behaviour of the CFT chiefly depended on the section slenderness (~s). Note, ~.~=(d/t).(fy/250), where d is the outside diameter of the CHS and fy is the yield strength. Three distinct types of behaviour can be identified in Figure 2. Specimen CBC9 was taken to represent the first type ~., < 20 (Type I), CBC6 for the second 20 < ~.s <-40 (Type II) and CBC2 for the third type 40 < Ls < 60 (Type 111). For Type I, the filled and unfilled specimens had a similar response. Therefore, only one curve for both CBC9 and CB9 was plotted. For Type II, the behaviour of the filled and unfilled specimens was similar until a point, where the filled curve separated from the corresponding curve of the hollow specimen. Strain hardening of the c F r specimen took place immediately after this point, while the hollow one exhibited a long plateau, followed by a formation of a kink and a drop in the curve. For Type HI, the behaviour was similar to Type II, but the point of separation occurred earlier, i.e. at a smaller normalized curvature. Also, the normalized bending strength of Type II specimens was larger than the corresponding one for Type M specimens. There is a considerable increase in bending strength of CFT over the hollow tubes, particularly for large slenderness. The ratio of the maximum moment obtained in the tests (Mu) for CFT and the corresponding one for hollow tubes is given in Table 1. The maximum increase is 37% for BC3 (~,s=54.92), while the minimal increase is 3% for BC9 (~=23.85). This level of increase in strength is consistent with the range of 10 to 30% reported by Lu and

466 Kennedy [13] for flextural tests on rectangular CFT, and 15 to 35% obtained by Zhao and Grzebieta [ 10] for square CFT beams.

Figure 2 Normalized moment-curvature response The maximum measured ovalization in the tests of CFT was 1.5% for CBC5, but generally less than 1% for the other 8 specimens. This is significantly less than the 10% uniform ovaling measured in bending tests of hollow tubes [ 11 ]. The radial deformation of the steel tube in the CFT tests was mostly outward, i.e. away from the tube's axial centre line. Unlike hollow tubes, the ovaling of the CFT was small, non uniform with loading and asymmetric due to the opposing effect of concrete dilation (Poisson's ratio effect). The inward deformation is caused by the longitudinal curvature. Therefore, ovalization can be neglected in any theory to predict the flextural strength of CFT sections. Table 2 lists the ratio of Mt~ to the corresponding strength determined using 4 design rules, CIDECT [14], AISC-LRFD [ 15], EC4 [16] and AIJ [17]. There are large differences in the plastic d/t-limits for CFT (grade C350) given by those design codes, i.e. 60, 68, 60 and 120, respectively. Interestingly, the AIJ [17] gives the most conservative values of the ultimate bending strength, but adopts the largest plastic d/t-limit of 120. Plastic bending tests of CFT in the range of 60 to 110 are currently running at Monash University to determine suitable d/tlimits for circular CFT.

2.2 Ductility and energy absorption The ductility of the CFT under pure plastic bending was determined using the maximum inelastic rotation 0max, which was the maximum stroke of the machine. 0m~ for the CFT corresponds to the maximum moment (Mu) obtained at the end of the test. The numerical values of 0max for the CFT and its normalised values using the corresponding values for the hollow tubes are given in Table 1.0m~x for the hollow tubes is measured up to the point when the moment capacity begins to fall, e.g. point D in Figure 2. It can be seen that the CFT have significant increase in the inelastic rotation over the hollow tubes, particularly for larger slenderness (larger values of d/t). The maximum increase of 6.87 is for CBC2 with ~=59.08. The absorbed energy of the CFT was determined from the moment-curvature response (up to 0max) and it was normalised using the energy absorbed by the hollow tubes (again up to 0max

467 for the hollow tubes). This ratio is given in Column 9 of Table 1. This ratio is larger for the CFT with larger slenderness, with a mean value of 3.59 and a coefficient of variation (COV) of 0.86. Column 10 shows the energy ratio of filled and unfilled tubes up to a specific angle of rotation (0p). Unfortunately, due to capacity limitations of the testing machine, specimens CBCI and CBC4 were not tested to the desired 0=600 bend angle. This explains the small ratio of energy absorption obtained for these specimens. Table 2 Comparison iof flexural Strength Specimen MgMt~o~y Mu]MCIDECT Mu]MAIsC CBC 1 CBC2 CBC3 CBC4 CBC5 CBC6 CBC7 CBC8 CBC9 Mean COV

1.14 1.13 1.15 0.96 0.99 1.20 1.10 0.98 0.96 1.07 0.08

1.13 1.14 1.16 0.96 1.00 1.19 1.11 0.98 0.96 1.07 0.08 i

i

i

M'u/~Ec4

MJMAU

i.09 1.10 1.12 0.93 0.96 1.16 1.07 0.96 0.94 1.04 0.08

1.24 1.22 1.24 1.03 1.06 1.25 1.16 1.02 0.99 1.13 0.09

1.11 1.12 1.14 0.95 0.98 1.19 1.09 0.98 0.96 1.06 0.08

3. U L T I M A T E S T R E N G T H M O D E L A simplified rigid plastic approach was used to determine the flextural capacity of circular CFT with d/t_<40. Since no sign of buckling was observed during the tests, the full section of the CHS was assumed effective. No slip was observed in the tests, perfect bond between the steel and concrete was assumed. The ultimate bending capacity of the composite section shown in Figure 3 can be written as: (1) Mthcory= Mcc + Mst + Msc "

fy

-

j_

fc

'"

-

I

~ -

1

(a) ~ i t e

sedion

I_

C~t

x

-1

ty (13) S teel lute stresses

(c) O:x~ete

s~resses

Figure 3 Model for ultimate moment capacity of circular CFT where, Mce ,Mst ,Mse are the moment due to: concrete in compression; steel in tension and steel in compression, respectively. By performing simple integration of the rectangular stress blocks

468 over the corresponding area of steel and concrete, the force and moment components about the centroid of the cross section are: Fcc =fc r i 2 ( r c / 2 - Y 0 - 0 . 5 s i n Y 0 ) fst = fyt rm(/1;+ 2%)

Fsc = fyt rm(/1;- 2~,0)

M==~2 fcri3 cos3 ~'0 Mst = 2fyrm2t cos Y0 M= = 2fyr2t cos Y0 where f~ is the cylinder strength of concrete, fy is the yield strength of the steel tube, t is the tube thickness, and rm and ri are the mean and inside radii of the tube, respectively. Force equilibrium results in

Fst = Fcc + Fse

(2)

The location of the plastic neutral axis (~'0) can be found from Equation 2. An iterative procedure can be used to determine "to. A closed form solution for ~'o can be obtained by assuming sinT~-)'0; thus

ri ;1 2+

~y rm t

The final expression for the bending strength (Mtheory) can be expressed as 2 fcri3COS3~0 + 4fyrm2tcos% Mtheory = -~

(3)

A comparison is made between the experimental ultimate moment (Mu) and the predicted ultimate moment using Equation 3 in Table 2. Good agreement is obtained with a mean ratio (Mu/Mtheory) of 1.07 with a COV of 0.08. The average concrete contribution in bending strength determined by Equation 3 is 8.7%. The maximum error resulting from assuming sin'y0---q'o in determining Mtheory is 2.8%. 4. C O N C L U S I O N S Based on the test results and comparison between the test and prediction, the following conclusions can be made and future directions suggested. 9 No slippage occurred for uncapped CFT under pure bending. 9 Concrete filling of compact steel tubes (d/t<40) induces more increase in flexural strength and ductility for thinner CHS than for thicker ones. 9 Concrete filling fully prevents local buckling of cold-formed steel tubes grade C350 when d/t= 13 to 40. 9 The maximum measured ovalization of CFT was 1.5%, but less than 1% in most cases.

469 A simplified formula to determine the ultimate bending capacity of circular CFT was derived based on plastic stress blocks and was shown to agree with experimental results. Plastic bending tests of CF~ with d/t=60 to 110 to determine plastic d/t limit are needed. Cyclic bending tests of empty and void-filled CHS are required. ACKNOWLEDGMENTS The writers are grateful to the Australian Research Council and Monash University for their financial assistance for the project. Thanks are given to Palmer Tube Mills for providing the steel tubes. The experiments were carried out in the Civil Engineering Laboratory at Monash University in Melbourne and the technical assistance of Mr Graham Rundle and Mr Jeoff Doddrell is gratefully acknowledged. REFERENCES 1. Hajjar, J. (2000), Concrete-filled Steel Tube Columns Under Earthquake Loads, Journal of Progress in Structural Engineering and Materials, Vol. 2, No. 1 (under press). 2. Webb, J. and Beyton, J. J. (1990), Composite Concrete Filled Steel Tube Columns, Proc. of Structural Engineering Conference, IEAUST, pp. 181-185. 3. Kilpatrick, A. E. and Rangan, B. V. (1997), Tests on High-Strength Composite Concrete Columns, Research Report No. 1/97, Curtin University of Technology, Australia. 4. Prion H. G. L. and Boehme, J. (1993), Beam-Column Behaviour of Steel Tubes Filled with High Strength Concrete, Canadian Journal of Civil Engineering, Vol. 21, No. 2, pp. 207-218. 5. Toshiyuki, F., Noguchi, T. and Mori, O. (1996), Evaluation of Deformation Capacity of Concrete Filled Tubular Beam-Columns, 3rd Joint Technical Meeting, Hong Kong. 6. Nethercot, D. A., Li, T. Q. and Choo, B. S. (1995), Required Rotations and Moment Redistribution for Composite Frames and Continuous Beams, JCSR, Vol. 35, pp. 131-163. 7. El-tawil, S., Sanz-Picon, C. F. and Deierlein, G. G. (1995), Evaluation of ACI 318 and AISC-LRFD Strength Provisions for Composite Columns, JCSR, Vol. 34, pp. 103-123. 8. ASCCS (1997), Concrete Filled Steel Tubes, A comparison of International Codes and Practices, Innsbruck, 18tn September. 9. Grzebieta, R, Zhao, X.L., Purza, F. (1997), Multiple Low Cycle Fatigue of SHS Subjected to Gross Pure Bending Deformations, Proc.5 tn ICSDSS, Vol. 1, pp 847-854,Japan. 10. Zhao, X. L. and Grzebieta, R. (1999), Void-Filled SHS Subjected to LargeDeformation Cyclic Bending, Journal of Structural Engineering, ASCE, Vol. 125, No. 9, pp. 1020-1027. 11. Elchalakani, M., Zhao X.L. and Grzebieta R. (2000), Plastic Slenderness Limit for ColdFormed Circular Hollow Sections Plastic Bending Tests of Tubular Steel Members, Australian Journal of Structural Engineering "Steel Issue"(submitted for publication). 12. Standards Association of Australia, "Methods for Testing Concrete," Methods for the Determination of Concrete Compressive Strength, AS 1012.9, Sydney, Australia. 13. Lu, Y. Q. and Kennedy, D. J. L. (1994), The flexural Behaviour of Concrete-Filled Hollow Structural Sections, CJCE, Vol. 21, No. 1, pp. 111-130. 14. CIDECT (1995), Design Guide for Concrete Filled Hollow Section Columns Under Static and Seismic Loading, ed. Pergmann, R., Matsui, C., Meinsama, C. and Dutta, D. 15. AISC-LRFD (1993), Manual of Steel Construction, AISC, Chicago. 16. Eurocode 4 (1992), Design of Composite Steel and Concrete Structures, Part 1.1, General Rules and Rules for Buildings, ENV 1994 1-1. 17. AIJ (1987), Standards for Structural Calculation of Steel Reinforced Concrete Structures, Architectural Institute of Japan, Tokyo.

This Page Intentionally Left Blank

Connections

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

473

Finite Element Modelling of Bolted Flange Connections Junjie Cao', Jeffrey A. Packer b and Shu Du c "Nuclear Engineering, Babcock & Wilcox Canada, 581 Coronation Blvd., Cambridge, Ontario N1R 5V3, Canada bDepartment of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, Ontario MSS lag, Canada. Tel: +1-416-978 4776. Fax: +1-416-978 6813. E-mail: [email protected], ca CCybersonics, Inc., 5368 Kuhl Road, Erie, PA 16510-5703, USA

This paper is concerned with finite element modelling of bolted flange connections for circular and rectangular hollow sections and pressure vessels. General modelling of the connections and simulation of contact behaviour and bolt pretension are disoassed.

1. INTRODUCTION Considerable experience has now been gained by researchers around the world with Finite Element (FE) analysis of hollow structural section welded connections, for the purpose of numerically modelling the connection behaviour under static or fatigue loading. BoRed hollow section connections, despite being very popular site joints in long-span structures and towers, have received relatively little research attention and particularly the FE modelling of such connections. The most popular method for member splices is the bolted flange-plate connection, using high-strength bolts located around the perimeter of a Circular Hollow Section (CHS) member or a Rectangular Hollow Section (RMS) member. Fig. 1 shows two typical connections for circular and rectangular hollow section members, under overall tension forces. The design of these connections requires the determination of flange dimensions and arrangement of bolts. Bolted flange connections are also used in pressure vessels. Although the circular bored flange connections used for pressure vessels and for tubular structures are totally different, they have some similar mechanics features. Similar techniques can then be applied in FE modelling of the two types of connections. FE modelling of both CHS and RHS flange-plate connections, as well as bored flange connections in pressure vessels, including non-linear material properties and non-linear geometric behaviour, is discussed. Issues considered include: suitable meshes, element selection for main member and flange-plate, weld modelling, bolt modelling, bolt pre-stressing and contact elements. The studies have utilised both ABAQUS [1] and ANSYS [2] finite element packages, and some of the virtues/drawbacks of these are discussed.

474

a L__ dTh~ T

-

IF

I

I

I

I

,h

Figure 1 Bolted Flange Connections in Tubular Structures 2. M O D E L L I N G OF CIRCULAR H O L L O W SECTION CONNECTIONS Most connections used for tubular structures, such as those shown in Fig. 1, are symmetrical about the middle surface between two connected flanges, so only one side of a connection needs to be modelled. For some connections in pressure vessels, two sides of a connection may not be the same. In such cases, both sides have to be modelled. For a circular flange connection under tension loading, as shown in Fig. 1, both axisymmetric and 3-D models may be used. Since the flanges, tubes and tension force are symmetric about the axis of the tube, it is easy to model the joint as an axisynmletric object. Fig. 2a shows an axisymmetric model for a circular flange connection. In this model, all discrete objects, related to bolts, are modelled as equivalent tings. Bolt heads (or nuts) are replaced by a ring, which has the same contact area with the flange as the discrete bolts. Bolt shanks are modelled as an annular linear spring and the stiffness of the spring is determined by the axial stiffness of the bolt shanks. The flange ligament is also modelled as a continuous ring and its properties are modified to consider the existence of bolt holes, which reduces the stiffness of the flange ligament. Contact between bolts and bolt holes is ignored. Fig. 2b shows a 3-D model for the circular flange connection, which includes only one slice of the connection. The plane through the bolt centre and the tube axis is the plane of symmetry for the slice of the connection. Symmetric boundary conditions are applied to all nodes on the surfaces connected to adjacent slices, i.e. the circumferencial displacements at these nodes are restrained. In this model, the bolt hole and the bolt shank are modelled and the contact behaviour between them is considered by arranging gap dements between them.

475

Tube :e 114.3 x51

Bolt : MI6 No.of Bolts:8

I I

JoinL Axis

I

No. of Elements: 1760

No. of Elements:984

No. of Nodes :

No. of Nodes : 1197

1857

Detail of Bolt Bolt

Boll Head

id Surface

80

105

(a) kxisymmetric Model

(b) 3-0 model

Figure 2 Axisymmetric and 3-D Models for a Circular Flange Joint For the two models shown in Fig. 2, a rigid surface is defined below the flange (the plane between the two connected flanges). Interface elements are used between the flange and the rigid surface to simulate the contact behaviour between two flanges. A prescribed displacement can be applied to the end of the bolt to simulate a bolt preload to the connection. Tension loading is applied to the end of the tube uniformly. The tension loading is balanced by the bolt forces and the contact force between the flange and the rigid surface. The two models have been used successfully for a wide range of circular flange connections [3, 4], in which material non-linearity and geometrical non-linearity are considered in the models. Both second order elements (with mid-nodes) and first order elements (without mid-nodes) can be used for the two models in Fig. 2. Elements with mid-nodes can provide better results, especially for the 3-1) model, but require more computer resources. Although the 3-D model provides more accurate results for this type of connection, the axisymmetfic model provides satisfactory results in most eases. If the external load on a connection is not axisymmetric (i.e., other than just axial loading) about the axis of the connection, a 3-D model with one side or half of one side of the connection has to be used. The model in Fig. 2b can be extended to such a model easily. For a bolted flange connection in pressure vessels, the major difference is that a gasket is used between the two flanges to prevent leakage. Contact elements are then needed between the gasket and the flanges. The vessel internal pressure causes axial tension loading in the tubes. Before internal pressure is applied, the bolts of the connection are tightened to exert an initial compression on the gasket. Hence, bolt pretension is a necessity for pressure vessels, whereas it is optional for structural steelwork (although pretensioning of high strength bolts subjected to tension is recommended in practice). Most features discussed above for modelling the connections in tubular structures can be applied to connections in pressure vessels.

476 3. M O D E L L I N G OF RECTANGULAR HOLLOW SECTION CONNECTIONS For a rectangular hollow section connection under axial loading, a quarter of one side of the connection, as shown in Fig. 3, can be modelled due to symmetry. In this model, there are two half bolts and one whole bolt, but this will depend on the bolting arrangement chosen. Brick dements are used throughout the model, except for a number of wedge dements for the weld, centre parts of the bolts and the edges of bolt holes. Both second order dements (with midnodes) and first order elements (without mid-nodes) may be used for the model and satisfactory results can often be obtained from the use of both types of dements, although in general the former type is superior. The shank part and thread part of the bolts are modelled separately.

~

Z

Weld Elements Bolt Elements for 8-noded Model Tube (RHS)

|

ANSYSVersion5.4 ] NodeNo."2221 ElementNo.: 1192 ElementTypes: 8-nodedsolid(SOLID45) 6-nodedsolid(SOLID45) Contact(CONTAC52) __

-~d II IIIIIIILiffllJ Bolt Elements for 20-noded Model

/'

~

1_

II

OR

Y

i

[.i.~qSVS Version5.4'

[

I odoNo.: 7744

I

| ElementNo.: 1238 I | ElementTypes: I | 20-nodedsolid(SOLID92)[ | 15-noded solid(SOLID92)I L,contact (CONTAC52) I

Weld

Bolt -Flange Figure 3 3-D Model for a Rectangular Flange Connection

477 Symmetrical boundary conditions are applied to the plane X-Z (Y=0) and the plane Y-Z (X=0) for the model. Gap elements are used between the bolts and bolt holes to simulate the gap existing there. Contact elements are arranged between the flange bottom and the symmetrical plane, which is modelled by some nodes. A prescribed displacement is assigned to the bottom of each bolt to simulate the pre-tension during the tightening procedure. The non-linear material curve, which was from material tests, is input for the model and large displacement effects are also considered in the model. The model has been used successfully for a pilot study of bolted flange plate connections for Pal-IS tension members [5]. 4. CONTACT BEHAVIOUR AND BOLT PRETENSION Under the action of external loading to a bolted flange connection, the flanges of the connection will bend and parts of the flanges may separate while other parts remain in contact. It is very important to simulate the contact mechanism in modelling a connection and investigate contact behaviour for the connection. The presence of contact conditions, nonlinear material and large displacement will cause high non-linearity in the analysis. If comparing the FE results to laboratory tests one should also be aware that the flange-plates in the experiments may be distorted due to the fabrication (welding) procedure, especially if the plates are thin. In the two meshes in Fig. 2, the symmetric plane between two flanges is modelled as a rigid plane. Interface dements are arranged between the bottom of the flange and the rigid plane. For the model in Fig. 3, nodes are defined on the symmetric plane between the two flanges and 3-D point-to-point contact dements are arranged between the nodes on the flange bottom and the nodes on the symmetric plane. Contact dements at these locations are essential for these models and have to be modelled properly. Contact dements may be also needed for a gap between a bolt and a bolt hole, as mentioned for the two 3-D models in Figs. 2 and 3. These elements are used to prevent any penetration between the bolt and the hole. However, a large number of contact dements may cause a convergence problem, so the number of contact elements should be limited. For a gap between a bolt and a hole, a few contact dements are enough. Different ways can be used to simulate pretension in a bolt. A common method is to set an initial lower temperature for the bolt shank. Because the bolt shank is cooler than other parts of the model, it tends to contract and becomes shorter, which will cause tension in the bolt shank. This method is simple and efficient. However, if thermal analysis, which may be required for many connections in pressure vessels, is involved in a model, this method can not be used. Another method is to stretch the bolt shank initially. In the models in Figs. 2 and 3, since the symmetric plane between the two flanges is around the middle of the bolt shank, the bolt end in these models, located on this plane, should be fixed. In order to apply a pretension, the bolt shank can be modelled a little shorter than the thickness of the flange and a prescribed displacement is set to the end of the bolt to stretch the bolt to the middle plane. This is similar to the initial tightening procedure and induces an initial tension in the bolt shank. The amount of the prescribed displacement depends on the required pretension in the bolt. This method is very simple and doesn't affect thermal analysis of the connection. However, if the bolt end is not at the middle surface, the end can not be fixed and this method can not be used.

478 There may be other methods to give an initial displacement to a bolt. The use of constraint equations is applicable to most cases. As shown in Fig. 4, a gap e can be left between the bolt head and the bolt shank end (or any cross section across the bolt shank). Then the following constraint equations can be used to define the relationship between the displacements of each pair of nodes at the bolt head and the bolt shank end:

u ,-ux

=0

Ur, - Ur2 + e = 0

-where:

1 and 2 represent the nodes at the bolt head and the bolt shank.

After the above constraint equations are applied, each node at the end of the bolt shank will have the same X displacement as its corresponding node at the bolt head and Figure 4 Constraint Equation for Bolt Pretension the Y displacement at the node at the end of the bolt shank will be always larger than the Y displacement of its corresponding node at the bolt head by the value of e. These constraint equations actually connect the two parts together. Since the bolt head and the other end of the bolt are restrained by other parts of the connection, pretension is induced in the bolt shank. Fig. 4 also shows the mesh with the constraint equations applied. Constraint equations can be applied by most commercial programs, such as ABAQUS[ 1] and ANSYS [2] 5. CONCLUSIONS Different types of models can be used for bolted flange connections and symmetric conditions can be used to reduce model size. Elements with or without mid-nodes can be used for the models. Contact dements are required to simulate contact behaviour between flanges or between flanges and a gasket. Temperature difference and prescribed displacement can be used to induce bolt pretension, but use of constraint equations for bolt pretension can be applied more widely. REFERENCES 1. Hibbit, Karlsson & Sorenson Inc., ABAQUS/Standard User's Manual Version 5.6, USA, 1996. 2. Swanson Analysis Systems Inc., ANSYS Release 5.4, Houston, USA, 1997. 3. Cao, J. J., Tension Circular Flange Joints in Tubular Structures, Ph.D. Thesis, Department of Civil and Structural Engineering, UMIST, Manchester, UK, 1995. 4. Cao, J. J., Finite Element Analysis of Circular Flange Joints under Tension Forces, Proc. Of Ninth UK ABAQUS Group Conference, Exeter College, Oxford, UK, 1994. 5. Packer, J.A., Du, S. and Willibald, S., Bolted Flange-Plate Connections for RHS Tension Members, CIDECT Report 8D-12/99, University of Toronto, Canada, April 1999.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

479

E x p e r i m e n t a l b e h a v i o u r o f m o m e n t c o n n e c t i o n s between concrete filled steel tubes and structural steel framing b e a m s Jason Beutel a, Nimal Perera b and David Thambiratnam c a Phd student, School of Civil Engineering, Queensland University of Technology b Director and Senior Engineer, Robert Bird & Partners Pty. Ltd., Consulting Engineers, Brisbane r Professor of Structural Engineering, Queensland University of Technology ABSTRACT: An investigation into the behaviour of composite column-to-beam connections using experimental and analytical techniques is currently being conducted, the main objective of which is the development of a deterministic procedure for the connections design. At the time of writing this paper, four large-scale connections have been tested under monotonic loading. All connection details tested consisted of a concrete-filled steel tubular column (circular), a compact universal beam (class 1 to Eurocode 3) and a shop fabricated connection stub. This stub directly connected the beam to the tube wall using flange plates, full strength butt welds (FSBW's), and a web cleat plate. Four reinforcing bars were welded to the top and bottom flanges of the beam and embedded into the concrete core, with the bar size increasing for each specimen. The purpose of these tests was to assess the connection's suitability for application into a moment resisting frame, determine its plastic capacity, and provide important deflection and strain data on the specimen, to be used during the analytical modelling stage of this investigation. It was found that the connection strength increased as the capacity of the embedded bars increased, to a stage where no connection failure occurred, and the beam formed a plastic hinge outside the zone of influence of the bars. 1. INTRODUCTION Concrete filled steel tubular columns (CFSTC) are becoming increasingly popular column alternatives, particularly in tall buildings. Their popularity is due to a variety of reasons, the most important of which are their inherent strength and ductility, and the advanced construction techniques that are used in their erection, resulting in lower column and overall structure cost. These advantages make them ideally suited to tall structures, particularly those located in regions of high seismic risk. When CFSTC's are combined with compact structural steel beams a potentially excellent elasto-plastic energy dissipating system can be created. Because these members are usually mated within a structural typology incorporating full strength connections, it is connection performance that will ultimately control how effectively the structure performs under seismic loading. A lack of understanding on connection behaviour has limited the use of CFSTC's in such applications. Due to the increasing popularity of CFSTC's a wide variety of different beam to column connections have been investigated quite recently. External connection details include: direct connection of the steel beam to the outside of the tube only (Shakir-Khalil, 1994), and using external stiffening rings at the level of the beam flanges (Kato et. al., 1992, Choi et.al., 1995).

480 It has been found that many of these details impose a severe elastic demand on the tube wall resulting in excessive deformation and poor cyclic performance. Modifications which have attempted to alleviate these problems have included through bolting and endplates (Kanatani et.al. 1987), continuing the steel beam through the column (Aziznamini, 1995) and the use of reinforcing bars welded to the top and bottom flanges and embedded into the concrete core (Alostaz, 1996, Ashadi, 1995). This research has provided vital information on the general performance of such details, however very little design information on such connections exists. A collaborative research project has been undertaken by the Queensland University of Technology and Robert Bird & Partners Pty. Ltd. to develop a deterministic design procedure, using experimental and analytical techniques, for an economically buildable joint configuration. This paper briefly discusses the first phase of experimental testing. 2. S P E C I M E N DETAILS AND E X P E R I M E N T A L P R O C E D U R E _

'

~

. CFSTC

:i 1 / i : : :j: :,1 : i

. . . . . .

1

4

. . . . . .

i !

.......

i

, i

o

I

i i i i i i i

/ FSBW flanges to plate and column

I

, f

! . . . . .

,_

. . . . . . . . . . . . . . . . . . . . . .

i

t i t

: '

/'-'-'-'

[~'- .....

~ ......

i i i ,

i 1 :

Bars fillet welded to flanges

/

~ ......

.~

[

l

I

I I

ion bolts and web cleat plate

Connection Elevation

i I

I I I I

Connection Plan

Figure 1. Specimen S M 1 2 - SM24. Connections such as these when incorporated into a moment resisting frame (MRF) are typically designed for a plastic hinge to form adjacent to or at the beam-to-colurnn connection. Therefore the primary interest of this research are connections that may generate the plastic bending strength of the beam, potentially provide good ductility and post-elastic response, and are relatively cheap and easy to construct. After a comprehensive review of the merits of various connection details, and upon consultation with the projects industry partner the following detail was selected for this project (Figure 1). Relevant background information regarding this review and the selection of this detail can be found in an earlier paper (Beutel, 1998). Each specimen was fabricated from a 360 UB 44 kg/m standard BHP 300 plus section, and a 406mm diameter pipe with a 6 mm wall thickness and 350 MPa nominal yield. The beam section was connected directly to the steel column using flange connection plates, FSBWs and

481 a web cleat plate. Four reinforcing bars were welded off the top and bottom flanges and embedded into the concrete core. Each of the specimens had varying diameter bars, Y-12, Y16, Y-20, and Y-24 for specimens SM12 to SM24 respectively, and where filled with 40 MPa concrete. Table 1.0 shows the theoretical and experimental connection capacity for each specimen. The theoretical capacity is based upon the capacity that can be generated at the face of the column via the through shear strength of the flange to column connection, and the tensile strength of the bars (using both nominal and actual material strengths). Also shown is the expected hinge location based on whether the connections strength at the face of the column can generate the plastic bending capacity of the framing beam (Mp = 290 kNm, calculated using actual material properties) past the end of the bars. All specimens were tested in the vertical plane, as a cantilever beam test. Specimen supports were designed to simulate beam and column inflection points, and therefore allowed in-plane rotations. The column axial load was applied through the top column support, which was designed to allow vertical movement. This load was 1000 kN for each specimen, being 12.5% of the squash load of the column, which is a typical loading level for a comer column in a moment resisting frame. This load was kept constant throughout the duration of the test. For this series of tests, a 30 tonne actuator was connected to the beam tip after the column load was applied and used to apply one constant push to the beam tip until specimen destruction. 3. TEST RESULTS The moment-connection rotation (M-O) and load-beam tip deflection response has been used to compare the performance of each of the connections tested. The moments are those at the face of the column, and are normalised with respect to Mp (being the plastic bending strength of the connection beam) calculated using actual material strength and cross sectional properties. t

1.4

.................

Beam Tip Deflection

-I~l--'--O-R - e l a t i ~

nS~

.................................................... !....................... :.........................! .......................-.........................i.........................!

I

o / ~ ~~:~*-~'~~ ~20- ! - ~ ~ - - ~ ~ . . - - - - - - ~ - ~ :

,-, I 0 0

.o.61

i~~'--""<-~','-------

9

~-'-

_

-

,

8o " 60

I!

04

.

L

/

'

Z

. . . . . .

~

,-i--

....

,,J

t --

40 ~

-

t

-

.

0.005

0.01

0.015

0.02

0.025

Rotation (Radians)

Figure 2. M-0response.

0.03

0.035

0.04

0

~

. . . .

I #: 20+-~--.-=--

0

~

20

I

i 40

~-

J

~.~ -"

......

~

i

~

___ i. . . . . . . . . . . .

.

~

.....

i

'

----~--------~-'-

-

i

s"~~

--K-:--------~,

~ i ~i ............... ~---C--~:

60

Deflection

80

100

120

140

(nun)

Figure 3. Beam tip response.

Figures 2 and 3 show the M-0response including connection classification limits according to Bjorhovde et al (1990), and the beam tip response of each specimen. In general, as bar size increased, so too did the ultimate strength of the connection, with significant improvements in initial rotational stiffness and the energy absorbed by the specimen. SM16, SM20 and SM24 had adequate rotational stiffness and capacity to classify them as rigid connections.

482 Table 1.0 Specimen Capacities. Specimen

SM12s SM 16s SM20s SM24s

Anchorage length (mm) 300 300 350 350

Column wall through shear strength (%)" 43 43 43 43

Bar strength (%)" 35 64 100 145

Calculated capacity (nominal) (kNm) 155 208 276 361

Calculated capacity (actual) (kNm) 161 220 294 390

Actual capacity (kNm) 174 236 304 Na**

Expected hinge location

Actual hinge location

Conn. Conn. Conn. Beam

Conn. Conn. Conn. Beam

Calculated as a percentage of the nominal flange tensile capacity. *'- Connection capacity not reached or exceeded. -

Specimens SM12 and SM16 were able to mobilise approximately 70 and 90 % of Mp respectively. These strengths correlate well with the calculated moment capacities shown in Table 1.0, but were only achieved after considerable connection rotation causing connection yield and strain hardening ie. at approximately .015 radians. Both of these specimens had a similar failure mode, with the initial yield of the top flange bars (occurring at 0.0018 and 0.0023 radians connection rotation respectively) causing the onset of connection softening. This softening resulted in excessive connection rotation under increasing load, producing a crack in the steel tube at the comer of the beam flange to column connection after which no further strength increase was achieved. The tube wall in the vicinity of the top bars underwent noticeable deformation (bulging) due to a combination of the flange-to-tube weld load transfer, and the spalling of concrete to a depth of approximately 50rran along the bars. This spalling was produced in the zone along the bar where necking of the bar was most evident ie. the bars yield zone. By the end of the test, the top tube-to-flange weld had also cracked along its full length. The bottom flange exhibited no signs of distress, although the bars did reach yield at a larger connection rotation than the top bars (at approximately 0.004 radians connection rotation respectively). No other signs of distress were evident at any other location along the beam, or at the connection. Both Specimens SM20 and SM24 had the capacity to achieve Mp, with better than 1.2Mp being generated at the connection for each specimen. The plastic capacity of the beam was achieved at relatively low connection rotation levels (approximately 0.006 to 0.008 radians), and both connections produced a degree of yielding in the connection beam away from the connection zone, in a region adjacent to the end of the bars. SM20 achieved full section yield at this location but only minor bottom flange buckling, after which yield zone stiffening transferred a larger moment to the connection causing failure in an identical fashion to specimens SM12 and SM16. Top and bottom bar yield occurred at connection rotations of approximately 0.003 and 0.004 radians respectively. Capacity predictions using actual material strengths again seemed accurate for both SM20 and SM24. SM24 had the capacity to form a full plastic hinge in the connection beam adjacent to the end of the bars, and because of this the connection's full plastic capacity was not achieved. This can be seen in Figure 2, which shows that the connection rotation halted and levelled out to a constant due to yield of the beam, and Figure 3, which shows its beam tip response was practically identical to SM20. Some minor cracks began to form in the surface of the tube at the comer of the tube-to-column weld, and stress levels in the tube around the connection did reach yield, as did both the top and bottom bars (at 0.0055 and 0.0045 radians connection rotation respectively). Connection integrity and its load transfer capability was not breached.

483 For all specimens tested, there was no evidence of bar fracture, pullout or cracking of the fillet weld connecting the bar to the flange. 4. DISCUSSION Reinforcing bars welded off the top and bottom flanges and embedded into the concrete were effective in transferring both compressive and tensile loads directly into the concrete core. Figures 4 and 5 show flange strains measured 25mm from the face of the column on the inside of the flange, and bar strain from the outside bar measured 15 mm from the face of the column, verses connection rotation. Please note that TF and BF stand for top flange and bottom flange strains respectively (Figure 4), and TBO and BBO stand for top outside bar and bottom outside bar strains respectively (Figure 5). All bars reached yield at relatively low levels of connection rotation. For specimens SM12 to SM20 the top bars reached yield before the bottom, with the rotation to cause first yield in the top bars increasing with increasing bar size (0.0018 to 0.003 radians for SM12 to SM20 respectively). The compression bars for each specimen all reached yield at approximately the same rotation (at approximately 0.004 radians). This sequence altered for SM24, where the bottom bars reached yield before the top bars due to the increased connection rotation required to produce top bar yield (0.0056 radians). F l a n g e Strain vs Connection Rotation 2 5 m m from Column Face 1000 1- . . . . . . . . . .. ,; ~ T F _ S M 2 4 800 4- . . . . . -- . . . . : _' -' I'i~ BF $M24 ,..,~ / .._.~A~..~~ -i - " " ~ TF-SM20

m~

400 J

.......

_~_+,t:._x~.._

,. ~..~..7,._~

'

i

: i - ! - BF. SM20

i

!'

!9

'

Outside BarStrain vs Connection Rotation 1 5 m m from Column F a c e [_.__ TBO SM24 .

5000..---

4000 4

.

if ~'

--

.

.

I i/ f~-

I - ~-~';,',

3000-[

.

.

.

.

.

.

i~

i ~--

',

BBO

SM24

i!_._m-s~2o ii-.-~o-~o

! _.,-~--~,~.."----qi~rso_sM~6

im

~,,

0-

.o -20(b.

~ -400

-6oo -800

t'~r~-

!

1

",

'

-

SMI6

14

i

9

,

I

n dna

-~.'-:'-4"'--~~:::: ::...... ~"'::" -"-:;'-~.;

- 1000 1200

._

Connection Rotation {fads)

Figure4(Fiange-str-ain2-5mm from column face.

~ 0.riO6

, -IZ008

.....

i

!

I' 0.010

"

i

\-~-. __'"-. i

Connection Rotation (rads)

Figure 5. Outer Bar Strain 15mm from column face.

Figures 4 and 5 show top flange and bar strains behaving in a linear fashion, until bar yield is reached. At this point flange strains begin increasing at a faster rate and a tear in the tube begins to form at the comer of the flange to tube connection at approximately 700 microstrain. Flange strain reaches a peak value of around 800 micro-strain, which corresponds to the maximum load that the flange to column weld is capable of transferring via its through shear strength (ie 214kN through shear strength using actual tube yield strength cf. 265kN flange load at 800 micro-strain). After top bar yield occurs very little strength increase is achieved (typically in the order of 20%), although the connection still undergoes significant rotation (typically two to three times top bar yield rotation). Bottom flange strains behave in a linear fashion until top bar yield. At this point they tend to level out, however bottom bar strains continue to increase in a linear manner. This is because very little extra tensile load is being produced in the top of the section, to be balance by the compressive zone. The larger plateau strain for each specimen in the bottom flange relates to

484 the higher load being supported by these specimens at the same connection rotation. 5. CONCLUSIONS The extemally stiffened bar connection detailed above was tested monotonically, with varying sized reinforcing bars used in each specimen. The following conclusions were made. 9 As bar size increased, so too did connection strength and stiffness. Increases occurred up to a point where the connection was strong enough to form a plastic hinge in the beam, outside of the influence of the bars and damage to the column wall was avoided. It was shown that such a connection can be designed with adequate strength and stiffness to classify it as a full strength rigid connection. 9 The bars were very effective in transferring both tensile and compressive loads directly into the column core. 9 The onset of top bar yield coincided with a reduction in the stiffness of the connection, and an increase in beam flange and column wall strains. Increasing bar size improved the tensile capacity and stiffness of the flange connection, increasing the connection rotation required to cause tension bar yield by pushing the neutral axis higher in the connection. This study will now lead into further tests on similar connections, using cyclic load application, to assess their suitability for application in areas of high seismic risk. Analytical testing will complement the experimental testing to assess the effects of such variables as concrete strength, tube thickness (D/t ratio), beam depth etc. Refer to Beutel et al (2000) for a full description of all of the experimental testing and results, conducted in this project.

REFERENCES Alostaz, Y. & Schneider, S. (1996) 'Connections to Concrete Filled Tubes', llth Worm Conference on Earthquake Engineering, Acapulco. Ashadi, H.W. & Bouwkamp, J.G. (1995) 'Behaviour of hybrid composite structural earthquake resistant joints', l Oth European Conference on Earthquake Engineering, Rotterdam, pp. 1619-1624. Azizinamini, A. & Shekar, Y. (1995) 'Design of through beam connection detail for circular composite columns', Engineering Structures, Vol. 17, No. 3, pp. 209-213. Beutel, J.G., Thambiratnam, D.P. & Perera,N.J. (2000) 'Experimental testing of steel beam to composite column connections under monotonic and cyclic loads', Physical Infrastructure Centre Research Monograph 2000-2, Queensland University of Technology, Brisbane, Australia. Beutel, J.G., Thambiratnam, D.P. & Perera, N.J. (1998) 'On the behaviour and design of composite column to beam connections under seismic loads', Tubular Structures V l l l Proceedings of the Eight International Symposium on Tubular Structures, Singapore, pp.615625. Bjorhovde, R., Brozzetti, J., and Colson, A. (1990) 'Classification System for Beam-Column Connections', Journal of Structural Engineering, Vol. 116, No. 11, pp. 3059-3076. Choi, S.M., Shin, I.B., Eom, C.H., Kim, D.K. & Kim, D.J. (1995) 'Elasto-Plastic Behaviour of the Beam to Concrete Filled Circular Steel Column Connections with External Stiffener Rings', Building for the 21st Century - Proceedings of the Fifth East Asia-Pacific Conference on Structural Engineering and Construction, Griffith University, pp. 451-456.

485 Kanatani, H., Tabuehi, M., Kamba, T., Hsiaolien, J. & Ishikawa, M. (1987) 'A Study on Concrete Filled RHS Column to H-Beam Connections Fabricated with HT Bolts in Rigid Frames', Composite Construction in Steel and Concrete - Proceedings of an Engineering Foundation Conference, New Hampshire, pp. 614-635. Kato, B., Kimura, M., Ohta, H. & Mizutani, N. (1992) 'Connection of Beam Flange to Concrete-Filled Tubular Column', Composite Construction in Steel and Concrete I I Proceedings of an Engineering Foundation Conference, Missouri, pp. 528-38. Shakir-Khalil, H. (1994) 'Beam connection to concrete-filled tubes', Tubular Structures VIProceedings sixth International Symposium on Tubular Structures, Melbourne, pp. 357-364.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

487

Strength a n d D u c t i l i t y of B o l t e d C o n n e c t i o n s in N o r m a l and H i g h Strength Steels A. Aalberg and P. K. Larsen Department of Structural Engineering, Norwegian University of Science and Technology N-7491 Trondheim, Norway This paper presents results from static tests on bolted connections in normal structural steels and in high strength steel grades up to $700 (yield strength 700 N/mm2). The primary objective of the study is to compare the behavior of connections in high strength steels with identical connections in conventional structural steels. Three types of tests are carried out. The first series is on a simple tension connection with splice plates and one row of bolts, where the failure was a fracture of the net section through a bolt hole. The other two series are on I-beam web connections loaded in shear (using a coped beam end) and in tension, respectively. These connections were designed to fail in block shear, i.e. with a tearing out of a block of material through the bolt holes in the web. The observed failure mechanisms involved a rupture of a net tension plane and significant inelastic deformations along a gross shear plane. Connections with the same geometry showed almost identical failure modes independent of the steel grade, but a noticeable reduction in connection ductility was found for the connections in high strength steel. The block shear test results are compared to the capacities predicted by the specifications in Eurocode 3 [1], AISC LRFD [2] and CAN/CSA [3]. The three different design procedures are shown to provide a quite reasonable and generally conservative level of accuracy for the connections in normal steel, while only the CAN/CSA procedure is adequate for the high strength steel.

1. INTRODUCTION Reduced material costs and improved mechanical properties and weldability of modern high strength steels have lead to increased use of such steels in Europe. Steel grades $420 and $460 are commonly used in bridges and other long span structures, and are also used in particular applications in the Norwegian offshore industry. Higher grades, $700 to S 1100 have been used for cranes, special purpose vehicles and mobile military bridges, and also in mobile jack-up platforms in the offshore industry. Today, plates of grades up to $700 and rolled sections in grades up to $500 are available for use in civil engineering structures. Current design rules are mainly based on experimental data for normal structural steels. Based on supplementary studies the rules for statically loaded structures in Eurocode 3 (EC3) have been extended to grades $420 and $460 with only minor modifications. When considering such extensions the main concern is the effects of reduced fu/fy ratio and elongation of the material on the structural ductility and safety, particularly in connection design. For that reason the intention of the present investigation is to provide some experimental insight to the behavior some typical connections in grades up to $700.

488 2.

TENSION

CONNECTION

WITH

SPLICE

PLATES

In order to study the failure modes of tension members, i.e. yielding in gross section vs. rupture in net section, tests were carried out on specimens with a coupon shape. Here, two plates with grip ends were connected using two splice plates and 3 bolts at each side of the splice. The geometry of the specimens is shown in the inset in Figure 2. In order to vary the ratio between gross and net section, specimens with bolt diameter 20 mm, 24 mm and 30 m m were tested, using a hole clearance of 2 mm. With a plate width of 110 mm this gave a ratio of the gross to net area Agro~An= ranging from 1.25 to 1.41. The threaded part of the bolts was not within the connection thickness. The bolts were tightened to a torque approximately 200 Nm. The loading rate was 2 mm/min, and the connection displacement was measured between the grip ends [4]. In all tests failure occurred in the net section of the main plate through the bolt hole closest to one of the grip ends, Figure 2. The steel grades $235, Raex 420 HSF, Weldox 460 and Weldox 700 were used, all delivered as plate material. A nominal thickness of t=12 mm was used for the $235 and Raex 420 tests, while t=10 mm was used for Weldox 460 and Weldox 700. All specimens of a given grade were manufactured from a single plate. It should be noted that the grades $235 and RAEX 420 are standardized according to EN10025 and EN10204 respectively, while Weldox is a trade name and the steels are produced by SSAB according to their own specifications. Weldox 460 is a TMCP steel while W700 is a QT steel. 900

I

800

-

700

-

..-, 600

-

rt

:~ 5 0 0

I

~

i

Steel

dox 700

Nominal

fy/fu

Weldox 460

r

300

~

$235

I

~--

~176

200

Raex 420

~_

0

0

i

I

I

J

I

5

10

15

20

25

Strain [ % ]

235 / (340470)

Raex

Weldox

Weldox

420HSF

460

700

420 / (490620)

460 / (530720)

700 / (780-930)

[N/mm2] Measured 290/441 401/495 472/556 820*/873 fy/f~ [N/ram2] Measured e, [%] 18 18 12 6 Ratio 1.52 1.23 1.18 1.06 f,/fy

-

t~

ca 4 0 0 4--,

$235

30

Figure 1. Stress-strain curves and values of mechanical properties (* = f02 ). The mechanical properties of each steel grade were determined from standard test coupons having a rectangular cross section with the full plate thickness and a width of approximately 2 times the thickness. Coupons were taken both parallel and normal to the rolling direction, but no significant differences in the strength parameters were observed for the two directions. Typical stress-strain curves for the four steels are shown in Figure 1. Somewhat unexpected, the stress-strain curves for the $235 grade displayed no pronounced yield plateau, while the Raex 420 and the Weldox 460 steels both showed a considerable yield point elongation, and an total elongation at fracture in the same order of magnitude as the $235 steel. The Weldox 700 steel had no distinct yield point and only half the elongation of

489

800 I . . . .

i .... i''"i

.... I .... I""

I

700

/~'~eldox

600

700

/,

Raex 420

(

'-' 5OO Zse

_ -

.s235

"'%t

\

o

u. 300

//,'1~1-1

.... I-o o o : - ; o .' t

o-l~-&~ l, I

m F

100

~ Net section failure

0

0

5

10

15

Displacement

20

[ mm ]

25

Bolt hole diam. [mm]

$235

400

200

Steel grade

30

$235 Raex 420 Raex 420 Raex 420 Weldox 460 Weldox 460 Weldox 460 Weldox 700 Weldox 700 Weldox 700

22 26 32 22 26 32 22 26 32 22 26 32

Mean ratio

Mean Mean displ, at displ, at ultimate failure force [mm] [mm] 23 23 20 19 17 15 15 15 14 11 10 9

27 27 26 23 22 21 20 19 19 15 14 13

o./fo

,

1.034 1.061 1.030 1.045 1.048 1.043 1.037 1.049 1.028 0.980 0.995 0.986 ,

.............

Figure 2. Force-displacement curves for connections with 26 mm holes (Weldox 700 and Weldox 460 in t=10 mm, Raex 420 and $235 in t=12 mm), along with table with test results. the others. The nominal values of the mechanical properties, as given by the producers, and the measured mean values, are summarized in the table in Figure 1. Three companion specimens were tested for each geometry and steel grade, giving a total test program of 36 tests. The repeatability of the tests was very good, and typical response curves are depicted in Figure 2 for connections with 26 mm bolt holes. The curves typically show some initial sliding, followed by a region with approximately linear connection stiffness. In the subsequent inelastic part the maximum force (ultimate load) occurred at a displacement (corrected for sliding) ranging from about 10 mm (Weldox 700) to 23 m m ($235). All tests gave failure at the net section as depicted in the figure. The initial stiffness varies slightly from one specimen to another due to the difference in bolt position within the holes, and the small misalignments of the connections, as no attempt was made to achieve perfect alignment of the bolts or the connection during bolt tightening and installation. The Raex 420 and Weldox 460 response curves display a plateau that corresponds quite well with the force needed to initiate yielding at the net section. Note that the two grades have almost the same yield force due to the fact that the difference in plate thickness almost balances the difference in yield strength. The results of the entire test program are given in the table in Figure 2. Here, the values in each row represent the mean values of the three companion tests, ou is the computed mean tensile stress in the fractured section at the ultimate load and fu is given by the coupon tests. The ratio o , / fu may be denoted the "efficiency" of the connection at failure, and varies between 1.034 to 1.061 for the $235 steel and between 0.980 to 0.995 for the Weldox 700 steel. The ratio remains almost constant for each grade, independent of the bolt size, which indicates that the connection efficiency is independent of the ratio Agross/Anet- The presence of a 3-D state of stress caused by the stress redistribution around the hole and the contact stress between bolt and hole explains why the mean stress for most materials exceeds the ultimate strength fu. As the axial strain is non-uniformly distributed across the width of the specimen, it is also expected that the connection "efficiency" will be reduced with increasing steel strength due to the reduced ductility of the material.

490

3. BLOCK SHEAR Block shear is a potential failure mode in bolted connections for tension members, coped beams and gusset plates. The block shear failure is characterized by a tearing out of a block of material in combinations of tension and shear failure planes through the bolt holes (Figure 4). The ability to redistribute stresses along the failure planes will determine the overall connection ductility, and the reduced fu/fy ratio and elongation of the high strength steels will be more important than for the simple splice connection described above. Previous studies on block shear are given in references [5] to [9], and include tests on thin gusset plates, beam web shear connections with coped I-beams and angles connected by one leg. These tests are, however, mostly limited to steels with a yield strength less than 310 N/mm 2 [8,10]. EC3 [ 1] considers a shear block failure mechanism involving "a tensile rupture along the line of fastener holes on the tension face of the hole group, accompanied by gross section yielding in shear at the row of fastener holes along the shear face of the hole group". In principle, the resistance is given as by the formula in Figure 3. The AISC LRFD [2] procedure is based on two possible failure mechanisms; either shear yielding combined with tension rupture or shear rupture combined with tension yielding. In the equivalent CAN/CSA [3] provisions the resistance is based on rupture in both shear and tension areas. Note that the three specifications employ different resistance factors; the partial safety factor )M0=l.1 in EC3 and the resistance factors ~ = 0.75 in AISC and 0.85. r = 0.85.0.9 = 0.765 in CAN/CSA. It should further be noted that the design models do not directly account for effects of possible eccentricity of the load acting on the bolt group or the shear block. EC 3

AISC LRFD

R = fu Aa,ne,

R = 0.6fy &,gross + fu Aa,net if fu Aa,ne, > 0"6fu &,net

fy + ~ At'gr~

R=O.6 fuAr.ne ' + f yAa, e,ross if 0.6 f uAr,ne t > fuAtr,ne,

CAN/CSA

R = 0.6fu At,net + fu Ao',net

Figure 3. Block shear resistance formulas. As rolled sections in grade Weldox 700 are not available, the present tests were carried out on welded beams with I-sections. The two beams had 20 mm thick flanges in grade $275, while the webs were made from 8.40 mm thick plate in grade $355 or 7.72 mm thick plate in Weldox 700 (thorn= 8 mm). The strength values were fy = 373 N/mm 2 and fu = 537 N/mm 2 for $355, and f02 = 786 N/mm 2 and fu = 822 N/mm 2 for Weldox 700. Two load situations were considered. Series I consisted of a shear loaded bolt group in the web of a coped beam end, while in Series II a bolt group was loaded in tension. A schematic view of the cross-section of the beam and the loading arrangements are given in Figure 4. All tests were carried out under displacement control. In Series I the test beams were connected to a stiff reaction frame by two 12 mm thick shear tabs welded to the frame, and loaded by a concentrated vertical load (P) by means of a hydraulic actuator. The resulting shear force (F) acting on the connection was calculated from the applied load P and the measured support reaction at the far end of the beam (beam span 2.1 m). The vertical connection displacement was measured between the frame and the top flange of the beam. In test Series 1I the load was transmitted to the beam end through two 12 mm thick splice plates, and the relative displacement between web and splice plates was measured be means of a displacement transducer on each side of the web.

491 1-20

I I I

P .

O

O

oo

.

.

.

_

I

i

6)

L A~-

I I

i

L Beam cross-section

!

I

Shear loading

Block shear failure (Test Series II)

Tension loading (Test Series II)

(Test Series I)

Figure 4. Beam section, loading arrangements and typical block shear failure geometry. Series I - Shear loading ,

j

2~

I

,

! ~FI ,.,-i-21 9

I

i

$11 9

"

I

t~

o

I

!

"

'

O

i "

I

! ,,,~-L8oi. I

40

I ,

I I

9

21 ,n

4

I

i .

I

I

Series 11- Tension loading

138

~7.s I o' o'~

i

i

i

L, Tlii

21

oool o ~ I ~

I

r41i

o

""- 191

TSl

........ t'-i

Figure 5. Specimen geometry for block shear tests. The geometry of the test specimens in both series is given in Figure 5. Four tests were carried out for geometry $3, two for each steel grade, to study the repeatability of the experiments. The chosen connection geometry of Series II (T1 toT3) allowed a wide range of A o to A t ratios to be studied. The cut between the two inner holes in geometry T4 was introduced in order to separate the resistance contributions from the shear area A~ and the tension area Ao. Geometry T5 was included in order to assess the restraining effect from the flanges. 20 m m and 18 mm bolts were used in Series I and II respectively, both with a hole clearance of 1 mm, Figure 5. All bolts were of grade 10.9, and were manually tightened to a snug tight condition by a torque of approximately 100 Nm. The threaded part of the bolts was not within the connection. The main test results are presented in Tables 1 and 2 respectively for Series I and II. In Series I the failure mode for all specimens was necking and rupture in the tension face along a horizontal line from the free edge of the web to the center of the bottom bolt hole. The tests were terminated before the shear block was completely torn off the web. For geometry S 1 and $2 there were large ovalizations of the bolt holes and excessive shear deformations along a vertical line next to the holes. The web block limited by the holes and the horizontal rupture line underwent a distinct vertical displacement, while the remaining part of the web below the bolt group showed no distortions. In general the shear deformations were more localized for the Weldox 700 specimens than those in $355. For geometry $3 (with both flanges coped) specimen S-7-$355 developed a 15 mm crack running upwards from the end of the coped bottom flange, but showed the same type of final failure as the others. This shows that for connection geometry $3 the failure mode might as well have been a vertical shear failure across the full height of the web. The force vs. displacement curves for the eight tests in Series I are presented in Figure 6. For all specimens the kink in the response curves corresponds to the fracture of the tension face, and the $355 specimens fracture under increasing load while the Weldox specimens fracture after the ultimate force is reached. Note that the displacement at ultimate force for Weldox 700 is only about 60% of that of $355, but that for geometry S1 and $2 the displacement at onset of failure is almost the same.

492 Table 1. Test data and comparison with design specifications, Series I. Test No.

Geometry

S-I-$355 S-2-Weldox S-3-$355 S-4-Weldox S-5-$355 S-6-Weldox S-7-$355 S-8-Weldox 800

Ultimate force on connect, F,[kN] 401 523 563 716 662 823 636 836

S1 S1 $2 $2 $3 $3 $3 $3

- ''''

i'

Displ. at Displ. ultimate at first force failure [mm] [mm] 11 12 7 11 12 12 7 11 18 18 10 13 21 26 12 15

' ' ' i ....

i ....

EC3

Ratio EC3/test

AISCLRFD

0.83 1.09 0.78 1.09 0.79 1.14 0.82 1.12

[kN] 337.8 541.6 498.4 690.1 565.1 861.4 565.1 861.4

[kN] 332.0 572.6 440.6 782.8 522.0 940.4 522.0 940.4

-

70O

0o(,00//

60O ,--,,

z

500 400

,o

CAN/CSA Ratio Ratio CAN/CSA AISCLRFD/ /test test [kN] 0.84 345.5 0.86 1.04 468.1 0.93 0.89 451.1 0.80 0.96 634.6 0.89 0.85 527.9 0.80 1.05 805.9 0.98 0.89 527.9 0.83 1.03 805.9 0.96

3oo 200 100 0

:

/

//.,::.

~

,

.......

-lie

Io

r

l

~' r,

0

,

I,

5

,

"

o

o

,,

,

I,

i

, , ,

I

=

~ , , -

10 15 Displacement [ mm ]

500

t" /t']'"

~

I:I ~ i"

300

-

!

6oo~ L !///. - - : . ~~" ' .... :~\

'~176

_

o i + Z,S-1-$355-

1 ___!

z

,,.,

i

I ,,

-

-,,/-s-3-s35~

s wo, ox

.

2oo ~

-

.,~

. . . . . . . . . .

_ ", -

,

-

S-7-$355 J

-

-:

100 0

20

0

5

10 15 20 Displacement [ m m ]

25

30

Figure 6. Series I, web connections in shear. Connection force vs. displacement. Table 2. Test data and comparison with design specifications, Series II. Test No.

Geometry

T-I-$355 T-2-Weldox T-3-$355 T-4-Weldox T-5-$355 T-6-Weldox T-7-$355 T-8-Weldox T-9-$355 T-10-Weldox

T1 TI "1'2 T2 T3 T3 T4 (cut) T4 (cut) T5 (cop) T5 (cop)

Ultimate force on connect,

Displ. at Displ. ultimate at first force failure

F~tkN] [mm] 551 730 751 994 925 1229 675 822 710 961

8 4.5 9 5 10 6 17 10 8.5 4.5

[mini 10 6.5 10 6.5 10.5 7.5 20 15 9.5 6.5

EC3

[kN] 437.9 779.9 609.7 1112.7 781.6 1445.6 481.2 * 931.9 * 609.7 1112.7

Ratio EC3/test

0.79 1.07 0.81 1.12 0.85 1.18 0.71 1.13 0.86 1.16

AISCLRFD

Ratio CAN/C~ CAN/CSA Ratio AISCCAN/CSA LRFD/ /test [kN] test [kN] 457.4 0.83 437.1 0.79 722.3 0.99 614.9 0.84 611.6 0.81 591.4 0.79 939.3 0.95 831.9 0.84 765.9 0.83 745.6 0.81 1 1 5 6 . 3 0.94 1049.0 0.85 462.8* 0.69 462.8* 0.69 651.1" 0.79 651.1" 0.79 611.6 0.86 591.4 0.83 939.3 0.98 831.9 0.87

* = contribution from shear areas only (EC3 9f l y / ~ )-A+.g~ms,AISC and CAN/CSA: 0.6"fu'A,,net ) The force-displacement curves for the specimens in Series II are depicted in Figure 7. For all specimens initial failure was due to necking and fracture in the tension face at the inner row of bolts, as indicated by the drop in the response curves. The remaining resistance was provided by the shear faces only. Inspection of the specimens showed that the shear failure occurred along a horizontal line "touching" the holes (Figure 4). For both the $355 and Weldox 700 specimens the ultimate load was reached prior to tension failure, but the displacement at ultimate load for Weldox 700 was only about half that of $355 (Table 2). The tear-out of the web block resulted in a splitting force in the web that caused bending in the

493

I000 9 0 0 I . . . . i .... . .- -'\~ / - T.- .5 -.$.3 5 5~'"' ~l' '' 51 /s S "

'-~ ,

800 700

ooo

!r

~176 i ,

100 0

. . . . . . ./-T-3-$355 " " - - - ~. *~ . . . . . . . . i '9,,;I ..-..-'" ........ 3__

oo ,

200

, , , , I 0

~. . . . . . . . .

.'-i .

. . 00~ i i

.... 5

.

... . ..

I ....

.

,

i i

._ T - 7 - $ 3 5 Z...with cut _~ -,,

...... .

.

o~

1300 1200 ~ 1100

~.

x, ,i i

ooo,

,

0~176i i

1000 900

~ F

~T-4-~Neldox J

,oo

25

0

3oo 200 100 0

i/_

-_-

\\

,oo'~176

1

I,~,~lt,,

10 15 20 Displacement [ mm ]

""

T-8-Weldox-

-

..

I=: i I:=:! I:==:i I~oo ~

i

5

Z I

10 Displacement

,

L~j

15 [ mm ]

,,

oo~ !1:::

L_~;

20

Figure 7. Series II, web connection in tension. Connection force vs. displacement. beam flanges as the applied load reached the ultimate value. In general, the largest bending deformations occurred for the $355 specimens, with inelastic deformations in the order of 3-4 mm. For beams with coped flanges, geometry T5, the transverse deformations were about twice this value. Tables 1 and 2 present the experimental results and the "characteristic" design resistance given by EC3, AISC LRFD and CAN/CSA. Note that the material factor 7 M0=l.1 (EC3) and the resistance factors~(AISC) and0.85~ (CAN/CSA) are not included. All calculations are based on measured values of web thickness, yield stress (f02 for Weldox 700) and ultimate strength. Significant discrepancy between test results and design resistance is observed, both with respect to steel grade, specimen geometry and design models. Disregarding geometry T4, the EC3 prediction ranges from 78% to 86% of the measured values for $355 specimens and from 107% to 118% for Weldox 700 specimens, i.e. a considerable overestimation for the latter grade. The corresponding ranges of AISC are 81% to 89% ($355) and 94% to 105% (Weldox 700). The CAN/CSA model ranges from 79% to 86% for $355 and 84% to 98% for Weldox 700, and does not in any case exceed the measured value. It should be noted that for all specimens the governing AISC equation combines shear fracture with tension yielding, which is in disagreement with the failure modes observed in present tests. Orbison [9] has made the same observation for specimens with single angles. It may be concluded that the CAN/CSA model most closely represents the actual failure mode, and that any model that uses f0.2is inappropriate for very high strength steels. A number of conclusions can be drawn from the results of Series II, Figure 7: 1. Fracture of the tension face occurred at approximately the same displacement, 10 mm and 6-7 mm for $355 and Weldox 700 respectively, independent of the number of bolts in the connection. 2. The reduction in connection resistance measured at tension face fracture agrees quite well with the computed value for the tension face resistance (f, "Aa, net), and it is here not feasible to account for the difference in connection "efficiency" as found in Section 2. 3. For geometry T4, which has a cut between the two inner holes, the shear faces provide the entire connection resistance, and the ductility as given by the displacement at ultimate load is almost twice that of the other specimens. The response curves show that at least 90 % of the ultimate shear strength is mobilized when tension face fracture occurs, a fact which justifies a design model that adds the full resistance in both shear and tension. It can be shown that the shear resistance not yet mobilized at this point is less than 6% of the

494 total connection resistance. Furthermore, the predicted resistance, taking the contribution along net shear area equal to 0.6"fu'A~,net, underestimates the actual ultimate force by about 20%. However, when replacing At,net in the model with the actual fracture area as measured on the failed specimens (Figure 4), very good agreements is obtained between experiments and predictions. 4. The response curves in Figure 7 show that coping the specimen results in a reduction in the ultimate force of 5.4% for $355 and 3.2% for Weldox 700. The shape of the curves is almost identical, and the percentage reduction is almost constant throughout the test. This implies that the effect of coping is the same both for the pure shear resistance and the ultimate load. A connection "efficiency" may be defined as ~=Fu/(fu. t), all based on measured values. By comparing ~ for specimens with identical connection geometry and loading situation, it can be shown that the efficiency is about 6-10% less for Weldox 700 than for $355 for Series I and II. For the tensile tests presented in Section 2 the efficiency reduction from $235 to Weldox 700 was approximately 5%. Outside the field of earthquake engineering the literature gives little specific information on reliability based requirements for the ductility of connections. In their investigation of bearing strength Kim and Yura [ 11 ] found that specimens with low fu/fy ratio had deformation capacities similar to those with a high ratio. They also noted that a displacement requirement of 6.35 mm (~/~ in) was used in the calibration of AISC bearing strength formula. However, this appears to be an ad hoc value more determined from practical consideration than from an overall assessment of structural ductility. It is the opinion of the authors that expanded use of high strength steel such as Weldox 700 requires a more firm basis for determining the ductility requirements.

REFERENCES 1. 2.

Eurocode 3, Design of Steel Structures, Part 1.1, ENV 1993- l- 1: 1992. Manual of Steel Construction, "Load & Resistance Factor Design (LRFD)", Vol. II Connections, American Institute of Steel Construction (AISC), 1995. 3. Canadian Standards Association, CAN/CSA-S16.1-M89 "Limit States Design of Steel Structures", 1989. Aalberg A. and Larsen, P. K., Strength and ductility of bolted connections in normal and high strength steels. NTNU report, Department of Structural Engineering, March 1999. 5. Hardash S. and Bjorhovde R., New Design Criteria for Gusset Plates in Tension, Engineering Journal, AISC, 22(2), 1985. 6. Birkemoe P. C. and Gilmor M. I., Behaviour of Bearing Critical, Double-Angle Beam Connections. Engineering Journal, AISC, 15(4) 1978. 7. Ricles J. M. and Yura J. A., Strength of Double-Row Bolted-Web Connections. ASCE Journal of the Structural Division, 109(ST1), 1983. 8. Gross J. M., Orbison J. G. and Ziemian R. D., Block shear tests in high-strength steel angles. Engineering Journal, AISC, 32(3) 1995. 9. Orbison J. G. et al., Tension plane behavior in single-row bolted connections subjected to block shear. Journal of Constructional Steel Research (49), 1999. 10. Cunningham T. J. et al., Assessment of American block shear load capacity predictions. Journal of Constructional Steel Research (35), 1995. 11. Kim, H. J. and Yura J. A., The effect of ultimate-to-yield ratio on the bearing strength of bolted connections. Journal of Constructional Steel Research (49), 1999. .

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

495

Evaluation o f b e a m - t o - c o l u m n connections with weld defects based on C T O D design curve approach K. Azuma a, Y. Kurobane a and Y. Makino b aDepartment of Architecture, Kumamoto Institute of Technology, Ikeda 4-22-1, Kumamoto 860 0082, Japan bDepartment of Architecture and Civil Engineering, Kumamoto University, Kurokami 2-39-1, Kumamoto 860 8555, Japan

This paper concerns the assessment of significance of weld defects in beam-to-column connections. Four full-sized beam-to-column connections with weld defects were tested under cyclic loads. When the unfused regions created by partial joint penetration groove welds were reinforced by fillet welds so that the welded joints have a sufficient cross-sectional area, ductile cracks grew stably and, in consequence, the connections showed sufficient deformation capacity. The connections with weld defects at the root of welds sustained a quick extension of ductile cracks and, eventually, failed by brittle fracture. Test results were reproduced well by non-linear FE analyses. Fracture toughness properties of numerically modeled weld defects were evaluated by using a recently developed fracture mechanics approach (Ref. 1, 2). The results of evaluation were found to correspond well with test results.

1. INTRODUCTION Brittle fracture occurred at welded beam-to-column connections in the steel moment resisting frames during 1995 Kobe Earthquake. Some of the failures were caused by cracks growing from the comer of cope holes, as was predicted prior to the earthquake, or weld tab regions close to the beam bottom flange. It was recommended to use improved profiles of cope holes or to avoid using steel weld tabs after the earthquake, but it is still difficult to eliminate weld defects. Therefore it is important that the influences of the weld defects on the integrity of welded joints are assessed to determine a tolerable flaw size in quantitative terms. This study place emphasis on the assessment of susceptibility to brittle fracture from weld defects. 2. CYCLIC TESTING OF BEAM-TO-COLUMN JOINTS WITH WELD DEFECTS

2.1 Specimens and Loading Procedures Four full-size beam-to-column connections, two with wide flange section eolurrms in grade SS400, designated as BH-1 and BH-2, and two with box section columns in grade STKR400, designated as BS-3 and BS-4, were tested. All the specimens were made of one-sided connections with wide flange beams in grade SS400. Each of the beams was reinforced by welding a cover plate on the top flanges. BH specimens had partial joint penetration groove welds at the ends of the beam bottom flanges, while BS specimens had those discontinuities at the roots of the welds to the beam bottom flanges, which were created by inserting steel plates into the grooves before welding. The configuration of the specimens and details of welded

496 joints are shown in Figure 1. Cyclic loads in the horizontal direction were applied to the end of the beams, while the both ends of column were fixed. The amplitude of the beam rotation was increased as 20p, 4~, 60p, .-., where Opis defined in section 2.3. 2.2 Material Properties and Charpy Impact Test

The material properties, in terms of engineering stress-strain, were obtained by tensile coupon tests for the beam, diaphragm and cover plate materials, which are summarized in Table 1. The fracture toughness was obtained by Charpy impact tests. Test pieces were taken from plates welded under the same welding conditions as those for the specimens. The positions of notch roots were at base plate, DEPO (deposited weld metal) and weld bond. The results of Charpy impact test are shown in Table 2.

16 19

'

k---

9

[--7

19

7

~ ~

X-X' Section - - ~ "

E

Weld Defect

[ o,! ~ !

I0 Wel, Deft

X

L-~mm to, BH-! S ~ i m ~ .

\11 I~ i

L=8mm for BH-2 Specimen ~.x~ ~

/'

--M -L

I

J rqr

, w

f

L

[ r~

..0 .

,,

•

3s%

,.0

~.._I . . . . . . . . . . . . . . . . . . . . .

L

J

,.o

-

.... 9

: i El-S00xS00x22 ;----; L--.---. ......... ,_..I._.~

•

J

..0

.a

BH specimens

BS specimens

Figure 1. Specimen configuration

Table 1 Result s of tensile coupon tests

Table 2 Results of Charpy impact test vEo

t Oy o, E.L. E (mm) (MPa) (MPa) (%) (GPa)

O)

vE~hclf

(J)

|

Beam (BH) Beam(BS) Diaphragm Cover plate

19.49 259.9 19.38 251.3 24.52 355.6 15.46 377.4

454.7 453.1 528.9 534.8

29.9 29.4 27.7 26.4

204.7 204.2 207.4 208.0

Note: t = Thickness of test pieces . . . . . Oy = Yield stress E.L. = Elongation

6. = Tensile strength E = Young modulus

vT,~

(~

vTr~

(~

,i

Base

45

99.4

3.6

23.6

DEPO

100

144.8

Bond

110

143.1

-8.9 - 15.2

-5.7 - 12.3

Note: vE0 - Absorbed energy at 0~ vEshelf= Shelf energy

vTr, = Energy transition temperature vT~ = Fracture surface transition temperature

497 Table 3 Cumulative plastic deformation factors

2 1.5

I],+

~,"

Eli,+

Eli,"

BH-I BH-2 BS-3

33.2 27.5 8.3

16.8 13.9 8.0

53.4 42.4 23.0

22.6 10.5 10.5

BS-4

5.5

8.3

6.7

10.7

Specimen

1

~

:

i

O

!..

i

:

:

,

,

:

i

,,

i

~

.

~

-1 -1.5 _~

:

.....

-0.5

Note: 11, = Total Plastic Rotation sVl, = Plastic Energy + = Tension Side - = Compression Side

:

~. 0.5

.

i

i

;

-20

-10

:

,,

-30

0

10

'

- - - BH-1 BH-2 - - - BS-3 BS-4,, , 20

30

40

O/Op

Figure 2. Moment vs. rotation skeleton curves

2.3 Failure M o d e s and D e f o r m a t i o n C a p a c i t y

BH specimens failed due to combined local and lateral buckling of the beams. Ductile cracks extended from the weld toes and defects stably, until the rotation of the beams reached 1/6 radian. BS specimens failed due to a tensile failure of the bottom flanges. Both ductile and brittle crack extensions were observed. Moment vs. rotation skeleton curves for all the specimens are shown in Figure 2, where full plastic moment Mp was calculated using the measured yield strength of the materials. The beam rotations at full plastic moment 0p, namely Mp divided by the elastic stiffness of the beam, were calculated. The elastic stiffness was determined by using the slope of unloading portions in the hysteretic curves. The cumulative plastic deformation factors were obtained from hysteresis curves for all the specimens and are shown in Table 4. BH specimens showed much stabler moment vs. rotation behavior than BS specimens. However, skeleton curves for BS specimens were quite identical to those for BH specimens until the former specimens reached the maximum loads at about 0.05 radians. 2.4 Fig A n a l y s i s A fmite element analysis and post-processing were carded out using the ABAQUS general-purpose finite element package. The models were constructed from 8-noded linear 3D elements. The plasticity of the material was defined by the yon Mises yield criterion. The isoparametric hardening law was used for this analysis. The material data in the analysis were calculated from tensile coupon test results. Mesh models were generated for half of the specimens because of symmetry in configuration. The weld defects were produced by the nodes in the defect area on the contact surfaces between the beam flange and the column flange. Static load was applied to the beam end and the load-deformation curves were compared with the skeleton curves that were obtained from experimental results. Figure 3 shows the test and analysis results. Each analysis reproduced test results well. Figure 4 shows the contour plot of equivalent plastic strain around the weld toes when the deformation reached the f'mal failure stage in BH-1 specimen. The strain concentrated at the weld toes at both edges of beam flange. Table 4 shows ultimate equivalent plastic strain at the defects and at the positions 175ram distant from the root surface. The strain obtained from FE models without defects are shown in parenthesis. For BH specimens, the strain at positions 175ram away from the root face are as great as four times the strain at the defects and are influenced by local buckling. Ductile or brittle failure occurred in BS specimens because the strain concentrated at the defects more significantly as compared with BH specimens.

498 Table 4 Ultimate local strain obtained from FE an.alysis results

E

....

_ _

BH-1

BH-2

BS-3

BS-4

Defect 175mm (X104~.1.) (x1041.1.)

Defect 175mm ..(x104~t) (x104~1.)

Defect 175mm (X104~1.) (X104~1.)

Defect 175mm (x104~1.) (x104~1.)

16.7 (2.39)

8.79 (1.31)

2.38 (1.02)

12.3 (11.8)

3.58 (1.02)

,

Figure 3. Moment vs. rotation curves

12.7 (11.8) ,,

4.31 (4.94) ,

,

,

,,1

,,,,,

,

,

2.33 (3.03)

Figure 4. Contour plot of equivalent plastic strain

3. ASSESSMENT OF WELD DEFECTS

3.1 Assessment Procedure Fracture toughness properties of the beam-to-column joints in full-scale specimens and of steel building structures, which sustained brittle fractures during earthquakes, were assessed by using a recently developed fracture mechanics approach (Ref. 1,2). The same approach was applied to the numerically modeled weld defects to evaluate the fracture toughness properties of four specimens. The assessment procedures are given as follows: 1. Evaluation of the equivalent CTOD using equivalent flaw size and local strain taking into account the effect of strain concentration 2. Determination of equivalent temperatures using skeleton strain, in which temperature elevation due to plastic strain cycling is also taken into account 3. Evaluation of absorbed energy at equivalent temperature and estimation of transition temperature based on Charpy impact test results 4. Evaluation of required fracture toughness 5. Comparison between required fracture toughness and fracture toughness of materials 3.2 Evaluation of equivalent CTOD Equivalent flaw size ~ was defined as the major radius of equivalent semi-ellipse for a surface crack, or as the half crack length for a through crack. ~ was calculated from J-integral at crack tip under small scale yielding. Critical CTOD was evaluated by using CTOD design curve given as the following equation (Ref. 3).

,~c=eya-~ 9 -Ey- - 5

)

(3.1)

499 in which e,y is the yield strain and 6 is the local strain, e is the average strain at points where assessment is made with the assumption that neither defects nor cracks exist. In this paper, local strains are defined as the skeleton strains around crack tips obtained by FE analyses. The material toughness 8c in equation (3.1) was obtained from three points bending test using SENB (simple edge notched bend) specimens. SENB specimens may be subjected much greater plastic constraint at the crack tips as compared with tips of surface cracks (Ref. 4). Therefore critical CTOD of a wide plate under tensile loading was over-estimated. Equivalent CTOD was defined by the following equation (Ref. 2). c~,q = 0.26~

(3.2)

3.3 Evaluation of equivalent temperature The equivalent temperature is obtained from the following equations.

=r-

L Sskeleton _<100,O00,U 100,000/Z _<e.~k~,,o,, <_ 200,000/Z

ATA = 3r sk~t.~. A T A = 1.65e~k,t.,o. + 13.5

(3.3)

in which T is the specimen temperature before test, ATA is the rate of increase in temperature during strain cycling, e.~l.to,, is the skeleton strain. Skeleton strains correspond to ultimate local strains obtained by numerical models without defects, which are shown in Table 4.

3.4 Correlation between fracture toughness and Charpy absorbed energy Correlations between fracture toughness and Charpy absorbed energy are obtained from the following equations (Ref.5). 6(T)= 0.001v E(T + AT) AT = 133 - 0.125try - 6 ~

(3.4)

in which 8(T) is the critical CTOD at T (~ ~ , @ + D T ) is the Charpy absorbed energy at T+AT (~ t is the thickness. Charpy energy transition curve are obtained from the following equation. vEsnet: e

(3.5) +1

in which vE,shelf is the shelf energy obtained from Charpy impact test results, vTE is the transition temperature, b is the material constant. Table 5 Evaluation results on brittle fracture Specimen Positions ~ t Oy I~L ~c ~eq •skelcton T (mm) (mm) 0Vff'a) (x104~t) (mm) (mm) ~x104~t) (~ 0.012 Defect 1 . 8 9 41.2 355 1.02 0.062 1.02 10 BH-1 0.062 Crack 0 . 9 1 19.5 355 17.8 0.571 0.114 17.8 10 i

BH-2 BS-3 BS-4

Defect

3.24

39.9

355

Crack Defect Crack Defect

1.14 19.5 8 . 0 0 19.4 50.0 8 . 8 7 19.4

355 355 355

1.02

0.106 0.021 1.02 0.106 17.4 0.697 0.139 17.4 2.39 0.649 0.130 2.39 1.17 1.898 0.380 1.17 1'31 0.381 0.076 1.31

i

ATA Teq vE(0) (~ (~ (J) 0.10 3.06 6.94 0.81 42.9-32.9 39.8

10 3.06 6.94 0.18 2.64 10 42.2-32.2 75.8 10 7.14 2.86 3.64 3.51 6.49 >200 10 3.93 6.07 0.46 |l

500

3.5 Assessment

The evaluation results on the susceptibilityto brittlefracture are shown in Table 5. Since the partialjoint penetration groove welds in B H specimens created discontinuities at the root of the welds and these discontinuities formed internal defects because the roots of the welds were reinforced by additional filletwelds, plastic constraint at defects is grater than that at surface flaws. Equivalent CTOD's obtained from both of equation (3-I) and (3-2) are shown in this table. The contour plot of FE analysis of B H specimen showed that the beam flanges sustained small average strains and small strain concentration at defects. ~E(O) is less than the fracture toughness of D E P O shown in Table-2. These results show that brittle fractures would not occur. Strains concentrated at weld toes at the edges of the beam flanges rather than at defects. The required fracture toughness at the tips of ductile crocks initiated from weld toes arc greater than that of the defect. These results again show that brittlefractures would not occur. Since the brittle fracture for BS-3 specimen occurred after stable ductile crack growth across the flange plate, the fracture assessment using initial size corresponds with the test result. However, fracture assessment using the local strain at the through crack created by ductile crack growth, where vE at equivalent temperature is larger than fracture toughness, suggests occurrences of brittle fracture. This result also corresponds with the test result. In this latter case, since the transition temperature exceeds the available range of the equation (3.5), it is assumed that ,~(0) is grater than 200J, which can be interred from a diagram in Rcf. 5. For the fracture assessment using the flaw size of BS-4, ~E(0) is less than fracture toughness of DEPO. The flaw size was large enough so that the ductile cracks grew stably resulting in reduction of applied loads; in consequence, ductile failureoccurred from defects.

4. CONCLUSIONS A new fracture assessment method was examined for four beam-to-column connections using strains obtained by FE analyses. The method was found to be applicable to the assessment of the brittlefracture. However, it is stilldifficultto assess other kinds of defects quantitatively. Further experimental verificationsto evaluate the fracture toughness of various joints with weld defects arc required to make this assessment method more reliable. REFERENCES

1. M. Toyoda, Problems to Materials for Avoiding Failure of Steel Framed Structures under Heavy F_aghquake. Document for IIW JWG on Brittle Fracas, Paris, France, 1998 2. H. Shimanuki, M. Toyoda and Y. Hagiwma, Fmctm'c Mechanics Analysis of Damaged Steel -Framed Stmcan~s in Recent Earthquakes. Proc. Int. Conf. Welded Construction in Seismic Areas, Hawaii, U.S.A., 15-26, 1998 3. JWES, Method of Assessment for Flaws in Fusion Welded Joints with Respect to Brittle Fracture and Fatigue Crack Growth, JWES 2805-1997, Japan Welding Engineering Society, 1997 4. E Minami, M. Ohata, M. Toyoda and K. Arimochi, ~ o n of Required Fracture Toughness of Materials Considering Transferability to Fmctm~ Performance Evaluation for Stmcaa~ Components, -Application of Local Approach to Fracture Control Design-, J1. Naval Archit. Japan, 647-657, 1997 5. JWES, Evaluation Criterion of Rolled Steels for Low Ternperaalrc Application, JWES 3003-1995, Japan Welding Engineering Society, 1997

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

501

Simulation of fracture failure of steel beam-to-column connections* Y. Chen, Z.D. Jiang and Y.J. Zhang College of Civil Engineering, Tongji University 1239 Siping Road, Shanghai 200092, China

ABSTRACT Fracture of beam-to-column connection is one of the damage pattems of steel frames in severe earthquake. In this paper a numerical analysis model simulating the phenomenon of the crack damage is introduced. Numerical computation results are compared with the laboratory tests and the frame response analysis considering fracture effect is performed.

I. INTRODUCTION Fracture is the most noticeable features of the damage of steel frame st~actures during the severe earthquake strikest~.2]. This kind of the damage was mainly concentrated on the beamto-column connections. Though the fracture on columnst21 shocked structural engineers much, for the lack of necessary data from investigation, the mechanism of the damage has not been understood clearly. Many research projects had performed on the issues of the fracture failure of steel frame building structures before this kind of damage occurred in Northridge and Hyogokan-Nanbu earthquakes. Since then, more related studies have being carried out on the mechanism of the cracking on steel as well its welds[3], on the practical details for beam-to-column cormections[4] and on the 'real behavior' of beam-columns with full engineering scale and under low temperature circurnstancest5J. Some researchers tried to establish analysis model to simulate the response of steel frames undergoing cracking at the beam end[61. It is important to develop a numerical analysis procedure for the earthquake response that can reflect the effect of cracking damage on the steel frame. Such a computing procedure can provide a tool to judge the dangerous of a damaged structure after earthquake. It will become a potential implement to diagnose the most possible location of the cracking when repair work is necessary. It is expect to help engineers to strengthen the "weak part in a frame in detail design stage. On the other hand, to build a useful numerical analysis program should solve many problems in advance. The tasks include the work to constitute a structural model, to set the criterion in the numerical simulation for cracking of steel or welds, to deal with the noncontinues change of the section or member where cracking occurred, and to develop the skills to keep the numerical response convergence. Recent efforts in these aspects by the authors are

* Supported by NSFC (59678037)

502 reported in the paper.

2. STRUCTURAL MODEL FOR ANALYSIS The analysis model proposed in this paper is based on a so-called multi-spring model[7l which is effective to simulate the inelastic behavior of steel members subjected cyclic loads including varying axial force and bi-directional moments. The model takes short segment of the beam-column end adjacent to joints and divides it into several elements. The elements are modeled as elasto-plastic spring bars with given area surrounding each spring bar (referring to Figure 1). The axial force and deformation relation of the spring can behave elastically or plastically by setting the skeleton and hysteresis loop with suitable parameters based on test data. The skeleton curves are multi-lines shown in Figure 2, and the hysteresis loop obeys Ramberg-Osgood function. The combination of these springs can behave different features such as local buckling, strain hardening, plastic deformation, deterioration of beating capacity and so on. Of course no cracking was involved in the original model in the previous research. For the study of the fracture in steel structure, a typical beam-to-column connection shown in Figure 3 is considered. The flanges of the beam are welded to the flange of column. As a detail, backing metal is used and there exists unavoidable narrow opening on the side from the scallop. This opening can be considered as an initial cracking on the beam flange. In the case that the lower backing metal locates under the beam flange, the situation is the same.

f

beam end segment ~

Tension Non-crack skeleton ~ .N.....

P~I

,

/ ~'~"" elast~ -plas[ic spring " *~'--

......

1

functiofi

od

Compression

Figure 1. Composed spring model

Figure 2. Non-crack skeleton curves and Hysteresis loop

,^

~~~ii~ column

.beam "/" " i backing metal ~

1

L Figure 3. Beam-to-column connection

503 Earthquake disaster investigation revealed that in many cases the crack happened after having undergone obviously considerable plastic deformation in memberstS]. Furthermore the research indicated that the crack developed in three phases, firstly a ductile crack initiation, secondly a stable growth of ductile crack, and finally transition to a fast brittle crack [9]. Following this process, the generalized stress-strain relation should extend to plastic range in the analysis model that is used to simulate the crack damage. The previous model shown in Figure 2 is revised to fit the characters of fracture failure. Skeleton curves for the modified model are described as shown in Figure 4. In the tensile side of the skeleton curves, a sudden drop line piece is supposed. When initial crack results in the rupture of part of the beam section, for example, the low flange, the springs representing it will 'break' and lost their loading capacities almost, while the others in the same segment can keep their functions still. By the rules of hysteresis loop, if the tensile deformation keeps developing after break, for instance, from point C to D in the curves of Figure 4, which is due to the rotation of the damaged section around its new neutral axis, the broken spring extends freely. And no axial force changes to the broken spring in this stage. When the broken spring deforms in opposite direction, no force is needed to compress it until the spring recovers original length before break. Once a spring experiences break damage, the tensile loading capacity loses evermore. As for the hysteresis rule, whatever non-crack range or crack range, Ramberg-Osgood function is adopted here (Figure 5), though the routines are something different, with different loading capacity levels in tensile and compressive sides respectively.

TensionPI

PT Tension D Compression Figure 4. Skeleton curves Considering fracturing

.........

A

Compression RambergfunctionOsgood Figure 5. Hysteresis loop in the fracturing model

3. CRITERION OF FRACTURE IN ANALYSIS MODLE In the present research, the crack of beam-to-column connection is taken the type as shown in Figure 6. The lower flange of the beam where crack is expected to occur is replaced by one or several springs. These springs will follow the skeleton curves illustrated in Figure 4. Where, the point B should be assessed. However, to fracture problem in building structures with complicated details, the precise theory to support a reasonable criterion has not yet founded. The possible way is to use approximate method based on some hypothesis to determine an acceptable value, thus the crack phenomena can be simulated and the frame response analysis can be carried out.

504

I

Figure 6. beam end fracture model Referring to Figure 4, the break point B should be the one where the material has developed its plastic strain to a certain extent. In order to conduct the criteria strain, therefore to calculate the criteria deformation of the spring, we use the equation of fracture mechanics 2nesadP_<6~c (1) as well an experimental formula = 2(e / e s - 0.25) (2) In equation (1), yield strain, e,, is a constant to given material. So, we should determine the initial opening, a, and the parameter 81c, criterial crack opening displacement. To the former, the initial opening not to exceed the thickness of the back metal in the real structural connections is supposed. To the latter, we relate 81c to G~c, the criteria of energy release rate, and suppose that the relation deduced under the condition of elastic range can be proximately used in plastic range. By the available data of these parameters, the criteria strain, e, can be computed. The ductile crack will have a stage of stable growth. With the growth of the crack the criteria strain will change, too. So the increment of initial crack is also considered in the model.

4. NUMERICAL PROCEDURE The numerical computation program for non-liner earthquake response of steel frames adopts centrally differential method with an increment procedure. The unbalanced forces due to the change of the stiffness and the geometrical position of members are removed in the subsequent step. However, some problems different from the ones in non-crack structures should be solved in the utility of the program to the fracture damage analysis. In the analysis model, the fracture is limited to occur on the ends of beams. Considering the effect of concrete slabs actually existed in steel frames, the fracture damage is reasonably supposed to occur on the low flange and the adjacent web area of H-shaped beams. The section geometry change is a main feature in the model, because the rupture of partial section makes the centroid of the section drift towards the side of upper flange. It is different from a non-crack section where the section geometry keeps even though uneven plastic deformation develops. So a 'monitor' is set up in the numerical program to adjust the section centroid after the break of an imaged 'spring' is detected and also when the broken spring is compacted again under compressive force. Referring to the skeleton curve of Figure 4, it is clear that partial section break will results in a large amount of the change of the recovery force of the spring. The procedure based on

505 small unbalanced forces becomes unreasonable here. Thus, a back-step strategy is tried here. The numerical procedure allows the response analysis to return to the status of the last step and to modify the member parameters in advance.

5. COMPARISON WITH TESTS AND FRAME RESPONSE The numerical analysis results are compared with the test data. The prototype of the test specimens with the same details as shown in Figure 2 comes from Ref. [5] (Figure 7). The loading procedure to these specimens based on repeated incremental displacement. The top displacement of the tested specimen was controlled by on-line system and the restoring force was measured during the tests. Specimens under different temperature circumstances were loaded, however, only the one under room temperature is shown here. Figure 8 is a simulation by using the numerical program. The imposed displacements to the specimen are the same as those in the tests. The determination of parameters apart from the ones concerned with fractures adopts the same method as described in Ref. [7]. The numerical result does not match the test curve exactly partly due to the lack of the precise measurement of material and the specimen section to the authors. However, the main features are simulated well in the case, especially the occurrence of the fracture after several repeated loading, the loading down tendency when the rupture happened, and the fracture point which is smaller than the experienced largest displacement. I ,

HI i O X 2 2 0 X 2 5 Y

I9

i

,--680 T

~-6CC ,J

~ - 6 0 0

Figure 7. Tested beam

,-- -600

Figure 8. Simulation

Figure 9. Experimental data

dis.(cm)

4r i

e(s)

-3

L__

No crack considered Crack damage considered

Figure 10 Frame response to earthquake

506 Figure 10 is an example of a five-storey steel flame. A very intensive ground motion with peak value of 950gal is imposed to the frame model. The story drift of the first floor is shown in the figure. The thin line is the model to which the fracture parameters are not given. The thick line is the results of considering the crack. Both of the models developed plastic deformation, however, the model considering crack demonstrated severe damage with the vibration continuance.

6. CONCLUSION Analysis model considering the effect of the fracture of beam-to-column connection is established and the related parameters are assessed by fracture mechanics. Approximate method for calibration of these parameters is tried. The numerical computation program should deal with some problem different from the non-linear response when the crack phenomena are not taken into account. The comparison with the test data shows the analysis model acceptable.

REFERENCES

[ 1] S. A. Mahin, Lessons from damage to steel building during the Northridge earthquake. Engineering Structures, Vol.20.No.4-6 (1998) 261 [2] AIJ, Damage and lessons of steel structures in the Hyogoken-Nanbu Earthquake. (1995) [3] H. Kuwamura and H. Akiyama, Brittle fracture under repeated high stresses. J. Constr. Steel Res., 29 (1994) 5 [4] K. C. Tsai et al, Experimental performance of seismic steel beam-column moment joints. J. ASCE 121 ST6 (1995) 925 [5] K. Ohi, K. Takanashi et al, Dynamic loading tests on welded beam-to-column joints. Seisan Kenkyu, 49-11 (1997) 581 [6] K. Uetani and H. Tagawa, Seismic response of steel frames including brittle fractures at beam-ends. J. Struct. Constr. Eng. 489 (1996) 77 [7] Y. Chen, K.Ohi and K. Takanashi. A study on inelastic behaviors of 3-D steel frames considering the effect of varying loads on columns. Bull. RES, IIS. Univ. of Tokyo, 26(1993) 103 [8] T. Nakagomi et al. Experimental study on the changes of properties of beam and column for the damaged building from Hyogoken-Nanbu earthquake. Japanese Monbusho Funds A0730502 (1997) [9] H. Kuwamura. Brittle fracture of beam-to-column welded connections in tall buildings. Japanese Monbusho Funds A0730502 (1997)

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

507

Failure analysis o f b o l t e d steel flanges P. Schaumann and M. Seidel Institute for Steel Construction, University of Hannover Appelstr. 9A, D-30167 Hannover, Germany

This paper presents results from nonlinear FE-calculation of bolted steel ring flange connections. Different FE-models for a segmental approach and calculation of the total flange are presented and the results are discussed in regard to the structural design calculation. 1. INTRODUCTION Ring flanges are used in standard connections in tubular structures, especially in towers for wind energy converters (WEC). Towers can require three or more flanged connections throughout their length, which contribute significantly to the total cost. The structure must be checked for both fatigue and failure conditions, and enhanced FE calculation methods are appropriate. 2. SIMPLIFIED CALCULATION METHOD In Germany the design of ring flange connections is generally performed by using an approach of Petersen [ 1]. This calculation model is based on the behaviour of the individual segment of flange which contains the bolt with the maximum tensile force (Fig. 1). The load bearing capacity can simply be calculated with the plastic hinge theory considering three failure mechanisms of the critical s e g m e n t (Fig. 2). This local concept ignores the fact that the opening of the connection and plastic deformation will lead to some redistribution of forces. Consideration of the total flange enables this redistribution to be taken into account.

Stresses in shell

Fig. 1: Typical ring flange with preloaded high-strength bolts

For that reason investigations are carried out with two different 3D-FE-models: Firstly a model of a single segment with one bolt to verify the simplified calculation method and secondly a model of the flange as a whole to take advantage of the global carrying capacity of the connection.

508

Fig. 2: Failure modes of the simplified calculation method acc. to Petersen [ 1] 3. FE-ANALYSIS OF THE SEGMENT

3.1. Description of the 3D-FE-Modei The Calculations were carried out with the commercial code ANSYS V5.5. Different variants concerning contact modelling as shown in Fig. 3 have been compared for the segment model. The first variant uses contact elements between washer and flange and between the flanges in the plane of symmetry. The connection between bolt head and washer is assumed to be rigid. As an alternative the second variant omits the contact elements, providing a rigid connection between washer and flange, along with Link-elements in the plane of symmetry. The computational time is less for this second variant. The elasto-plastic analysis uses a bi-linear stress-strain relationship with characteristic values for the yield strength of the flange and the shell and a modified yield strength for the bolt. As the FE-Model for the bolt uses a constant cross-section over the total length including the thread, the yield strength is modified as such that the plastic resistance of the bolt equals the characteristic tension resistance according to Eurocode 3 [2, Table 6.5.3]. Safety factors are not taken into account because the results are to be compared with experimental data. Variant 1:

Variant 2:

Contact elements between washer and flange

Rigid connection between washer and flange

Contact elements with target surface in plane of symmetry

Link-Elements in plane of symmetry

Fig. 3: 3D-FE-Models of the flange (segment model)

509 The FE-Model with contact elements has been verified by comparison with experimental results from Wanzek [3] and Petersen [1]. The T-Stub-tests presented by Wanzek serve as an example for thin flanges up to 20mm. The tests from Petersen [1] cover medium flange thicknesses in the range of 20 to 50mm. The FE-Model with contact elements shows good agreement for all flanges, so that model can be rated as reliable.

3.2. Parametric Study for different flanges (Segment Model) A parametric study has been carried out to show differences between simplified calculation and the two FE-variants for the segment. The geometry of a WEC-flange was used as a basis from which different flanges were derived. Fig. 4 illustrates the flange geometry. The original flange has a thickness of 65mm. The results from the study are summarized in Table 1. It can be seen that the simplified model is always safe and gives good results for flange thicknesses above 50mm when failure mode B (see Fig. 2) is decisive. However the load carrying capacity is underestimated for thinner flanges. For a flange thickness of 25mm the prediction of the simplified model is less than 50% of the result from the nonlinear FE-calculation. The failure modes obtained from the simplified calculation method agree with FE-results for thick flanges. In the medium thickness range the failure modes and load carrying capacity differ significantly. Comparing the two FE-variants it can be stated that the calculated ultimate loads from the variant without contact elements are too high for thin flanges, while results agree almost exactly for thick flanges.

Flange

Simplified method

FIE-calculation

thickness t

Failure load

Failure mode

[mini

[kN]

[-]

[kN]

[kN]

25

94,4

C

203

210

30

127,7

C

253

280

35

166,7

C

297

299

45

260,4

B

299

299

65

304,8

B

299

300

100

304,8

B

312

300

with contact without conelements tact elements

Table 1: Ultimate load acc. to different calculation methods

Fig. 4: Parameters of example

510 3.3.

Enhancement of the simplified calculation method

In order to improve the calculation of carrying capacity for thin flanges with the simplified method as described in chapter 2 the failure mechanisms of the FE-calculations were analyzed. The detailed investigation of the stress distribution in the ultimate limit state shows that the bending resistance of the plastic hinge in the flange is supported by the moment contribution of the contact pressure resulting from the bolt's tensile force (Fig. 5). The prediction of ultimate carrying capacity from the simplified calculation acc. to Fig. 2 is too low, because only the remaining part of the flange (width c' in Fig. 5) is assumed to contribute to the plastic moment resistance. A better approximation is obtained if the contribution of the bolt's contact pressure is included in the calculation of the moment resistance. This additional share adds considerably to the overall resistance because of the dimensions of flanges for WECs with many closely arranged bolts. Formulas for an improved simplified calculation method are being derived and will be published soon.

Fig. 5: Stress distribution in flange at ultimate limit state 3A. Influence of model accuracy on deformation behaviour The deformation behaviour of the two FE-variants for the segment is important for the generation of the total flange model. In section 3.2 it was shown that the prediction of ultimate loads of thick flanges can be performed with variant 2. For the total flange it has to be made certain that the model without contact elements predicts deformations accurately. Fig. 6 shows load-deflection-curves for the different variants. It can be seen that the model without contact elements reacts too stiffly for the thin flange with t = 25mm, assuming that the deflections from the model with contact elements are correct. As this variant has been verified with experimental data, this assumption is acceptable. However, the deflections are in excellent agreement for the flange t = 65mm. The same applies to the bolt force so that the segment model without contact elements can be used to build up the FE-model of the total flange with a thickness of t = 65mm.

511

Flange t=25mm

250,0 ~, 200,0

300,0 ~ 250,0

150,0

200,0

0~ 100,0

":

~ 150,0 I JIr

0,0

;

i

:

!

:

~Tension

"= 100,0~-.-,--! ..... ~---..Force F i~. 50,0 _~ _.____ With' conkacielemenis

~ 50,0 0,0

Flange t=65mm

350,0

0,5

1,0

1,5

2,0

2,5

Gap [ram]

0,0

3,0

a) Comparison for flange t = 25mm

0

0,2

0,4

0,6

0,8

Gap [ram]

1

b) Comparison for flange t = 65mm

Fig. 6: Influence of model accuracy on gap widths in dependency of tension force 4. FE-ANALYSIS OF T H E T O T A L F L A N G E The total flange is generated by duplication of the segment model along the circumference. A length of 5000mm of shell is attached to the flange, so that redistribution of forces can take place. This model is used to determine bolt forces and stresses in shell and flange under pure bending for the total connection. Fig. 7 illustrates the bolt forces for the segment model and the highest loaded bolt from the total flange model. The bending stress from the total flange has been converted to a tension force by multiplying the maximum bending stress with the appropriate area from the shell wall. It can be seen that the bolt forces are in excellent 575-r . . . . . . ~ ......... -~..............,-......... ,-. . . . . . - , - - - - - - - r - - - - - - ~ agreement for low tension | ~ ~ Tension , I Failuro of I ,,~ (resp. low bending 525 stress). For any given ,/ = r--'H-~ , ,I ,I /r/ .-v~1 'I ~ I I.I I I , , ,/_.." I, , opening of the connection, 4751/ "" ~ I~ ? _11.1B I "!...... i ..... -; .......... ] the bolt force in the total m ~ olt Force C~L;O!I i i

I

1

-I-

forces

I

i

'"

i

,i

I

i i

.'"

1

! i

I

/~ ~'9 ~t~ Force :_,t segment, ....mode=,'/~'~!-a-"--'~:-'~t, oo

flangefromtheiSbelOWs...~en, ~that ~., model.f~ This _behavi~ iSthetObe eX-of _oected parts the flange which are opening have a lower stiffness than the parts that are still in contact.

425 t - - I

375/

~---~

~Z:ZlI U

: . . . . . . ~i/me th o d : . . . . L: ', 2/'I ... 4--'-Total flange C

~

.,_ S __ ~ _ _-i ' ' _ _ L--.-I---

,

,

I

I

,j--

, i

i

,

.

egment ,

Tenston force

, .

[kNl:

i

325 0

50

100

150

200

250

Fig. 7: Bolt force in dependency of tensile force

300

350

512 The opening of the flanges also causes some redistribution of stresses in the shell. The stresses on the tensile side are lower than according to linear theory, while the stresses at the compressed side remain virtually identical. Therefore, the neutral axis is offset to the compression side, but the differences regarding the decrease of the maximum stresses on the tensile side between FE-Model and beam theory are rather small (about 10% for the calculated geometry). A significant decrease of stresses and therefore bolt forces on the tension side, as determined by Ebert and Bucher [4] from a spring model, does not occur. The can'ying capacity of the total flange is considerably higher than that of the segment. This increase of load bearing capacity is accompanied by significant plastic deformation in the bolts. The magnitude of strains depends on the geometric variables of the flange, such as number of bolts and flange dimensions. The actual c a n i n g capacity will therefore not be limited by strength criteria but by allowable deformations and strains, particularly in the bolts. Details on required plastic deformations will be made available by future research works at the University of Hannover. 5. CONCLUSION Different methods to calculate the ultimate carrying capacity of bolted ring flange connections are discussed. It could be shown that a simplified design method proposed by Petersen [1] for flange segments gives good results for thick flanges, but underestimates the capacity for thin flanges. A modification of the calculation method has been derived which attains better results for thin flanges. The carrying capacity estimated on the basis of a linear distribution of loading around the flange, taking the attainment of ultimate capacity at an individual segment as defining failure, is significantly less than when the flange is fully modelled, due to redistribution. The increase in calculated loading capacity is in general limited by the allowable plastic strain of the bolts. It is considered that the work described in this paper will lead on to significant economies in flange joint costs, which will assist in making wind power generation more profitable. REFERENCES 1. Petersen, C.: Stahlbau (Steel Construction), 3. Edition: Vieweg-Verlag. Braunschweig 1997 (in German) 2. DIN V ENV 1993-1-1: Eurocode 3: Design of Steel structures; Part 1-1: General rules and rules and buildings. European Committee for Standardisation CEN. Brussels 1992. 3. Wanzek, T.: Zu Theorie, Numerik und Versuchen verformbarer AnschluBkonstruktionen. Berichte aus dem Konstruktiven Ingenieurbau der Universit~it der Bundeswehr MUnchen 97/7. Mtinchen 1998 (in German) 4. Ebert, M.; Bucher, C.: Stochastische nichtlineare Untersuchung vorgespannter Schraubenverbindungen unter Windeinwirkung (Stochastical nonlinear study of preloaded bolts under wind load). Braunschweig 1997 (in German)

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

513

Ultimate Capacity of Bolted Semi-Rigid Connections to the Column Minor Axis L. R. O. de Lima a, p. C. G. da S. Vellasco b and S. A. L. de Andrade c a Civil Engineering Department, PUC-RIO - Catholic University of Rio de Janeiro, Brazil. [email protected] b Structural Engineering Department, UERJ - State University of Rio de Janeiro, Brazil. [email protected] c Civil Engineering Department, PUC-RIO - Catholic University of Rio de Janeiro and Structural Engineering Department, UERJ - State University of Rio de Janeiro, Brazil. [email protected]

The traditional non-sway frame design usually adopts flexible connections in the column's minor axis. Unfortunately, when sway frame design is required, rigid stiffened connections have to be used. On the other hand, rigid connections have higher associated fabrication costs and give rise to a number of questions about their real structural behavior. To overcome these difficulties the semi-rigid connection fit as a natural solution, reducing the final cost and presenting a more realistic structural behavior. These facts were the main motivation for the development of an experimental program [1], [2], comprising beam to column minor axis connections. The experimental results enable the assessment of structural parameters like: the moment versus rotation curve, stress patterns, and collapse mechanisms. A non-linear finite element analysis of minor axis frame connections was also performed using the ANSYS [3] computer program. With these results in hand, a semi-rigid design model for minor axis frame connections was proposed. The developed model is based on the mechanical model concept adopted in the Eurocode 3 Annex J [4].

1. INTRODUCTION The sway frame design generally adopt the fully rigid connection concept, while employing connections that actually behave as semi-rigid. This assumption underestimates the beam's maximum positive moment and overestimates the moment transmitted to the columns. On the other hand, if the design is carried-out using flexible connections, the columns will be subject to additional bending moments, usually not taken into account in the structural design, while the beams become over designed. These facts can lead, in extreme cases, to structural collapse, or possibly to a narrower design safety margin. When column minor axis connections are used, the great majority of connections are designed as flexible. On the other hand, when the use of bracing is not allowed, rigid connections have to be utilized. In this case, the belief that these connections will behave rigidly about the minor axis is highly unlikely. This background served as the main

514 motivation for the present investigation on the structural response of beam-to-column semirigid steel connections to the column minor axis. This paper introduces a simple mechanical model, in accordance with the Euroeode 3 Annex J, [4], in order to evaluate the connection's structural behavior. The connections investigated were made with double web angles, a support angle or a transverse web stiffener welded to the column web. A validation study of the proposed model was accomplished through a comparison with the acquired experimental results and finite element simulations using the ANSYS [3] computer program.

2. THE EXPERIMENTAL PROGRAMME The experimental investigation, [2], comprised of a set of three tests performed in column's minor axis using a cantilever loading beam, Figure 1. The beams and angles were made of rolled steel sections (S shape 10"x37.7, L 76x76x9.5 (first test), L 127x76x9.5 (other tests)). The column used was a three-plate welded steel section, CVS 300x56.5 (similar to W 310x60). All the test dimensions are presented in Figure 7. The measured steel yield and ultimate stress measured in the beam, column, first test angles and other angles were: 364MPa, 497MPa, 309MPa, 419MPa, 325MPa, 472MPa, 417MPa, and 528MPa, respectively. In the first test, due to the web plate flexibility, large deformations occurred. The maximum applied load reached 25.3kN (corresponding to a 38kNm connection moment) when the load cell slipped out due to the large rotation present in the cantilever beam end. Figure 2 presents a top view of the residual deformations present in the column web. The yield line configuration due to the column web plastic deformation together with the column web point location where lateral displacements were measured, can be observed in Figure 3.

3. THE FINITE ELEMENT SIMULATIONS The finite element used to model the column web and flanges were the eight node shell. It leads to much lighter models than solid elements, and may cope with the relevant deformations: bending, and membrane as stated in Gomes [5]. The ANSYS, [3], finite element program used, enabled the material and geometric non - linearities to be taken into account. The finite element mesh used as well as the load application points and boundary

515

conditions are represented in Figure 4. The principal stresses at the last load step, calculated through a Von Mises yield criteria, is depicted in Figure 5 where its possible to observe that a large part of the web panel has yielded.

Figure 5. Principal Von Mises stress configuration (MPa).

Figure 4. Adopted finite element mesh.

A validation of the finite element model simulation was performed through a comparison with the experimental results, Figure 6. In this graph, load displacement curves for three points located in the column web, Figure 3, depicted a good agreement between experimental and finite element simulations. The finite element simulations proceeded focusing on the column web thickness, the most relevant stiffness parameter for minor axis semi-rigid connections. Three column web thickness: 6.3mm, 8mm and 10ram (corresponding to 43.65, 37.37 and 27.5, column web slendernesses) were simulated with the ANSYS program keeping all the other experiment dimensions, Figure 7. These results will be presented in the following sections of this paper.

160 t ' " 140

.- .o NODE A

.,"9 NODES B and C

120 100 z 80

9

.0""

.,J 60

~

lrl~'--:-- Fintte"&lementResults i

0.000

0 005

0.010

0.015 Displacement (m)

Figure 6. Finite clement model validation.

Experimental Results

0

020

0 025

516 4. T H E P R O P O S E D C O N N E C T I O N

MODEL

The investigated connections can be idealized by a mechanical model illustrated in Figure 7. The adopted model uses the semi-rigid design model concept adopted in the Annex J of the Eurocode 3 for major axis connections, [4]. Three independent load paths and the connection's rotation center, R. C. can be identified in Figure 7. The connection's rotation center was determined by the presence of a welded stiffener positioned on the back side of the column web. The first and second paths represent all the components in tension that influence the connection's rotation rigidity (the definition of each individual component is present in table 1). The last path takes into account the influence of all the components along the connection's compression path. The application of this model to minor axis connections required the introduction of a new component, kl3. This component takes into account the column web panel stiffness subjected to out of plane loads imposed by the double web angle bolts. A detailed description of this component is presented in the next section of this paper. Sim6es da Silva et al [6-7], extended the scope of the Euroeode 3 model from a pure elastic to an elasto-plastic analysis by using the equivalent elasto-plastie spring depicted in Figure 8. In this simple model the connection rigidity is divided in two parts: the elastic K e and the plastic K p. This simple model can be adapted according to the ductility of the component being modeled. When the component related to the column web panel subjected to out of plane loads is considered the bilinear model illustrated in Figure 9 can be utilized due to the component's high ductility. With the values of K e and K p in hand, a simple procedure based on energy methods, [6-7], can determine the connection's response and associated rotation.

1

96 ,,,

8 96 ~1'

_~ 12.5 11

1"]'

1

,

2

275

!2-5 toll / / /

9

,2L

/ /

~

.

.

.

.

.

.

/

M

.

.

.

.

.

.

//

,

...............

R.

Figure 7. Connection geometry and equivalent mechanical model (mm). Table 1 Component stiffness description Eurocode 3 Annex J Coefficient k5 K7

ks k9

k13

Component Description Bolts, one bolt row in tension Flange cleat in bending Plate in beating Bolts in shear Column web panel subjected to out of plane loads

517

Kp F (Pa)

/~

Q2

. . . . . . . . . . . . . . . . . . .

Q!

kp F

> A Figure 8. Equivalent elasto-plastic spring (after SimSes da Silva [7]).

Figure 9. Typical force-displacement diagram (after Simfes da Silva [6]).

5. THE COLUMN WEB SUBJECTED TO OUT OF PLANE LOADS COMPONENT The column web panel subjected to out of plane loads component stiffness was assessed through the finite element simulations described in the previous sections. A welded profile CVS 300x56.5 (similar to W310x60) was initially simulated and compared with experiments, Figure 6. To investigate the influence of the column web thickness, the experimental column web thickness, 8mm was changed to 6.3ram and 10mm keeping all the other profile test dimensions (corresponding to 43.65, 37.37 and 27.5, column web slenderness). The bilinear model described in Figure 9 was applied to the finite element results generating K e and K p stiffness values for the three investigated column web thickness. The column web subjected to out of plane loads component stiffness, kl3, was determined dividing the K e stiffness determined in Figure 10 curves by the Young's Modulus, E.

t. (ram)

6.30

8.00

10.00

(XOTlm)

hlt:w

~

43.65 34.3"/ 27.50

3"/15.0 6529.4 11933.3

~

(1011m)

2312.1

208"/.5

1"/86.4

k (m)

0.018 0.032 0.058

~~ /

/

~ J

,,,..

~ . ~

t,.= I0 m m

(IVtw : 2750) 9

tw : 8 mm (h/tw : 34.37)

tw = 6.3 rru'n 43 6s)

. 0,000

.

0,005

.

.

0,010

.

0,015

0,020

0,025

Displacement(m)

Figure 10. Finite element simulations (column web thickness of: 6.3ram, 8mm and 10mm and slenderness of 43.65, 34.36 and 27.50).

518 Based on the fmite element simulations a preliminary design stiffness, k13. w a s developed, equation 1. At present the scope of this equation is restricted to columns similar to the welded profile CVS 300x56.5. The finite element simulation is currently being extended to widen up the range of application of the proposed design stiffness equation. 42. (tw) 2"5 kt3 =

E

(k13 in m)

(1)

where tw is the web thickness in mm and E is the Young's Modulus in N/mm 2.

6. CONCLUSIONS The use of semi-rigid connections has been significantly increased over the last few years. In the attempt of representing the connections true behavior, many models were proposed, mainly for the major axis. When the minor axis is considered, the knowledge is still very limited. New experimental investigations [1-2] evaluated the structural behavior of bolted semi-rigid connections in the column minor axis. A non-linear finite element analysis of minor axis connections was also performed using the ANSYS [3] program. The finite clement simulations focused on the column web thickness, the most relevant stiffness parameter for these type of semi-rigid connections. A semi-rigid design model, based on the spring model concept adopted in the Eurocode 3 Annex J [4] and on SimSes da Silva et al investigations [67], for minor axis frame connections was conceived. A preliminary formula, at present restricted to columns similar to the welded profile CVS 300x56.5, for the stiffness of column web panels subjected to out of plane loads, k13. was also proposed.

REFERENCES

1. Lima, L. R. O. de, Vellasco, P. C. G. da S. and Andrade, S. A. L., "Bolted Semi-Rigid Connections in the Column's Minor Axis", Eurosteel, Second European Conference on Steel Structures, Praga (1999). 2. Lima, L. R. O. de, "Avalia~ao de Liga~6es Viga-Coluna em Estruturas de A~o Submetidas a Flexao no Eixo de Menor In&cia", MSc Dissertation, PUC-Rio, in portuguese (1999). 3. ANSYS, Version 5.4, Basic Analysis Procedures Guide, Second Edition. 4. Eurocode 3, ENV - 1993-1-1, "Revised Annex J", Design of Steel Structures, CEN, European Committee for Standardisation, Document CEN/TC 250/SC 3 - N 419 E, Brussels (1997). 5. Neves, L. and Gomes, F., "Guidelines for a Numerical Modelling of Beam-to-Column Minor-Axis Joints", Numerical Simulation of Semi-Rigid Connections by the Finite Element Method - COST C 1 (1999). 6. Silva, L. A. P. S. da, Santiago, A. and Vila Real, P., "Ductility of steel connections", Canadian Journal of Civil Engineering, submitted for publication (1999). 7. Silva, L. A. P. S. da, Coelho, A. G., "A Ductility Model for steel connections", Journal of Constructional Steel Research, submitted for publication (1999).

Buckling

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

521

T h e Effects of Fabrication on the B u c k l i n g of T h i n - W a l l e d Steel B o x Sections Martin Pircher a, Martin D. O'Shea b and Russell Q. Bridge a a School of Civic Engineering and Environment, University of Western Sydney, Nepean, PO Box 10, 2747 Kingswood, NSW, Australia b Hyder Consulting, Australia

Steel box sections can be found in a wide range of structural applications including bridges, buildings, industrial plants and resources equipment. The sections are usually fabricated from fiat plates which are welded at the corners. The welding process can introduce residual stresses and geometric imperfections into the sections which are known to have an influence on their strength. For some thin-walled sections, large periodic geometric imperfections have been observed in manufactured sections. Subsequent investigations have indicated that the imperfections are in fact buckling deformations i.e. the box section has buckled due to welding residual stresses prior to any application of external load. The behaviour of the box sections has been modelled using a finite element analysis that accounts for both geometric and material non-linearities. The welding procedure has been modelled including the cooling process around the heated area due to welding. Tests have been carried out on box sections with a range of width to thickness ratios for the plate elements. Using calibration with the test results, simulation studies have been performed to determine the initial "stress-free" imperfections prior to welding. Modelling was shown to give good correlation with the test results. The conditions for buckling to take place as a result of the welding process have been established. The influence of welding buckling deformations on subsequent loading strength has been examined theoretically and experimentally and design recommendations have been made. I. INTRODUCTION The strength of steel tubes is influenced by local buckling of the tube walls whic h is a function of the slenderness of the plate elements forming the tube. For rectangular or square bare steel tubes, the local buckling pattern can consist of inward and outward buckles and the influence of this local buckling on the column strength has been included in all major steel design specifications. Bridge & O'Shea [ 1] have reported the results of a series of tests on thin-walled square steel tubes with varying plate slenderness and varying length to width ratios. Residual stress measurements and plate imperfections resulting from the manufacturing process were measured. A number of these measurements indicated that some specimens displayed much greater deformations after fabrication than others. These imperfections were obviously introduced when the four flat plates representing the four sides of the box section were welded

522 together to form the box. Stresses induced during the cooling of the four welds were enough to exceed the buckling resistance of the steel plates. Due to the great care that was taken to record the deformations and residual stress patterns after welding a Finite-Element model of one of these specimens could be built where the recorded data was used for calibration. This FE-model and the influence of various parameters on the local buckling under axial load were investigated. 2. THE FINITE E L E M E N T MODEL

2.1. Tube Geometry The measurements indicated that deformations and stress patterns were symmetric about the longitudinal plate centre-lines. Therefore only one quarter of the specimens had to be represented in the FE-model as shown in Figure 1. Figure 1 also illustrates the dimensions of specimen 'B27' [ 1] which was used for the case study presented in this paper.

k___

Symmetry

Symmetry

1.4

.....

"~1

b Figure 1. Welded Square Steel Tube - Cross section and FE-Model

2.2. Modelling the weld In a preliminary study, the cooling process of the weld was modelled taking into account the highly non-linear temperature dependent material properties of steel. The numerical results proved to be in close accordance with experimental results (Figure 2) but computing proved to be very expensive. An adaptation of a simplified procedure described by Rotter [2] proved to yield similar results. When using this method, a residual strain field is applied to the area around the weld. The applied strains cause deformations and residual stresses to form in a similar way as in the detailed model.

2.3. Material properties The average steel properties in the test series [1] were found to be t = 2.142 mm, fy = 282 MPa and Es = 199400 MPa. In the computer analysis two different simplified models were

523 used and compared: firstly an elastic-plastic model; and secondly a bi-linear model which took strain hardening into account. :9r

~

...~o

1

i

i

Measured stresses [ 1 ]

,sol

9

='|

-- Stresses

FE-rnodel ..

-so .loo o

~_ 2o

-

~

'

~

4o

Distance a~r0ss Width

[.m m . ]. .

Figure 2. Weld-induced residual stresses.

3. TEST SPECIMENS

3.1. Manufacture of test specimen and imperfection measurement Four plates were cut from the steel sheet, tack welded into a box shape and then welded with a single bevel butt weld at the comers as shown in Figure 1. The out-of-plane geometric imperfections of the tube walls after welding were measured on a grid on each face of the tubes using a Wild NA2 automatic level with micrometer. The results for one face of specimen B27, 282x928BS are shown in Figure 3. Residual stresses induced by the welding process were measured on a representative box using the sectioning technique. The complete set of measurements for geometric imperfections and residual stresses has been reported by O'Shea and Bridge [3].

] [ ~ i m ~

a~=tl:ss

Box

( mm

)

Figure 3. Plate Imperfection Side 1, B27, 282x928BS

3.2. Testing Procedure For the axial compression tests described in [1] the ends of the specimen were rotationally and laterally fixed by low temperature metal in a milled groove in custom built end plates. The specimens were tested under stroke control in a DARTEC 2000 kN testing machine. Axial shortening of the specimen was measured between the thick machined plates using four linear displacement transducers evenly spaced around the specimen.

524 4. NUMERICAL STUDY

4.1. Geometric Imperfection Only geometric imperfections were considered in a first series of analyses to separate the influences of geometric imperfections and weld-induced residual stresses on the buckling behaviour. The shape of the first eigenmode under axial stress was determined and superimposed onto the perfect shape of the box section. The amplitude of this imperfection was scaled to up to 2.0 wall-thicknesses and a buckling and post-buckling analysis was performed. As expected the initial stiffness and the ultimate strength depended on the amplitude of the imperfection as shown in Figure 4. 4.2. Weld Shrinkage before Load Application In a second series of analyses residual stresses were taken into account. To produce a residual stress field matching measured results, strains were applied gradually along the welded zones. As the applied strains increased, the box sections responded linearly- tension stresses developed at the welded, inducing compressive stresses and small bending moments in the areas away from the weld which subsequently caused the sides of the box section to buckle. Figure 5 shows the lateral deflection of Point "A" (Figure 1) in relation to the applied strains at the weld. During the pre-buckling phase only small deflections could be observed while the residual stress field developed to levels up to yield in tension. Strains applied after buckling had a great influence on the lateral displacements of the buckles but did not alter the residual stress patterns.

4.3. Residual Stresses and Geometric Imlmrfeetion The amplitudes of the measured weld induced buckles averaged 1.2 wall-thicknesses (120%). This value is reached at result point "2" (Figure 5). At result point "1", the stress field is already fully developed but the amplitude of the geometric imperfection is only 10% of a wall-thickness. These two points were used as a basis for ensuing analyses of the box under axial compression. In Figure 6 the influence of the residual stress field becomes apparent. The stiffness and the ultimate strength of the models including residual stresses are considerably lower than in the initially stress-free models. IIowever, the amplitude of the geometric imperfection hardly influences the results when residual stresses are considered concurrently. Using simply supported instead of clamped boundary conditions led to slightly lower ultimate loads and less stiffness in the pre-buckling phase.

4.4. Strain Hardening Elastic-plastic material behaviour was compared to a bi-linear material law which assumed first yield at 282 MPa and an ultimate stress of 374 MPa at a plastic strain rate of 0.6. These values correspond to those measured in the tension tests [ 1]. Both material models resulted in the same weld-induced residual stress fields and displacements. Differences occurred in the post-buckling behaviour of the box where plastic hinges are developed and strain-hardeningreserves result in a significant gain in post-buckling strength (Figure 7). Again the amplitude of the initial geometric imperfection had hardly any effect on the load-deflection path of the structure.

525 300 250 -

10% ~._

- 100%~ ' , / "

~

i~

150

~ ~

TestResult

50

o~ 0

,, 0.5

1

1.5

2

2.5

3

Axialdisplacements[ram] Figure 4. Load deflection curves - purely geometric Imperfections of varied amplitudes w/t

j

~

Resultpoint2

0.00125

~Result point 1

0. 0

.

Yieldstrain 0.00125 =

. 0.5

.

1 1.5 Lateral displacement at

.

A

2

2.5

Figure 5. Load - displacement curve for box sections under weld-induced strain loading

4.5. Comparison with laboratory test results As can be seen in Figure 4, 6 and 7 some differences between the measured results and the computer models exist. The closest match was achieved when using the bi-linear material model. Pre-buckling stiffnesses still differed. An explanation can be found in the boundary conditions used in the tests. Rotations might not have been fully restricted and the low temperature metal holding the specimen in place might have had a softening effect. However, ultimate loads and post-buckling behaviour in the FE-model closely resembled the results gained from the experiments [ 1].

526

300

250

250

~

200

~

150

Stresslf0~~

~

120Z

~,_. ~

~

I

21111

~p~&~~,~,~~

Z

150

/ ~ ~ ~ //~~

(clamped,

~' 0

0.5

"~ lOO <

elastic-plastic

\ Test Result

Point"1" & Point "2" (simply supported)

_

.

.

Test Result

1

1.5

2

bi-linear

f

o

"d .< 100

//

2.5

3

Axial displacements [rnm]

Figure 6. Load deflection curves- Influence of residual stresses and boundary conditions

/ 0

1

2

3

4

Axial displacements

5 [mm]

6

7

Figure 7. Load deflection curves - Influence of strain hardening

5. CONCLUSIONS In thin-walled box-sections buckling prior to load application can be caused by weldinduced residual stresses. This has been accurately modelled using the finite element method. The influence of these residual stresses and weld-induced deformations on the subsequent axial load behaviour has been studied and accurately modelled. 6. REFERENCES 1. Bridge, R.Q. and O'Shea, M.D. Behaviour of Thin-walled Steel Box Sections With or Without Internal Restraint, Journal of Constructional Steel Research, Vol. 47, No. 1/2, July-August, 1998, pp. 73 - 91 2. Rotter, J.M. Buckling and Collapse in Internally Pressurised Axially Compressed Silo Cylinders with Measured Axisymmetric Imperfections: Imperfections, Residual Stresses and Local Collapse, Proc. Imperfections in Metal Silos Workshop, Lyon, France, 1996, pp. 119-139 3. O'Shea, M.D. and Bridge, R.Q. Behaviour of Thin-Walled Steel Box Sections with or without Internal Restraint, Research Report, The School of Civil Engineering, The University of Sydney, 1997.

Structural Failure and Plasticity (IMPLAST2000)

Editors:X.L.Zhaoand R.H. Grzebieta 9 2000ElsevierScienceLtd.All rightsreserved.

527

Inelastic Dynamic Instabilities of Steel Columns Tetsuya Yabuki a, Yasunori Arizumib, Carmelo Gentilec and Le-Wu Lu d a, b Department of Civil and Architectural Engineering, University of Ryukyu, Nishihara, Okinawa, 903-0213, Japan. CDepartment of Structural Engineering, Politecnico di Milano, 20133 Milano, Italy. dCivil and Environmental Engineering, Lehigh University, Bethlehem, Pennsylvania, 180153045, U.S.A.

This paper deals with inelastic dynamic instabilities of steel columns subjected to dynamic axial loads. It is shown by a nonlinear dynamic finite element analysis that, in some cases, the presence of nonlinearity makes the instability regions different from those given by the solution of Mathieu's equation of linear, elastic columns.

1. INTRODUCTION When dynamic horizontal forces due to earthquake or wind are applied to the braced system of a structure, all members in the system will be subjected to dynamic axial loads. For certain combinations of member stiffness and magnitude and frequency of the axial load, it is possible for a member to be self-excited and vibrate laterally even6 if the applied load is less than the Euler buckling load. The amplitude of self-excited vibration could reach very large values. This phenomenon is referred to as dynamic instability (1), (2), (3). In this paper, a general method of analysis to determine the load carrying capacity of spatial structures subjected to dynamic motions, allowing for overall instability is first described. Several characteristics of the inelastic dynamic instability of simply supported steel column with initial crookedness subjected to a steady axial force and an alternating component of axial force as shown in Figure 1 are then presented from an inelastic dynamic instability analysis. Finally, based on the results of a parametric study of the simply supported columns regions of stable and unstable vibrations are established and compared with those of linear dynamic instability. F = P a( l + ots i n OT)

B

L

0 . 8 B ~ Y

\\

Outside~ X

Cross-Section v

Figure I. Column Model and Its Cross-Section Studied

528 2. METHOD OF ANALYSIS The instantaneous dynamic equilibrium equation of a structural system at any particular instant in the short time increment t from any reference time TOcan be generally written in an incremental form as follows;

[g]{i~,}+[C]{ili}+[K,]{rl,}={Fo}+{f}-{Fi},

{Fi}--[g]{Xi}-[-[C]{Ai}-~-{ei}

(la), (lb)

in which [MI= structure mass matrix; [Cl=viscous damping matrix; [Ki|= instantaneous tangent stiffness matrix; {/li}, {r/i}, {t/i }'= "instantaneous increments of~n&lal acceleration, velocity, and displacement vectors; IF0[ = applied load vectors at the reference time; {f}= increments of the applied load vectors; ~F.1,= summation of internal forces at the particular instant;. {,} }~' {i/~i ~" nodal acceleration and'velocity v e c t o r s ; {i}R = the corresponding internal restoring torces. The instantaneous tangent stiffness matrix [Ki] and the internal restoring force vectors {Ri} in the spatial structural analysis have already been established by finite element formulation that includes the effects of both geometrical and material nonlinearities mentioned earlier (4). The equation of equilibrium Eq.(1) is solved numerically, using incremental displacement method combined with the Newmark-[i procedure. The short time increments are assumed to be equal for computational convenience and one 32nd of the natural period of the column is adopted herein as the initial increment. In this approach, the tangent stiffness method and the NewtonRaphson iterative procedure are applied to the nonlinear analysis for each increment (4). Since the excited system keeps in equilibrium condition at the reference time T0, {~} is equal to {F,.} in Eq.(1) when loading at T0. However, there is no assurance for {F0}= {F/} at a particular instant in the time increment because of the nonlinear response and the iteration process is necessary. As the collapse state is approached a large number of iterations are required and finally no convergence is achieved. When the divergence is achieved, the time increment reduces to half of the previous time increment and the nonlinear response is again traced by numerically solving the equations of motion for the incremental displacements according to the above-mentioned iteration procedure. When the time increment at the reference time reduces to less than one 64th of the initial time increment, it is assumed that the collapse state has been achieved. In this study, the viscous damping is not taken into account for the forced vibration. 3. NUMERICAL MODEL The numerical model adopted in this paper is shown in Figure 1. The external force is given as;

F=Pd(I +otsinOT),Pd=A~. /I.7

(2), (3)

where 0=load frequency, A = cross-sectional area of column. In Eq.(3), ~ . is the column strength and herein evaluated by Japanese Design Code for Steel Structures (J~CSS; Nishino, F. 1997 edited) as;

~,,g=tYrfor ~.<_0.2;~. =(l.109-0.545~,)tyrforO.2<~.<_l; ~. =ff/(O.773+A,2)forI<~ where tyr = yield stress of steel. The ~, is the slenderness ratio parameter given as

(4)

529

;t = 4 a / 1 4 a ,

(5)

/ e x ~ , , / ~r

where I = moment of inertia of cross section, L=length of column, ~r theoretical effective length factor for elastic buckling. In the following numerical models, mild steel with modulus of elasticity E=2.1 XIO aMPa, yield stress err=240 MPa and yield strain er=0.00114 is used and the long column with Z=2.0 is considered. The constitutive law of the material is assumed to be elastc-peffectly plastic. The component plates of the cross sections of the columns are assumed not to fail prematurely by local buckling. The initial geometrical imperfection w o adopted herein is sinusoidal in the direction normal to the axial coordinate of the column as follows; w0 = w0 sin n:x / L

(6)

The ~0 =L/1000 is specified as the initial geometrical imperfection according to the DCSS (5). In the parametric study, the elastic dynamic stability coefficient v similar to that given in Ref.(1) is used. That is,

v=(ao/ O)2(1-P a/Pe); I2o=(EIL/m)'a ne/L2; P e= ~ EI/L z

(7), (8), (9)

in which m = mass of column. The load frequency 0 is decided by substituting a parametric value of v, Eqs.(3), (8) and (9) into Eq.(7). Then the alternating external force is given by substituting the value of 0 and the value of the coefficient of the periodic load intensity r into Eq.(2) and the dynamic responses for the various values of the external force corresponding to the values of 0~ are analyzed. Eventually, the maximum load carrying capacity for the dynamic stability a,~ is traced and the dynamic stability parameter q/,,~xis obtained as follows (1) ;

~. =(~o/OY ot,.= P a/P e

(1 O)

4. NUMERICAL RESULTS

4.1. Inelastic Dynamic Instability Behavior Figure 2 examplifies the aspect of the simply supported column oscillated by the minimum instability load for a coefficient v= 1.2 given by Eq.(7), until instability occurs during the

t l [ .... ;' '

~,,/~

' '

.............

[/_Simplysupported steel column

I

[~,o=2.0 ,a~=1.2 a.,..=0.395, V"x=0.460~

Figure 2. Time-Histories Aspect of Oscillating Column

530 desired time. Herein, ten times of the natural periodic time of the column 10T is proposed as the desired time. The minimum instability load for the ultimate stability condition of the column is defined by ~ of the column subjected to combination of the static and periodic axial forces given by Eq.(2), where 0 is decided by substituting a parametric number of o= 1.2 into Eq.(7). The minimum instability load for this case is 0t,,,,==0.395 (W,, =0.460). In the Figure 2, the time history of the transverse deflection at the mid-height of the column is also shown by solid curves. Generally, the amplitude of oscillation of a system acted upon by external excitation at the resonance will increase exponentially to the strength-limit amplitude by keeping the initial neutral position of the transverse deflection-amplitude of the column. In the circumstance investigated herein, however, it has to be noted that not only the amplitude but also shifting of the neutral position from its initial position increase gradually in the direction of the initial geometrical imperfection as proceeding to the applied loading time. This shifting gives residual deformation to the column. At the critical time, the vibratory response disappears and then the residual deformation becomes suddenly infinite. That is, the inelastic dynamic instability occurs for the column. Figure 3 records strain time-history on extreme fiber that yields initially in the column, which is chosen as the subject in Figure 2, from 5.2T (just before the initial yielding) to 10.25T (at the ultimate state). The extreme fiber is located on the convex side of the deflection curve in the same direction as the initial crookedness, i.e., positive direction of y-axis shown in Figure 1

Inside ExtrimeFiberStrain ~,. : 5 - - - - - - 6 - " - - - - - - ' Z T - ' - - " - - 8 - ' ~ ' ~ v 9 - - ' ~ k 10 J T f f n Ill'gO

-20

Outside ExtrimeFiberStrain

-30 -40

SimplySupportedColumn Loadingcase of aJ=:1.2

-50

Figure 3. Time-Histories of Extreme Fiber Strains

Tfrn=5.28 +0.4B

/

J

/

T/Tn=6.34

/

j

/

T/Tn=7.44

/

/

/

Tfrn=8.56

-

,/

/

T/Tn=9.91

T/Tn= 10.25

, , ,

,..--

-0.4B

-1

0

1 -1

0

1 -1

0

1 -1

0

1 -1

Figure 4. Stress Distributions

0

1 -1

0

1

531 at the mid-height of the column and which is the outside extreme fiber. In the figure, simultaneous strain time-history on the opposite side from the outside extreme fiber (or the inside extreme fiber) is also recorded. Figure 4 shows time-history of stress distribution of the cross section in the inelastic range corresponding to the strain time-history shown in Figure 3. In the figure, 0.4B and -0.4B show the outside extreme fiber and inside one, respectively. The stress distribution at T=5.28T in Figure 4 shows the stress condition just after the initial yielding and indicates that the initial yield occurs in compression. Thus, it is seen that the transverse deflection mode in the reverse direction of the initial crookedness brings the initial yielding for the column. T=8.56T causes initial yielding at the inside fiber in the column, thus the initially combined yielding in both regions of tension and compression in the column. Progressively, the combined yielding ranges spread in the cross section and immediately after T=9. 91T, yielding occurs at 28/31= 90% of the cross-sectional segments. At the moment just before collapse, combined strains due to the bending moment and axial thrust increase very rapidly and the critical section eventually becomes fully plastified. n

4.2. Parametric Study Figure 5 compares the results on the parametric study on the interactive relationship between the nonlinear dynamic stability parameter g , ~ by Eq.(11) and the v-parameter defined by Eq. (7). In the figure, the Mathieu functions defining the first two instability regions known as linear dynamic instabilities (1) are shown for comparison purpose. Based on the calculated results of the steel columns, it has been found that the interactive relationship between ~,~ and v in the region of v<0.6 is almost same as the Mathieu functions and that in 0.6< o<1.0 is similar to. However, the nonlinear g,,~ in the region of v>1.0 is smaller than the linear g,.,.

2.5

••max 0

....

9 Nonlinear Analysis ,

,,

./\

/"x

1.5

-'7.. b O

0.5

33

0

ii

0

0.25

0.5

0.75

1

1.25

Figure 5. Nonlinear ~r,~-v Relationship compareing with Mathieu Functions

1.5

532 5. CONCLUSIONS A widely applicable nonlinear dynamic analysis considering both geometrical and material nonlinearities of spatial structures has been derived by using a finite element approach. The analytical method was used to determine the load carrying capacity of steel columns subjected to the combination of the static and periodic axial forces by allowing for overall dynamic instability. The parametric study was performed in order to examine the behavior of column characterized by the nonlinear dynamic instability. Based on the calculated results of the steel columns, it has been found that in the cases of o> 1.0, the presence of nonlinearity makes the instability strength smaller than those given by the solution of Mathieu's equation of linear, elastic columns. An area of research that is needed in order to advance understanding of the dynamic instability is that related to the effect of nonlinearity on this behavior for simply supported columns in the region of v>l.0. Experimental data on the dynamic instabilities are urgently needed. Considerations of the effects of local buckling on this behavior are also needed.

ACKNOWLEDGMENTS This paper is based on research sponsored by the Research Foundation under the Japan Kozai Club. Such financial aid is gratefully acknowledged. The findings and conclusions of this paper, however, are those of the writers alone. The analysis was performed using the computer facility of the Information Processing Center in University of Ryukyu.

REFERENCES 1. T. V. Galambos (editor), Guide to Stability Design Criteria for Metal Structures, Structural Stability Research Council, 5th ed., John Wiley and sons, New York, N.Y. (1998) 2. B. Kato and Le-Wu Lu, Instability Effects under Dynamic and Repeated Load, Proc. of 1st International Conference on Tall Buildings, Lehigh Univ. (1972) 3. S. Kuranishi and A. Nakajima, Failuar of Elasto-Plastic Columns with Initial Crookedness in Parametric Resonance, Proceeding of JSCE, No.356/I-3, 207-210 (1985). 4. T. Yabuki, S. Vinnakota and S. Kuranishi, Lateral Load Effect on Load Carrying Capacity of Steel Arch Bridge Structures, Journal of Structural Engineering, ASCE, 109(10), 2434-2448 (1983). 5. F. Nishino (editor), Design Code for Steel Structures Part A; Structures in General, Committee on Steel Structures, JSCE (1997).

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

533

Plastic buckling of circular sandwich plates S. C. Shrivastava Department of Civil Engineering and Applied Mechanics McGill University, 817 Sherbrooke Street West Montreal, Quebec, Canada H3A 2K6 A n analytical study of the bifurcation buckling of circular sandwich plates, stressed by radial pressure beyond the elastic limit of the face material, is presented. The analysis is based on the plastic behaviour according to both the J2-incremental and J2-deformation theories of plasticity. As is usual in the analysis of sandwich structures, the theoretical formulation accounts for the effect of transverse shear deformations. General and exact solutions are obtained for the governing differential equations, and buckling loads are calculated for various types of edge conditions. A surprising result is that for simply supported plates, the buckling loads predicted by the incremental theory turn out to be nearly the same as those from the deformation theory.

1. INTRODUCTION The first paper dealing with the buckling of plates of rectangular or circular geometry was that of Bryan [1]. Since then, while buckling of rectangular plates has been investigated extensively, comparatively little work is available on the buckling of circular plates [2]. The present work deals with both elastic and plastic buckling of circular homogeneous and sandwich plates. Usually, since the core shear modulus is m u c h lower than that of the face plates, the consideration of transverse shear deformations is of crucial importance in the analysis of sandwich plates. Reissner [2] was the first to devise an engineering theory to account for these effects on the elastic bending behaviour of plates, including sandwich plates. Bijlaard [3] dealt with plastic buckling of sandwich plates, including circular plates, on the basis of 3'2- deformation theory. His results are approximate, being based on a simplified theory. The present work follows the approach used in [4, 5, 6] for accounting the transverse shear effects.The analysis is exact, based on the constitutive relations of both the J2-incremental and J2-deformation theories of plasticity. Numerical values of buckling loads are obtained for sandwich plates of 24S-T3 aluminum alloy faces and balsa wood core. Supported by the Natural Sciences and Engineering Research Council of Canada.

534 2. C O N S T I T U T I V E R E L A T I O N S The usual assumptions of small strains, isotropy, and no plastic volume change are made. The yield condition employed is t h a t of yon Mises: J2 = a2/3 where a is the stress in a uniaxial compression test. Figure 1 shows the plate configuration where a = radius, h = core thickness, t = face thicknesses. Radial, circumferential, and axial coordinates are denoted by r, 8, and z respectively. Subscripts or superscripts f and c distinguish face and core properties respectively.

I

!

~T'~r ~

~

.

\..

] "

~

---r

~ I

\

@-------~

r

~

i t, face ,

~

~

er~ _.._~. ~h. core ~

r

~

~--~

.,

" ~ - ~ - ~ ' ~ - ' ~ ' ~ ' ~ ' ~

~

~

~ ' - -

=r

~

~

~.~--

@

a'rr

'"

Figure 1. Plate Configuration and Notation.

The plate is loaded by an axisymmetric compressive radial displacement of the perimeter. For a full circular plate this implies a uniform and equibiaxial prebuckling state of stress, a~ = a~ = constant, although the constant values are different for the faces and the core. The strain compatibilities require t h a t a~ kc/a ~ where kcI is in general a variable factor. W e assume kc/~ Ec/E/. The relationship between the increments of stress da~jand those of strata de,j,for the case of loading (dJ2 > 0) from an equibiaxial state of stress are [6]:

da,.~ = B'de~ + C'd~o, daoo = C'd~r~ + B'd~oo, daij = 2F'd~# (i ~t j) where

(2.1)

B ' - E(A + 3 + 3e)/{(1 + A- 2u)(2 + 2u + 3e)} (2.2) C' - E(1 - A 4- 4v 4- 3e)/{(1 4- X - 2u)(2 4- 2v 4- 3e)}, F ' = E / ( 2 + 3e 4- 2u). The p a r a m e t e r s A = E / E t and e = ( E / E , ) - 1 are obtained from the uniaxial compression test of the material. E is Young's modulus, and Et and E, are respectively the t a n g e n t and secant moduli at the (yon Mises)effective stress (2.3) In the equibiaxial loading, considered here, a~ = aoo = a, and a~ff = a = the prebuckling compressive radial or circumferential stress.

535 The relations (2.2) hold for the J2-deformation theory of plasticity. However, p u t t i n g e - 0 yields the J2-incremental theory relations. With e - 0 and = 1, the resulting relations are those for the linear isotropic elastic behaviour. 3. G O V E R N I N G E Q U A T I O N S A N D T H E I R S O L U T I O N The effect of transverse s h e a r deformations is accounted for by modifying the conventional kinematic hypothesis of Kirchhoff for elastic plates; a line normal to the undeflected middle plane r e m a i n s straight, but not necessarily perpendicular to the deflected middle surface. Denoting by r and r the components of the rotation of the normal in the r-z and O-z planes respectively, the (out-of-plane) buckling displacements of the plate are t a k e n as: =

- zr

0), , =

- zr

(3.1)

0), ~ = w ( r , 0).

The equilibrium equations and the b o u n d a r y conditions, appropriate to the above kinematic assumptions follow from the principle of virtual work: OM,.,. OM,.o M,.,. - Moo o--P- + t o o + r -Q~=O, OM~o O M o o 2M~o o--;- + t o o + ~ - Qo = o, OQ,. Q,. OQo cO2w 1 0 w 0---~- + - - + -Per( +-

(3.2) 02w +

)=0

with the conjugate pairs of b o u n d a r y conditions, at the edges r = constant:

OW Mrr - 0 or 6r = 0, M,.e = 0 or 6r = 0, Q,. - P c r ~ r = 0 or 6w = 0, and

(3.3a)

OW Moo = 0 or 5r = O, M,.o = 0 or 5r = 0, Q0 - Pcr-~-~ = 0 or 6w = 0

(3.3b)

at the edges 0 - constant. Pc,- is the uniform compressive load per unit circumference of the plate at which it begins to buckle, and is given by Pcr = (ha~ + 2ta~)

(3.4)

where a~ and ac~ are the stresses at buckling. In the present theory, a simple support m a y be defined in two alternative ways, for say r - constant boundaries: Type 1: w = M~r = r = 0, or Type 2: w = Mr,- = M,.o = O. Type 2 is a more realistic condition, and leads to lower buckling stresses.

(3.5)

536 We accept Shanley's concept that plastic buckling occurs under increasing load. Thus we assume there is no unloading from plasticity when the plane plate bifurcates into a buckled shape. The relations (2.2), therefore, apply throughout the thickness of the plate, and the buckling stress resultants can be expressed as

M.= f do~,d,=

-r

-{Br162176

Moo= f d~oo~dz= - {CCr + B(lr

r

(3.6)

M~o= f d~o~dz =

- F{-r-OS + ( 0r

r

G Ow }, Qo = f daozdz= ~--~{ G - r q ~ = f d a r z d z = ~--~{r +-~r-r

Ow r---~}

where the integrals are over the faces and core thickness, and consequently B = B~ch3 B} 12 {1+ ~-~/(t)},

C=

ci f(t)} C'cha {1 + ~-7 12 Cc ' (3.7a)

D'f "h-7/(t)},

D = D'~h3 {1 + 12 Dc

'

F =

F'h~ 12

{1 +

F~ -b-;/(t)} Fe

with

(3.rb)

(7 -- kFtch 3 and f(t) = 6(t/h) + 12(t/h) 2 + 8(t/h) 3.

The coefficients B}, C~, D ) , and F} for the face material, and Bc~, Cc', D~ and Fc~ for the core are defined by using appropriate material constants and stress levels in eqns (2.2). The symbol k stands for a correction factor which takes into account the non-uniform distribution of transverse shear stresses through the plate thickness. It can be shown that k ~ 1 for sandwich plates of moderate face/core thickness ratio (t/h <_ 0.25), whereas it is well known that k = 5/6 for homogeneous elastic plates. Substitution of relations (3.6) into the equilibrium equations (3.2) leads to the following set of governing equations BO BO -ff~r2 -t rOr

B Fc92 r2 4 r2002

G 02 0 G Ow =0 h2 }r + {(B - F) rOrOo-(B + F)r--~00}r + h2 Or

02 0 FO 2 FO {(S - F) rOro0 + (S + F) r200 }r + { ff~-r2 -t rcgr

a 0r

r

aOr

V ( ~ +-,. )+ h2~oo

F BO 2 r 2 ~ r2002

G 02 1 0 02 (h-2 - Pcr)(O~r2 + -r Orr + r2002)w=0.

G G Ow =0 h 2}r + --~ h r0---~

(3.8a, b, c)

537 The boundary conditions for the stress resultants, eqns (3.3), may also be expressed in terms of r r and w by means of eqns (3.6). The above system of equations, together with the homogeneous boundary conditions, constitutes a linear eigenvalue problem for the parameter Pc,.. We note that eqns (3.8) cease to represent an eigenvalue problem when Pc,- = G/h 2 ~ F~h. This value of Pc,- is associated with the pure shear ('crimping') instability of the core. We restrict the analysis to plates for which Pc,- < G / h 2. A general solution separating the independent variables r and 8 in eqns (3.8) has been derived, but is not presented here due to length limitation. The solution is suitable for analyzing plastic buckling of homogeneous and sandwich plates of full, annular, or sectorial configurations. The results presented here are only for full circular plates. The bifurcation conditions for the various cases of edge supports at the periphery, r = a, are summarized in Table 1, where m is the number of circumferential lobes and "7is the buckling parameter related to Pc,-. Table 1" Bifurcation Conditions For Full Plates with Various Edge Conditions. Edge Conditions at r = a (a) Fixed Edge " (b) Simply Supported Type 1

Bifurcation Condition, m = 0, 1, 2, 3, ... 7J~('7) - mJm('7) = 0 ....... 2 "TB (m ~-~--)Jm(7)- '7J~('7) = 0

(c) Simply Supported Type 2

'7: B (m + 1){mJ, n('7)- '7J~(7)} - ~ J m ( ' 7 )

(d) Free Edge, Center Supported .

.

.

.

.

.

.

.

.

.

.

= 0

Same as Simply Supported Type 2 .

.

4. SOME P A R T I C U L A R R E S U L T S Numerical values of buckling stresses are presented for sandwich plates with aluminum (24S-T3) ahoy faces with a stress-strain (a-O curve 9 U

U )7.

E = 11, 100 + 0 . 0 0 2 ( ~

(4.1)

where Ef = 11, 100ksi, and a is in ksi. This relation is valid up to a - 45ksi (310 MPa), with initial yield ~ 25 ksi (172 MPa) and a Poisson's ratio vf = 1/3. The core is assumed to remain elastic, balsa wood in this case, with Ec = 53.2 ksi (367 MPa) and vc = 0.4. The analysis yields the nominal buckling stress act as

Pc,- "72B '72 B h2)_1 act= 2"T = 2ta 2 (1+ ~ a---~

(4.2)

where '7 denotes a root of the appropriate bifurcation equation. This is a general formula applicable with proper interpretation to homogeneous or sandwich plates, in the elastic or plastic range, and with or without the shear effects. We

538 consider only axisymmetric, m = 0, buckling. In this case the difference between Type 1 and Type 2 simple supports disappears. Figure 2 shows the nominal buckling stress as a function of the a/h ratio, for simply supported plates with t/h ratios of 0.1 and 0.15. We note that between the two theories, the results from the incremental theory are higher, but only slightly (less than 0.5%). This is surprising since, generally, the incremental theory yields significantly (and some times absurdly) higher bifurcation loads for plates, see [4]. The only case where the two theories lead to identical results is for buckling of homogeneous columns. No experimental results are available to evaluate the present theoretical results. 45

-~

\i .

40

.

.

. . --0.15 .... i _t~__'~~t/hii_-i~-~~---- "i.

,i,,i

r~ r~.

r~

.I; t/h--O,1 .

.

.

.

.

.

.

',.........

::::::D.:

!

:"i

:i;iiiiiiiii, ii

35

.~ -P,4

-~ 30

ISimply Supp~

II

)

b~ 25

20

.

-'-t

6

plates[

!

Ii

....... ......... "i

""~_~

/ D ~ Defornmtion The~

....

.........

!

i i

, ,

| ,

t ,

~ ,

, ,

i ....

, ....

i . . . . . . . . . . . .

; ....

i ....

8

10

12

14

16

a ! h ----- R a d i u s t o C o r e - T h i c k n e s s

18

20

22

Ratio

Figure 2. Buckling Stress for Circular Simply Supported Sandwich Plate. REFERENCES

[1] G.H. Bryan, On the Stability of a Plane Plate under Thrust in its own Plane, with Application to the "Buckling" of the Sides of a Ship, Proceedings of the London Mathematical Society, 22 (1891), 54-67. [2] S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, 2nd Edition, McGraw-Hill, New York, 1961. [3] P.P. Bijlaard, Analysis of the Elastic and Plastic Stability of Sandwich Plates by the Method of Split Rigidities-I, J. Aero. Sc., 5 (1951), 339-349, and -II, J. Aero. Sc., 12 (1951), 790-796, 829. [4] S.C. Shrivastava, Inelastic Buckling of Plates Including Shear Effects, Int. J. Solids Structures, 7 (1979), 568-575. [5] S.C. Shrivastava, Inelastic Buckling of Rectangular Sandwich Plates, Int. J. Solids Structures, 32 (1995), 1099-1120. [6] S.C. Shrivastava and M.H.F. Nadir, Plastic Buckling of Simply Supported Rectangular Sandwich Plates from a Biaxial State of Stress, Submitted to the Journal of Strain Analysis for Engineering Design, July (2000).

Structural Failure and Plasticity (IMPL4ST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

539

Buckling instability of a curved-straight pipe configuration conveying fluid A.M. Al-Jumaily Diagnostics and Control Research Centre, Auckland University of Technology Private Bag 92006, Auckland 1020, NZ

This paper considers the buckling instability of a pipe configuration conveying fluid, which consists of a circular-arc tube of a radius R with an intermediate support located at an angle 9, rigidly connected to an inclined straight tube of length L and angle of inclination a. The time dependent terms in the flowinduced vibration equation are deleted to obtain the governing equations. Clamped conditions are imposed on the outer boundaries and simply supported conditions at the intermediate support. The effects of the design geometrical parameters L / R - ratio, a and 0 on the critical flow velocity are thoroughly investigated. A design formula is proposed for the critical velocity in terms of L/R-ratio. General design recommendations are given for other parameters.

1. I N T R O D U C T I O N Flow induced buckling (divergence) instability of tubing systems consist of curved-straight tube combinations conveying flowing fluid has practical merits as well as some academic interest. The necessity of using these combinations is normally encountered in many engineering applications such as relaxing the vibration and thermal effects in fuel lines of high power engines [1]. Space and size limitations may oblige designers to use these elements. Furthermore, hydraulic lines in control systems find good field of application of such items. Most of the effort spent in this area has been devoted to study vibrations and stability of straight [2] and curved [3] tubes conveying fluid, separately, less attention has been directed toward those of systems of combination. However, in a previous work [1], the vibration characteristics of an intermediately supported curved-straight tube configuration with a = 90 ~ Figure 1, subjected to a constant thermal force was investigated by the author. The non-conservative nature of the modes of vibration as well as the effects of different design parameters on these characteristics were considered. This paper is concerned with the critical flow velocities that initiate buckling of a curved-straight tube configuration conveying steady fluid. Emphasis is made on the effects of some design geometrical parameters on these velocities. Approximate empirical formulas are proposed for design purpose.

540

B

L,, \

Figure 1. Curved-straight tube with clamped ends. 2. FORMULATION OF THE PROBLEM In this work the tube conveying a steady fluid with a velocity U is considered to be composed of an intermediately supported semicircular segment rigidly connected to a straight segment at junction C with an angle of inclination a, Figure 1. The tube has flexural rigidity E l , modulus of rigidity G, polar second moment of area J and mass per unit length mr, transporting a stream of steady incompressible fluid of mass mf per unit length. The X-coordinate coincides with the centre line of the straight segment, which has a length L, and the coordinate coincides with the centre line of the semicircle of radius R. 2.1 G o v e r n i n g E q u a t i o n s The general equation of motion for flow induced vibration in pipes conveying fluid consists of a centrifugal force term which represents the force necessary to change the direction of the fluid to conform the change in the curvature of the pipe during lateral motion. This force is equivalent to a compressive force which causes buckling instability when the flow velocity reaches a critical value known as the critical flow velocity vc. This phenomenon is similar to a bar subjected to a compressive load. Conservative systems lose stability by buckling and normally dealt with by using a static method [4] based on deleting all the time-dependent terms in the governing equations. For the straight segment this leads to two independent equations; one for the out of plane displacement s (displacement/R) and the other for the twist angle ~,, namely d4~. + . 2 d2~. _ 0

d2~" = 0

dp4- "..4,----i

(1)

du----i-

However, for the circular segment two coupled equations are generated in terms of the non-dimensional transverse displacement ~eand the twist ang]e~e, namely d4~c dO 4

d2~c dO 2

K(d2~c 9d 2#~, ~d~ dO 2 -t- . - ~ ) + o; - ~

d~o =0

~d~o d ~

-'-----dO 2 - fka + K , d02 + dO 2 . =

0

(2)

where u = x / R , K=6J/F_1, v~ =RU~m//E1 and the subscripts s and e refer to the

541 straight and circular segment respectively. The solutions of the first two equations are: ~:~(,u) = D1 + D2,u + D 3 cos(vJz) + 0 4 sin(vs,tt )

(3)

~s = 195 + D6P

However, for the last two equations, two sets of solution are generated, namely for vc
C3cos(p~0)+ c4 sin(p~0)+ C5cos(p~0)+ C6sin(p20)

~ =

__+K)cos(plO)+C4

C10+C2 +

prI_~+K)sin(plO

+ C, (--~-+ K) cos(p20) + C6(--~-+ K) sin(p,0)} P2 P2

(4)

and for vc>K r

C3cos(p10)+ C4sin(p~0)+ C5cosh(p30) + C6sinh(p30)

1 { C,O + C 2 + 6,3(_Pl~ + K)cos(plO) + C4( l_l_ C e= I -+ K p2 + K)sin(pr + Cs(K - -~-)cosh(p30 ) + C6(K -

P3

1 )sinh(p30)t P32

(5)

where 2 + v~2 "~v,,~lv,,2 + 4(1 + I l K

P~:

=

I[

2

v,:~/vcz + 4(1 + ll K i - 2 - v, 2

P3 =

2

(6)

2.2 B o u n d a r y C o n d i t i o n s In this work the two extreme ends of the tube-configuration, points A (01 = 0) and D (X = L) Figure 1, are assumed clamped with the boundary conditions

~,(O)=~,(L)=O

~c(0) =~(L) = 0

d--7(0) = a/z (L)=0

(7)

However, continuity of displacement, twist angle, slope, bending moment and twisting moment, respectively have to be imposed at junction B (0 = 01, 02 = 0) and junction C. For a simple support located intermediately along the curved segment the continuity conditions are: d~cl

d~o~ - dO -T-

~,~=

ab 2

-~o~

where the subscripts respectively. If the segment is neglected continuity conditions

.~ +_ - = _ -, ao ao ao

ao

d~,2

(8)

1 and 2 refer to the left and right portion of the support, curvature of smoothness between the arc and straight (ie. direct connection is assumed at X = 0 and 0 = 180~ the at junction C are:

542

[

[dO 2 -r

d/a2

-

-

dta d/z3

~, = ~, cosa + d~c dO sinai

a/u-= ~osin,~ + dO + s , : J1

r162

=

LLdO

dO3

+

COSa

dO dB

-

dO

LdO

~o cosa+

[dO 2

-~

sina

(9)

dO

Substituting equations (3-6) in the boundary conditions, Eqs. (7-9) leads to a matrix of the form

[Aj.]{c.}=o

(lO)

where j,m=l,2...12 for curved-straight tube combination, and j,m=l,2...18 for the same system with an intermediate support. The critical velocities are obtained from the vanishing d e t e r m i n a n t of the m a t r i x Aim.

3. D I S C U S S I O N O F R E S U L T S The present work is concerned exclusively with the effects of different geometrical p a r a m e t e r s on some of the dynamical characteristics of an intermediately supported curved straight tube configuration. More precisely, the effects of the angle of inclination a, L / R - r a t i o and the middle support positioning are of design interest. The critical flow velocities are determined for different non-dimensional input data using Eq.(10). These variables depend on the stiffness ratio K; for a circular tube K = l / ( l + v ) . In this work a Poisson's ratio v = 0. is considered. Figure 2 shows the effect of the angle of inclination a, at the junction C, on the first three critical velocities. It is clearly indicated t h a t the critical flow velocity increases gradually with a and reaches m a x i m u m at a = 45 ~ after t h a t it gradually decreases within the range of investigation 0 _
543

6

> .--, == 3 ~

2 1

a 0

,

,

. . . . . . . . . . . . . . ,

,

,

15

30

45

60

75

,

90

0

,

15

,.

30

.

.

.

.

.

.

.

45

,

60

,

-,. . . . . . .

75

,

90

Figure 2. Variation of vc with a for L / R =1 and for various modes, (a) with an intermediate support, (b) without an intermediate support.

However, for an a larger t h a n 45 ~ the junction between the two segments will be more constrained and the n a t u r a l frequencies of the curved segment of a length nL, which are normally lower t h a n those of the straight segment of a length L, are most likely to be excited. Obviously this is reflected on the vc values. Further, the above behaviour is more pronounced, may be tripled (depends on the support position), with the existence of an intermediate support at ()1 = 90 ~ as indicated in Figure 2a. The effect of L/R-ratio is depicted in figure 3. It is clear t h a t as this ratio increases slower fluid will initiate buckling. This is attributed to the fact t h a t a longer structure is produced, which implies earlier occurrence of divergence. Figure 4 shows the effect of the middle support positioning on the critical flow velocity. A fluctuating behaviour is observed. This is attributed to the fact t h a t a middle support might suppress lower or higher modes of vibration depending on the particular geometry and position [1]. Evidently this fact is reflected on the corresponding critical velocities.

4. D E S I G N R E C O M M E N D A T I O N S Equation (10) represents an 18x18 matrix for the tube with an intermediate support and 12x12 matrix without support. This leads to the fact t h a t it is extremely difficult, if not impossible, to determine a closed form formulae for vc in terms of the dependent variables which are of concern in this work. Therefore attempts were made to fit the theoretical results with appropriate functions to propose some design recommendations. Disregarding the small variation of vc around a = 45 ~ in Fig. 2, the following approximate formula represents the best fit which could be used for design formulation wherevo is the critical velocity at a=O, and a and b depend on the buckling mode

544

5

7.5

~4

6

~

8

'~3

4.5

;52

~3 1.5

0

o o

0.5

1

1.5

2

2.5

L/R.ratio Figure 3. Effect of L/R-ratio on vc for various modes.

'"

3O

60

90 #z

120

~50

Figure 4. Effect of 01 on vc.for various modes.

and could be determined from the two extreme conditions at a=O and a=90. However, the behaviour of vc with L / R , Fig. 3, is nonlinear of a different nature from that of a. Thus an attempt was made to estimate this nonlinearity and to come up with an appropriate design formula. This formula is proposed to be of the form L

v---~-C= e - - ~

(12)

~Jo

where vo is the critical velocity for a semicircle. This formula fitted the results of equation (10) with R2-value varying between 0.97-0.99, which indicates the good efficiency of the curve fitting technique. As a matter of fact the values were so close, it was very difficult to show the values generated by Eq.(10) and Eq.(12) together on the same graph. In fact the curves shown are those from the latter equation. Therefore, this formula is recommended for any structure with a Poisson's ratio of 0.3.

REFERENCES 1. A.M. A1-Jumaily and Y.M. A1-Saffar 1990 American Society of Mechanical Engineers, Pressure Vessels and Piping Conference, Nashville, Tennessee 189, 245-252. Out-of-plane vibration of an intermediately supported curvedstraight tube conveying fluid subjected to a constant thermal force. 2. R.D. Blevens 1977 Flow Induced Vibration, Van Nastrond Reinhold. 3. A.K. Misra, M P Paidoussis and K K Van 1988 Journal of Fluids and Structure 2, 221-261. Dynamics and stability of fluid conveying pipes. 4. S.S. Chen Flow induced vibration of circular cylindrical structures, Hemisphere Publishing Corporation, 1987.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

545

Axial crushing of frusta between two parallel plates A. A. A. Alghamdi, A. A. N. Aljawi, T. M.-N. Abu-Mansour and R. A. A. Mazi Department of Mechanical Engineering~ King Abdulaziz University, P.O. Box 9027, Jeddah 21413, Saudi Arabia

Axial crushing of frusta as impact energy absorbers has been investigated by researchers for decades [ 1-3]. However, effort is made in this paper to classify the modes of deformation of frusta when crushed axially between two parallel plates. Tens of Aluminum spun capped end frusta of different semi-apex angles (15~to 60~ and thicknesses (1 mm to 3 nun) are crushed at quasi static loading conditions using Universal Instron Machine. The resulting modes of deformation can be classified into: 1) Outward-inversion, 2) Limited inward-inversion and outward-inversion, 3) Full inward-inversion and outward-inversion, 4) Limited extensible crumpling and outward-inversion, and 5) Full extensible crumpling. An explicit version of ABAQUS 5.8 Finite Element fiE) program is used to model the crushing modes. Good agreement between the FE predictions and the experimental work is obtained.

1. INTRODUCTION Energy absorbers are systems that convert kinetic energy into other forms of energy, such as plastic deformation energy in deformable solids. The process of conversion in plastic deformation depends, among other factors, on the magnitude and method of application of loads, transmission rates, deformation displacement patterns and material properties. The predominant domain of applications of collapsible energy absorbers is that of crash protection. Such systems are installed in high-risk environments with potential injury to humans or damage to property. The active absorbing element of an energy absorption system can assume several common shapes such as circular tubes [4], square tubes [5] and frusta [2]. Axisynunetric~ and circular shapes provide perhaps the widest range of all choices for use as absorbing elements because of their favorable plastic behavior under axial forces, as well as their common occurrence as structural elements. In this paper the selected absorber has a truncated capped fi~stum shape which is employed over a wide range of applications.

2. AXIAL LOADING OF THIN FRUSTA Johnson and Reid [6] identified the dominant modes of deformation in simple structural elements in the form of circular and hexagonal cross-section tubes when these elements are

546 subjected to various forms of quasi-static loading. They described the load-deformation characteristics of a number of these elements. Thin-walled tubes absorbers having symmetrical cross sections may collapse in concertina or diamond mode when subjected to axial loads. The collapsing of such components by splitting or by inversion is also reported [5]. The behavior of thin tubes (large diameter D/thickness t), with circular and square cross sections, when subjected to axial loads, has been of particular interest since the pioneering works of Alexander [4]. One of the first study of frustum (truncated circular cone) was carried out by Postlethwaite and Mills [ 1] in 1970. In their study of axial crushing of conical shells they used Alexander's extensible collapse analysis [4] to predict the mean crushing force for the concertina mode of deformation for fi-usta made of mild steel. Mamalis et al. [2] investigated experimentally the crumbling of aluminum and mild steel frusta of small semi apical angle (5 ~ and 10~ when subjected to axial compression load under quasi-static conditions. They proposed empirical relationships for both the concertina and the diamond modes of deformation. They concluded that the deformation modes of frusta can be classified as a) concertina, b) concertina-diamond, and c) diamond. Mamalis and associates in a number of papers, published between 1983 and 1997, refined the work of Postlethwaite and Mills [ 1] in using the extensible collapse analysis for predicting the mean and the progressive crushing loads, and fair agreement with the experimental results was reported. Axial crushing of thin PVC frusta of square cross-section and thin-walled fiberglass composite [3] is reported by them. The above studies deal with axial crushing (or crumbling) of frusta with small semi apical angle (15 ~ maximum) between two parallel plates. However, this paper investigates the quasistatic crushing of aluminum frusta with large range of semi apical angles (15 ~ to 60~) and classifies the deformation modes into 5 categories.

3. FINITE ELEMENT MODELING In the present study, ABAQUS Explicit FE code (version 5.8) [7] is employed to investigate the axial deformation modes of frusta under quasi-static loading. An axisynunetric four-nodded element, CAX4R, is used for modeling the frustum shown in Fig. 1. About 300 elements are used for the model. Material properties of the model were taken as rigid perfectly plastic with yield strength Sff 125MPa, and density 0=2800 Kg/m3. All nodes at the centerline of synunetry were selected to move only in the vertical direction. Both upper and lower surfaces were set in contact with rigid body surfaces. These rigid surfaces were modeled using two nodal axisymmetric rigid elements, RAX2. A coefficient of friction of Ix-~. 15 was incorporated between the contact surfaces. A reference node was introduced at the top end surface of the model. This node was set to move at a velocity of 0.01 m/s representing quasi-static case. The upper small capped end of the frustum was in contact with a rigid body moving at a constant velocity. The lower end was restrained from moving in vertical direction as shown in Fig. 1. The axisymmetric elements were chosen to model the axisymmetric collapse of the frusta, and most of the experimentally observed deformation modes were of this type especially at large semi apical angle and/or large thickness.

547

mtAQtm

3.6 mm

6.3 mm

20. mm

22. mm

iA I I & t m B

18. mm

AIUmtB 28. mm

24. mm L

~

31. mm

L

~

Figure 1: ABAQUS deformed plots for crushing of aluminum frustum (Specimen 60101).

4. RESULTS AND DISCUSSIONS A large number of frusta, featuring different thicknesses and semi-apex angles were subjected to various loading conditions. The program involved the use of 50 different sizes of aluminum frusta (10 different semi-apex angles and 5 different thicknesses) in crushing tests. Tests were conducted by the use of a 10-ton Instron Universal Testing Machine (U . Table 1 gives the details of the crushing tests. The table fists experiment number, specimen number, semi-apical angle (r in degrees, thickness (t) in nun, mass (m) in kg, average crushing force (P,v) in Newton, absorbed energy (E) in Joule, energy density (E*) in Joule~g and the deformation mode. The large and small diameters of the frusta were kept constant and equal to 75mm and 20ram, respectively. The average force is calculated over the whole range of the displacement and the elastic contribution is ignored as a common practice in metallic energy absorbers [7]. It was noticed that the deformation modes can be classified into: i) Flattening the lower end of the frusta and then curling up or upward inversion of the lower end (FLE then UI). This mode is limited to frusta with semi-apical angle = 60~ b) Partial inward inversion of the upper capped end followed by flattening of the lower end and then cuffing up or upward inversion of the lower end (PII and FLE then UI). This mode is observed to cover wide range of angles from 55~ to 30 ~. c) Full inversion of the upper capped end till the small end touches the lower plate. Then the deformation mode cha-qges into flattening of the lower end and then curlLqg up or upward inversion of the lower end (FII and FLE then UI). This mode is limited to few cases, mainly specimen 55101, 60101 and 60151.

548 d)

e)

Limited extensible collapse mode followed by flattening of the lower end and then curling up or upward inversion of the lower end (LEC and FLE then UI). This mode is limited to angles 25 ~ and thick 30 ~. Full extensible collapse mode. This mode is the only mode investigated in details in the open literature, see for example [2,3]. This mode is seen at small semi apical angles 20 ~ and 15 ~

Table 1" Details of the Experimental Work. NO.w 84

1 2 3 4 5 6 7 8 9 IO 11 12 13 14 15 16

~

ml,i

ten t en t en t en t en

nio.nLo~m mm~~ ~ m NN,,~K,t~

....

18 19

20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

145 46

ten

~

im,i

ONio.~L.O~T~4NNN~..,~i;,,~IN~

ten t en ten t en ten ten t en

i~..~.~.~.IN.' o

en

t en t en

L L L [ [ [ L [

L L L L L l l l l l I' l l I I l I I l l !

:NNN ~

~

oNIo,.~Lg.,~.,~.NN~

~ '~

,~HE,~L.~N

~

nm,~ i oNI~IL.~i o ~JNNI.I.I.I.~,9.]Nn m ~ s :N~,~[~.~,]N

~

~

ten t en t en t en t en t en ten ell t en t en t en t en t en

oNIo,.~L.o~..L~IN,N~,,~_.,,L.~NI , mm~ m l N~_,~.~I.L.~N

t~

-PlI-igfiicEg-R-~t,

~

nm 9m ~

i!b:~

ItE~NN ~

NB~,~91.Lgl

mmIIKt:

t t. t t t t

en en_ en en en en

549

20.

i

|1111

[xZ0 3]

I

-i'

'='

=

=

3.5. " 'EXP

I

A

z

I~.

10.

W

2 r

,:

"...'" ...-

...........

o.

o.

s.

zo.

9 =o.

zs.

, I 2s.

' 30.

3s.

Displacement (ram)

[ x Z 0 -3 ]

Figure 2" Experimental and F E load-displacement curves for Specimen 60101.

X 10 4

3.5

"Or

...2.5

z

,o

t-0.9 t=1.3 t=1.8 t=2.2

- - ...... ........

2

mm mm mm mm

.i-9~ !!i

I

i

9

/

,,"

I: tr

t=2.7 m m

o

~ . ~.,~,,

ii

"

e,, r

2 fo 1.5

-

.,.,,.-

1

, 0.5

CA .= 0

.-. . . . . . .

......

..-.

. . . .

:.-<. . . . .- _ . . . . . .

5

10

:,,

***

,,***,~..~ ~

tr

,...,..

/

f..~

~;.-,.'-"J

_ ""~..-~..-.. . , _ , - y

1

Displacement (ram)

20

Figure 3" Effect of thickness change on the crushing force.

30

550 .." ...: .. .."

- - ......

i

Angle = 55 degl A n g l o ,, 5 0 d o g I

.' _

,,'-"/t

........ Angle- 4s degI

/,,"

:

I

i

I

!

.

9. . . . . . 2 . i

.........

...: . . . . . . . . . . . .

;-

,'o

,'s

;0

Figure 4: Effect of semi-apical angle change on the crushing force.

The FE details of the crushing process can be seen in Fig. 1 that gives the crushing mode of specimen 60101 in 9 stages. These stages were captured at the following axial intervals: 2.4, 3.6, 6.3, 18, 20, 22, 24, 28 and 31 ram. The first stage shows the initiation of the inward inversion, and the fourth one shows the upper end touching the lower plate. Flattening of the lower end is illustrated in the sixth stage. Figure 2 gives the experimental and the finite element load-displacement curves for specimen 60101. Fair agreement between the two curves is shown. Figure 3 and 4 show the effect of changing the thickness and the semi apical angle of the frusta on the experimental progressive crushing load, respectively.

REFERENCES

I. H. E. Postlethwaite and B. Mills, "Use of Collapsible Structural Elements as Impact Isolators with Special Reference to Automotive Applications," J. Strain Anal., Vol. 5, pp. 58-73, (1970). 2. A. G. Mamalis and W. Johnson, "The Quasi-Static Crumpling of Thin-WaUed Circular Cylinders and Frusta Under Axial Compression," Int. J. Mech. Sci., Vol. 25, pp. 713-732, 1983. 3. A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, and M. B. loannidis, "Analytical Modelling of the Static and Dynamic Axial Collapse of Thin-Walled Fibreglass Composite Conical Shells," Int. J. Impact Engng., Vol. 19, pp. 477-492, (1997). 4. J. M. Alexander, "An Approximate Analysis of the Collapse of Thin Cylindrical Shells Under Axial Loading," Quart. J. Mech. Appl. Math., Vol. 13, pp. 10-15, (1960). 5. G. Lu, L. S. Ong, B. Wang and H. W. Ng, "An Experimental Study on Tearing Energy in Splitting Square Metal Tubes," Int. J. Mech. Sci., Vol. 36, pp. 1087-1097, (1994). 6. W. Johnson and S. R. Reid, "Metallic Energy Dissipating Systems," App. Mech. gee., Vol. 31, pp. 277-288, (1978) 7. HKS, Inc, ABAQUS/Explicit User's Manual, Theory and examples manual and post manual, Version 5.8, Explicit, (1999).

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

551

Strength analysis of buckled thin-walled composite cylindrical shell with hydrostatic loading J. B r a u n s

Department of Structural Engineering, Latvia University of Agriculture 19 Academy St., Jelgava, LV-3001, Latvia This study focuses on the characterisation of failure of a thin-walled composite cylindrical shell due to hydrostatic pressure. On the basis of the experimentally fixed buckling form of the shell the stress state analysis was performed. The full stress state was examined excluding normal stress to the shell surface. Failure criterion permits predicting the sites of fracture and maintenance of the shell upon the loss of stability as well as development of the inner structure of the material. I. INTRODUCTION An investigation on the forms of buckling of cylindrical shells has been the subject of many experimental and theoretical studies [1-4]. On the basis of analysing the forms of equilibrium states it is possible to determine the stability of a shell on the whole, especially if a statistical analysis is used [5]. It is also known that the form of buckling determines the stress state of the shell. The purpose of this study is to analyse the stressed state in the case of short-time loading of orthotropic glass-fiber reinforced shells by a hydrostatic load. The radial displacement field for a series of shells during loading was investigated. The displacement function lends itself to a mathematical description and is characterised by the sum of the harmonics of a Fourier series. The analysis discussed is intended to be used in the development of the inner structure of the shell material. For strength related properties, a phenomenological failure criterion can supply the feedback in material permutations and provide guidance in the fabrication process of composite structures. Failure is interpreted here as the occurrence of discontinuity in the material. 2. DETERMINATION OF STRESSES Let us consider the cylindrical shell of orthotropic glass-fiber reinforced plastic due to hydrostatic pressure q (Fig. 1). We denote the length of the shell by L, the radius R, mad the thickness h; we direct axes xl, x2, x3 along the generatrix, along the circumference, and along the normal to the center of curvature; these axes coincide with the axes of the symmetry of the material. Using the Kirchhoff-Love hypothesis, we determine the total strains and stresses as the sum of the strains and stresses in the middle surface and of the flexural strains and stresses in the form

552

~, (x, ,x2 ,x~)-8~,0 (x, ,x~_)+ x~ [K:=,(x, ,x~ )-~:o~, (x, ,x~ )];

(i)

o ~ ( x ~ , x, , x 3 ) - o

(2)

~~,13(x, ,x ~_) + x3A,~tr,8 [K,a (x, ,x ~-) - K ors (X' 'X 2 )]; I

q l

x31

,I,

xl

X2 . i

- f

Figure 1. The shell under action of hydrostatic pressure and geometric dimensions. Here e~ o~ are membrane strain and stress; K0 ~, r,~ are the distortion tensor of initial geometrical imperfection and deformation of the shell due to hydrostatic pressure q; A ~v~ are the stiffness components of the material; (x, 13, y, 8 = 1,2. In the case of the Kirchhoff-Love kinematic model the stress function of the middle surface of the shell ~(xl, x2) is related uniquely with the displacement function w(x~, x2) by the equation of strain compatibility. In determining of the stress function we use the equation of strain compatibility for the cylindrical shell of orthotropic material by using compliance components Sijkl:

S~ll;

c~(x,,x:) ~4

+

S 2,-'-2

0~ ~(x, ,x: ) c3~r ~,x2 ) + (2S1122 + 4S!21, ) ~4 c~2c~2

(3)

1 c32w(x, ,x,_ ) R

~2

,

We solve differential equation (3) by expressing the function of radial deflection w(x~, x2) and the Airy stress function ~(xl, x2) by complex double Fourier series: M

N

w(x, ,x2 )= Y', ~[o~oo-'~omo]~m (x, ,x,_ );

(4)

m=- M n=- N

(I)(Xl

M

'X2

N

)= E E(DmnkI'/mn(Xi m=-Mn:-N

m'/tXI

'

X2

).

qRx .

2h

qRx 2 .

.

4h

'

(5)

nx .~

where Wren (x ,, x _, ) = e i( --C- +-# ) (i - imaginary unit). The relation between coefficients of the stress function of the middle surface and the deflection function can be expressed by

553 1

[q~=]=-~fm

([Ot=]--[Ot0m.]).

(6)

n4L 2 n2Rm -" (2S1~2.- +4S1:12 )n 2 ~ Here the coefficients B m. = S ~11~-n 2 m" R OR s + So_-e.2 L2 + ,

m

and n

are the number of half-waves along the generatrix and the number of waves along circumference, respectively. According to the experimental data obtained we take maximum number of harmonics of the series along the generatrix M = 4 and along circumference N = 6. Here and bellow ~ and oto m are the coefficients of displacement imperfection function. The final membrane stresses in the middle surface were determined as o0 c~(x,.x~) o ,, (x, , x._ ) &~ o 2 2 (x, , x 2 ) :

O0(x,.x~) ,, & ;~ "o;2 ( x , ' x ," ) =

80(x,.x:) & , 0x 2

the the the and

.(7)

The flexural stresses for x3 = + h/2 by using the stiffness tensor Aijkt of material were expressed in the form f(x x )=+A /~2h ~'~ a~ ~, 2 ~2L 2

(a,.~--~0m~)m'~m~(X~,X 2 ) +

m=- M n=- N

(8)

h ~t N +A1,22 2R 2 1 2 2 2 (o~m. --OtOm. )n-~Pm. (X~ ,X: ); " m=- M n=- N

(3' 22 f (X I ' X 2

) = +A

220_2

h ~t N 2 R 2 EZ (~

mn -- ( Y " 0 m n ) n "

"tPn~ (xl

,X2

)q-

m=- M n=- N

/i;2h M --+- - A"2= -2L 2

N

E E (l~'mn m=-Mn=-N

--l~'0mn

of nh ~ 12 (Xl ,X2 ) - +A121.., LR

(9)

)m2qJm. (x, ,x 2 );

(arm, --~0mn )mntXJrnn (Xl ,X2 )"

(lo)

m=- M n=- N

Figure 2 shows a scan of a part of the surface of the shell with lines of the level of stresses 622 and o~.~. The dashed lines indicate the sites of the actual fracture of the shell. It is seen that at the sites of maximum compressive stress 022 the shear stress is minimum and vice versa. At the places where the curvature ~:~p of the deformed surface changes sign, the level of the shear stress is rather high (0.25-0.35 of the breaking stress). The stress component o~ is small in comparison with 022, which plays the main role in fracture of glass fabric reinforced shells. In order to take into account delamination process in buckling process of a shell, the interlayer shear stresses a~s and a2s were determined:

554

X)

-201+10+20 " I

it

~-2(3

IlL

+10+20 , II,

I32

~

X2

130 o

150 o

170~

190~

210~

230 ~

-

\ X2

250 ~

130~

150 ~

Figure 2. Lines of constant level of stresses: a) O22(Xh X 2 )

0,3 (x, ,x z ,x s ) -

fiii011

h2 _4x23 A ~3 ~,~ 8 ,,,, -~-

--

170~

190~

210~

const; b) t~z(Xl,

X2)

230 ~

~-"

const.

(Otto,, -- Ct Or,,, )mSiW,~,, (x, ,x 2 ) +

m=- M n=- N

h: -4x23 n: + 8 (A'~'~ v"' + 2A'z~2 ) LR:

250 ~

M N ~'~ ~"~ (~ ~ - ~ 0ran ) m n 2 i ~ m ( x ' 'x2 );

(ll)

m=-Mn=-N

023(x,,xz,xs)=

h 2 -4x~ 1 M N 8 A .... r~n3 ~ ~ ( o t m - a o m ) n 3 i W ~ ( x , , x z ) + m=-

h 2

+

4X 2

8

3 (A

M

l,ll v 2 1 + 2 A I - , I -., .) L e R~

Mn=-

(12)

(Otto --OtOmn )m zni~IJmn (x I ,xz.).

E ~ In= -

N

N n= -

3. STRENGTH ANALYSIS OF THE SHELL

In order to characterise the stress level in a shell depending on the co-ordinates the in-plane (membrane) stresses, couple stresses as well as transverse shear stresses were taken into account. The description of the surface of strength by a quadratic polynomial usually is proposed. The equation in a general form is given in [6-8]: f ( o i j ) = F~joi~ +F,juo,jok~ +F,;,~m. OuOklOm. +...= 1

(13)

i , j , k , l , m , n = 1,2,3.

In the case of glass fabric reinforced plastic we consider the strength of the shell in the plane stress state. The equation (13) with retention of terms of the first and second order depicts the surface (ellipsoid) in three-dimensional space of stresses and takes the form F))o)) +F22022 +2F~20~: + F ~ o ~ +2F1122(~11(~22 + 4 F ~ l p C 3 1 1 0 1 2

+F2222(y~,2 +4F~2~20~2 +

+4F2212(322(~12

=

(14)

1.

The coefficients of equation (14) are components of tensors Fij and

Fijkl.

They are

determined by the use of the strength tensor R~av, where ot = 0, 11, 1 1 13 = 0, 22, 2 2 y = 0,

555

12, 12. The index 0 denotes that the given stress component is absent; the bar over the index shows the presence of a compressive component. Using the strength values obtained experimentally, we represent the coefficients in the following form: Rrioo - R~oo -

FII- R~ooRi_io.

Ro~ o , F22 =

Ro22o

Ro,_2oRo~o

,

Fi111 =

Fll - F2, 1 =~ + FI~ + F .... - ~ ; 2Fl122 Rl12_~o .... RZ**,-5_o

1

1 " F .... = R~looRi-ioo .... R,m:,R,:,~,:'

4F1212 =

1

(15)

Rool2Roo E

Regarding the geometry of the fracture surface, the axes 1 and 2 of the ellipsoid are in the plane 0~1 - 622 and axis 3 is parallel to axis o/2. The components of the tensor of the surface of strength Fix and F=2 express the displacement of the center of ellipsoid in the direction of axes 1 and 2, respectively. The angle of rotation of the ellipsoid relative to axis 1 is a function of the component F1122. We assume that F i l l 2 = F2221 = F12 = 0. Substituting into the equation (14) the coefficients of (15) obtained by the use of the numerical values of the ultimate strength of reinforced plastic a quadratic equation was obtained. Solving the equation we can get the theoretical values of the boundary components of the ultimate strength (surface of failure). The analysis was performed with consideration of the following experimental strength (MPa): R~oo = 180; Ri-ioo = 170; Ro2~_o = 300; Ro2-~o = 230;

Rom 2 = Roo ~

95 and characteristics of rigidity of the material (MPa): El = 1.2 • l04,

=

E2 = 1.9 • 104, G l 2 = 0.25 x 1 0 4, Poisson's ratios v12 = 0.2, v21 = 0.16. By using the stress components into the criterion (14), we establish the level of stresses over the whole shell. Figure 3 shows the lines of the constant stress level according to the criterion (14). XI

L/2

X2 130 ~

150 ~

170 ~

190 ~

2 0~

230 ~

250 ~

Figure 3. Lines of constant level of stresses f(%, Okt) for glass fabric plastic shell. It is fixed that after buckling of glass fiber/epoxy shells made by winding method the delamination of material took place. The influence of transverse shear stresses on the stress level is shown in Figure 4. It should be noted that at the moment of fracture the form of deflection could not be recorded completely, and in the analysis the results of the last measurement preceding fracture were used.

556

/

f(fl'ii, flkl)

ffaii,~kl) I 0.4

0.6 0.4

1

0.2

0.2

L

x3 0

h/4

h/2

x3 0

h/4

h/2

Figure 4. Distribution of stress level f(aij, au) for glass fiber/epoxy shell at the sites of maximum stress a23 (a) and a ~ (b): without (1) and with (2) the effect of stress a23. The boundary surfaces, determined by the criterion (13), depend considerably on the loading rate. The strength characteristics of material are established on loading at a certain constant rate. In the case of a shell the loading was stepwise, since at each step of loading the form of buckling was measured for 5 min. Consequently, an additional study of fracture toughness of the shell with consideration of time factors is necessary. 4. CONCLUSIONS The algorithm developed permits calculating stresses in a shell by means of an experimental deflection function. It was established that the type of fracture depends on material properties. The fracture of glass fabric reinforced plastic shells occurs at points of maximum couplestresses. In the case of glass fiber/epoxy shells made by winding method the delamination took place under action of transverse shear stress. Failure criteria permits predicting the sites of fracture and maintenance of a shell upon loss of stability. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

B.O. Almroth, A.M. Holmes and D.O. Brush, Exptl. Mech., No. 9 (1964) 69. J.A. Brauns and R.B. Rikards, Polym. Mech., No. 4 (1972) 575. P. Montague, Mech. Eng. Sci., 11 (1969) 103. H. Showkati and P. Ansourian, J. Constr. Steel Res., No. 1 (1996) 53. B.P. Makarov, Mech. Solids (in Russian), No. 1 (1970) 97. I.I. Goldenblat and V.A.Kopnov, Polym. Mech., No. 2 (1965) 54. A.K. Malmeister, Polym. Mech., No. 4 (1966) 324. S.W. Tsai and E.M. Wu, J. Compos. Mater., 5 (1971) 58.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

557

I m p e r f e c t i o n sensitivity function in d y n a m i c response and failure of 1-D plastic structures* F.L. Chen and T.X. Yu Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (E-mail: [email protected], [email protected]) This paper proposes and elaborates a novel concept imperfection sensitivity function from the viewpoint of energy dissipation to describe quantitatively the effect of imperfections on plastic response and failure of structures under intense dynamic loading. Case studies are then conducted to demonstrate the imperfection sensitivity function in various types of 1-D structural members, such as cantilever beam, circular ring, simply-supported beam and freefree beam, each containing a crack or a notch. It is found that an imperfection sensitive region exists for each case. Fundamental features of the imperfection sensitivity function are explored. 1. INTRODUCTION Engineering structures are usually imperfect in their geometry and/or in containing cracks, notches, holes, defects and voids [ 1]. For example, in the Duane Arnold nuclear plant a recirculation inlet nozzle was found to have an intergranular stress corrosion crack that penetrated over 50 per cent of the wall thickness all around the circumference [2]. Engineering practice and relevant investigations [3-10] indicate that when the structures suffer impact loads arising from working environment or accidents, these imperfections trap a significant part of the work done by the dynamic loads, which dramatically alters their dynamic behaviour. Taking a 1-D cracked or notched beam as an example, the characteristics arising from an imperfection include: (i) reduction effect at the imperfect section (see Section 2.1); (ii) complex energy dissipating mechanism (e.g. fracture may be involved); (iii) concentration of permanent deformation and plastic dissipation at the imperfect section; (iv) local 2-D effect, etc. To quantify the effect of the imperfection on the dynamic response and failure of imperfect structures, this paper will propose and elaborate a novel concept imperfection sensitivity function and explore its fundamental features. 2. DEFINITION OF IMPERFECTION SENSITIVITY FUNCTION 2.1. Reduction factor The material of structures is assumed to be rigid, perfectly-plastic and rate independent in * The work described in this paper was conducted as part of CERG research project HKUST 811/96E funded by Hong Kong RGC. The authors wish to express their gratitude for this support. Also, the authors would like to thank Prof. J. L. Yang for valuable discussion.

558 this study. In view of the characteristics arising from imperfections, to quantify the effect of the imperfections, such as cracks/notches in 1-D beams, a convenient way is to introduce a reduction factor ), on the fully plastic bending moment at the imperfect cross-section. The nonlinear relationship between ), and physical crack size depends on geometry of the crosssection and the crack. For a beam of rectangular cross-section with height H, it is found [5] from a statically admissible stress field over the uncracked ligament that )' = (1- c) 2 , with c CIH and C being the crack depth. However, a more exact solution based on slip-line field [11], which partly reflects the 2-D effect arising from the crack, leads to a gently larger It: it = (0.63/0.5)(1-c)2 = 1.26(1-c) 2 . For the annular cross-section of a pipe penetrated by a crack subtending angle 2 0 , ~' = c o s ( r

(1/2)sin O, see [5].

2.2. Imperfection sensitivity function To quantify the influence of an imperfection, such as a crack or a notch present over a cross-section of a 1-D structure, on its dynamic response/failure, from the viewpoint of energy dissipation we define an imperfection sensitivity function as

(I)

S=Ei/E t

where Et is the total energy dissipation in the whole structure, which is equal to the input energy to the structure, and Ei is the energy dissipated over the imperfect cross-section during the entire dynamic process, by plastic flow as well as other possible energy dissipating 9mechanisms (e.g. fracture), in other words, it is the energy trapped by the imperfect section. Apparently, S ranges over interval [0, 1], with 0 and 1 representing two extreme cases. A larger value of S indicates a greater effect of the imperfection. S = 0 implies completely insensitive; while S > 0 sensitive, in particular S = 1 completely sensitive. It can be easily conjectured that S is dependent on the location of the imperfection, then an imperfection sensitive region (where S > 0) exists for each case. The imperfection sensitive region may depend on the geometry of the structure, the size of the imperfection (value of ),) and sometimes the magnitude and variation of the dynamic load. 3. I M P E R F E C T I O N S E N S I T I V E R E G I O N

3.1. Cantilever beam (i) Static loading. Fig. l(a) shows an initially straight cantilever of length L, while Fig. 1(b) depicts the same beam but having a crack located at distance b from its tip. Consider a static load P applied at the tip of the cantilever. If b < ~ , with increase of load P, the bending moment at the root reaches its fully plastic limit Mp before the bending moment at the cracked section reaches ~/p, thus the failure mode will be a plastic collapse at the root section. Therefore, the imperfection sensitive region is b > ),L. (ii) Step loading. For a perfect cantilever, a plastic limit load Pc = Mp/L exists. Based on the distribution of the bending moment given in [12] with inertia force being taken into account, the imperfection sensitive region in a cracked one is found as follows: lower load (P < Pc):

b > '--P P

(2)

moderate load (Pc < P < 3Pc):

b >Xl

(3)

559

(a) A ,

........

~1211\~ 1 4 y=019.................... sensitive 101 \~k~Y/=0"8 region

PIPc 8

(b) ~

21 r e g i ~ (c)

t~"' 'IT

0.0 0.2 0.4 0.6 0.8 1.0 b/L

C

b

L'I' "1

Fig. 1 A cantilever: (a) perfect, (b) & (c) with a crack at section C intense load (3Pc < P)"

~

Fig. 2 Imperfection sensitive region in the cracked cantilever on P - b plane

,4,

_

Here Xl is the root of cubic equation - 1

X

P - 3 +

1--~-

- 1

- 1

= 7' within

interval [0, L]. The imperfection sensitive region described by (2-4) is graphically depicted in Fig. 2, which exhibits some basic features: (a) the smaller the reduction factor, the larger the sensitive region; (b) the larger the magnitude of the load, the larger the sensitive region. (iii) Rectangular pulse. For the perfect cantilever, after the load is removed, the beam segment will decelerate and finally cease to move. It is found that the bending moment at any cross-section is smaller than that in the loading period. Therefore, the imperfection sensitive regions for rectangular pulse are the same as the step-loading cases of lower load, moderate load and intense load discussed above, respectively. (iv) Impulsive load. Impulsive loads are usually much more intense in comparison with the static collapse force; they are, however, of brief duration. An impulsive load can be modeled by setting the magnitude of a rectangular pulse be infinity with the total impulse remaining a constant I. Letting P --~ ~, in Eqn (4), it is evident that the imperfection sensitive region covers all sections of the beam.

3.2. Circular ring A series of low-velocity impact tests on thin-walled metallic notched circular tings with arc-shaped supports were reported in [9]. The results showed that the exterior notches at some regions had no effect on the deformation of the rings, but those at the remaining regions did have. By employing the Equivalent Structure Technique the condition that a plastic hinge is formed at the notched section was theoretically derived in [9] in terms of central angle a: cost~ > (y + 2.24)/3.24

(5)

On the o~- y plane, there are two distinct regions (refer to Fig. 5 in [9]): Region (I) where inequality (5) holds is imperfection sensitive; outside this region, i.e. Region (II), is imperfection insensitive. For the tested samples, y = (1 - 1.2/4) 2 = 0.49, then inequality (5)

560

Y X I)~, ~

1.0 0.8 ~.0.6 ~ 0.4 0.2 0.0

C b'

j "-i

P(t L

14

.........

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3 A cracked free-free beam subjected to a transverse step load at one end Fig. 4 Imperfection sensitive region on the fl- ?'plane results in - 3 2 . 6 ~ observations.

<32.6 ~

This result agrees favorably with the experimental

3.3. Free-free beam A free-free beam with a crack at C subjected to a step load P at one of its ends, as shown in Fig. 3, is examined. The results show that, with increasing magnitude of the step-load, if t, < fl =- b/L

< f12

(6)

the cracked section will become a plastic hinge earlier than the perfect section that has the largest bending moment. Herefll and ~ (0 < fll < ~z <1) are two roots of cubic equation fl(1- fl)2 = 47'/27. Thus inequality (6) can be regarded as an imperfection sensitive region. For a given value of reduction factor y, this imperfection sensitive region covers an interior portion along the beam, but closer to the impact end, see Fig. 4. With a severer crack (i.e. a smaller 7'), the imperfection sensitive region becomes larger. This agrees qualitatively with the experimental results reported by Woodward and Baxter [4]. 4. VARIATION OF I M P E R F E C T I O N SENSITIVITY FUNCTION W I T H R E L E V A N T PARAMETERS As mentioned above, analyses on dynamic response of various 1-D cracked/notched structures have been conducted by several authors, such as Petroski [6], Yang and Yu [7,8] and the authors [ 10]. These cases will be taken as examples to study the dependence of the imperfection sensitivity function on relevant parameters.

4.1. Location of imperfection A modal solution for a cracked cantilever impinged by a mass at its tip (Fig. l(c)) was given in [6]. If fl = blL > Z only one stationary plastic hinge at the cracked section occurs, resulting in S = 1. If fl < 7', two plastic hinges at the cracked section and the beam root develop. Based on Eqn (29) in [6], the imperfection sensitivity function is obtained as

S=2py((1-fl)2(l+fl)]( 6 ]2 (9'- t) 6 + 4pfl - pile

(7)

561

~(~ .2

1.0

1.0 . . . . . . .

........................................

0+

-~

0.8

0.6 0.4"

S 0.6 0.4 0.2 0.0 ...... 0.0

0.2 !

0.2

:

I

....

0.4

I I !

0.6

1 -

~"

" ~ 3~=0"95 -'~'NI.8 "~'NI.6

I =-

0.8

1.0

Fig. 5 Variation of imperfection sensitivity function with the crack location

0.0 ..................,......... +...........,~ 0 20 40 60

,

80

, ....... ,

100

120

Fig. 6 Variation of imperfection sensitivity function with the load parameter

where p - m L / 2 M is mass ratio. Eqn (7) indicates that S depends on three parameters t , 7' and p, i.e. the location of the crack, the crack reduction factor and the mass ratio. Of course, this modal analysis is valid only for a heavy striker (p ---> 0); a complete solution [7] may be adopted to improve the results. It is seen from Fig. 5 that with increasing t , S increases slowly first (double-hinge mechanism), then quite rapidly up to 1 as fl approaches 7; and remain 1 once fl > ),(single-hinge mechanism). 4.2. M a g n i t u d e o f l o a d i n g

For uniform rectangular pulse loading of an imperfect simply-supported beam with a crack at mid-section, it is found [8] that a load parameter # - q o L 2 / M p controls the deformation patterns and the energy partition. The dependence of the imperfection sensitivity function S on the load parameter # is depicted in Fig. 6. In case of 27' < # /ZA, S depends on both the load parameter/z and the reduction factor y. 4.3. I m p e r f e c t i o n r e d u c t i o n f a c t o r T

Figs. 5 and 6 both indicate that imperfection sensitivity function S varies with the reduction factor 7. As ~' decreases (i.e. the crack depth increases), S increases. In addition, for impulsive loading of the simply-supported beam with crack at the mid-section [8], S is merely a function of reduction factor ~ independent of the impulsive velocity V0 and other parameters. It is seen from Fig. 7 that S ---> 113 as ~'---> 1, implying that even a very small imperfection at the mid-section may lead to remarkable consequence. As 7 decreases from 1, S increases rapidly to around 0.7 at T= 0.8, whereafter S increases rather gently. 4.4. M a s s ratio

Mass ratio, in general, plays a significant role on the partitioning of energy dissipation in a structure after impact. Based on a complete analysis by the authors [ 10] of impact on a fully clamped beam with cracks at the supporting ends, imperfection sensitivity function S is determined and depicted in Fig. 8 against mass ratio g = GImL. It is seen that, in this problem a larger g (heavier striker) leads to a larger value of S.

562 1.0

.5

Q8

~

A

.

0.3

06

0.2

t34 (~20

~,L

0.4

0.1 _

Q0

_

0.0

_!

O2

'

(14

'

7

(16

'

(18

~"~

1.0

Fig. 7 Variation of imperfection sensitivity function with the reduction facttor

-15.

!

-1

-!

-0.5

I

I

0

0.5

..... 1

1

1.5

logg Fig. 8 Dependence of imperfection sensitivity function on mass ratio

5. CONCLUSIONS A novel concept of imperfection sensitivity function has been proposed and elaborated from the viewpoint of energy dissipation. Case studies indicate that this concept can successfully describe quantitatively the effect of the imperfections present in a 1-D structure on its dynamic plastic behaviour under intense dynamic loading. Without essential difficulties, this concept may be extended to 2- and 3-D situations. An imperfection sensitive region exists for each case. Both the imperfection sensitive region and the imperfection sensitivity function are found to depend on the severity of the imperfection, structural geometry, supporting conditions, and sometimes the magnitude of the loads. REFERENCES 1. 2. 3. 4. 5. 6. 7.

T.X. Yu and F. L. Chen, AEPA'1998, Metals and Materials, 4 (1998), 219-226. Pipe Crack Study Group, NUREG-0531, US Nuclear Regulatory Commission, 1979. H.J. Petroski and A. Verma, ASME J. Engng. Mech., 111 (1985), 839-853. R.L. Woodward and B.J. Baxter, Int. J. Impact Engng., 4 (1986), 57-68. H.J. Petroski, Int. J. Pres. Ves. & Piping, 13 (1983), 1-18. H.J. Petroski, ASME J. Appl. Mech., 51 (1984), 329-334. J.L. Yang and T.X. Yu, Acta Scientiarum Naturalium Universitatis Pekinensis, 27 (1991), 576-589. 8. J.L. Yang, T.X. Yu and G.Y. Jiang, Int. J. Impact Engng., 11 (1991), 211-223. 9. Y.P. Zhao, J. Fang and T.X. Yu, DYMAT J., 2 (1995), 135-142. 10. F.L. Chen and T.X. Yu, ASME J. Pres. Ves. Tech., 121(1999), 406-412. 11. A.P. Green, Qu. J. Mech. Appl. Maths., 6 (1953), 223-237. 12. W.J. Stronge and T.X. Yu, Dynamic Models for Structural Plasticity, Springer, 1993.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

563

Straightening effects of steel I-beams failed by lateral-torsional

buckling

M. Kubo ~ and N. Sugiyama b ~Department of Civil Engineering, Meijo University 1-501, Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan bCivil Engineering Design Section, P.S. Corporation 1-17-19, Marunouchi Naka-ku, Nagoya 460-0002, Japan

This paper presents experimental results on the gag straightening effects of steel beams failed by lateral-torsional budding. Eight different welded I-sections were tested under a central concentrated load at a simply supported beam. In the first buckling tests loads were applied until unloading region of approximately 60% of the maximum moment for each beam. After the residual deformations were straightened by press working, the second buckling tests were carried out under the same loading conditions. The ultimate strength and the rotation capacity were compared with the two cases in order to investigate their recovery.

1. INTRODUCTION For bending forming and deformation straightening of steel members, the mechanical working method by use of rollers and press and the gas heating method are used. Any of these methods brings about change in the material properties of steel and the initial stress of section because it utilizes the effect of plasticity. Generally, cold working deteriorates the ductility although the yield strength and ultimate stress are increased. Deformation straightening during fabrication of members causes redistribution and reduction of the residual stresses in addition to improved straightness, and therefore, the buckling strength is increased. This fact has been verified from the experiments [1, 2] on columns. However, researches on straightening effects on ,beam members in the past have hardly been found. In this paper, lateral-torsional buckling tests were carried out under a concentrated load at the mid-span of simply supported beams. The steel beams greatly plastically deformed are straightened within the fabrication tolerances by means of press working, and thereafter similar buckling tests are made again to investigate the recovery in the ultimate strength and the rotation capacity.

564

2. TEST PROGRAM

2.1. Test s p e c i m e n s For the test specimens, there are used beams to which loads were applied until unloading region of approximately 60% of the maximum load for each beam in the previous buckling tests [3, 4]. Figure 1 shows the state of the beams failed by the initial loading, indicating that they are beam members rather greatly deformed by lateral-torsional buckling. Then, about eight months later, these buckled beams were returned by means of press straightening. The beam specimens are eight types of welded I-sections as shown in Figure 2. Four types of compact sections have small width-thickness ratio of plate elements at beam height of d =250 mm and another four types of non-compact sections have relatively large width-thickness ratio at beam height of d =300 mm. Table 1 shows the actually measured section dimensions and the widththickness ratios b/tc, 2hc/t~, of compression flange and web plate of the specimens in each series. The numeral in the specimen name represents the width-thickness Table 1 Dimensions of beam specimens Type of section

P

d (mm)

bc (mm)

bt (ram)

tw (mm)

t~ (ram)

tt (mm)

b/t~

2hJtw

DS07

0.50

250.3

MS07 MS07T ML03T DSll MSll MSllT ML04T

0.26 0.33 0.67 0.50 0.29 0.26 0.74

250.7 250.1 250.2 300.2 300.9 300.3 300.5

84.1

84.1

3.J8

6.05

.....6 . 0 5

6.95

74.9

sa.o 84.1 84.2 100.0 100.0 99.9 100.1

120.1 84.1 84.1 99.9 134.9 100.2 99.9

3.18 3.17 3.20 3.19 3.20 3.19 3.20

6.05 6.01 12.03 4.31 4.31 4.31 12.05

6.05 12.03 6.04 4.35 4.30 12.04 4.32

6.95 83.6 7.00 9O.9 3.5O 55.1 11.60 91.5 11.60 98.6 11.59 118.0 4.15 60.3

565

Table 2(a) Statistical results of initial and residual deformations for compact section ' of beam

N=16 (1)Initialstate After unloading (2)Aft~straightening Allowablelimits

Crookedness of

~kedn~

con~ressicmfl.~.m,.6es

.tension flanges

about z-axis

about y-axis

about z~axis

of

~ ~ about y-axis

k

webplate

IJuo IJvo ]./uo L/Vo h/wo 3730(co=0.69) 2668(co=0.63) 2887(o~=0.38) 2272(o~=0.70) 684(co=0.37) 131(~o=0.61) 1513(o0=0.42) 882(~o=0.95) 2119(co=0.67) 330(co=0.40) 2849(oo=0.49) 2228(o~=0.38) 3742(o~=0.35) 2~7(co=0.52) 619(~o=0.41) 1000 1000 250

Table 2(b) Statistical results of initial and residual deformations for non-compactsection " Crookednessof Crookednessof Deflecti~nof compm~n flanges tension ~ webDliite Numberof beam about z-axis abouty-axis about z-axis abouty - ~ N=16 IJuo I~r Uuo L~o bJwo (1)Initialstate 31~3(o~=0.44)3490(~o=0.62) 2887(~o=0.38) 2560(co=0.40) 490(~o=0.65) Aftexunloading 141(co=0.77)1329(co=0.59) 1259(co=0.86) 1426(co=0.79) 243(o~=0.44) (2)Afterstraightening 3(}61(co=0.39) 2396(co=0.64) 4207(co=(}.47) 3117(00=0.73) 429(co=0.4Z) Allowablelimits 1000 1000 250 ratio of the compression flange. Also, monosymmetry of the section is indicated by using a parameter of p =Ic/(Ic + I~) in which Io, L = the second moments of area about weak axis (z-axis) of the compression and tension flanges. The span length of beams was varied to four types: L=l.5, 2.0, 2.5 and 3.0 m for each series. Material properties of the original plates were determined by tensile coupon tests. The yield stress Fy is 331 Mpa for web plate and the values of the flange plates are 270, 305 and 263 Mpa for the nominal thickness of 4.5, 6 and 12 ram, respectively.

2.2. Test p r o c e d u r e As experimental equipment [3, 4], bearing supports and a horizontally moving tension jack (capacity: 200 k l ~ were used. The test beam is simply supported against lateral-torsional deformation at the both ends as shown in Figure 2. A vertical concentrated load is applied at the position 25 mm above the top surface of the compression flange at the mid-span section. The deflections and strains of the beam during loading were measured at the mid-span section. Also, the rotation angle due to in-plane bending was obtained from vertical deflections in the arm member overhanging from the ends of beam.

3. T E S T R E S U L T S

3.1. Initial deformation and residual deformation As regards the initial deformation and the residual deformation after unloading of the test beams, the crookedness of flanges and the flatness of web plates were measured. The mean value m (with coefficient variation co) obtained from the

566

1.O

Elastic theory

",,,./

loadi~8

Elastic/

Initialloadin$ ]

N/" .

~',.-"

./j'm= ./~'x.

"\ I .-'-P-

,.,

k.\.

!

0.5 f [

. . . . . MS07 ---MSOTT

0.5 h ~ ~

I I

\ " " la ~,,._ ~/~

1.0

2.0 0

1.0 2.0 0/0 p Figure 3(a). Moment-rotation curves for compact section.

0

~

~r

[ _

0

,..

'/ \ /

1.0

I

_

2.0

.

.

.

.

DSll MSll

---Mslrr -

0

.

.... MLO4T

1.0

2.0

O/Op

3.0

Figure 3(b). Moment-rotation curves for non-compact section.

maximum value in each beam can be expressed in the form of ratio of the beam length L or the web depth h as shown in Tables 2(a) and (b). In the magnitude of residual deformation by initial loading after unloading, the crookedness of the compression flange about the weak axis (z-axis) is overwhelmingly great. Also, the magnitude of out-of-plane deflection of the web plate remained approximately twice the initial deformation. From Table 2, it can be seen that the residual deformation due to lateraltorsional buckling could be straightened within the allowable limits of misalignment required in the specification [5]. In the crookedness of the flange, the strong axis (y-axis) having high rigidity is inferior in returning. The noncompact sections having larger beam height have about 1.4 times larger in the flatness of web plate. 3.2. M o m e n t - r o t a t i o n

behavior

Figures 3(a) and (b) show the moment-rotation curves in the beam having L-1.5 m. The coordinates are expressed by the full plastic moment Mp and the corresponding rotation 0 p=MpL/(2E]y), where ]y - the second moments of area about strong axis. Each section also behaves in accordance with the elastic theoretical value after straightening. After straightening, the recovery of the ultimate load differs depending upon the sectional shape. In the doubly symmetric sections DS07 and DS 11( o -0.5), the ultimate load noticeably rises, but the load reduction after the maximum load is quite rapid as compared with the initial loading. The monosymmetric sections MS07 ( o -0.26) and MS 11 ( o =0.29) with increased tension flange width have also inferior rotation capacity at the initial loading and after straightening. The monosymmetric sections ML03T (p =0.67) and ML04T (o =0.74) with increased compression flange thickness are given plastic rotation capacity after the straightening.

567

"~ 1.6

o 9 9 n

DS07 MS07 Ms07r ML03T

o DSll 9 MSll 9 M.SllT o ML04T

"~1.6

II

II

1.2

o

0 A

o

O

~ 1.2

O&

[] m

1.0

0

0

A

1.0

,Ak .

i

0.8 I

I

I

1.0

1.5 2.0 i b=~Mp/M.

Figure 4(a). Ultimate load for compact section.

~

t

.

DS07 MS07 Ms07r ML03T

0

.

.

o i

ii i

I~1

0.5

1.0

1.5

2.0

Figure 4(b). Ultimate load for non-compact section. -

II

o DSll 9 MSll 9 MSllT o ML04T

1.6 1.4

,.,o 1.2

1.2 0

9 9

0

.

I

0.8 0.5

o -' 1.6t A 9 eq 0::, 1.4 []

A

o

0

I

0.5

I

1.0

o m m~'~--

nA ]~l-. - - 0 . . . . .

1.O

A

0.8 I

1.5 2.0 i b=~MJMo

Figure 5(a). Rotation capacity for compact section.

i. . . . .

0

0.5

t

1.0

~..

o

1.5

2.0

Figure 5(b). Rotation capacity for non-compact section.

4. S T R A I G H T E N I N G E F F E C T S ON B U C K L I N G C A P A C I T I E S 4.1.

Ultimate

strength

Figures 4(a) and (b) show the recovery of ultimate strength by straightening with the ratio 6 p=Pu2/P,1 of the ultimate load after straightening to that at the initial loading. The abscissa is represented by modified slenderness ratio-~b =~Mp/~l, of the beam, where M, is elastic buckling moment. The ultimate strength after the straightening increased m= 1.14 (~ =0.14) Vanes in average for the compact sections, and m=l.06 (~o=0.15) times in average for the non-compact sections, respectively. The monosymmetric sections with large tension flange thickness have inferior recovery to other sections. Particularly,

568 the section MS11T (o =0.26) in which lateral-torsional buckling occurred early has ultimate strength decreased to m=0.89 ~mes in average. 4.2. Rotation capacity Figures 5(a) and (b) are obtained by plotting the ratio 6 0= 0 2/0 1 of rotation capacity at the maximum load in initial loading and after straightening. As compared with the figures, the rotation capacity after the straightening increased m=1.11 (~o=0.16) times in average for the compact sections, and m=1.03 (~o =0.23) times for the non-compact sections, respectively. As described above, the rate of increase in the rotation capacity is slightly lower than that in the ultimate strength.

5. CONCLUSIONS In the case of welded steel I-beams failed by lateral-torsional buckling, the recovery of the buckling capacity after press straightening was experimentally investigated. The residual deformation due to lateral-torsional buckling could be straightened within the fabrication tolerances. The recovery of the rotation capacity is slightly inferior to that of the ultimate strength. The monosymmetric section with increased tension flange thickness has inferior recovery to other sections. Even for the beam to which loads were applied until unloading region of approximately 60% of the maximum load, it could be verified that it is possible to obtain considerable buckling capacity by means of press straightening. In the future study, it seems to be necessary to pursue in detail the cause and effect relationships with the unloading level, the change in sectional stress state by straightening and the strain aging.

REFERENCES

1. Alpsten, G. and Shultz, G. W., Cold-Straightened Column, Introductory Report, 2nd Int. Colloquium on Stablity of Steel Structures, ECCS (1977) 91-97. 2. Galambos, T.V. (ed.), SSRC Guide to StabiliW Design Criteria for Metal Structures, 4th ed., John Wiely & Sons, New York, 1988. 3. Kubo, M. and Kitahori, H., Buckling Strength and Rotation Capacity of Monosymmetric I-Beams, Proc. 5th Int. Colloquium on StabiliW and Ductility of Steel Structures, Nagoya, Vol.1 (1997) 523-530. 4. Kubo, M. and Kitahori, H., Lateral buckling Capacities of Thin-walled Monosymmetric I-Beams, Proc. 2nd Int. Conf. on Thin-Walled Structures, Singapore, (1998) 705-712. 5. Japan Road Association, Specifications for Highway Bridges, Part 2; Steel Bridges, Tokyo, 1994.

Ductility/Constitutive Models

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

571

Investigation o f damage accumulation using equal channel angular extrusion / drawing R. Lapovok, R. Cottam, R. Deam CSIRO Manufacturing Science and Technology, Melbourne, Victoria 3072, Australia

The damage accumulation in continuously cast A1 6061 due to plastic deformation is investigated for different stress histories. The density variation is related to the damage parameter, where the density is measured using Archimedes' principle. The processes of equal channel angular extrusion and drawing are used to introduce some damage in the specimen for a specified stress history. The amount of plastic deformation is determined by the angle between the two intersecting channels, while the stress ratio is varied by applying different back-pressures.

I. INTRODUCTION The major concern for engineers is to avoid the ductile fracture in metal processing. It has been shown [1,2] that ductile fracture is governed by the nucleation, growth and coalescence of voids. These processes involved in the formation of defects intensify with increasing plastic strain [3]. The experimental facts underpinning damage mechanics theories consider damage accumulation to be directly proportional to the extent of plastic shear strain. However, the intensity of damage accumulation also depends on stress history. A variety of damage evolution laws has been proposed [4,5,6] where different stress characteristics have been used to evaluate the loading path. Microstructural models, [7], have shown that two of these stress parameters are most important: the mean stress responsible for the radial growth rate of cavities and equivalent stress responsible for shape change of cavities. Further, in this paper the stress history will be characterised using a stress index which is the ratio of the hydrostatic stress (mean stress) to the equivalent stress, [8]. The processes of equal channel angular extrusion (ECAE), [9], and equal channel angular drawing (ECAD), [10], have been chosen to investigate the damage introduced into a material as a function of plastic strain at a specified stress index. The amount of plastic strain introduced into a material is defined only by the angle between two intersecting channels, while the stress index can be varied by applying different back-pressure but can be kept constant during deformation. A definition of the damage parameter using density change has been shown to be efficient in the case of ductile fracture, [11, 12]. Therefore, the damage accumulation during deformation can be determined by density measurements using Archimedes' apparatus, [ 13].

572 2. THE DAMAGE MECHANICS APPROACH The phenomenological description of damage accumulation in isotropic materials can be characterised by a scalar function, to, of stress and plastic strain tensors and temperature introduced by Kachanov, [3]. The damage, co, is equal to zero for undamaged material and increases to the value 1 at the point of fracture. 2.1 Damage variable

The basic concept of damage mechanics is that damage, co, is proportional to the extent of accumulated plastic shear strain, 6, which characterises the path and the extent of plastic strain. The dependence of damage on shear strain for monotonic loading can be described by the power law, where the coefficients of the power function depend on stress characteristics and temperature, [4]. However, for ductile fracture, the density change has been shown, [12], to be a measure of the damage variable in Kachanov's definition, [3]. Following Lemaitre, [11], the relative variation in density (p-Po)/Po between the damaged state p and the initial undamaged state P0 can be related to the accumulated damage co by the following formula:

aJ = O - p / p o ~ "

(1)

For undamaged material p = P0 and to = 0. At fracture, when p goes to the critical value of zero, the damage goes to the limit value of 1, (co = 1 ). Because the material investigated has some initial damage due to porosity, the following formula has been introduced to calculate damage imparted into the material by plastic deformation:

co= O- plp,~A - O - polp,<,~

(2)

where P~a is the density of ideal material without porosity. 2.2 The stress index

It has been shown that the stress state can be characterised by invariants of stress tensor, where the hydrostatic pressure and equivalent shear stress are the most important. These two invariants will be used in the form of dimensionless quantity called the stress index, [8]. The stress index is the ratio of hydrostatic pressure to equivalent shear stress:

7

(<':, -

(3)

were ~j is the Kronecker delta and summation is done by repeating index.

3. EQUAL CHANNEL ANGULAR EXTRUSION (ECAE) ECAE is the process developed and patented in Russia by Segal in 1977, [14]. The general principle of this process is shown in Figure 1a. The tool consists of a die with two channels of

573 the same cross-section intersecting at an angle ~. A well lubricated billet of almost the same cross section is placed in the first channel and pushed by the punch through the intersection to the second channel. In the vertical channel, the billet moves as a rigid body while all deformation is localised in the small area around the channels' meeting line, [9]. As a result of this process a large uniform plastic strain is imposed in the material without reduction of the initial cross-section. The metal is subjected to a simple shear strain under relatively low pressure compared to the traditional extrusion process. As the properties of a deformed material are dependent on the stress history, the extrusion of material in the equal channel die can be done against a punch producing some prescribed backpressure, as in Figure la. In this case, the shear deformation is accompanied by some level of compressive stresses. The use of square section channels for the ECAE as used in these experiments, results in only plane strain deformation. The extruded material can be considered as a rigid-perfectly plastic isotropic medium. Following Segal, [9], the deformed state is defined by two angles: angle ~ between the axis of the two intersecting channels, and an angle determined by the curvature of the comers and level of friction in the channels. In this work, a low level of friction and sharp comers have been considered. The metal flow for this case is shown in Figure 1b. In this case the deformation is located in a thin layer around the bisector line.

L

r

__J

Punch

Entry C h a n n e l

.'< .,",,."i

:;;;;'r 9 N

i / .,: i . " / " / / " / /

i J

v~

Exit C h a n n e l

a

b

Figure 1. A sketch of a cross section of ECAE. a - ECAE with an angle r = 90 ~ b- material velocities during extrusion through the die with an angle ~ = 90* Using the expression for stress and strain components obtained in [9] the stress index during ECAE with back pressure o"2 , is defined by the following expression: ~/o-<, = (o-2/r~ + cot(~/2))/~f3

(4)

where ~ is a constant hydrostatic pressure and r.,. is the yielding shear stress. The equivalent shear strain is, [9]"

c. = 2cot(~/2)

(5)

574 Negative values of the stress index are obtained in ECAE, and they can be kept constant during deformation by controlled back pressure.

4. EQUAL CHANNEL ANGULAR DRAWING (ECAD) Suriady & Thomson, [10], were first to deform material by drawing rather than by extrusion through equal channel angular dies. They studied the change in microstructure of a specimen after multi-drawing it through a die with 135 ~ angle according to different schedules. The stress-strain state during this process was assumed to be nearly the same as for ECAE. The general principle of this process is shown in Figure 2a. In the equal channel angular drawing, Figure 2a, material undergoes a plastic deformation in which the process can be presented as a bending under tension, Figure 2b. The specimen has to be initially pre-bent to fit the channels. Therefore, the metal flow in the width direction during the following drawing is restricted by a significant large bulk of rigid material. For this reason, the strain in the width direction is neglected and the problem is considered as a plain-strain problem. The stress and strain distribution will be obtained using the assumption that transverse cross sections that were originally plane and normal to the center line remain so after deformation. The case where the pure bending is combined with tension is considered by Hill, [ 15]. If the bending is carded out while the bar is stressed by tension cr2 applied to the ends, it results in uniform pressure p = cr2h/ / r a ~ applied over the inner surface of the bar. Here h is the thickness of the specimen. The stress index can be calculated, [ 15]:

~/cre = O - 2.1nrb/r)/xf3 for

(R < r < rh)

~/tro = (- p / r , - 1 - 2. lnr/ra)/xf3 for

(6)

(7)

(r~ _
where R satisfies:

R 2 = rat b exp(-p / 2r,.)

(8)

The positive values of the stress index can be obtained by controlled back tension within those areas of materials were the tensile specimen are cut off. The equivalent shear strain can be defined as: (9)

575

Tension

--

]

!

Exit 8o

i/'/',~!: / ~ i/ : j i

BIr

0 I ~O 1

9 Ii"I,I

i z i ! f l l,.,N{~ Eltly r

a

b

Figure 2. A sketch of the cross-section of ECAD. a - ECAD die with angle r = 90 ~ b - presentation of ECAD process as a combination of bending withtension.

5. EXPERIMENTAL PROCEDURE

5.1 Sample preparation The original billet of AI 6061 used in these experiments was made by continuous casting process and has an equiaxed grain structure with the grain size changing from 80 ~tm in the center of the billet to 70 ~tm near the surface, and the corresponding hardness changing from 50 HV to 53 HV respectively. T4 pre-heat treatment was given to all specimens to ensure that the material had maximum ductility and a homogeneous microstructure particularly in terms of the precipitate state, and to eliminate the segregation effects resulting from the casting process. This heat treatment involved solution heat treatment (530~ for 2hr and quenching in water) followed by a full anneal (415~ for 2hrs and cooling in the furnace). The microstructure of specimens after heat treatment was analysed for the distribution of the second phase, grain size and grain orientation. The consistency and homogeneity of the microstructure in the transverse and longitudinal directions has been verified by optical microscopy, and also by hardness testing. The hardness after the heat treatment became relatively constant (32.7 - 33.5 HV) across the section of the billet in both directions compared to the as received material which showed an increase in hardness towards the surface of the billet. 5.2 Equipment for extrusion and drawing Three dies, with angles 90 ~, 120 ~ and 150 ~, were designed and constructed to conduct the laboratory tests using an INSTRON machine to apply the load, Figure 3a. A frame for applying back pressure and back tension to the samples during processing was designed, made and fitted to the INSTRON, Figure 3b.

576 The square bars of 10xl0 mm cross section were prepared and extruded/drawn through the dies at different speeds with and without back pressure/tension. Molybdenum di-sulphide grease was used as lubricant to minimise friction during extrusion and drawing.

C,

C

.

C

0

a

b

Figure 3. The dies and machine used for ECA extrusion and drawing a - three dies with angles 90 ~ 120 ~ and 150~ b - INSTRON set with die and hydraulic cylinder for back pressure/tension 5.3 Density measurement technique The density of samples was measured and recorded before and after Equal Channel Angular Extrusion / Drawing. The samples were cleaned by grinding with 320 grid silicon carbide paper to remove any contaminates from the surface. The apparatus used for density measurements is of the standard type were the frame is equiped with "Metier 200" scales capable of measuring of weights up to 200g to four decimal places in air or immersed in a liquid. However, the apparatus has a dual walls with a vacuum in-between to reduce temperature change during experiment. An extraction fan was fitted to the lower side wall to draw away vapor from the immersion fluid. The immersion fluid used was Tetrachloroethylene with a density of 1.62 g/m 3 closer to the density of material compared to water, in order to increase the accuracy of measurements. The liquid in container and samples were allowed to stand in the environment for at least twenty minutes to avoid thermal gradients. The tweezers were used to place the sample on the holder, which was then dipped in the fluid several times to eliminate air bubbles on the surface of the samples that could introduce error into the measurements. Once the scales had stabilised the measurement was taken and the density calculated as follow, [13]: p - p~w~/(w, - w2)

(10)

where p~ is the density of fluid, and wl, w2 are the weights of the sample in air and in the fluid respectively. The temperature of the fluid was recorded to correct the fluid density for temperature change.

577 6. RESULTS OF DENSITY MEASUREMENTS Two specimens have been extruded and two specimens have been drawn through each of the three dies (~ = 90 ~ 120 ~ and 150 ~ with different velocities of the ram equal to 4, 40 and 400 mm/min to provide three different strain rates equal to 0.002, 0.02 and 0.2 s"l. The density of specimens were measured before and after processing according to technique described in section 4.3. The back tension in ECAD was varied between 0.3 and 5.9 kN and the back pressure in ECAE was varied between 3 and 26.5 kN, providing a variation of stress index between -6.96 and +0.49 according to formulas (4), (6), (7). Variation of strain was between 0.02 and 2.00 according formulas (5), (9). As has been shown, the damage is a function of both the equivalent plastic strain and the stress index during deformation, [8], and can be expressed as: co = a6e exp(b 9~/cre)

(11)

Over seventy data sets were used to calculate the damage, equivalent shear strain and stress index. These values fitted by this function give best values for coefficients of a =0.00353, b = 0.112136. Function (11) is presented in Figure 4. It can be seen that the damage introduced into the material increases with the plastic strain, but the intensity of damage growth depends on the stress index. Negative values of stress index can lead to a suppressing of damage accumulation while positive values of stress index intensify the process of damage accumulation. As expected, the damage increases for stress index changing from negative values to positive values and with increase of strains. The difference between damage data for different strain rates does not seem significant, which reinforces the hypothesis that damage accumulations at room temperature is independent of strain rate.

~176i

Figure 4 Damage introduced into material during plastic deformation as a function of plastic strain and stress index.

578

7. Conclusion The damage accumulation in continuously cast A1 6061 due to plastic deformation has been studied using the processes of equal channel angular extrusion and drawing. The damage accumulation has been shown to be proportional to the amount of plastic strain, while the intensity of accumulation is dependent on the stress index. As the density variation is related to the damage parameter, this work has developed a new experimental technique for the determination of porosity development in materials during plastic deformation.

References 1. Dodd, B., Bay, Y., Ductile Fracture and Ductility, Academic Press, Harcourt Brace Jovanovich, Publishers, 1987. 2. Atkins, A.G., May, Y-W., Elastic and Plastic Fracture, Ellis Horwood Series Engineering Science, Chichester, 1985. 3. Kachanov, L.M. Introduction to Continuum Damage Mechanics, Kluwer Academic Publishers, Dordrecht, 1986. 4. Lemaitre, J., Chaboche, J. A Non-Linear Model of Creep-Fatigue Damage Cumulation and Interaction, In: J. Hult, editor. Mechanics of Visco-Plastic Media and Bodies, 291-300, Springerverlag, Berlin, 1975. 5. Gurson, A.L., Continuum Theory of Ductile Rupture by Void Nucleation and Growth, Journal of Engineering Materials and Technology, Transaction of the ASME, Vol. 99, 1977, pp. 2-15. 6. Golos, K.M., A Hypothesis of Cumulative Damage for Fatigue Crack Initiation and Propagation, In: Proceeding of 9th Intern. Congress On Fracture, Vol. 3, Pergamon Press, UK, 1997, pp. 1353-1360. 7. Tvergaard, V., On Localisation in Ductile Materials Containing Spherical Voids, International Journal of Fracture, Vol. 18, 1982, pp. 237-252. 8. Lapovok R., Smirnov S., Shveykin V., Damage mechanics for the fracture prediction of metal forming tools, Intemational Journal of Fracture, Vol. 100, 2000, in print. 9. Segal V.M. , Reznikov V.I. , Drobyshevskiy A.E. ,Kopylov V.I. , Plastic working of metals by simple shear, Translation. Russian Metallurgy, (1), 1981, pp. 99-105. 10. Suriadi, A.B., Thomson, P.F., Control of Deformation History for Homogenizing and Optimizing Mechanical Properties of Metals, Proceeding of Australiasia-Pacific Forum on Intelligent Processing & Manufacturing of Materials, IPMM, 1997, pp. 920-926. 11. Lemaitre, J., Dufailly, J., Damage Measurements, Engineering Fracture Mechanics, Vol. 28, No. 5/6, 1987, pp. 643-661 12. Chaboche, J.L., Continuum Damage Mechanics, Journal of Applied Mechanics, Transaction of the ASME, Vol. 55, 1988, pp. 59-71. 13. Aitcin, P.C., Density and Porosity Measurements of Solids, Journal of Materials, Vol. 6, No. 2, 1971, pp. 282-294. 14. Segal V.M., The Method of Material Preparation for Subsequent Working, Patent of the USSR, No 575892, 1977. 15. Hill R., The Mathematical Theory of Plasticity, Oxford University Press, 1986, pp. 355.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

A simplified constitutive loading conditions

579

m o d e l for s t e e l m a t e r i a l

under

cyclic

S. M u r a k a m i a , S. Nara a, Y. S h i m a z u b a n d T. Konishi c a D e p a r t m e n t of Civil Engineering, Gifu University, 1-1, Yanagido, Gifu, Gifu 5 0 1 - 1 1 9 3 , J a p a n b O k u m u r a Corporation, 29-8, T a k e h a s h i - C h o , N a k a m u r a - K u , Nagoya, Aichi 4 5 3 - 0 0 1 6 , J a p a n c G r a d u a t e d School of Gifu University, 1 - 1, Yanagido, Gifu, Gifu 501 - 1193, J a p a n This p a p e r p r e s e n t s a simplified c o n s t i t u t i v e m o d e l for s t r u c t u r a l steel material. This m o d e l c o n s i s t s of p a r a m e t e r s r e l a t e d to b o t h m a t e r i a l p r o p e r t i e s a n d cyclic b e h a v i o u r . The former p a r a m e t e r s are o b t a i n e d by m a t e r i a l c o u p o n t e s t u n d e r m o n o t o n i c loading a n d t h e l a t t e r ones c a n be given as a c o n s t a n t with no relation to m a t e r i a l classification. N u m e r i c a l s i m u l a t i o n s r e l a t e d to plate b u c k l i n g p r o b l e m are c a r r i e d o u t a n d the validity of t h e p r e s e n t c o n s t i t u t i v e m o d e l is investigated. 1. INTRUDUCTION Cyclic plasticity a s s o c i a t e d with the m a t e r i a l plastic b e h a v i o u r u n d e r cyclic loadings is i m p o r t a n t in s t r u c t u r a l a n a l y s i s . Dafalias-Popov[1] p r e d i c t e d s t r e s s - s t r a i n r e l a t i o n s h i p s u n d e r cyclic loading c o n d i t i o n s t a k i n g a c c o u n t of the yield-surface a n d t h e b o u n d a r y - s u r f a c e . S h e n Mizuno-Usami[2] a n d N i s h i m u r a - I k e u c h i [ 3 ] i m p r o v e d this c o n s t i t u t i v e model p r o p o s e d by Dafalias-Popov, respectively. However, the above c o n s t i t u t i v e models d e m a n d m a n y series of e x p e r i m e n t u n d e r b o t h of m o n o t o n i c and cyclic loading to o b t a i n t h e p a r a m e t e r s t h o s e c o n s i s t of each constitutive model. Therefore, it is difficult to predict elastoplastic r e s p o n s e of existing s t r u c t u r e s u n d e r s e i s m i c loading. 2. CONCEPT OF CONSTITUTIVE MODEL 2.1 Conditions Figure 1 shows a n e x a m p l e for s t r e s s - s t r a i n r e l a t i o n s h i p u n d e r cyclic loading condition. Figure 2 s h o w s the c o n c e p t of yield or m e m o r y

580 2( ~*+d ~ *)'LG 8

memory

surface

rfsce

7,y ,

li_. Figure I.

E

"+d ~ ") An e x a m p l e of s t r e s s -strain relationships

F i g u r e 2.

Deviate s t r e s s s p a c e

Table 1 M e c h a n i c a l p1'operties of steel g r a d e ( r e g u l a t e d in JIS) Steel G r a d e SS400 SM490 SM570 .... E(GPa} 207 209 " 203 oy(Mpa} i " 270 ..... 379 . - 489 EP, t(Gpa) 5.19 .~ 5.32 . 2.84 ..... eP.t 0.0125 0.0185 ...... 0.0057

-

LYRS6O

|...

,

.

.

206 460

4 66 0.0094

s u r f a c e in d e v i a t e s t r e s s s p a c e , xo, r a n d r* is the r a d i u s of initial yield s u r f a c e , t h a t of c u r r e n t yield s u r f a c e a n d t h a t of m e m o r y s u r f a c e , respectively. The p r e s e n t c o n s t i t u t i v e m o d e l t a k e s a c c o u n t of e l e m e n t a r y c r i t e r i o n a n d r u l e s in e l a t o - p l a s t i c region. T h e y a r e yon M i s e s ' s yield criteria, Ziegler's h a r d e n i n g rule, P r a n d f l - R e u s s p l a s t i c flow r u l e a n d h y b r i d h a r d e n i n g rule [4]. S t r e s s - s t r a l n c u r v e u n d e r cyclic l o a d i n g c o n d i t i o n is b r o k e n d o w n into t h r e e regions. F i r s t a n d t h i r d region, s u c h a s t h e region from p o i n t 3 to p o i n t 4 a n d t h a t f r o m point 5 to p o i n t 6 etc. a s d r a w n in Fig. 1, are d e t e r m i n e d a c c o r d i n g to s t r e s s - s t r a i n r e l a t i o n s h i p s u n d e r m o n o t o n i c l o a d i n g c o n d i t i o n . On t h e o t h e r h a n d , s e c o n d r e g i o n is p l a c e d a s t r a n s i t i o n a l region b e t w e e n first a n d t h i r d region. T h e s e t h r e e r e g i o n s a r e n a m e d a s elastic one, t r a n s i t i o n i n g one a n d h a r d e n i n g one, respectively. 2.2

Parameters Most of p a r a m e t e r s of t h e p r e s e n t m o d e l are d e t e r m i n e d by t h e m a t e r i a l c l a s s i f i c a t i o n of steel g r a d e a n d c a n be o b t a i n e d b y m a t e r i a l c o u p o n t e s t u n d e r m o n o t o n i c l o a d i n g condiUon. E x a m p l e s of t h e s e p a r a m e t e r s for 4 k i n d s of steel g r a d e r e g u l a t e d in J a p a n I n d u s t r i a l S t a n d a r d (JIS} a r e listed in T a b l e 113].

581

1,

9 SS400(Ref.3)

i

O. o ~t

o

0.6:

"0.5-

q

200 |

A

Sl1490(Ref.3)

180

m

SIB70(Ref.3)

160 140 G

o

LYR590(Ref.3)

120

SS400

100

0.3-

~,-- SM490

60

0

41--- SM570

O. 1

0 - LYR590

0 4-

O.

0.0

e--

0.5

Figure 3.

1.0

'

1.5 20 C Pmon(%)

u

2.5

SS400(Present) ~

SIJ490(Present) SII570(Present) ~

LYR590eresent)

- - - -

3.0

Varying of K/Ko

0

"' . . . . .

mmumi [...................................

0 0.5 1 1.5 2 2.5 3 3.5 4 4 5

5

eP

Figure 4.

Varying of E P

The p r e s e n t model c o n s i d e r s more t h r e e significant p a r a m e t e r s , K/Ko, a n d E P. Figure 3 s h o w s the v a r y i n g of K/Ko i n v e s t i g a t e d by N i s h i m u r a et al[3]. K is the c u r r e n t size of elastic region of e l a s t o - p l a s t i c h y s t e r e s i s c u r v e a n d Ko is the initial one. ePmo, is the effective plastic s t r a i n on s t r e s s - s t r a i n curve u n d e r m o n o t o n i c loading condition. K b e c o m e s less t h a n ~o as ePmo, glows. Finally, it will converge on a c e r t a i n value. In the p r e s e n t model, the ratio of r to Ko is given as a c o n s t a n t of 0.6. According to h y b r i d - h a r d e n i n g rule, t h e ratio of isotropic h a r d e n i n g to k i n e m a t i c h a r d e n i n g is defined as (l-x): X. In t h e p r e s e n t model, X is given as a c o n s t a n t of 0.5. Figure 4 s h o w s the v a r y i n g of the t a n g e n t i a l coefficient in p l a s t i c i t y (EP). E P d o m i n a t e s e l a s t o - p l a s t i c b e h a v i o u r in t r a n s i t i o n i n g region. In the p r e s e n t model, e s t i m a t i n g f o r m u l a of E p is given as follows,

1.0 + ~-/ey

' ,,.,.,..

w h e r e E is the elastic m o d u l u s , ey is yield s t r a i n a n d eP is a c c u m u l a t e d effective plastic s t r a i n . 3. NUMERICAL SIMULATIONS 3.1

Material Coupon Tests Under Cyclic Loading Cyclic loading t e s t s for 4 k i n d s of steel m a t e r i a l s , t h o s e arc listed in Table 1, are s i m u l a t e d with u s i n g the p r e s e n t c o n s t i t u t i v e model. The n u m e r i c a l r e s u l t s are c o m p a r e d with e x p e r i m e n t a l d a t a a n d d r a w n in Figs. 5(a)-(d). Black s y m b o l s a n d solid lines s h o w e x p e r i m e n t a l d a t a a n d the n u m e r i c a l r e s u l t s , respectively. T h o u g h m e c h a n i c a l p r o p e r t i e s of SM490, S M 5 7 0 a n d LYR590 are different from e a c h other, t h e p r e s e n t

582

4OO

400

~'~f/X,,dI ! ~o

~

~

200

{o

H'I! !1

-21111 9 Ref. 3

-400

_6001 .... I .... I .... I .... -3

-2

-1

I

0 1 ePo;)

- ~ - - Present 2

3

Present -600

-3

.

-2

(a) SS400

.

.

-1

.

0

.

.

1

.

.

e P(IQ) (b} SM490

2

.

.

3

600

600

/~' Iii tflt III

200

~0 b

-3

~oo

~o

Iil II I~

-200

Ii1 !~~,

Ref. 3

-1

3

..... ~ . . I -2

0 1 P (%)

((fAlllll,

400

~IIIIIil

-400

i -3

-2

-1

0

~P(%)

1

9 Ref. 3 2

3

(c} SM570 (d} LYR590 Figures 5. C o m p a r i s o n b e t w e e n n u m e r i c a l a n d t e s t r e s u l t s model gives the n u m e r i c a l r e s u l t s t h a t are in good a g r e e m e n t s with e x p e r i m e n t a l data. R e m a r k a b l e points are t h a t the model n e e d s only m e c h a n i c a l properties of m a t e r i a l a n d does not require a n y cyclic loading test to o b t a i n the constitutive model. However, in case t h a t steel grade is SS400, the n u m e r i c a l r e s u l t is m u c h different from t h e e x p e r i m e n t a l data. We are investigating the r e a s o n of this difference.

3.2

Plate Buckling Analysis U n d e r Cyclic Loading C o m p u t e r model is simply supported square plate subjected to unlaxial cyclic loading as s h o w n in Fig. 6. The relative slenderness of plate is 0.767 and aspect ratio of plate is 0.7. Both initialplate deflection {W o} and residual stress, those are illustrated in Fig. 6, are considered as initial imperfections of plate. Initialplate deflection is given as follows, W~

b

xX sin nY" cos a b

(2}

583 Values of tensile a n d c o m p r e s s i v e r e s i d u a l s t r e s s e s (a, a n d arc] are given as ay a n d - a v / 3 , respectively. Steel grade is S S 4 0 0 a n d m a t e r i a l p r o p e r t i e s of t h i s steel are listed in Table 1. Figure 7 s h o w s t h e loading p a t t e r n . P a r a m e t e r m(-4) is the m a x i m u m ratio of average s t r a i n of plate to yield s t r a i n of m a t e r i a l . Figures 8(a)-(c) s h o w s t r e s s - s t r a i n h y s t e r e s i s c u r v e s a n d F i g u r e s 9(a)-(c) show s t r e s s - d e f l e c t i o n r e l a t i o n s h i p s of plate. The s u b t i t l e s of (a), (b) a n d (c) m e a n the c o n s t i t u t i v e model, s u c h as t w o - s u r f a c e model[2], the p r e s e n t m o d e l a n d h y b r i d - h a r d e n i n g model[4], respectively. B r o k e n lines are t h e n u m e r i c a l r e s u l t s w i t h o u t r e s i d u a l s t r e s s e s a n d solid lines are those with r e s i d u a l s t r e s s e s . The p r e s e n t m o d e l gives the r e s u l t s t h a t are in b e t t e r a g r e e m e n t s with those p r e d i c t e d by t w o - s u r f a c e model t h a n t h o s e done by t h e h y b r i d h a r d e n i n g model. The differences in average s t r e s s b e t w e e n b o t h r e s u l t s of t w o - s u r f a c e model a n d the p r e s e n t model are a b o u t 5 % in t e n s i o n a n d 6 % in c o m p r e s s i o n . S u c h r e s u l t s do n o t d e p e n d on residual stresses. 4. CONCLUSIONS A simplified c o n s t i t u t i v e model for steel m a t e r i a l s is p r o p o s e d a n d its validity is investigated. N u m e r i c a l s i m u l a t i o n s of steel p l a t e s s h o w t h a t the r e s u l t s p r e d i c t e d by the p r o p o s e d m o d e l give very good a g r e e m e n t with r e s u l t s b a s e d on t w o - s u r f a c e model. An i n t e r e s t i n g c o n c l u s i o n is t h a t m e c h a n i c a l p r o p e r t i e s of steel m a t e r i a l o b t a i n e d by tensile c o u p o n test m a k e it possible to predict the cyclic plasticity b e h a v i o u r of t h e plate buckling. Three significant p a r a m e t e r s of x / x o, ~ a n d E P are defined t e n t a t i v e l y referring to e x p e r i m e n t a l d a t a at p r e s e n t . Therefore, it n e e d s to e x a m i n e s u c h experiential v a l u e s a n d to r e s e a r c h t h e i r t h e o r e t i c a l s o l u t i o n s . REFERENCES [1] Y. F. Dafalias a n d E. P. Popov: A Model of Nonlinear H a r d e n i n g Materials for Complex Loading, Acta Mechanics, (1975) 173. [2] C. S h e n , E . Mizuno a n d T. Usami: Generalized Two-Surface Model for S t r u c t u r a l Steel u n d e r Cyclic Loading, P r o c e e d i n g s of J S C E , S t r u c t u r a l E n g i n e e r i n g / E a t h q u a k e Engineering, Vol. 10, No.2, (199?) 23. [3] N.Nishimura, T.Ikeuchi, :Hysteretic Behavior of Steel Plates u n d e r Uniaxial T e n s i o n a n d C o m p r e s s i o n , Technology Reports of t h e O s a k a University, Noi.45, No.2232, (1995) 221. [4] S. Nara, M. H a t t o r i a n d Y. Moriwaki: E l a s t o - p l a s t i c Analysis of Steel Plates U n d e r Cyclic Uniaxial Loading, Proceedings of S y m p o s i u m on C o m p u t a t i o n a l Method in S t r u c t u r a l Engineering, Vol. 19, (1995) 177. (in Japanese)

584

T/ev

e8

OOmm )mm tO0

Figure 6.

Computer model

Figure 7.

Loading pattern

>,, L

E'y =0.0 =1/3 (a) Two-surface model (b) Present model (c}Hybrid-hardening (Ref.2) model (Ref. 4) Figures 8. Stress-Strain hysteresis

,~1[ Ib

0.5

:"-..

I~

.i -(

-0.5

-

(a) Two-surface model (b) Present model (c)Hybrid-hardening (Ref.2) model (Ref. 4) Figures 9. Stress-deflection relationships of plate

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

585

Acceleration waves and dynamic material instability in constitutive relations for finite deformation P.B. B d d a ~ and

Gy. B 6 d a b

"HAS-TUB Research Group of the Dynamics of Machines and Vehicles, Technical University of Budapest, Bertalan L. 2, H-1111 Budapest, Hungary bDepartment of Applied Mechanics, Technical University of Budapest, Mfiegyetem rkp. 5, H-1111 Budapest, Hungary

ABSTRACT Acceleration wave in a generalized sense is considered to be the case when there is a jump in the derivatives (with respect to time and space coordinates) of the acceleration field. The constitutive equation should satisfy the assumption that in general such waves have at least four wave speeds. Finally, a one-dimensional form of that constitutive equation will be used to find the conditions of material instability for finite deformations.

1. INTRODUCTION In the recent years material instability problems have been studied with increasing interest [1,2,3]. These cases may even lead to rapid catastrophic damage of structures. However, most of the investigations published dealt with small deformation and static or quasi-static loading conditions. To perform work involving together dynamic effects and high rate loadings we need appropriate constitutive equations. This kind of materials were studied by postulating the existence of a (second-order) acceleration wave with finite wave speed [4,5,6]. Unfortunately, the constitutive theory based on the second-order waves cannot consider the so-called second gradient materials [3,7,8] widely used for numeric investigations of postlocalization. The effect of including second gradient terms and the difficulties in dynamic studies can be described by applying the theory of dynamical systems [9]. Dynamical systems are widely used for (Lyapunov) stability investigations of various cases [ 10]. Quite recently we have obtained results for the forms of possible constitutive relations by combining acceleration wave dynamics and the theory of dynamical systems [ 11]. In this paper we assume that a generalized wave (a jump) exists in the derivative of the acceleration field and this singular surface propagates forwards and backwards with finite velocities. From that assumption conditions are obtained for the second-order derivatives of the variables of the constitutive equations. To find further conditions we prescribe that loss of

586 stability should be genetic in terms of the theory of dynamical systems, which is essential in dealing with instability problems. All the studies are performed for finite deformations. The resulting constitutive equations are suitable for solving material stability:instability problems with large deformations at high rates of stress. The set of fundamental equations of continuous bodies consists of the Cauchy equations of motion, the kinematic equation (for large displacements) and the constitutive equations in generalized form f~(...)= 0, where a = 1,...,6. Notations are as usual: x' - r is time, /9 mass density, v' - velocity, qi . the body force, go "the metric tensor, t '~ - the Cauehy stress tensor, v o. - the deformation rate tensor, v:j. k . the velocity gradient and a~. - the Euler strain tensor. Semicolon denotes eovariant derivative, while overdot indicates material time derivative:

~'

=

.... + V k i ~cT~r v;k. In the constitutive functions f~(...) the variables are still

indeterminate. In the next part, the necessary variables to fulfill the condition of the existence of a third-order wave with finite speed will be shown.

2. SECOND-ORDER CONSTITUTIVE EQUATIONS In constructing constitutive equations we use physically objective quantitites such as -the Lie derivative of the stress gradient tensor L~(t;~), - the Lie derivative of the (Euler) strain gradient tensor ~(ao.;! ), the second covariant derivative of stress t ~v and of strain a t. tensors. The constitutive equation reads -

f~(L~(t~;,),t~.,,l~(aij:k),ao.:u)=O,

(1)

Lie derivative is used as in [ 12], i.e.

L~(t~';I):(t~';,)" - t ~ ;lViqk t~ ;lv~q+ P t ~ :q:i v q and L~(ao~) = (a#;,)'+a~:,v:~+a,q~v:j+ao.qv q q . ~.. Along wave surface ~Xg~x,,x4)= 0 the second derivatives of the stress and of the strain tensors have a jump [(t~j)']=y~(q~4+v'q~)q~.Then and [v:]=O. Similarly, [L~(ar

IL~(t~;j)']=I(t~;~)" ] because [(t~';~)]=O because [(ar

and

[v;~q]= O. Note that while (t e:,)" * (te): , for the jump I(t kp.,)'] = [(t~'): ] and it is satisfied in other cases. Now, the equations of motion for the Lie derivative of stress are

587

(LJ~')p+ (t+V~q+ t vi,),,+

v,, q v;,

+q

"''

t ~' = t "~ .

(2)

The Lie derivative of the kinematic equation is L,,

,j = ii o. + a, k9~ + %.V~ +

. + 2g%.V~ + a,kv:~v;j + %.V;eV~ + aavijv:,

.

(3)

From (1), (2) and (3) we can calculate an equation for the propagation of the generalized wave. First, the wave speed can be expressed as c = the wave front reads

nn = ~

~g

~4~,, +

~o4 + tPk~O~ v"~o. and the normal unit vector of 4gk~

..... Then the wave propagation equation is ~Ok~O! (4)

r l

Using notations ~(L~te; )_= T ~ ,

O~/ra

4L~a~., )

_

ijk

A:

G

G = E~~ for the

coefficients of (4), we get

O:~a r~' ~kp(vr(pl .l.(..+v o.)~,)+A~k(vr(pk..~(~04. +..n,.)6;) ++,"

+S.'r"(e'.q,.

-

9

q,.e:)+ E y 2~,~,~,,.

Then by considering the compatibility equations [6] the wave propagation equation reads

+Eo .~.:.((~,,g,,)~,+

(5)

For ~'~ r 0, (5) should have a solution, thus, the matrix expression coefficient in bracket {..} should have a zero determinant, hence, det{..}= 0 is called the wave speed equation. In onedimensional case, scalar variables and parameters are used instead of matrices. Then the wave speed equation is

2pr: - 2::

- A2(2~- 0c + E2(2~- 1) = o.

A constitutive equation is possible if (6) has three real roots c representing the wave speeds.

(6)

588 3. MATERIAL INSTABILITY IN ONE-DIMENSIONAL CASES

For the stability investigation of state S O of the continuum, the set of the fundamental equations consists of the Cauehy equation of motion, the kinematic equation for large displacements = 1 a~

a

~

2 a 03,

(7)

and the constitutive equation. Let us assume that there is a simplified constitutive equation of the same type as in part 2 and it can be transformed into

=~~_

~

~:

~,~.

(81

Assume that state S O is described by ao,to,Vo, thus, the system of equations (7), (8) is satisfied by these values. The Lyapunov stability investigates the response of the system to sufficiently small perturbations, thus, the perturbed quantities a 0 + Aa,t 0 + At, vo + Av should be substituted into the fundamental equations. Expressing also the material derivatives and linearizing them in S ofrom (7), (8), a set of differential equations is obtained for perturbations tT~V ,~v ~ A, ~v ~a ,%w = C, ,, + C~ - ~ + C~,~+ C~ -~ + C, -~r + C~ --~- + C, - - - -

~a~r

(9)

- ~ = D, v + D2 - ~ + D3a + D4 --~ where A is omitted for the sake of simplicity and C~ = - 2 ~176 - 2 ~~v~

2K~ tri o ,C3 = 2K~ O~Vo,C4= 2K~ a2~o

p

K,,

C5 = -VZo_ K , + 2K, ao,C6 = _ K 3 C7 = 2vo,D, = ----~-,D2 = -vo,D3 = -26'3"~ P

P

p'

~2~ + 1

~'

With new variables y~ = a,y2 = v, y3

69r and vector y

[Y~,Y2,Y3] a dynamieal system can

be formed from (9)

8 ~_y= tyr

D~~ + D~

D, + D. ~

0

0

0

1

G -sa + c~ + c , ~- ~

a + c , -a~~ c ,a c, + c4 --s

.

(lo)

589 The stability investigation is based on the eigenvalues of the operator defined by the fight hand side of (10) (see [9] or [13]). A characteristic equation can be obtained

Xy=

8 + D3 D2 --~ 0

8 D, + D4--~ 0

c,--g+c~ +c6--~ c, +c, a~

0

1

.

(11)

c,~-r c,

Equation (11) is of generic type. State S O is stable if Re2 < 0 for all the eigenvalues of (11). Let us study the case of divergence instability, which is closely related to the strain localization. Then on the stability boundary 2 = 0. From (11) we obtain the localization condition _vo__ ~ . - 2-~--y, e;'3~'o ---~-y2 ~o + (1 - 2a)03"=0

21(1 ~ #__!~+ ..1(! O~Vo K3~y.___ &2 L -2 (O~Vo , &2 o)Y2 p & o5c z - - ~ - - ~ - y , - p o5co%

+(2-K, - - -,~o -K2 p&

+-Vo~ K ' - 2 P

(12)

ao -~=0

Assuming, for example, that v 0 = 0. ~a,o .

0. from the first equation of (12) y, = y, (Y2) can be

expressed and substituted into the second equation of (12), thus, the localization conditions are ~o + K~ -a~o a~o + 2 p .&o ~~Vo = 0, -4aop(K, - K~)--~- ~ = 0,K, ~v0 &--T--~--

K,(6~o'~o ~v~ ))2RK~-~~ 0,K,(1- ao)0. a3ca3r ' + 2 - ~ ~ - a o -~= =

(13)

From (13) we can see, for example, that the constitutive equation Lv(t) = 0 does not imply instability (as in the classic small deformation case). Now ~V0 = 0 should also be satisfied. Finally, let us remm to the wave speed equation. With notations of part 3, (6) reads

(14) We require real solutions c of (14), thus

590 should hold, moreover, for the signs of the consecutive terms in (14) the Cartesian rule should be satisfied: either twice the opposite and once the same signs, or twice the same and once the opposite signs. These conditions should be added to the stability condition ReX < 0 for the solutions of (11 ). The divergence instability studied before results zero wave speeds, c = 0 in (14), and this shows that the classic material instability condition [ 1] and the divergence instability definition ,;L= 0 describe the same phenomenon.

ACKNOWLEDGEMENTS This work was partly supported by the National Scientific Research Fund of Hungary under contract T022163. REFERENCES 1. J.g. Rice, The localization of plastic deformation, Theoretical and Applied Mechanics, Ed. W.T. Koiter, North-Holland Publ., Amsterdam, (1976) 207. 2. L.J. Sluys and R. De Borst, Wave propagation and localization in a rate-dependent cracked medium - model formulation and one-dimensional examples, Int. J. Solids Structures, 29, (1993) 2945. 3. H.M. Zbib and E.C. Aifantis, On the localization and post-localization behavior of plastic deformation, I. Res Mechanica, 23, (1988) 261. 4. Gy. B6da, Methode zur Bestimmung der Materialgleichung, Publ. of the TU-Miskolc, 12, (1962) 271. 5. Gy. B6da, Possible constitutive equations of the moving plastic body, Advances in Mech., 10, (1987) 65. 6. Gy. B6da, Constitutive equations and nonlinear waves, Nonlinear Analysis, Theory, Methods and Appl., 30, (1997) 397. 7. R. de Borst, L.J. Sluys, H-B. Mfihlhaus and J. Pamin, Fundamental issues in fufite element analyses of localization of deformation, Engineering Computations, 10, (1993) 99. 8. I. Vardoulakis, Potentials and limitations of softening models in geomechanics, European Journal of Mechanics, A/Solids, 13, (1994) 195. 9. P.B. B6da, Material Instability. An application of dynamical systems in continuum mechanics, Dynamics of Continuous, Discrete and Impulsive Systems, 5, (1999) 123. 10. H.Troger and A. Steindl, Nonlinear stability and bifurcation theory, An introduction for scientists and engineers, Springer, Wien, New York, 1990. 11. Gy. B6da and P.B. B6da, A study on constitutive relations of copper using the existence of acceleration waves and dynamical systems, Proc. of Estonian Academy of Sci. Engin., 5, (1999) 101. 12. J.E. Marsden and T.J.R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, 1983 13. P.B. B6da, Material instability in dynamical systems, European Journal of Mechanics, A/Solids, 16, (1997) 501.

Structural Failure and Plasticity (IMPLAST2000)

Editors: X.L. Zhaoand R.H. Grzebieta 9 2000 ElsevierScienceLtd. All rightsreserved.

591

Plastic D e f o r m a t i o n and Creep o f P o l y m e r Concrete with Polybutadiene Matrix

O.Figovsky ~ D.Beilin b, J.Potapov c " Polymate LTD, MigdalHaemek,,Israel t, SchwarzStephan-CivilEngineeringLTD, Haifa, Israel c CivilEngineeringUniversity,Voronezh, Russia

Abstract Composite material based on polymer binder mad mineral fillers are widely used as structural chemically resistant, vibration and impact proof materials for industrial construction and chemical machinery. Up to the present hetero-chain polymers: unsaturated polyesters, polyepoxy and polyurethane are applied as a binder of such concrete-polymer composites. A new type of conglomerate composites - polymer concrete based on polybutadiene belonging among the liquid rubbers (RubCon) has been enveloped and investigated. Such a rubber is used as a binder hardened by sulfur in the presence of special admixtures. Quartz sand and fly ash may be used as fillers and fine-grained granites and basalt chipping as coarse aggregate. The resulting material has very acid and alkali resistance, toughness and adhesion to metal reinforcements, low water absorption and remarkable compression strength (80-90 MPa). Plastic deformation and creep of RubCon samples were investigated under compressive load applied for 180 days at standard conditions and in aggressive media. Based on experimental results rheological characteristics and creep curves were obtained. Formula, which enables to determinate of material deformations in any time of material work under load, was derived as well. 1. Introduction Composite materials on oligodiene base hardened by the standard sulfur system are successfully used in anti corrosive technology. In order to improve physical and technical properties, to decrease the compositions cost and consequently to expand an application spectrum of these composites, the fillers and aggregates of organic and inorganic nature were used. We have designed and investigated an effective new type of composites: rubber concrete based on polybutadiene, belonged to liquid rubber. Such rubber is cured through double bonds of polymer chains in the presence of sulfur accelerated, oxidizing- restorative or peroxide systems. The curing system consists of vulcanizing agents (e.g. sulfur), accelerators and activators and special additives.

592

For the purpose of inner stress, creep and shrinkage decrease we used the fillers as finely dispersed inorganic and organic matters. Such fillers were quartz sand and fly ash. The mixture consists of single component package for hot curing (150-180 ~

with a shelf life of three months and a two-component package for cold (20-

25~

and semi-hot (70-100~

curing with a shelf life of six months. The

components can easily be formulated on-site m a non-toxic and completely safe manner. 2. Experimental Results

Accomplished investigations made it possible to suggest a number of RubCon compositions for chemically resistant products and structures and to reveal the main physical and mechanical characteristics (Table 1)1~ Table 1

Com~n

Indices

1 Density. ( kg/m3)

2

3

4

2100-2200

2100-2200

2100-2200

2150-2300

70-80 12-15

75-85 20-25 12-15

80-90 25-30 13-19

70-75 18-25 12-15

Modulus of dastieily (MPa xl 04)

1.2-2.5

1.2-2.5

1.9-2.7

1.2-2.5

Poison ratio

0.20-0.26

0.26-0.28

0.26-0.28

0.26-0.28

Thermal conductive' coefficient (W/m~

0.3-0.5

0.3-0.5

0.3-0.5

0.3-0.5

Heat stabili~" (Oc)

80-100

80-100

80-100

80-100

Water absorption (%)

0.05-0.06

0.05-0.06

0.05-0.06

0.05-0.06

0.97

0.98

0.975

0.97

lactic acid

0.95

0.965

0.96

0.95

-20% solution of caustic potash water

0.995

0.995

0.995

0.995

Strength (MPa) at compression at bending at tension

18-25

Coefficient of chen~cal resistance (after 360 days x 24 hours exposition) -20% solution of sulphuric acid -10% solution of

593 We investigated the process of RubCon deformation at short-term action of compressive load (T) as well. Poisson's ratio was obtained on the basis of the experimental data. During the experiment we controlled a process of

RubCon samples

destruction. It took place in working zone along filler grains and binder. The destruction surface was a well-marked cone-shaped form, which is an inherent feature of the polymer concrete. We did not detect the rupture along boundary; consequently for RubCon the strength of adhesion connections with filler is higher of latter strength and cohesion strength of polymer binder. In the investigation framework we studied creep of RubCon as well. Samples 40x40x160 mm was loaded with help of special lever system and ball supports. As results the samples were subjected to axial compression. During the test we analyzed RubCon stress-strain state on dependence of value and time of compressive load action. The sample series was tested under constant compression is equal to 40,50,60,65,70.75,80,85 and 90 % of mean ultimate strength (Tn), which was received fi'om short time tests. Creep deformation was determinate through the test time. Creep curves are shown on Fig. I and 2.

1 __l

|i Ill I | ]

]F~I~r

4g.7 m,a

~ JtJ ! ! ~ T - - - - - ~

0.8

~

,

I III

i

I

I ,,.8

0.6 ' i" ....

0.4

i

i / , t z-!

~-- 40.0 vpa~ a~.o~Pa'~

-

t

0.:~

0

30

60

90

Fig.l. Creep curves at compression

120

160 days 180

594 8o wo |

'

'

<1

,f

/

79

m

o oN

v

0

30

60

90

120

160

180

days

Fig.2. The RubCon dependence of exposition time on aggressive media: 1-water; 2-solutionof sulfuric acid; 3- solution of caustic potasl~ 4- solutionof lactic acid

Analysis of creep tests results revealed that at compression load action (40, 50 and 55 % of mean ultimate strength TR) creep deformation are stopped after 30 days and is not changed for year. Creep of samples under 65 and 75 % of ultimate load was terminated for 2 months. Rupture of samples at 85 % of T R was detected after 20 days. Convex-concave character of creep curves

(Fig.l) is specified to collapsed

samples. During the early period of the test we detected the steep rise of the deformation and thereafter with time the drastic increasing of the creep deformation. Considering that points of reflection are one level it may be deduced that maximal elastic deformation at all limit loads is equal. Stress-strain relationship of RubCon samples is illustrated on Fig.3 60

/ a 40

/ m o

m

20 I

I I

1 0.6 0.8 deformation,~ Fig.& The strain dependence of stress for RubCon samples at compressive load 0.2

0.4

595 Dashed line corresponds to this relation at loading of samples whereas the stress dependence of full deformations (having regard to creep) conforms to continue line. Curvature of the line was detected on 180 day. Diagram of RubCon behavior at compressive load 3 is presented on Fig.4

O.O

......

0.6 I

0.4

0.2

0 c--O.O

! 120

( o - 5 ~ f "6

,~ 120

,,,

L20

(~,)o.~

Fig.4. Working diagram of RubCon at compressive load g. -creep deformation, E' =(g +c)2/2~. Using the diagram as the base we derived the formulae, which makes possible to determinate RubCon deformations in any time of material work at compressive load. = ~ ' - c +_ !(~' ) ~ - 2 ~ ' ( c + a J E

)AI ~

where g',c - constant (see Fig.5), ao -initial stress, E- modules of elasticity, A = 1= l ( c + a J E ) / 2 ~ ' ] E X P - { ( E t / r l ) + ( E / E O I I = E X P - ( E T / r ) ] } coefficient that takes into consideration structural heterogeneity of RubCon consisting fTom viscous and elastic phases, E 1 - modulus of elasticity at high elasticity deformation, rl and r- inner friction coefficients of viscous and high elasticity phases correspondingly.

596 The numerical value of RubCon prolonged strength factor equals the area of working diagram to the circumscribed rectangular area ratio 3. In the mean it is equal to 0.75. The limit of prolonged strength depending on time can be determined by experimental curve (Fig.5).

Ktn 1.o 0.9

l

I

120

160

0.8

0.7

l

I

I

I

0.6 0

30

60

90

160

days

Fig.5. Prolonged strength factor of RubCon

References

1. Figovsky, O., Beilin, D., Potapov, J., New type of polymer concrete based polibutadiene matrix. Proc. of VIII Int. Cong. on Polymers m Concrete, Oostende, Belgium, 1995, p.427-433 2. Figovsky, O., Beilin, D., Blank, N., Potapov, J., Chemyshev, V. Development of polymer concrete with polybutadiene matrix. Cement and Concrete Composites, vol. 18, No 6, 1996, pp.437-444. 3. Ivanov, A. Potapov, J. Working diagram of furferal-acetone concrete at compression, Mechanic of Polymers. Riga, 1968, pp.454-461.

Structural Failure and Plasticity (IMPL4ST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

597

Study o f influence o f loading m e t h o d on results o f the Split H o p k i n s o n Bar test A.D. Resnyansky Weapons Systems Division, A M R L , Defence Science an Technology Organisation, PO Box 1500, Salisbury, SA 5108, Australia This paper considers the method of loading of the impactor as a factor affecting the output results of the Split Hopkinson Pressure Bar (SHPB). The present study consists of two parts: (i) a study of the influence of loading methods associated with several popular SHPB systems on the shape of input pulse and the recorded output and (ii) an analysis of the influence of nonlocal strain-rate material properties on the flow stress recorded at a specified strain rate. For the SHPB systems analysed it is shown that 1-, 2-, and 3-wave SHPB analyses [1,2] give a discrepancy between the estimates of the yield stress that is associated with the pulse shape. It is also shown that the flow stress obtained with SHPB systems at a specified strain rate is determined only by the local stress-strain rate behaviour of material.

1. INTRODUCTION The Split Hopkinson Bar (SHB) device is amajor instrument for testing solid materials at high strain rates. Assumptions made in the analysis of the experimental data have been discussed for decades. Relevant references can be found in many surveys (see, eg, [1,2]). Significant attention is paid to non-one-dimensional effects in SHPBs, especially the Pochhammer-Chrec oscillationsof input and reflectedpulses recorded in the tests.This effect has bccn studied extensively both in experiments [3] and by numerical analysis [4]. S H P B modifications, such as those with the intermediate damping disk, have been suggested [5] and presently are widely used. The present paper deals with one-dimensional numerical analysis of the testing process. W c demonstrate that the observed discrepancy of test data between I-, and 3-wave analysis may bc caused by one-dimensional effectsassociated with the loading of the striker bar by a real launcher. W c also analyse the strain-rateresponse of materials. A strain-ratesensitivemodel is employed for the directmodelling of SHPBs. 2. SPLIT HOPKINSON BAR ANALYSIS In classical S H B analysis,the flow stress of a sample is deduced from information about the generated stress pulse travellingthrough the sample sandwiched between two rods (scc diagram of S H P B in Fig. I). Material properties of the rods and conditions of the loading arc selected so that pulses in the rods are guaranteed to bc elastic. The primary assumptions of the S H B analysis are uniform deformation of the sarnplc and

598

(/)

(2)

(3)

(5) j /

(4)

f__>,,

(4)

(5)

(6) .

(6)

L

I

Figure 1. Basic elements of Split Hopkinson Pressure Bar. 1 - chamber with compressed gas; 2 - impactor; 3 - buffer disk; 4 - input bar; 5 - sample; 6 - output bar; 7,8 - gauges. the absence of stresses in the transverse direction. Other assumptions include a constant strain rate while testing and quick equilibration of stresses in the sample. In the analysis, solutions for elastic left- and right-going waves in rods are used [2]. The incident pulse, induced by the impact of the striker bar, propagates via gauge (7). During the propagation this gauge is recording strain 6.. After the pulse is reflected from the sample the t same gauge records reflected pulse c and gauge (8) records transmitted pulse 8 t . Denoting r cross-sectional areas of the bars and sample as A and A the SHPB analysis gives: S'

o'=

AE A

-6 t,

s

2c

~=--6 . l r

(1)

s

Here rr is the stress in the sample within the assumption o f the stress uniformity (rr L = rr R in Fig. 1), ~ is the strain rate in the sample, l is the current length of the sample, E , p , and c s are Young's modulus, density and the velocity of sound in the bars. This method of derivation of flow stress is called one-wave analysis because it involves only the transmitted pulse s t . The analysis is based on equilibration of forces F L and F R (Fig. 1) in the bars at the ends adjacent to corresponding sides of the sample where

F, = AE(6 i + 6 r), F, = AE6 t.

However, the validity of putting F L = F R has been under discussion for a long time and attempts have been made to use additional information about the incident and reflected waves. Use o f the information about two waves involved in the determination of F L is known as 2wave analysis and the averaging of F L and F R gives formulas of 3-wave analysis with the following replacements for a in (1):

rr=

A

$

i

+c r

'

rr=

2A

s

i

+6

r

+c t

"

(2)

3. M O D E L L I N G O F T H E L O A D I N G M E T H O D S Several SHPB configurations, which are analysed in the paper, are stated below (Fig. 1). Configuration (I), launching the impactor by a stud gun [6], contains a zone of compressed

599 gas (1) pushing impactor (2) (other methods of generating the compression pulse are possible [7]), input bar (4), sample (5), output bar (6). Configuration (II), loading the impactor by a gas gun, bears the same elements as in ease (I). In case (III) a stud gun accelerates the striker bar hitting the sample directly: this is a modified SHPB system [1,6,12] with a reduced number of elemems (1), (2), (5), and (6) (input bar is eliminated). In case (IV) the impactor is loaded by a stud gun and hits a buffer disk (Fig. 1). We also consider an idealised case (V) with initially stress-free impactor (2) hiring the sandwich (4), (5), (6) at a constant velocity. For modelling, we employ a strain-rate sensitive model [8] which was used earlier for the shock-wave and impact studies [9]. The model is complemented by assuming that the transverse stresses in the bars and sample are zero. A one-dimensional code of second order accuracy was developed on the basis of the Godunov scheme [10]. For the gas-dynamic calculations in the chamber zone (1) (Fig. 1) the Euler model with the polytropic equation of state is used. The effect of a stud gun is simulated by an applied pressure of 2.5 kbar at the left side ofthe gas chamber for 2 /t see. A gas gun is modelled similarly by holding a pressure of 0.4 kbar for 1.3 msec. Regarding the geometrical parameters of the SHPB setups, the lengths of the input and output bars are 1 m, the length of the striker bar, sample and buffer disk are 20 cm, 1 cm and 1 mm, respectively. The initial ratio of the cross-sectional area of the sample and buffer disk to that of the bars is chosen to be 0.5. In the present paper the numerical analysis involves the following steps. We start from the choice of input data, which are presented by a set of flow stress points, o"r , at different strain rates k. This step is finalised by incorporation [11 ] of the data into constitutive equations of the model. The next step is direct numerical modelling of the testing process (including the loading and launching of the striker bar by compressed gas) at a specified loading condition, providing a desired strain rate c0. This step involves numerical solution of the conservation laws and constitutive equations. While modelling, we acquire stress data ('experimental' records) at the measurement stations (midpoints (7) and (8) of the bars in Fig. 1). The third step is direct output of the state of the sample during the calculation. The fourth stage involves the processing of the collected data by the SHB analyses and output of the flow stress crr and strain rate k as a function of strain c. The final stage is a comparison of the three sorts of o"v ; GPa 0.16

i-

[] 2-

II

9 3- []

40

9

q~

o.12 ~ O.OSt~ 0.04 l

[1

mm -1

mm

0

L 51 ,.

L [ ~ ,,,

,,J II

lm

-"

1

2

3

log ~

Figure 2. Flow stress input to the analysis. Points 1, 2, 3 correspond to Materials I, II, III, respectively.

O

X

Figure 3. Velocity profiles in the impactor accelerated by a stud gun (case (I)) at the nine ascending moments of time.

600

Figure 4. The incident, reflected and transmitted pulses (stress versus distance along the SHPB setup) for the SHPB systems of configurations (a) - (II); (b) - (IV); (c) - (V). data: (i) an input (or- z) curve at the local strain rate ?-0, (ii)inferred (or- z) and (?-- z) curves for the 1-, 2-, and 3-wave analyses, and (iii) (or- c) and (?-- c) curves resulting from the direct tracing of deformation of the sample. The materials chosen for modelling are steel for the bars (Cry ~ 1.8 GPa) and aluminium for the sample and the buffer disk (case (IV)). The input (err -?.)-dependence for aluminium is presented by points 1 in Fig. 2. We shall refer to this material as material I. For the study of the strain-rate sensitivity effect we generate two other hypothetical materials: material II has low strain rate sensitivity (points 2 in Fig. 2), and material III is very strain rate sensitive. First, let us observe how the pulse shape del~nds on the SHPB design. In Fig. 3 the velocity versus distance is a result of the direct modelling. It is seen that the impactor does not accelerate instantly to a constant velocity. Instead it is loaded gradually by small amplitude increments seen as steps on the top of the profiles (Fig. 3). The velocity gain by the striker bar htIPa / 't/-31 320 240

I I I / l'Jll,-,! 1 I h .J.~3al i I. I7-t:~.--L---JI/"1',,,'-31 - I ~-7-l~b4XI J4------V--G~I I II1 I\[ li:_._~ lY !

aol ~_~~!

L_..---~ ~ ~

w'-

7a-

~

-'i:b i

iam ""-~

\-7

160

80 MPa

~'~

~f--3

320 240 160 80

-7,

:/

MPa 120

:1

40

3->---+ - L;_-7, ,

/:/ ~//-~-4 .U.r3 6

~---- r 2

:"2 1:

'

r

:7,)

....

'

6

;,%

:lvll'J

6

1',

....

120

.... 40 8,%

Figure 5. SHPB analysis and direct traces of the stress state in the sample for cases (I-V) figures (I-V), the scale of the strain rate (curve 7) is 1000 s -l to 80 MPa. Figure (VI): stress in the sample made of materials I, II, and III (curves 1, 2, and 3, respectively) from the 1-wave analysis (a) along with the direct output (b) for the configuration (I).

601 with a gas gun is similar but it takes a much longer time. The disturbances in the velocity and stress are passed on to the input pulse. Examples of the incident stress pulse (denoted by 'I') at a moment of time overlayed by the reflected and transmitted pulses ('R' and 'T') at a later time are shown in Figs. 4 (a-c) for cases (II), (IV), and (V), respectively. The pulse shapes are quite different for different systems. We shall clarify how the differences affect the results of the SHPB analysis. To do this we apply the wave analyses to the data collected for each of the systems. The results and the data traced directly are summarised in Figs. 5 (I-V). The data for the materials with varying strain rate sensitivities are shown in Fig.5 (VI - a, b). Let us analyse the results. The flight velocity of the impactor obtained from the direct calculation approaches 37 - 39 m/s in each of the eases (I-IV). For ease (V) the velocity of the impactor is exactly 40 m/s. These conditions provide a strain rate ~0 close to 4000 s -~ . All drawings in Figs. 5 (I-V) contain curve 1, which is the input stress-strain curve at /" = 4000 s -~ for material I. Curves 2-4 are the output of the 1-, 2-, and 3-wave analyses. For the configuration (III) without an input bar the 2-, and 3-wave analyses are not applicable. Curves 5 and 6 are the result of the direct tracing of the stress crR and the difference o"L -o" R . Curves 7 are plots of the strain rate versus strain obtained from the SHPB analysis (1). The results of 2-wave analysis can be understood from the reflection of a pulse from the sample in the rod-sample-rod sandwich. Numerical modelling of the problem for a strain rate sensitive sample shows the following. An incident pulse with constant conditions behind its front will equilibrate sooner or later (F R = F L). However, for a real SHPB the state behind the front is not constant. In this case F L does not converge to F R and the force difference and its sign are determined by the gradient of change of the stress state behind the front and its sign. Larger gradients mean larger divergence of the forces. Confirmation of this can be found by comparing the incident pulses in Fig. 4 with the results in Fig. 5. The constant state behind the front of the incident pulse for case (V) (Fig. 4(c)) results in close convergence of the 1and 2-wave analyses in Fig. 5(V) after the stress relaxation to equilibrium. For case (II) (Fig. 4(a)) a gradient behind the front of the incident pulse is clearly seen to explain the force difference in Fig. 5(II). The SHPB system (IV) produces a highly non-stationary state behind the front (Fig. 4(d)) accompanied by the sign change of the gradient that results in the change of sign of the difference between (or- 6) curves in Fig. 5(IV) produced from 1-, and 2-wave analyses. The zones of disagreement appear as oscillations; their extension can be reduced by decreasing the buffer thickness but cannot be excluded completely if the damping disk is present. The reflected pulse is quite sensitive to the numerical viscosity but that has no effect on the minimum magnitude of the force difference. Curve 3a in Fig. 5(I) illustrates the result of 2-wave analysis for calculation with very fine numerical mesh. It takes a longer time for the material to reach 'material equilibrium' (the state corresponding to a given strain rate) for the 'modified' SHPB (III) than with conventional systems (curve 5 in Fig. 5(III)). The cause is the shock-wave character of loading. In contrast, the quasi-isentropic loading in the conventional systems results in the sample achieving the material equilibrium faster. It is interesting that the stress equilibrium (or L = crR) inside the sample is achieved much more quickly (curve 6) than the material equilibrium. Regarding the choice of relationships for the calculation of current length of the sample used in [12], the modelling demonstrates that the contact velocity at the left side of the sample is nearly constant for a rather long time. Therefore, it is reasonable to perform the velocity correction due to the change of the contact force for the right side of the sample only. Strain-rate curves

602 7 and 7a in Fig. 5(111) correspond to the two- and one-side velocity corrections, respectively. Evidently, the ( ~ - 6) curve 7a is closer to the directly traced strain rate 7b. Nevertheless, the both methods of correction give (tr - 8) curves, which are very close to each other (curve 2). Finally, we analyse the influence of the strain rate sensitivity of material on the SHPB results. We selected the generic materials I, II, III in such a manner that they have the same flow stress, 170 Mpa, at k = 4000 s -1 (Fig. 2). The 2-wave analysis gives very close results for the materials. Results of the 1-wave analysis are shown in Fig. 5(VI-a). Fig. 5(VI-b) is tracing directly the stress state in the sample. It is seen that after the stress equilibrium is attained the results are identical. The material equilibration lasts much longer than the stress equilibrium, resulting in crL = o"R, for the more rate sensitive material. That has just been illustrated for the modified SHPB system (III).

4. CONCLUSION It is concluded that (i) For 2- and 3-wave SHB analysis the influence of the launching devices should attract more attention than has been the case. Oscillations and divergence of the SHB analyses may be caused not only by the Pochammer-Chree oscillations but by the launching conditions as well. (ii) The flow stress obtained with SHPBs is determined by local stress- strain-rate properties of the material. However, a careful interpretation of data on the initial part of the stress-strain curve should include the possibility of stress relaxation inside the sample for highly strain rate sensitive materials.

REFERENCES 1. J.E. Field, S.M. Walley, N.K. Bourne and J.M Huntley, Review of Experimental Techniques for High Rate Deformation Studies, in 'Proe. Acoustics and Vibration Asia 98', Singapore, 1998, pp. 9-38. 2. G.T. Gray III, High-Strain-Rate Testing of Materials: The Split-Hopkinson Pressure Bar, LA-UR-97-4419, Los Alamos National Laboratory, 1997. 3. P.S. Follansbee and C. Frantz, J. Eng. Mater. Technol., 105 (1986) 61. 4. L.D. Bertholf and C.H. Kames, J. Mech. Phys. Solids, 23 (1975) 1. 5. S. Ellwood, L.J. Griffiths and D.J. Parry, J. Phys, E: Sci. Instrum., 15 (1982) 280. 6. G.L. Wulf, Dynamic Stress-Strain Measurements at Large Strains, in 'Inst. Phys. Conf. Ser.', No. 21 (1974) 48. 7. M. Quick, K. Labibes, C. Albertini, T. Valentin and P. Magain, J. Phys IV France Colloque C3, 4 (1997) 379. 8. S.K. Godunov, E.I. Romensky, Elements of Continuum Mechanics and Conservation Laws [in Russian], Novosibirsk, Nauehnaya Kniga Publ., 1998. 9. L.A. Merz~evsky and A.D. Resnyansky, Int. J. of Impact Eng., 17 (1995) 559. 10. S.K. Godunov, J. Comp. Phys., 153 (1999) 6. 11. A.D. Resnyansky and L.A. Merzhievsky, Fizika Gorenia i Vzryva [In Russian], 28 (1992) 123. 12. S.J. Cimpoeru and R.L. Woodward, J. Mater. Sci. Let., 9 (1990) 187.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

603

Enhanced ductility of copper under large strain rates D.R. Saroha, Gurmit Singh and M.S. Bola Terminal Ballistics Research Laboratory, Sector-30, Chandigarh-160020, India The metallic jets produced from explosive-driven conical copper liners, called shaped charges, exhibit extraordinarily high dynamic ductility. The shaped charge jet stretches under very high strain rate due to velocity gradient along its length. The jet is eventually partieulated preceded by quasi-periodic ductile necking along the length of the jet starting from its front end. The physical mechanism responsible for such a large strain in a shaped charge jet before onset of particulation process is still not thoroughly understood quantitatively. The present paper is an experimental study of the ductile fracture mechanism of the metallic shaped charge jets. The processes of jet-elongation and particulation were recorded by multi-channel Flash Radiography. The jet length and diameter, jet break-up time, number of fragments and their size were calculated from the experimental data. The value of I-Iirseh velocity parameter Vpl which is a material property parameter defined as the average velocity difference between adjacent jet particles, was calculated and compared with its values given by various analytical models and experimental data reported in the literature. The effect of strain rate on different jet l~zameters was also studied. 1. INTRODUCTION Under intense dynamic conditions such as collapse of explosive driven conical metal liners, called shaped charges, certain metals e.g. copper exhibit extraordinarily high ductility. These metals under ambient conditions, however, do not show same degree of ductility. The cor.ical metal liner is collapsed around the liner's axis of symmetry by the very high pressure from a detonating explosive charge resulting in a metallic jet. The high velocity metallic jets thus produced have received a considerable amount of attention in the past due to their application in industry as well as in military for penetrating thick and hard targets. The target penetration capability of the jet is mainly limited by the length of the continuous jet. The jet is plastically stretched under very high strain rate due to velocity gradient along its length. The process of ductile stretching is eventually arrested by the break up of jet into discrete particles which limits the jet length. Therefore, an understanding of break-up mechanism and methods of delaying its occurrence are the important areas of interest for the designers of shaped charges. The physical mechanisms which enable copper shaped charge jets to exhibit high ductility under dynamic condition are still not thoroughly understood. In the past, several computer codes i~ and theoretical models 3,4,5have been developed to explain the necking phenomena and particulation process of the stretching jet; but there is still very little quantitative work done on this problem. In fact, the problem becomes complex as the material properties of liner are not well known under intense dynamic loading conditions of jet formation and elongation.

604 In the present paper the ductile fracture mechanism of the copper shaped charge jet has been investigated experimentally by employing the technique of Hash Radiography. Various parameters which affect the elongation of jet have been determined and compared with other analytical and experimental data available. 2. JET BREAK-UP MECHANISM Several attempts have been made in the past to evolve the methods of delaying the onset of particulation process in the shaped charge jet to achieve a longer continuous jet. Consequently, several break-up mechanisms have been suggested and calculations of break up time have been made by following analytical as well as empirical approaches. I-Iirsch4'5 has suggested a very simple break-up mechanism for homogeneous, ductile metals under high strain. This model has been applied to the stretching metal jets. In this model it is assumed that vacancies formed at the jet surface due to elongation process are gradually increased until the break-up of jet occurs. The following formula was given by Hirsch to calculate the breakup time tb of the jet,

(~)

t~ =---

v,,

where do is the initial diameter of the jet and Vpl is the velocity difference between consecutive elements of the stretching jet. The break-up time is measured from the start of the elongation process of the jet. This mechanism of break-up has been supported by the porosity found in the particles recovered from the elongated jet ai~r its break-up. Very recently Curtis et.al. 6 have proposed an empirical jet break-up model and calculated break-up time for the shaped charge jets. The break-up time has been shown to be inversely proportional to the Hirsch velocity parameter VO and the strain rate. rokl K 2 tb =----- + ----

v,,

~o

(2)

Here, ro is the initial radius of the jet, ~0 is the strain rate and K~ and K2 are the arbitrary constants. The equation (2) is of general nature and can be reduced to many models for breakup time calculations as reported in the literature ~ simply by changing the two constant parameters. If the values of constants K~ and K2 are taken 2 and O, respectively, this formula is reduced to the I-Iirsch formula as given by equation (1). In this paper, the Hirsch velocity parameter, which characterises the material property of the jet has been calculated from the experimental data for the jets produced at different strain rates. The jet break-up times have also been calculated from experimental data by using Hirsch break-up time formula. 3. EXPERIMENTAL SET-UP Flash Radiography is the most widely used experimental technique to record the shaped

6O5 charge jets. In the present study a three channel flash x-ray system was used to record the formation, necking and particulation process of the jet. The metal lined shaped charge was placed at the crossing point of the beams from the three x-ray tubes placed inside the small holes in thick concrete walls. The radiograph of the jet was recorded at three different times on the separate x-ray films. The x-ray films and the tubes were protected from the blast of detonating high explosive by providing metallic and low density material sheets in front of them. A steel fiducial was placed parallel to and near the emerging jet. The length and velocity of the jet were measured by using time of exposure and the position of the jet tip relative to the position of the fiducial.

4. MEASUREMENT OF JET LENGTH AND BREAK UP TIME

The copper jets were produced from the shaped charges of cone angles 30~, 60 ~ and 90~. These jets were recorded during and after completion of particulation process. In Figure 1 records of the jets taken after completion of particulation process have been shown with reduction in their size. The enlarged view of tip and tail regions of a jet are also given in this Figure to demonstrate the breaking pattern of the jet particles. The lengths of individual jet particles and their velocities were calculated from the actual records. The cumulative length of each jet was obtained by adding lengths of individual particles. It may be mentioned here that the cumulative jet length depends upon the slowest particle available in the record. The jet is assumed to stretch at uniform rate and break simultaneously from tip to tail in a number of particles at a time when it acquires its maximum length. This time of particulation tb, called cumulative break-up time, is calculated by dividing the cumulative jet length L by the difference of the velocities of jet tip particle (V~p) and the slowest particle (V~a) recorded in the experiment. L

tb =(Vap 'V~t)

(3)

The value of cumulative break-up time also varies with the velocity of the slowest jet particle included in the calculations. This is due to the reason that the cumulative jet length is not a finear function of jet velocity. 5. ANALYSIS OF EXPERIMENTAL DATA The cumulative length of the jet was computed from the experimental data as a function of velocity of the jet particles. The variation of cumulative jet length with the velocity of jet particles is shown in Figure 2. The particle velocity in this figure indicates the velocity of the slowest particle included in the calculations. The two curves shown here are for the jets obtained from shaped charges of 60 ~ and 90~ cone angles. The cumulative jet length is found to vary exponentially with particle velocity. The length of individual particles increases from the tip towards the tail of the jet. In general, thin particles with stretched ends were observed near the tip region showing high ductility, whereas, near the jet tail thick particles with blunt ends were found indicating brittle break-up behaviour of jet particles.

606

Figure 1 Records of jets produced from conical copper liners of different angles

Figure 2. Variation of cumulative jet length with velocity of jet particles

The average cumulative break-up time for the entire jet length is a single value obtained by dividing the length of the jet by the difference of velocities of jet tip and tail particles. The variation of cumulative break-up time as a function of the velocity of jet particles is shown in Figure 3 for the shaped charges of 60 ~ and 90 ~ cone angles. This cumulative break-up time was taken to be the sum of the lengths of n number of particles counted from tip towards the tail divided by the velocity difference between the tip particle and the nth particle. The particle velocity shown in this figure is the velocity of nth particle. The dashed lines show the average cumulative break-up times for the entire jet lengths which are the values corresponding to the slowest velocity point plotted here. Very large break-up times were observed towards the tip end of the jet. This is due to the small velocity difference between successive particles following the tip particle. Similar trends in cumulative break-up times were also observed by Waiters and Summersv for copper jet. The Hirsch cumulative break-up time for the entire length of the jet was calculated from equation (1) by putting experimental values of jet diameter,do, and Hirsch velocity parameter, Vp~ in this equation. A deviation of 4 to 12% has been observed in the values of cumulative break-up time calculated from Hirsch formula as given in equation (1) and the break-up time formula of equation (3). The initial strain rate of the jet was varied by changing the angle of the conical liner of the shaped charge. The difference in velocities of the tip particle (V~p) and the slowest particle (Vt=0 is divided by the initial length (1o) of the jet to calculate the initial strain rate (So).

lo

(4)

The initial jet length was calculated from the relation s to =

- v=)

(5)

where I~is the slantheight of the con/cal liner.The initialstrainrate decreases as the angle of conical lineri s / n ~ .

607

Figure 3. Variation of cumulative breakup time with velocity of jet particles

Figure 4. Variation of V# with velocity of jet particles

The Hirsch velocity parameter, which is characterized by the material of the liner, was calculated as a function of velocity of jet particles from the relation V,~ = (V~ - V.)

(6)

where Vn is the velocity of nth particle and n is the number of particles considered along the jet starting from the tip particle. The variation of Vpl with the velocity of slowest particle included in the calculations, is shown in Figure 4. In this figure Vpl values for the jets obtained from the shaped charges of three different cone angles have been plotted to see the effect of strain rate on Vpl. The average values of Vp] for the entire lengths of the three jets have been shown by the dashed lines. These curves indicate that the value Of Vp] is not very sensitive to the change in initial strain rate. This is in agreement With the experimental data as well as model calculations reported in the literature s showing a weak dependence of Vpl on the strain rate. The values of Vp~are found to vary from 111 m/sec to 121 m/sec for the three strain rates considered in the present study. The earlier experimental and analytical data for copper jets reported by different authorss'9 also suggest the value of Vpl in this range. However, more experimental data is required to be generated to see the dependence of Vpmon strain rate. This study is in progress. In Table 1 various parameters calculated from the experimental data have been given for the three shaped charges used for the present study. The initial strain rate, Vp~,average particle length, cumulative break-up time, initial jet diameter and jet break-up time calculated from Hirsch formula have been listed in this table. It is observed from this table that the average particle length and the diameter of the jet are decreased as the strain rate increases.

608 Table 1 D!ffe~n t ~ t e r s Angle of Conical Liners (Degree)

0f_.shapedch~gejets calc~at~ from e x p e ~ e n t ~ da~ Initial Initial Jet Cumulative Hirsch Vpl Average Strain Rate Diameter Break-up Break-up (Km/sec) Particle (xlO4/sec) (mm) Time (gsec) Time (gsec) Length

...........................................................................................................................................................................

30 60 90

2.63 2.05 1.57

4.2 5.0 7.8

36.27 38.69 62.58

37.84 41.32 70.27

0.111 0.121 0.111

( ~ )

..........

4.37 4.57 6.79

6. CONCLUSIONS The d~t~ for the jets produced from the shaped charges with copper liners shows the ductile behaviour of copper under high strain. High ductility near the tip region of the jet was observed with thin long-necking particles whereas relatively brittle behaviour was observed near tail region with thick particles without necking. The length of the jet particles and the jet diameter increase from the tip towards the tail of the jet. The Hirsch velocity parameter V~ also increases from tip towards the tail of the jet; but it is not much sensitive to the change in the jet strain rate. The initial diameter of the jet decreases as the jet strain rate is increased. ACKNOWLEDGEMENT The authors are thankful to Shri V.S. Sethi, Director TBRL, for granting permission to publish this work. Thanks are due to Shri Dileep Kumar and Shri Balwinder Singh for their assistance in carrying out experiments. The help given by Smt Pankajavally in putting the paper in the present format is also acknowledged. REFERENCES

1. P.C. Chou andJ. Corleone, J. Appl. Phys. 48 (1977)4187 2. P.C. Chou, M. Grud~ Y. Liu, andZ. Ritman, Proc. Ofthe 13~ Int. Syrup. On Ballistics, Stockholm, Sweden, 1-3 June, 1992. 3. J.M. Walsh, J. Appl. Phys. 56(7) (1984) 1997 4. E. Hirsch, Propell., Explos., Pyrotech., 4 (1979)89 5. E. Hirsch, Propell., Explos., 6 (1981) 11 6. J.P. Curtis, M.Moyses, A.J. Arlow and K.G. Cowan, Proc. Of the 16th Int. Symp.on Ballistics, San Francisco, CA, USA, 23-28 Sept 1996 P-369 7. W.P. Waiters and R.L. Summers, Propell., Explos., Pyrotech., 18 (1993)241 8. W.P. Waiters and R.L. Summers, Proc. Of the 14th Int. Syrup. On Ballistics, Quebec, Canada, 26-29 Sept 1993 P-49 9. J.E. Backofen Jr. and E.Hirsch, Proc. Of the 13~ Int. Symp. On Ballistics, Stockholm, Sweden, 1-3 June, 1992 P-359.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

609

Kinematics of large deformations and objective Eulerian rates A. Meyers, O. Bruhns and H. Xiao Institute of Mechanics Ruhr-University Bochum D-44780 Bochum, Germany In recent times the eigenprojection method has been revealed to be a powerful tool in the formulation of large deformation kinematics. With this tool it can be shown that from all objective corotational Eulerian rates only the logarithmic rate of the Hencky strain is equal to the rate of the deformation tensor D. Moreover, only the logarithmic rate is exactly integrable in the case of a hypoelastic material of grade zero. 1 INTRODUCTION The concept of objectivity is essential when we describe large inelastic deformations in Eulerian frames. The material time derivative of an objective quantity does not need to be objective, too. A variety of objective time derivatives has been proposed. The question arises if the objectivity requirement is sufficient for the formulation of such rates, since unexpected results have been seen in specific computations (see e.g. Lehmann(1972)). It has been discussed whether the additive decomposition of the deformation rate (Green and Naghdi (1965)) or the multiplicative decomposition of the deformation gradient (Lee (1969)) has to be used; both descriptions seemed to be incompatible. With introduction of the logarithmic rate (Xiao et al. (1997)) we will show that the descriptions may be related to each other and the corresponding relations will be shown. We will restrict our reflections on the same vector space. The symbolic notation is used. Let a, b, c and d be first order tensors and A, B second order tensors. A' is the transposed, trA the trace of A. Also (a | b) : (c | d) = (a . c) (b . d)

-+

A:B=tr(AB')

(a|174174 = c|174174 g (s) = (A + A ' ) / 2 , g (a) = ( g - A ' ) / 2

(1) (2) (3)

2 OBJECTIVE COROTATIONAL RATES In the past a large variety of objective rates of symmetric Eulerian tensors has been presented and their applicability has been discussed. To our opinion there are good reasons to confine to corotational rates. Therefor let us have a look at a general form of objective rates of a symmetric second order Eulerian tensor A A ~ = A ' + LA + A R ,

(4)

where L and R are second order Eulerian tensors related to the considered rate. We develop three ideas:

610 Chain rule: The material time derivative of a scalar is objective, i.e. (f(A))" = (f(A)) ~ We apply the chainrule and get with F = a f ( A ) / a A in respect of the general form (4)

(f(A))~ = ( f ( A ) ) , tr(F(A" + L A + AR)) = tr(FA'), tr (F (LA + AR)) = 0.

(5)

A F = FA, since F may be expressed by a Taylor series of A. After permutation and transposition we find tr(FA (L + R)) = tr (FA (L + R r ) ) = 0 .

(6)

This equation should hold for arbitrary symmetric A and arbitrary f (A). Therefore we gain the main result that form (4) is generally fulfilled for L=-R

or

L=-R

T.

(7)

Identity tensor test: Let I be the second order identity tensor. We compare arbitrary derivatives (marked by a diamond) of I and 12, i.e.

(I~)~176176176176176

~

r=I~

(8)

(4) should also hold for the identity tensor I. From this and the foregoing result we find that only the first form of (7), i.e. L = - R , is valid. Symmetric increments: A symmetric remaining tensor A should have symmetric increments, i.e.

A ~ (A~ = (A') ~ . We apply latter relation to eq. (4) and get A(R-

(9)

L') + (L - R ' ) A = 0.

(10)

The relation holds for arbitrary symmetric A; hence R = L' + cl

>

R (a) + L (a) = 0

and

R (s) - L (s) = c l .

(I I)

Subtracting R = - L from the left side of (11), we have L (s) = - ( c i ) / 2 , wherefrom R ('~ = ( c i ) / 2 and A ~ = A + L(a)A- AL (~) . (12) Corotational objective rates are of the form (12), i.e.

A ~ = A'+ An-

f~A,

(13)

where f~ is a spin tensor. In the following we will focus our attention to this rate type. It should be noted, however, that not every corotational rate is objective. Let F be the deformation gradient. It relates the position vector x in the actual or Eulerian configuration to the position vector X in the reference or Lagrangean configuration, i.e. F=o"x/0X,

detF>0.

(14)

The deformation gradient may be multiplicatively decomposed into the double field rotation tensor R and the symmetric Eulerian stretch tensor V as F = VR.

(15)

611 Table 1. Examples of objective corotational rates

f~

Authors Zaremba (1903), Jaumann (1911) Green and Naghdi (1965) .. Xiao et al. (1997)

h(x, y, z)

~'~(J) ---- W

.....

f~(R) = ( a ' ) a f/oog)

0-

'

( y = x ) / ( y + x) ..... (y2 + x2)/(y2 x 9) + 1/(lnx - In y)

The particle velocity v and the velocity gradient L are denoted by v=x',

L=0v/0x=F'F

-x.

(16)

The deformation rate D and the vorticity W are the symmetric and antimetric parts of the velocity gradient respectively: D = L ~s), W = L


(17)

Let V/be the m distinct eigenvalues of V. The m eigenprojections Vi are given by m

V=~V~V~, i=1

m

~V,=I,

V~=Vi

(n>0),

V, V k = 0

(i#k).

(18)

i=1

The left Cauchy Green tensor B = V 2 shares the eigenprojections with V; its m distinct eigenvalues are Bi = Vi=. Xiao et al. (1998) showed that the most general form of objective corotational rate is related to the spin m f~ = W + ~ h(V~, V~, trV)V, DVk. (19) Herein the summation is meant as double sum over i and k, excluding the terms where i = k. The sum vanishes for m = 1. The spin function h(x, y, z) obeys the rule

h(x,y,z) = -h(v,x,z)

.

(20)

Some well known objective corotational rates are defined in Table 1. In the following we will motivate to use the logarithmic rate. 3 OBJECTIVE EULER/AN STRAIN RATE The strain e is a function of of the left stretch tensor V (Hill (1968), (1970), (1978)) m

e = f ( V ) = ~ f(V/)Vi.

(21)

i=1

In particular the logarithmic strain (Hencky (1928)) is expressed as m

h = l n V = ~ ln(V~)V~ = ( l n B ) / 2 .

(22)

i=1

There is no strict relation between the measure of deformation e and D, a measure for the rate of deformation. We assume that D may be equalled to an objective strain rate, i.e. e~ = D.

(23)

612 Xiao et al. (1997), Meyers (1999) showed that in special consideration of (13) this leads to Vie'Vi

=

Vie~176

=

=

V/-IViV'Vi

(24)

-

f~~176

(25)

ViDVi

and

D = h ~176 = h + with the logarithmic spin (see Xiao (1995))

hn

(l~

fl ~176= W + ~ { 2 / ( l n ( B i l B k ) ) + (1 + BilBk)(1 - BilBk)} ViDVk 9 i,~r

(26)

By (...)~176176 we denote the logarithmic rate, which is defined as A ~176= A + A l l ~176- f l ~ 1 7 6 1 7 6

(27)

4 ELASTICITY AND THE LOGARITHMIC RATE Hypoelastic materials have a constitutive relation of the form

~o= (n(tr)): D,

(28)

where tr is the Cauchy stress and H = H ' the fourth order hypo-elasticity tensor, which is symmetric in the first two indices, too. Let us assume the hypoelastic model T~

= H 0~ : D = d e t V (tr ~176176 + trD tr) ,

(29)

where T = d e t V tr is the Kirchhoff stress tensor. Xiao et al. (1997) showed that this constitutive equation fulfills Bemstein's integrability conditions (Bemstein (1960)) to be Cauchy- and Greenelastic. For an initially natural body state (VIt=o = I, Tit=0 = 0) moreover it turns out that

(H~176

= Vh.

(30)

We conclude that hypoelastic models based on the logarithmic rate are integrable to deliver an isotropic elastic constitutive equation. Sim6 and Pister (1984) showed that for any of the commonly known objective stress rates, the corresponding rate type model for the elastic response is in general not integrable and thus inconsistent with the notion of elasticity, in particular hyperelasticity. Let E(T) be any given differentiable isotropic scalar function. Bruhns et al. (1999) proved that the rate equation

is exactly integrable to deliver an isotropic elastic relation if and only if the stress rate T ~ is logarithmic. The unique integrable-exacfly rate equation defines the hyperelastic relation

h = (0]E) / (Or).

(32)

5 ADDITIVE AND MULTIPLICATIVE DECOMPOSITION IN ELASTOPLASTICITY Two different decompositions, namely D = D e + D q' additive decomposition of the deformation rate F = FeF p

multiplicative decomposition of the deformation gradient

(33) (34)

613

are widely used in elasto-plasticity kinematics. The determinants of both elastic and plastic parts of F are positive. Furthermore we assume a natural, stress-free initial state, i. e. Felt=o -- FeP[t=o = I ,

~rlt=o = Tit=0 = 0.

(35)

With (16) we find for the velocity gradient L = (Fe)'F e + Fe(FP)'(Fp)-I(Fe) -1

(36)

Furthermore F e may be multiplicatevely decomposed as Fe =

veR

(37)

e ,

where the elastic rotation R e and the elastic stretch V e can be consistently and uniquely determined from F e. Comparing (36) with (17) and (33) we propose De=

((Fe).F e)(s)

,

;

Dep = (Fe(Fp).(Fp)-l(Fe)-l)(s);

(38)

From this results the elastic Green tensor B e = F e F e'

and a general elastic relation

r

(39)

e) = (0~) / (aT).

(40)

In a purely elastic process this relation coincides with (32). Therefore it is straightforward to propose the elastic relations h e = (0E) / (0T) = ( l n B e)/2 and with (25)

D e = (lnBe)~176

(41) (42)

With (25) and the initial condition (35) we finally determine V e = exp(he).

(43)

The rotation is obtained by integrating ( R e ) ' = h e r e, where

Relt=0 = I,

(44)

m

12e= f~Oog)_E ((2VicV~)/((V~)2-

(Vie)2) + 1/(In Vie - In Vff)).

(45)

i#k Then we find L e = (Fe)'Fe-1,

D e = (Le)(s),

W e = ((Fe).(Fe)-l)(a) = ((ve).(Ve)-i + vef~eve)(a) F p ___ (Fp)-xF,

(46) (47)

L p -- (Fp).(Fp) -1 = (Ve)-l(L -- D e - W e ) F e

(48)

D p = (Lp)(s),

(49)

W

p --

(Lp)(a)

CONCLUSION Based on the assumptions that (1) the objective rate is corotational; (2) the deformation and the objective rate of the strain tensor are identical (D = e~ (3) the elastic part D e of the additively decomposed deformation rate (D = D e + D ep) is identical with l~e(Fe)-l, where F e is the

614 elastic part of the multiplicatively decomposed deformation gradient (F = FeFp); (4) the elastic strain is of Hencky type (e e = h e = t In Be); a set of consistent kinematical relations has been determined, where (1) the logarithmic rate is an essential measure for the objective rates of the total stress and the total strain; (2) the total strain is of Hencky type; (3) the hyperelastic strain part is self-consistent, i.e. exactly integrable; (4) the elastic stretch V e from the decomposition F ~ = V e R e is expressed as function of the Hencky elastic strain e e (43,42); (5) D e as well as D ep from the additive decomposition of the deformation rate and F e as well as FP from the multiplicative decomposition of the deformation gradient can be uniquely assigned to each other; both decompositions are equivalent. REFERENCES LEHMANN, TH. Anisotrope plastische Formiinderungen. Romanian J. Techn. Sci. Appl. Mechanics 17 (1972), 1077-1086. GREEN, A. E. and NAGHDI, P. M. A general theory of an elastic-plastic continuum. Arch. Rat. Mech. Anal. 18 (1965). LEE, E. H. Elastic-plastic deformation at finite strains. ASME J. Appl. Mech. 36 (1969), 1-6. XIAO, H., BRUHNS, O. T. and MEYERS, A. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124 (1977), 89-105. XIAO, H., BRUHNS, O. T. and MEYERS, A. On objective corotational rates and their defining spin tensors. International Journal of Solids and Structures 35 (1998), 4001-4014. ZAREMBA, S. Sur une forme perfection6e de la th6orie de la relaxation. Bull. Intern. Acad. Sci. Cracovie (1903), 594--614. JAUMANN, G. Geschlossenes system physikalischer und chemischerDifferentialgesetze. Akad. Wiss. Wien Sitzber. IIa (1911), 385-530. HILL, R. On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16 (1968), 229-242; 315-322. HILL, R. Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. Soc. London A 326 (1970), 131-147. HILL, R. Aspects of invariance in solid mechanics. Advances in Appl. Mech. 18 (1978), 1-75. HENCKY, H. Uber die Form des Elastizita'tsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys. 9 (1928), 215-220. MEYERS, A. On the consistency of some eulerian strain rates. ZAMM 79 (1999), 171-177. XIAO, H. Unified explicit basis-free expressions for time rate and conjugate stress of an arbitrary Hill's strain. Int. J. Solids Structures 32 (1995), 3327-3340. XIAO, H., BRUHNS, O. T. and MEYERS, A. Hypo-elasticity model based upon the logarithmic stress rate. J. Elasticity 47 (1977), 51-68. BERNSTEIN, B. Hypoelasticity and elasticity. Arch. Rat. Mech. Anal. 6 (1960), 90-104. SIMO, C. and PISTER, K. S. Remarks on rate constitutive equations for finite deformation problem: computational implications. Comp. Meth. Appl. Mech. Engng. 46 (1984), 201-215. BRUHNS, O. T., XIAO, H. and MEYERS, A. Self-consistent eulerian rate type elastoplasticity models based upon the logarithmic stress rate. Int. J. Plasticity 15 (1999), 479-520.

Structural Failure and Plasticity (IMPLAST2000)

Editors:X.L.Zhaoand R.H.Grzebieta 9 2000ElsevierScienceLtd.All rightsreserved.

615

A study of the large deformation mechanisms of weft-knitted thermoplastic textile composites" P. Xue a, T.X. Yu a and X.M. Taob a Department

of Mechanical Engineering Hong Kong University of Science and Technology, Hong Kong

b Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hong Kong The investigation on the large deformation tensile properties and the relevant mesoscale mechanisms of weft knitted polyethylene terephthalate (PET)/polypropylene (PP) textile composites is presented. The correlation between fabric structure, matrix damage and material properties are described. The results show that all PET/PP co-knitted samples along the wale, course and 45 ~ directions are all significantly non-linear. The tensile behavior is superior in the wale direction to those in the course and 45 ~ directions. The deformation mechanisms in meso-scale were identified experimentally by in-situ observation of large deformation process along the wale, course and 45 o directions. The inelastic properties of this material are attributed to the damage evolution in the matrix, sliding between the wales of the knitted fabric, as well as the change in the configuration of the fabric structure during loading. 1. INTRODUCTION In recent years, knitted fabric reinforced composites are of increasing interest due to the possibility of producing net-shape/near-net-shape performs and the excellent formability of the fabric which allows forming over a shaped tool of complex shape. The basic mechanical properties of knitted textile composites have been extensively studied. The in-plane stiffness and strength of knitted fabrics were found to be inferior to woven, braided, and unidirectional materials with an equivalent proportion of in-plane fibers, but to be superior to continuous or short fiber random mats composites [ 1]. However, in-plane mechanical properties of knitted textile composites may also undergo profound changes upon adjusting the fabric structure. Leong et al [2] tried to improve the tension and compression properties of the composites by increasing the number of knitted fabric layer. In addition, knitted textile composites are superior in terms of energy absorption, damage tolerance, bearing and notched strength and fracture toughness. Ramakrishna et al [3] investigated the tensile properties and damage resistance under static and low velocity impact of knitted glass fiber reinforced thermoplastic polypropylene composites. Yu and Tao et al [4,5] studied tensile properties in large deformation for the nylon/polyester and PET/PP co-knitted textile composite. Attempts were made to characterize the energy absorption behavior of the griddomed textile composites under compression and impact [4,6]. 1 Notwithstanding the tensile properties of textile composites have been an attracting * The authors wish to acknowledge the f'mancial supports from the research Grants Council of Hong Kong (Project No. HKUST6017/98E).

616 topic, however, so far the studies on the failure mechanisms of knitted textile composites have been limited to small deformation, e.g. see Ramakrishna [7], Ruan and Chou [8] and Rios et al [9]. In order to develop textile composites with high energy-absorbing capacity, this paper will focus on the tensile properties and deformation mechanisms of welt knitted textile composites in large deformation. The correlation of large deformation tensile properties and damage evolution, the change in the configuration of the fabric structure during extension will be revealed. 2. SAMPLE PREPARATION AND EXPERIMENTS

2.1 Sample Specifications Polyethylene Terephthalate (PET) and Polypropylene (PP) co-knitted interlock fabrics were produced in our laboratory. Schematic diagram of the fabric structure is shown in Fig. 1. The flat composite panels were fabricated by the compression molding technique. The PET/PP coknitted fabric, with metal boards and frame was put into the Hot Press at a maximum temperature of 180 ~ maximum pressure at 45 tons, and 38 minutes for the whole pressing process. During compression molding the PP fibers melted and impregnated the knitted fabrics. At the end of impregnation, the complete set-up was cooled by water to room temperature. Fig.1 Schematic diagram of the Tensile specimens were cut into narrow strips of welt-knitted interlock structure 20mm • 150ram parallel to the course and wale directions, as well as along 45 ~ with respect to the course direction, respectively. 2.2 Tensile Test Tensile tests were conducted by Universal Testing Machine (UTM). The experimental set-up is showed in Fig. 2. The loading speed was 2 ram/rain. During tensile test, one end of the tester fixed, the other end moved at the loading speed. The gauge length was set to 80rnm. Together with the test machine and the data recorder system, a digital video and a stereo microscope (Olympus SZH10) were installed to observe in-situ the deformation process and to identify large deformation mechanisms of those textile composites. Digital video

I sto,oo crosco !

Material Tester "

I

Observed

I I

~

L_

l| V "'

L+8 ]i

Fig. 2 Experimental setup of tensile test. (a) System, (b) Material tester

617

2.3 Tensile Properties In the loading process, tensile deformation of the specimen distributed unevenly. It propagated wale by wale or course by course from the loading end to the other end of the sample. The tensile curves for pure PP and PET/PP co-knitted textile composites along the wale, the course and the 45 ~ directions are given in Fig. 3. The pure PP broke at a strain of 0.025. It shows an absence of ductility. .

80 I '

.

.

.

.

.

alongwalodirection

[ - - al~ 45 directi~ d '~" 60] ,--.alongcoursedirectio//" [ ~

~"

A

-

4o 2o

O, 0

0.3

0.6 0.9 1.2 1.5 True strain I Fig. 3 Tensile curves for pure PP and the ET/PP co-knitted textile composites along the wale, course and 45~

It is evident that the PP/PET co-knitted samples exhibit strong non-linear behaviors and the tensile properties and material constants are all orientation-dependent. When extended along the loading direction, PET/PP co-knitted samples also deformed obviously along the transverse direction. The Poisson's ratio of the material was determined as (d - d l ) / d v= =0.5 (l I - 1 ) / l where l, ll, d and dl are as defined in Fig. 2. From Fig. 3, it can be seen that Young's modulus and the yield stress in the course direction are the smallest among those in the three directions. The maximum strain to fracture in the course direction is the largest comparing with those along other two directions. 3. THE L A R G E D E F O R M A T I O N C O M P O S I T E IN MESO-SCALE

MECHANISMS

PET/PP

THERMOPLASTIC

Fig. 4 The deformation process of the PET/PP co-knitted textile composite under tension along the wale direction, c denotes the average true strain.

618 A series of images were picked up at different moments as the samples were extended to identify the deformation characteristics and the damage evolution. Figures 4-5 present the deformation process of the samples along the wale, course and 45 ~ directions, respectively. At the initial state, the sample deformed elastically and the structure of the composite almost remained intact, while the stress-strain displayed a linear relationship. With the

Fig. 5 The deformation process of the PET/PP co-knitted textile composite under tension (a) along the 45 ~ direction; (b) along the course direction increase of the load, the relative displacement occurred between the courses along the wale direction, or between the wales along the course direction. Meanwhile the loop shape changed in different manners depending on the loading direction, as shown in Fig. 6. Along the 45 ~

(a)

(b)

(c)

(d)

Fig. 6 The sketch showing the change of the loop shape after extension. (a) original loop shape; (b) extended along the wale direction; (c) extended along the course direction; (d) extended along the 45~direction direction, the sliding between the wales and the relative displacement between the courses appeared simultaneously as the samples were elongated. The relative displacement between the wales and the courses, as well as the sliding between the wales, resulted in changes of the configuration of the fabric structure. As the relative displacement between the wales and/or courses and sliding between the wales occurred, cracks were initiated in the matrix, then evolved into holes. The location and

619 the configuration of holes on the extended samples along the three directions when the samples were nearly fractured are shown in Fig. 7. The relative displacement between the wales or courses and the damage in the matrix are main deformation mechanisms for the composite samples pulled along the wale or the course directions, whilst the sliding between the wales and the damage in the matrix play the major roles for samples pulled along the 45 ~ direction. Following the cracking in the matrix, the load would then be redistributed to fiber bounds, and the cracks were involved into holes, whilst these fiber bounds were further elongated. Because the proportion of fibers oriented in the wale direction was higher than that in the course direction of the knitted fabric, the PET/PP co-knitted textile composite displayed superior tensile properties in the wale direction compared to the other directions. When the change in the configuration of the fabric structure

Fig. 7 The location and the configuration of holes on the samples pulled along (a) the wale direction; (b) the 450 direction; (c) the course direction and the damage in the matrix occurred, the stress-strain relationship deferred from the linear path and demonstrated a nonlinear feature. Most of the holes appeared in the shadowed areas marked in Fig. 6(a), which were indeed the regions of high stress and minimum fiber content. The evolution of holes is demonstrated by Fig. 8 for the sample extended along wale direction. It can be seen that the size of holes (i.e. the dimension along the loading direction) approached a constant. The spacing between the subsequent holes was almost constant (28mm), too, which was just the initial height of the loop. However, the growing speeds of the holes varied from a hole to the next one. This speed increased progressively and the time interval between the initiation of two subsequent holes decreased gradually, as a result of the damage accumulation in the material. 1 ~.

0.8

4.5

~

4

,~ r

3.5

g 0.6

~ ~ ~ i P ~ ~ 3 C ' - . - e - Hole-1

0 N

"G 0.4

e o 0.2 ,,!-

r

/

!

100

150

200

--e--- Loop height

2.5

idth

~ 2 ~ 1.5 G) =: 9 1 ~" O.5

o 50

e-~

250

300

350

rime (s)

Fig. 8 The evolution of the hole's size with time

'-I

0

,

0

0.2

'

0.4

--

0.6

0.8

Strain

Fig. 9 The evolution of the loop height and width with the tensile strain

The change in the loop shape was another source contributed to the large deformation of

620 the PET/PP co-knitted samples. From Figures 4-5, it is evident that the shape of the loops changed significantly during the tension process. Fig. 9 depicted that the loop height increased and the loop width decreased when the sample pulled along the wale direction. In the early stage, the loop almost kept its original shape; but then it became longer and narrower. Therefore, the fiber bound experienced a straightening process during the large deformation. At last, the fiber bound could not be extended any more, so the loop height and width approached respective constants before the sample was fractured. 3. CONCLUSIONS The large deformation inelastic tensile properties of the weft knitted PET/PP textile composites are experimentally investigated, and the meso-scale mechanisms are identified. The results show that the tensile curves of PET/PP co-knitted samples along the wale, course and 45 ~ directions are all significantly non-linear. The tensile behavior in the wale direction is superior to those in the course and 45~ directions. By in-situ observation of deformation process along the wale, course and 45 ~ directions, it reveals the inelastic property of the material is attributed to damage evolution in the matrix, sliding between the wales of the knitted fabric, as well as the change in the configuration of the fabric structure during loading. It can also be seen that the size of the holes developed in the matrix approached a constant, whilst the spaces between the holes almost remain as a constant which is just the initial height of the loop. However, the growing speeds of the holes increase progressively and the time interval between the initiation of two subsequent holes decrease gradually. The loop shape changes significantly during the tension process by increasing the loop height and decreasing the loop width. The loop height and width approach respective constants before the sample is fractured. REFERENCES

1 I. Verpoest, B. Gommer, Gert Huysmans, Jan Ivens, Yiwen Luo, Surya Pandita, Dirk Philips, ICCM-11, 1997. I:108-133. 2 K.H. Leong, P.J. Falzon, M.K. Bannister and I. Herszberg, Composite Science and Technology, 58(1998), 239-251. 3 S. Ramarkrishna, H. Hamada, N.K. Cuong, Z. Maekawa. ICCM-10, 1995, IV:245-252. 4 T.X. Yu, X.M. Tao and P. Xue, Composite Science and Technology, 60(5), 785-800. 5 S.W. Lain, P. Xue, X.M. Tao, in Advances in Engineering Plasticity (Ed. T.X. Yu, Q.P. Sun & J.K. Kim), Key Engineering Materials, Vols, 177-180 (2000), 339-344. 6 P. Xue, T.X. Yu and X.M. Tao, in Advances in Engineering Plasticity (Ed. T.X. Yu, Q.P. Sun & J.K. Kim), Key Engineering Materials, Vols, 177-180 (2000), 745-750. 7 S. Ramarkrishna, N.K. Cuong and H.R. Hamada, Journal of Reinforced Plastics and Composites, 1997, 16(10), 946-966. 8 X.P Ruan and T-W Chou, Journal Composite Materials, 1998, 32(3), 198-222. 9 C.R. Rios, S.L. Ogin, C. Lekakou and K.H. Leong, ICCM-12, 1999, 1035-1041.

Fire Loading

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

623

Nonlinear Analysis of T h r e e - Dimensional Steel Truss in Fire P.Fedczuk" and W.Skowrofiski" 'Faculty of Civil Engineering, Technical University of Opole, ul.Katowicka 48, 45-061 Opole, Poland

The paper presents the concept of analysis of 3-D static loaded steel truss (with or without string) till failure during fire with using modified method of forces [ 1, 2]. Failure of the steel trusses in fire is based on criterion of stresses Behaviour of steel is described by nonlinear constitutive model [3] (based on hypo-elastic Ramberg-Osgood formula and Dorn creep theory) and Plem proposition [4] (for string). Both models are approximated in calculations by hyperbolic Norton-Bailey rule. Fire simulates thermal forcing being an action of high temperature that increases linearly up to some level. The complete formulation of this method contains presentation of calculation algorithm with two-stage method of model parameters identification [5]. Analysis of results for 3 versions of specific truss made of ASTM A36 and A421 steel in fire is presented.

1. INTRODUCTION Of crucial importance in the field of fire protection designing is the problem of fire resistance of construction, equivalent to the engineering task of searching for a temperature at which the elements of a building structure under fire are destroyed. The studies are being carried out focusing on theoretical modelling of fire tests and, in particular, establishing an engineering procedure of calculations of structural steelworks in fire [6].

2. TENTATIVE ASSUMPTIONS 2.1.. Model of steel Increase in temperature causes essential changes in the structural steel properties. The proportional limit and the yield stress of steel decrease monotonically with the gain in temperature whereas the strength grows with an increasing temperature up to about 250~ and then drops rapidly. Cold drawn steels lose their strength at elevated temperature faster than mild steel. Elasticity modulus of steel decreases at elevated temperatures but, as it was observed, it decreases slower than the yield stress. At elevated temperature, steel strains due to creep can be considerable. Total construction steel strains at elevated temperature is obtained as a sum of thermal strains and mechanical strains described by the equation of Ramberg-Osgood and time-

624 dependent strains (thermal creep strains) according Dom theory [3, 7]: T + OWl(T) + a[o lit(T) -1 xg2 (T) +

ala[ m-1 w3(T', t ) = (1)

1 ~+CY T + o E(T)

[t~ [It (T)-1

0.002 trt~y ','JJt'ra'~t(T)

+ 0

[t~ Im - 1 B

exp -

dt

e denotes strain, o - stress, AH- activation energy of creep, R = 8.3183 - gas constant, Joule/moleK, B and m - material constants, t - time, min., T - temperature, ~ T ' temperature, K. Young's modulus E(T), yield stress oy(T), strain-hardening coefficient ~t(T) and material constants W, , W2 , ~g3 are temperature-dependent. The model was worked out under the following assumptions: steel is a homogeneous and isotropic continuum, no repeated load is considered, the strain process is slow or is a static one, the strains are small. Total prestressing steel strain (string) at elevated temperature is obtained as a sum of thermal strains and strains described by the equation of Plem [4]:

e(o,

T)=E 0

O'

I~

0

(2) e(o,

T)=

e

T+~ [1+ Z(~)0] O

E:

for 0 > 0 O'

O

where: Z(o) - Zener-Hollomon's parameter, eo - stress-dependent strain, 0 - Dom's parameter, 0o - limit value of Dorn's parameter. A simplified, but sufficiently accurate for fire engineering purposes, Bailey-Norton formula can replace the equations (1) and (2)

=A(T,T)o n(T)

+aT,

(3)

where A and n denote temperature-dependent material functions. Two stages method of identification [3] is applied for determination of the pair of parameters A and n from Bailey-Norton equation (3) that approximates the programmed nonlinear constitutive relations (1) and (2). An application of that method requires: 1) generation of the set of the pair of value "stress o i - strain el" calculated from constitutive equation (1) (or (2)), 2) linearization by the two-sided finding the logarithm of Bailey-Norton equation (3) and assessment of the initial values of the parameters A and n by the linear least squares method, 3) determination the final values of the parameters A and n using gradient method of

625 Marqurdt-Levenberg [8, 5, 2]. 2.2. Modified method of forces

For an analysis of the statically indeterminate space steel trusses, the modified method of forces [1, 2] is applied. That method considers approximation by equation (3) of nonlinear constitutive relations (1) and (2) for steel. It is assumed that a truss consists of steel bars (connected jointly in nodes) treated as one-dimensional dements. Fire simulates thermal forcing being, generally, an action of the temperature that increases linearly up to some level under an assumed rate of increment, individually, for the particular truss member. Static load in a form of the system of forces is applied to the joints. All system and its particular components (struts) do not loose stability at elevated temperature. Failure of the system occurs in case of exceeding a mean value of stress (in a section of any bar) that is limited by a yield stress at elevated temperature. Analysis of such defined problem by modified method of forces requires solution of the system of an algebraic nonlinear equation: F(X (j)) = 0,

(4)

where components of the vector of function F(X (J)) have a form K

n

{ Z [Zs(X k)- x k ] + Zs (P)} s. Zs(Xi) 1 +5 s

n

iT

+5

iA '

( A s ) - I " fs s

(5) 8iT=Y,[Zs(Xi)eTls S

],

5iA = - E [ Z s ( X i ) A l s ] . S

System of such equation is created routinely, by reduction of indeterminate truss system to determined one by means of selection of K redundant forces X i and establishing forces Zs(Xi) and Zs(P ) in bars for states X i = 1 and for external load P. Coefficients 5 i T are displacements along direction of redundant X i induced by changes in temperatures T. Coefficients 8iA are displacements induced by assembly errors (shortenings of strings A1s presstressed to stress level ~ = 0.8~y(T-20~

). Length 1s and area of section fs characterize geometry of a bar or

string. Constants A s , n s and thermal deformation e T define steel on every of the considered level of temperature T under the rate of their increment. Solution is achieved by calculation of an algorithm that requires: I) identification of the parameters of Bailey-Norton model by two-stages method, II) solution of the system of equation (4) in question using iteration Newton method by: a) assignment of the forces Z s in truss bars for states X i = 1 and for external load P and assessment of the initial values of the vector components redundant X (J),

626 b) calculation of the vector of function F(X (J)) from relation (6) and matrix of derivatives F ' (X (j)) by means of finite difference method, c) corrections of the redundant values according to the formula: X ( j + 1) _ [ F ( X ( j ) ) ] - 1 F ( X ( J ) ) ,

(6)

d) checking of the condition of calculation interruption for all components of vector F(X (J)) (and in case of not satisfying above condition- continuation of iteration from point (b))

F/(X~ ")) < "t (x

(7)

= 10-4),

e) assignment of the real values of forces Z s in truss bars for determined vector F(X ( j ) ) , III) checking of the failure condition of the truss structure according criterion of stresses -c

Y

( T ) <__ g =

Z s f ___ ~ Y ( T ) .

(8)

S

Presented algorithm is used by application made of a set of seven integrated computer programs SKOFET3D that communicate by a common database.

3. NUMERICAL EXAMPLE

Three variants of the same space steel truss (Fig. 1) made of typical American steel ASTM A36 and ASTM A421 (for string) was exposed to calculation analysis of thermal influences. All bars were made of tubes. Area of horizontal bars equals to f~ = 4.01 cm2. 4 x 2 . 5 m = 10m G3

~P

/p

~/

1.5

Figure 1.3-D steel truss. Areas of the cross braces and truss posts are equal f~ = 3.65 cm 2. Area of string (7 stranded

C/3

~,

<,~

II

o

o

O~

I

"-

o

O0

I

.

o I

~

'

,

o

I

'6

r

,

,0

9/

"

0 .~ij :r,3 ,,- j

ic :t~, ,

:,...

~

I

c~

~ / B o

,~

o

9"

x:

,,

o

u~ ~

~

T

~

~

,

0~.

-

,---

"~l

I

0 :u'3

,0

oE

c~

0 in

,~

n

i~ , ~oV

,

~Z

I

u4~

:~

a)

i~ e

Q.

o

c ~-

:C r---, ~o(,.)

0 .ur}

U'%l'-

--

~ o

~

~'- ~

o

I

o

.,,

~) ssan, S

~

ur]

0

ff'j

C~

I

un

I

[Od)l~+Ot x ]

I

!oV

: I-~C~

!~S

:%,

ff o

[oa~,+0tx] ~ ssaJ~s

" u~

:r" I--

!c u~ ~ m

I

o~

o

B

iu" ~

.

0

u')""

'r..,

:C,

.,-- #--.

~,0~11,=1c

I|

;,:

~b

c~ I

|

I'~

O~l

/

~

..-." .............

0

cn e0 ~

.~.-'~

[Od>l~;+OI, X] _o ssaa%S

o

[Od~+OtX] ~ ss~J~s

_~ 0

~.}

.,..~

627

628 wires) equals to f~ = 1.06437 cm2. This string was prestressed to stress level g = 0.8ay(T=20~ = 1.71998x106 kPa. Upper joints of the structure are loaded by set of 6 (for scheme (A) and (B) from fig.2) or 4 vertical concentrated forces. Total 10ad is equal to 96 kN. Action of fire is simulated using rate of temperature change equal to 12 ~ Three variants of truss structure presented in Fig.2: truss with string (scheme (A)), truss without string (scheme (B)) and inverted truss without string (scheme (C)) was analysed. The results of calculation for a chosen bars marked as in Fig. 1 are presented in Fig.2 in a form characteristics "stress a - temperature T". Calculation was performed for 3 levels of temperatures T. Failure of the truss with string occurs as a result of exceeding, in bar K~, criterion of stresses (8) at temperature T = 428 ~ The same failure criterion is exceeded in bar D3 at temperature T = 114 ~ for truss without string and at temperature T = 312 ~ for inverted truss without string. The best form of truss structure is scheme (A).

4. CONCLUSIONS The analysis performed indicates that it is possible to calculate precisely the fire resistance of steel trusses on the basis of: (1) the known properties of steel at elevated temperature, (2) the details of a cross-section dimensions, loaded shape of a construction, (3) the data concerning the temperature increase in a construction, determined by the efficiency of the fireproof insulation. The problem is important for people's safety.

REFERENCES 1. P. Fedczuk, W. Skowrofiski, Effects of fire temperature changes on the flat steel truss (in Polish), Technical University of Opole, Zeszyty Naukowe, No. 224/96, Z.41, (1996) pp.109-121. 2. P. Fedczuk, W. Skowrofiski, Fire study of steel trusses in interacting systems of tall buildings, 5th International Conference on Tall Buildings, Vol.1, Hong Kong, (1998) pp. 1712-1732. 3. W. Skowrofiski, Load capacity and creep problems of structural steelworks in fire (in Polish), Studia i Monografie Z.62, Technical University of Opole, (1992). 4. E. Plem, Theoretical and experimental investigations of point set structures, Document D9, Swedish Council for Biulding Research, Stockholm, (1975). 5. P. Fedczuk, W. Skowronski, Two-stages method of parameter identification of BaileyNorton model for fire heated structural steel (in Polish), Technical University of Opole, Zeszyty Naukowe, No. 222/96, Z. 40, (1996) pp.51-67. 6. W. Skowrofiski, Material characteristics in the analysis of heated steel beams, Fire and Materials, Vol. 14, No.3, (1989) pp. 107 - 116. 7. W. Skowrofiski, Buckling fire endurance of steel columns, Journal of Structural Engineering, Vol. 119, No.6, (1993) pp.1712- 1732. 8. D.W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Industr. Appl. Math., Vol.11, No.2, (1963) pp. 431-441.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

629

M o d e l l i n g o f plastic strength o f composite tubular m e m b e r s under elevated temperature conditions M. B. Wong a, J. Ghojel b and N. L. Patterson a a Department of Civil Engineering, Monash University b Department of Mechanical Engineering, Monash University The use of composite member made of steel tube filled with concrete is prevalent in building construction due to its favourable construction efficiency and fire resistance characteristics. When using such members, it is necessary to determine their elastic stiffness, plastic strength and their corresponding characteristics under fire conditions for design purposes. The determination of such data becomes more complicated when non-uniform temperature distribution in the cross-section exists. It is also difficult, if not impossible, to obtain those data for every structural element through fire tests because of economic constraints. Therefore, calculation procedure to obtain numerical solution is an attractive alternative in terms of economy as well as efficiency. Tiffs paper details the calculations of the elastic stiffness and plastic bending strength of a concrete-filled steel tubular member under a fire scenario. Temperature distribution of the cross-section of the member is calculated on the basis of heat transfer software and the resulting properties are compared for cross-sections modelled with and without thermal contact resistance elements at the steel-concrete interface.

1. INTRODUCTION Apart from construction efficiency, one distinct advantage of using structural members made of steel tubes filled with concrete (termed composite members herein) in building construction is its favourable fire resistance characteristics. When using such members in construction, it is necessary to determine their elastic stiffness and plastic bending strength for design and their corresponding characteristics under fire conditions in order to satisfy the serviceability requirements. The determination of such data becomes more complicated when non-uniform temperature distribution in the cross-section exists. This is frequently the case when only part of the member is exposed to a fire source in a compartment. It is also difficult, if not impossible, to obtain those data for every structural element through fire tests because of economic constraints. Therefore, calculation procedure to obtain numerical solution is an attractive alternative in terms of economy as well as efficiency. The calculation process of determining the elastic stiffness and plastic bending strength of these composite members can be separated into two steps. The first step is to evaluate the temperature profile of the crosssection on the basis of a specified fire curve in the furnace. This is usually carried out by subjecting the member to a fire scenario simulated by computer modelling and the temperature profile is obtained using appropriate heat transfer models. The second step is to calculate the values of elastic stiffness and plastic bending strength of the cross-section on the

630 basis of the temperature profile. The accuracy of the results in both steps of computations is sensitive to the parameters adopted in carrying out such computations. This paper gives details on the process of determining the elastic stiffness and plastic bending strength of such composite members and the parameters involved in obtaining accurate results for those data.

2. TEMPERATURE PROFILE OF CROSS-SECTION When a composite member is subject to fire, heat is transmitted from the outer steel tube to the inner concrete core mainly by conduction. Existing mathematical models for the prediction of temperature response of composite structural columns (Lie, 1994; Lie and Irwin, 1995) ignore an important aspect of the physical model, namely the presence of thermal contact resistance at the interface between steel and concrete. There is always contact resistance to heat conduction across solid-solid interfaces caused by the relatively few points of contact between the surfaces (surfaces can have some degree of roughness) and the presence of entrapped gases in the voids present at the interface. The latter factor is often dominant because the thermal conductivity of the gas is normally lower than that of the solid. In steel-concrete composite elements exposed to fire the voids will increase as a result of differing thermal properties while both air and water vapour can be present at the interface. The dominance of the gaseous interface can be gauged by considering the fact that the thermal conductivities of concrete and steel, for example, are respectively 16 and 550 times greater than that of steam. Nowadays, computer programs can be used to predict temperature distribution very accurately for solid members provided that appropriate thermal properties of the crosssections are used. While the thermal properties of both steel and concrete are well known and can easily be established, the effect of the thermal contact resistance at the interface between steel and concrete on the temperature distribution of the composite member has not been investigated. In fact, some discrepancies between theoretical prediction and experimental results of temperature distribution have been observed when the interface effect is ignored. In the current work on composite members under fire conditions, temperature analysis was conducted using a software package called SINDA/G (Network Analysis Inc., Tempe, Arizona, USA). SINDA/G is a finite difference network analyzer with 3-D graphical modeller and post processor. The estimated heat flux densities were used as the time-dependent input boundary conditions. The material properties were entered as temperature-dependent arrays. The four flux densities, which act on four quadrants of the cross-section of the composite member, were obtained during the tests in an electric furnace using inverse heat conduction analysis (Ghojel, 1999). The analysis was carded out with and without the thermal contact elements. (Details of the characteristics of the thermal contact resistance elements are outside the scope of this paper.) It has been found that by including the thermal contact resistance elements, the results of temperature distribution prediction obtained both from the experiments and SINDA/G match very well.

2.1. Results of analysis Figure 1 shows the 2-D finite element model used by SINDA/G for the composite section used for the current investigation. The outside diameter of the section is 150ram and the steel tube thickness is 6mm. The concrete core is divided into 10 concentric layers and the whole section is further divided into 24 identical sectors. Each element formed by the boundaries of

631 the lines of divisions is assumed to have a constant temperature equivalent to the average of the temperatures at the corners of each element. Figure 1 also shows the temperature response of the two eases (with and without thermal contact resistance) along the perimeter at time = 6000 seconds. It can be seen that the ideal composite structure (no thermal contact resistance) exhibits better fire resistance when compared with the realistic one. When thermal contact resistance is accounted for, two things happen: 1. The presence of contact resistance slows the rate of heat transfer from the steel casing to the concrete causing an increase in the temperature of the steel casing at the initial stages of the exposure 2. The temperature profile across the interface and through the concrete core follows consistently at a level above that observed when thermal contact resistance is absent. Experimental results confirm this trend. Steel tube

6afl~

2oncrete core

T ~ - temperature with contact resistance (T ~ - temperature without contact resistance

623oc (5990C) Figure 1. Model and results of temperature analysis of a steel/concrete composite member

559~ 1531~

sSI~

381oC ." (360~

458~ (437~

3. ELASTIC STIFFNESS The calculation of elastic stiffness (EI, E = modulus of elasticity, I = second moment of area) is important in assessing the elastic deflection of the member. For a cross-section with non-uniform temperatures, an effective second moment of area can be calculated using the transformed area method. This is based on bending about the principal axes x'x' and y'y' at an angle 01 with the horizontal axis as shown in Figure 2. y',.

Y

\

Figure2. Transformed area method elastic stiffness calculation

for

,

...x__ x

x ~-

. .............. ......... 9....... .., .......

~.............--'."..": .. " ' " / [ x'

y

\ ~

" - Y'

Centroid

632 For pure bending in a cross-section, ~Eri ydA i = 0

(1 a)

~Eri xdA i = 0

(1 b)

where ETi, given in AS4100 (1990) for steel and AS3600 (1994) for concrete, is the modulus of elasticity at temperature T for an elemental area Ai. It should be noted that Ai = 0 for concrete elements below the neutral axis as concrete is assumed to have no tensile strength. When Ai is transformed into an equivalent area A~i at 20~ Equation (1) can be written as

E2o J'ydAei = 0

(2a)

E2o IxdAei = 0

(2b)

The principal second moments of area Ix'x' and Iy'y' are calculated from Ixx, Iyy and Ixy which, by trial and error method to establish the neutral axes, can be evaluated using spreadsheet programs. The results of such calculations for the composite section shown in Figure 1 are: With thermal.contact resi.stance, Ix'x' = 1.92xl 07 mm 4 Iy,y, = 1.83xl 07 mm 4 0z = 1 4 ~

Without thermal contact resistance, Ix,x, = 2.07xl 07 mm 4 Iy,y, = 1.98x107 mm 4 01 = 18 ~ The difference in Ix,x,, and hence the deflections, for these two cases is about 8%.

4. PLASTIC BENDING M O M E N T CAPACITY The calculation of the plastic bending moment capacity of a section under non-uniform temperature distribution can be found in a way similar to the previous section. However, the plastic neutral axis for such a section is usually not the same as the centroid of the elastic section and therefore needs to be found first. The plastic neutral axis is calculated by using

f, dA, =0

(3)

633 where fyT, given as yield stress in AS4100 (1990) for steel and as factored characteristic strength 0.85 f~ in AS3600 (1994) for concrete at temperature T, is a linear function of T. The plastic neutral axis can be found by trial and error method for which Equation (3) is satisfied. Again, Ai = 0 when concrete is in tension. Once the plastic neutral axis is found, the plastic bending moment capacity MpT is calculated by Mpr = E fyr dAi

(4)

where the summation is carried out about the plastic neutral axis. The results for the two cases are given below. With thermal contact resistance, (MpT)xx = 28.2 kNm (MpT)yy= 26.3 kNm Without thermal contact resistance, (MpT)xx = 30.0 kNm (MpT)~ = 28.0 kNm

5. CONCLUSIONS When carrying out temperature analysis for a concrete-filled steel tubular member, it has been observed that there are always discrepancies between experimental and analytical results. The analysis is usually performed by computer software where an ideal heat transfer model for heat to be directly transferred by conduction between steel and concrete is used. In fact, thermal resistance exists at the interface between steel and concrete, resulting in higher temperature distribution, and hence weaker sectional properties. The weaker sectional properties will lead to increased deflection and reduced plastic bending strength of the member. The difference is about 8% for deflection and 6% for plastic bending strength vehen the temperature is the highest in the section. A useful outcome of this investigation is that the conductance characteristics of the thermal contact resistance elements obtained from the tests can be applied to other heat transfer analysis when such steel-concrete interface exists.

REFERENCES

1. Lie, T.T., Fire Resistance of Circular Steel Columns Filled with Bar-Reinforced Concrete, Journal of Structural Engineering, Vol. 120, No. 5 (1994). 2. Lie, T.T., Irwin, R.J., Fire Resistance of Rectangular Steel Columns Filled with BarReinforced Concrete, Journal of Structural Engineering, Vol. 121, No. 5 (1995). 3. Ghojel, J.I., Contact Thermal Resistance in Composite Structural Elements Exposed to Fires, The 1999 Australian Symposium on Combustion, Newcastle, 30 September-1 October (1999). 4. Standards Australia (SA), AS4100 - 1990, Steel Structures, SA (1990). 5. Standards Australia (SA), AS3600 - 1994, Steel Structures, SA (1994).

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

635

The experimental and theoretical behaviour of composite floor slabs during a fire C.G. Bailey Building Research Establishment Garston, Watford, WD2 7JR, United Kingdom. The results from six previous fire tests on a full-scale steel-framed building have shown that composite floors have a greater inherent fire resistance than that suggested by current codified design methods. Following these tests, it is generally accepted that composite floors support the applied load during the latter stages of the fire by tensile membrane action. A smaller scale independent test, which included only those components that maintained significant strength during a fire, was conducted at ambient temperature to investigate the behaviour of tensile membrane action. The test was designed such that nominal horizontal restraint was applied to the edges of the slab. The results from this test, together with the previous fire tests, were used to develop a simple design method. This method is shown to produce accurate estimates of the load-carrying capacity of composite floors at elevated temperatures. 1. INTRODUCTION During 1995 and 1996 six localised fire tests I were conducted on the full-scale eight storey steel-framed building at the Building Research Establishment (BRE) Cardington Laboratory. The test building consisted of steel beams supporting a composite floor system comprising steel decking, lightweight concrete and anti-crack A142 mesh reinforcement. The tests were conducted by BRE and British Steel and are summarised in Table 1, with the position of the tests (on plan) shown in Figure 1. In each test the steel columns were protected using ceramic fibre, with the steel beams and underside of the composite floor totally unprotected. The maximum steel temperature recorded during each test is shown in Table 1. Current codes of practice suggest that the structure would collapse at the high temperatures recorded in the tests. However, the applied load was adequately supported during and after each test, although in some instances vertical displacements were very large. In addition, in Test 6 the floor slab was supported by the loadbearing blockwork, which formed the compartment wall The results from the tests imply that the current codified design method is not addressing the true structural behaviour of the building during a fire. In particular it was felt that the composite floor system was far stronger than that suggested by the current design method, which limits its design to bending theory. Although simple, this design method ignores any beneficial effect of tensile membrane behaviour of the floor slab at large displacements.

636

Figure 1. Location of fire tests. Although it was generally felt that tensile membrane ar~on occurred in the composite slab during the fire tests, it was not possible to obtain conclusive evidence of this behaviour due to the complicated naane of full-scale fire tests. In particular it was difficult to show from the results that the floor slab near an edge of the building formed a compressive ring in the concrete around its perimeter, allowing tensile membrane action to occur at the centre. This led to an additional independent test carried out at BRE Garston, on a 9.5m x 6.5m composite slab (of similar construction to that used on the Cardington frame), with nominal horizontal restraint around its edges, to investigate if tensile membrane action can occur at elevated temperatures and large displacements. Due to problems with using measuring devices on structural tests at elevated temperatures, it was dezided that the only possible method to allow a full investigation of tensile membrane action was to conduct the test at ambient temperature. Table 1 Stop.mary of fire tests Test No.

Organisation conducting test

Description

Floor area (m2)

Location

Max steel temp. (~

1

British Steel

One beam

24

level 7

875

2

British Steel

Slice across the building

53

level 4

850

3

British Steel

Comer compartment

76

level 2

954

4

BRE

Comer compartment

54

level 3

903

5

BRE

Large compartment

340

level 3

691

6

British Steel

Large compartment

136

level 2

I 150

637 2. TEST TO INVESTIGATE TENSILE MEMBRANE ACTION To allow the results from the test to be compared to those obtained from the six previous fire tests, it was necessary to represent as closely as possible the behaviour of the composite floor slab during a fire. The previous fire tests have shown that the steel deck reaches temperatures in excess of 1100~ and thus has nominal strength. Therefore in the ambient temperature test, the deck was removed before load was applied to the slab, leaving the lightweight concrete (which was east into a trapezoidal shape) and the anti-crack mesh, as shown in Figure 2. These are effectively the only components of the composite slab that retained most of their strength during the previous fire tests.

Figure 2. Test procedure to investigate tensile membrane action. The test duration was 3 days (8/06/99 to 10/06/99), with the structural response of the slab continuously monitored over this period. The test can be split into three main stages, as follows, 9 removal of the steel deck from the concrete, 9 placing the loading system onto the slab, 9 applying additional load until failure of the slab. The central vertical displacement of the slab over the full test period is shown in Figure 3.

638 Vertical displacement at the centre of the slab 800

A E

6oo

Removing steel deck

Plating loading

system onto test slab

500

,,,., rE

/ / II

Testing slab t o , . destruction

700

L" Preparing loading system

400

L," ~

L,"

m

a. 300 .~_ 1:3 2OO

t

lOO 0

.

08/06/00:00

.

J

.

.

08/06/12:00

.

.

.

09"/06/'0 0 : 0 0

.

.

.

09/06/12:00

.

10/06/'0 0 : 0 0

,,,

.

.

.

10/06/12:00

.

11/06/00:00

Date/time

Figure 3. Maximum vertical displacement of slab during the test.

Removal of the steel deck After the steel deck was removed, inspection of the underside of the concrete slab showed that the trapezoidal shape of the concrete was maintained and the pattern of the cracks was similar to the classic yield-line pattern for simply-supported rectangular concrete slabs. The slab reached a vertical displacement of 59ram, and the horizontal displacements around the perimeter of the slab were less than lmm. It is worth mentioning that the classic yield-line theory predicts a failure load of 2.3kN/m 2, which is equal to the actual self-weight of the slab. Therefore, theoretically, tensile membrane action must occur to support any additional applied load. Placing the loading system onto the slab Following removal of the steel deck, the loading system was constructed and placed onto the slab. In the first instance, steel plates were placed horizontally on a bed of mortar at each of the 16 loading points. Once this was complete, the slab was left overnight to allow the mortar to set. During the second day of the test, the remaining components of the loading system were placed onto the slab. The weight of this system caused the vertical displacement of the slab to increase to 113mm, with nominal horizontal movement of the slab edges (less than 1.5mm). Loading the slab to failure A uniformly distributed load was placed on the slab. Failure of the slab occurred at a load of 4.81kN/m:, which is slightly higher than double the load capacity of the slab calculated using classic yield-line analysis (2.3kN/m2). As shown in Figure 4, failure occurred due to a large central crack forming through the full depth of the slab across most of the shorter span. After the formation of the central full depth crack the slab was effectively split into two with each half supported on three sides. This led to a large crack forming across one diagonal of the slab as shown in Figure 4, leading to ultimate failure.

639 The maximum horizontal displacement at the edge of the slab was 57mm. The measured horizontal displacements suggest that the strains in the mesh reinforcement in the shorter span direction were relieved by the 9.5m edges of the slab being pulled towards the centre. In the other direction the strains were increased as the vertical displacement of the slab increased. Observation of the slab during and after the test showed compressive failure of the concrete at the centre span of both the 9.5m slab edges (Figure 4). This suggests that the compressive ring formed, as expected, and thus provided support to the central area of the slab which was in tensile membrane action.

Figure 4. Location of cracks on the slab during the test. 3. THEORETICAL BEHAVIOUR

A simple design method has been developed based on the behaviour of the slab in the previous test. This uses an equilibrium method and calculates an enhancement factor based on the vertical displacement of the slab, which is applied to the load calculated using the yield-line theory. Limited space excludes the opportunity of presenting the theory, however Figure 5 shows the comparison between the load-carrying capacity using the design method and that recorded in the previous test. It can be seen that, up to the point at which the mesh in the centre of the slab fractured, the design method predicts accurate loads for a given displacement. Limits based on maximum vertical displacement and the slab aspect ratio need to be defined to represent the point at which the mesh fractures. Research work at BRE is currently being conducted to def'me these limits, both at ambient and elevated temperatures. The design method was used to predict the load-carrying capacity of the slab and heated steel beam in Test 4 at Cardington. The comparison between the design method and test results is shown in Figure 6. It can be seen that as the steel temperature reaches 900~ the applied load in the test is supported by tensile membrane action of the slab.

64O

9

Load carrying capadty using design method Limits need to be applied to define fracture of reinforcement

8 7

~

s

3

2

"

\ TestResults

100

0

200

3(~0

400

5()0

600

700

800

Displacement (mm)

Figure 5. Comparison between design method and ambient temperature test result. 1200

Design meSxxJ

1000 o "-"

800

t_

~. 6o0

vE

,oo

~Test

oo

resuR

/

200 0

'

0

'

"

,

,

50

.

.

.

.

' l

,

,

"~'"

,

i

,'

7-

,

=

w

100 150 200 Vertical displacement (ram)

,

,

~

,

,

250

,

~

.... ,

,

300

Figure 6. Comparison between design method and fire test (Test 4). 4. C O N C L U S I O N

Following the full-scale fire tests, a smaller independent ambient temperature test was conducted to investigate the tensile membrane behaviour of the composite floor slab during a fire. This test showed that the slab could develop tensile membrane action with nominal horizontal restraint at the edges of the slab. The load-carrying capacity was slightly greater than double that calculated using the classic yield-line theory. Using the results from the tests, a simple design method has been developed. This has been shown to give accurate estimates of the load-carrying capacity of the slab at both ambient and elevated temperatures. REFERENCES

1. Bailey C.G., Lennon T., Moore D.B. The behaviour of full-scale steel-framed buildings subjected to compartment fires. The Structural Engineer. Vol 77/No.8. April 1999 pp. 1521.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

641

T h e r m a l c o n t a c t resistance at the concrete/steel interface of concrete-filled steel columns J.I. Ghojel Department of Mechanical Engineering, Monash University 900 Dandenong Road, Caulfield East, Vic 3145, Australia

The thermal contact resistance at the steel/concrete interface of circular steel tubular column filled with concrete under transient and high temperature conditions was estimated using IHCP (Inverse Heat Conduction Problem) numerical modeling in conjunction with measured temperature. Direct heat conduction analysis showed that the empirical correlation thus obtained can improve the accuracy of predicted temperature response of composite structural elements under conditions similar to fire environment.

1. INTRODUCTION Tubular steel columns filed with concrete are widely used in construction because they combine the advantages of structural steel and concrete. Compared with traditional forms of column construction, their use can also lead to 60% savings in total column cost in tall buildings (Uy, 1996). Additionally, filling tubular steel sections with concrete provide high fire resistance. Typical fire resistance, defined as the time to failure of a structural element to support the test load under standard fire conditions, for a concrete-filled tubular steel column is 1-2 hours compared with 15 minutes for a hollow steel column (Lie and Chabot, 1990). Some of the well known existing mathematical models for the prediction of temperature response of composite structural columns (Lie, 1994; Lie and Irwin, 1995) ignore an important aspect of the physical model, namely the presence of thermal contact resistance at the interface between steel and concrete. Contact at solid-solid interfaces occur at relatively few points forming voids that can be filled with gas. Since the thermal conductivity of the gas is normally lower than that of the solid, thermal resistance to heat conduction increases at the interface. In steel-concrete composite elements exposed to fire the void spaces will increase as a result of differing thermal properties and both air and water vapour can be present at the interface. The dominance of the gaseous interface can be gauged by considering the fact that the thermal conductivities of concrete and steel, for example, are respectively 16 and 550 times greater than that of steam. A better understanding of the effect of the contact resistance and possibly better estimate of the numerical values of this resistance as a function of temperature can improve the accuracy of numerical models designed to predict the temperature response of steel tubular columns filled with concrete under fire conditions. Since little is known about the exact geometry of the contact area and the properties of the gas in the voids at the interface, the problem is best treated as an Inverse Heat Conduction

642 Problem (IHCP). Unlike direct modeling, which is concerned with the determination of the temperature field inside a solid body when boundary temperatures and/or heat flux densities are specified, inverse modeling can estimate the surface temperature and heat flux fields from the temperature-time histories at any number of locations in the solid. These boundary conditions can be obtained by coupling the temperature measurements with heat transfer numerical models and minimizing the differences between the calculated and measured temperatures at given locations and times. Contact resistance can then be estimated from the temperature gradient across the interface and the flux densities determined by the IHCP analysis. 2. EXPERIMENT Circular tubular steel tube (140 mm outside diameterx6 mm steel thickness) filled with non-reinforced concrete (33 kg cement, 10 kg water, 31 kg sand, 80 kg aggregate and 0.5 litre super plasticiser) was heated in a modified 15 kW electric furnace from the ambient temperature until a maximum furnace temperature of 900 ~ was reached then cooled down for 15 minute to allow the heat flux to peak before data acquisition was terminated. Both ends of the tube were insulated to reduce the analysis to a 2-D heat transfer problem (Figure 1). The concrete core temperatures were measured using K-type grounded junction thermocouple in steel protective sheaths of 3 mm diameter and 300 mm length. Steel surface temperatures were measured using special K-type bolt-on high-temperature ceramic fiber insulated thermocouples. The temperature measurements were taken at 2-second intervals using Datataker 500 and DeLogger Plus (Data Electronics, Rowville, Victoria). Temperaturedependent thermal properties (thermal conductivity and specific heat) for steel and concrete reported by Lie (1994) were used throughout the investigation.

Figure 1. Schematic diagram of the experimental setup

Figure 2. 2-D model for INTEMP showing temperature measurement locations

3. INVERSE HEAT CONDUCTION PROBLEM (IHCP) ANALYSIS Inverse heat conduction software INTEMP (TRUCOMP CO, Fountain Valley, California, USA) was used for the IHCP analysis of the measured data. INTEMP solves linear or nonlinear inverse heat conduction problems using either the Crank-Nicolson or the fully

643 implicit nonlinear heat conduction models (Trujillo and Busby, 1997; Busby and Trujillo, 1985). The input data consists of the known (measured) temperature-time histories at any number of nodes in the model (Figure 2). INTEMP uses Dynamic Programming to solve for the unknown flux densities that minimize the general least square error. It also allows material properties to vary with temperature, be orthotropic or vary for each element. Figure 3 shows the flux-time histories for the circular column estimated from INTEMP. Four different flux densities, over different quadrants, were assumed for the model: top (T), fight side (R), left side (L) and bottom (B) as shpwn in Figure 2. There is no heating element on the floor of the furnace, hence the negative flux at the bottom location in Figure 3. 16000 . . . .

~

............

concrete Ts

-~ 9 8000 L

a

4oooj.

~ :r:

q

,.,

, ,

Tc

o - 4 0 0 0 I'

q

2000

' 4 i 00 Time, sec

steel

60D0

Axs

~- Axe ~ Figure 3. Heat flux histories at four locations estimated by INTEMP

Figure 4. Schematic diagram of interface for steel-concrete composite column

3.1. E s t i m a t i o n o f i n t e r f a c e c o n d u c t a n c e

Contact resistance (or contact conductance) can be estimated at any point of the interface if the heat flux (Figure 3) and temperature drop across the interface at that point are known. 1000

'r 10(I)

E 8OO O t"O

9

T

. . . .

L

400

~ 400 ~ 2OO

O 0

0

i

i

I"

J-

~

100

200

300

400

500

1::::: O

-7

600

700

TenI~rature, ~ Figure 5. Calculated thermal contact conductance at four locations

0

0

';

,

,'"

,

0 lOO 2o0 3oo 4o0 5o0 6oo 7o0

T ~

~

Figure 6. Averaged thermal contact conductance for circular column

The temperature on the steel side is taken as the surface temperature, which varies little for thin walled tubes. The relation for contact conductance from Figure 4 is:

644 = AT Ax~ Axe hc q ks kc where hr - is the contact (or interfacial) conductance, W/m 2 K

(1)

A T = Ts-Tc

q - heat flux across the interface, W/m 2 k~., kc -thermal conductivities of steel and concrete, respectively, W/m K ztx~ a n d A r c - the distances to the nodes adjacent to the interface in the steel and concrete, respectively, m. Figure 5 shows the computed values of the contact conductance hc for a composite circular column as a function of temperature at three locations: at the top, fight-hand side and lefthand side of the cross section of the circular column. The conductance is not the same for all quadrants indicating that the gaps at the steel/concrete interface can have different physical and thermal properties depending on the incident heat flux and the corresponding gap temperature. In order to make the use of this data easier it was decided to average the three values and fit a curve to the resulting data. Figure 6 shows the fitted curve to the average values of thermal conductance hc for the circular column over the test temperature range 30600 ~ The average contact conductance hc as a function of temperature t (~ can be expressed by the following correlation: c h C = a + bt + - T

W/m2 oc t where a = 423.11441, b = --0.29932678, c = 733672.57

(2)

Figure 7.2-D thermal model generated by SINDAJG showing the interface elements The value of hc is quite large at low temperatures when there is good contact between steel and concrete dropping sharply around 100 ~ then slowly afterwards. The decrease in conductance could be explained by the increase in the size of the gap as the temperature increases as a result of the difference between the values of the coefficient of thermal expansion of steel and concrete. The initial sharp drop could be due to evaporation of the moisture in the gap forming steam which has lower thermal conductivity than water.

645 4. DIRECT HEAT CONDUCTION PROBLEM (DHCP) ANALYSIS To test the validity of correlation (2), direct heat conduction analysis was conducted using the thermal analysis software SINDA/G (Network Analysis Inc., Tempe, Arizona, USA). oO800

.4oo

E

0 0

1000 2000 3000 4000 50(X) 6000 7000 "lima,

Figure 8. Measured (markers) and predicted (solid lines) temperature-time profiles at three surface locations in the circular column SINDA/G is a finite difference network thermal analyser with 3-D graphical modeller and post processor. The model generated by the graphical modeller was identical to the INTEMP model as far as number of elements was concerned with the computed heat flux densities by INTEMP serving as the time-dependent boundary conditions. The material properties were entered as temperature-dependent arrays. To simulate the contact resistance, two-dimensional elements of 2.54 mm thick in the radial direction were inserted at the interface between the steel and concrete. The radial thermal conductivity at the interface was then taken as the estimated average contact conductance multiplied by the thickness of the interfacial element. The values of thermal conductivity in the other two directions were taken equal to zero to allow for one-dimensional heat conduction across the interface. Figure 7 shows the 2-D model generated by the graphical modeller in SINDA/G for the circular composite column.

=a 400

300

t ....

~. 200 ! E 100 I-'

0

,

0

~

==

199

-

-,

"

' , '

, .............

,

,

, ..............

,

1000 2000 3000 4000 5000 6000 7000 Time, see

Figure 9. Measured (markers)and predicted (solid lines) temperature-time profiles at three interior locations in the circular column 5. DISCUSSION Figures 8 and 9 compare the temperature-time histories of the measured and predicted (by SINDA/G) at different nodes (temperature measurement positions). These temperature-time

646 histories are for the locations corresponding to the node numbering shown in Figure 2. Comparison is made for three surface nodes (11, 77, and 143) and three interior nodes (105, 199 and 204). The agreement between the results obtained by measurement and direct heat conduction modelling is quite good for all the nodes with the exception of the core node. This may be explained by errors in temperature readings and/or assumed concrete properties. Other factors affecting accuracy of the results may include: 9 The difficulty in simulating 2-D heat conduction problem in the tests since insulating materials are imperfect thermal insulators. 9 The difficulty in determining the exact location of the thermocouple probes. The probes were inserted axially in holes drilled in dry concrete at a depth of about 200 mm, which could have caused deviation from their intended locations. 9 The distortion in the actual measurement of surface temperature due to the effect of radiation heat exchange with the heating elements on the thermocouple reading. 9 Deviation of the actual thermal properties of the concrete from the assumed properties. Canadian data (Lie, 1994) for specific heat and thermal conductivity of concrete were used for Australian aggregate. 6. CONCLUSIONS 1. The work demonstrates the feasibility of using IHTP techniques to determine interfacial contact resistance in composite structural columns. 2. The estimated values of the contact conductance can be used in the simulation of temperature response of concrete-filled steel tubular columns under high temperature conditions such as fires. 3. The validity of the obtained correlation for contact conductance has been verified by direct heat conduction modelling using the heat flux output from INTEMP and the obtained correlation as an input into a well established general thermal modelling software. 4. The accuracy of the results can further be improved by better knowledge of the temperature measurement locations in the concrete. One approach being considered is to pre-install the thermocouples in the concrete during the preparation of the test samples. 5. There is a need to develop thermal properties of concrete aggregates prepared from local products in Australia. REFERENCES Busby, H.R., Trujillo, D.M (1985) Numerical Solution to a Two-Dimensional Inverse Heat Conduction Problem, Int. J. for Numerical methods in Engineering, Vol. 21,349-359. Lie, T.T., Chabot, M. (1990) Concrete Filling: Fire protection for steel columns, Canadian Consulting Engineer, May/June, p. 39-40. Lie, T.T. (1994) Fire Resistance of Circular Steel Columns Filled with Bar-Reinforced Concrete, Journal of Structural Engineering, Vol. 120, No. 5. Lie, T.T., Irwin, R.J. (1995) Fire Resistance of Rectangular Steel Columns Filled with BarReinforced Concrete, Journal of Structural Engineering, Vol. 121, No. 5. Trujillo, D.M., Busby, H.R. (1997) Practical Inverse Analysis in Engineering, CRC Press LLC, Boca Raton, Florida. Uy, B. (1996) Behaviour and design of thin-walled concrete-filled steel box columns, Australian Civil/Structural Engineering Transactions, Vol. CE39, No. 1, Sept. 1996, 31-38.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

647

M a t h e m a t i c a l m o d e l for the p r e d i c t i o n of t e m p e r a t u r e r e s p o n s e of steel c o l u m n s filled with c o n c r e t e and e x p o s e d to fires J.I. Ghojel Department of Mechanical Engineering, Monash University 900 Dandenong Road, Caulfield East, Vic 3145, Australia

The paper summarises the development of a mathematical model for heat transfer in fires. The distinguishing features of the model are the radiation heat flux and steel-concrete contact resistance components. The former improves the accuracy of the model by treating the combustion products from the fire as a participating medium which both emits and absorbs radiation energy. The latter uses an empirical correlation, based on experimental and inverse heat conduction analysis, to estimate the contact conductance at the interface between steel and concrete as a function of temperature. The predicted and measured temperature-time profiles in circular and rectangular steel columns filled with concrete correlate well. 1. INTRODUCTION Heat transfer models most often used in fire engineering practice to predict the temperature of composite structural elements exposed to fires in compartments adopt a radiation component which assumes that the fire is separated from the surrounding surfaces by perfectly transparent media. Also, the effect of thermal contact resistance at the interface between steel and concrete is ignored. As a result the predicted temperatures usually differ significantly from the measured values. The model described below attempts to improve the prediction accuracy by treating the combustion products as a non-grey participating medium which is both emitting and absorbing radiation energy and accounting for the thermal contact resistance as a function of temperature. 2. DEVELOPMENT OF THE M O D E L The model comprises the following modules: heat flux, resistance-capacitance formulation, moisture content and steel/concrete interfacial (contact) thermal resistance. 2.1 Heat flux module This module predicts the total radiation and convection flux from fires to the surfaces of the structural element. Detailed description of the model and its application to steel columns exposed to standard and real fires are given in Ghojel (1998) and Wong et al (1998) and it can be summarised as follows. The model treats the combustion products as participating media that emit and absorb radiation energy. It is based on the assumption of a single grey (partially absorbing and partially emitting) enclosure (emissivity or) filled with isothermal non-grey gas yielding the following radiant heat flux

648

q = Fgt~T2 - Fst~T4

(1)

The solution for this model accounting for one reflection from the enclosure yields the following transfer factors (Edwards and Matavosian,1984)

EsEgl 1-(1-Es)[(Eg 2 --Egl)/Egl] (2)

Es~gl

Fs=

Where eg is the total gas emissivity at temperature Ts over the mean beam length of the enclosure, and ag is the total gas absorptivity for radiation from a enclosure surface at temperature Ts absorbed over the mean beam length by a gas at temperature Tg. Total emissivity is a function of gas temperature and total absorptivity is a function of both the gas and wall temperatures. Subscript 1 denotes properties for the mean beam length of the enclosure and subscript 2 the properties for two mean beam lengths including the effect of one reflection. All radiative properties for the current model are calculated for the gas mixture (10% CO2, 10% H20, 80% N2) at a total pressure of 1 atm in an enclosure measuring 5.1mx2.5mxl.7m giving a mean radiation beam length of 1.69 m. The convective heat flux is given by

qc =hc(Tg -Ts)

(3)

The recommended values of the convection heat transfer coefficient hc for structural fire safety calculations range between 20 and 25 W/m 2 K (Barthelemy 1976, Smith and Stirland 1983). A value of 20 W/m 2 K was used in the current study.

2.2 Resistance-capacitance 0R-C) formulation Figure 1 shows an elemental representation in Cartesian and cylindrical co-ordinates of a multidimensional unsteady conduction without internal heat generation. The internal node i is surrounded by the finite control volume A Vi equal to riArAq)de.. The surrounding nodes denoted j are also surrounded by similar control volumes denoted AVj. The heat transferred by conduction from the surrounding volumes to volume AVi can be represented in terms of thermal resistances R 0 and temperature gradients Tj~ - T~~in time step x, i.e.

q= ~(T~-T~)/Ro. The---

thermal resistance is defined as the material thickness through

which heat is transferred divided by the product of the thermal conductivity and cross sectional area perpendicular to heat flow. The energy balance for volume AVi over the time interval At leads to the following explicit formulations (A/~~) T~TM = 1 - ~

T/r+l

= 1- ~

At~Tj-

T~~ + ~

J R0

+-~i " R~ +--~i qi AAi

for internal nodes

(4)

for surface nodes

(5)

Ci in these equations is the product of the mass of element and its specific heat (thermal

649

Figure 1. Control volume and resistances for R-C formulation in Cartesian (a) and cylindrical (b) co-ordinates capacitance), qi is the heat flux incident upon the elemental area z~i. Using specified initial temperature T~~ at time t=O, the new temperature T~1 is determined. The process is then repeated for progressively increasing time. Heat transfer in structural members exposed to fires is normally treated as a twodimensional problem as a result of which a unit length is assumed (Az=I) and the elemental volume AVi becomes equal to rizarAqg. Resistances Ri5 and Ri6 in Figure 1 then become equal to zero. A distinctive advantage of the R-C formulation is the versatility in handling complex problems such as composite and partially exposed structural members. Examples of the former include steel columns filled with concrete and steel columns with protective insulation layers. Examples of the latter include partially insulated members and imbedded members (comer columns). 2.3 Effect

of moisture

content

Water content in a typical concrete mix can be as high as 7% by mass. Analysis of published results of temperature measurement in both concrete columns and steel columns filled with concrete (Lie and Celikkol 1991, Lie and Irwin 1993, Lie 1994, Lie and Irwin 1995) indicate that water evaporation inside the concrete occurs at a temperatures close to 100 ~ In the present study the saturation temperature Tsa is taken equal tol00 ~ which corresponds to a saturation pressure of 1 atmosphere. The moisture can exist in three distinct states: - Liquid state for temperatures below 100 ~ The heat input during this state into an element will cause an increase in temperature of both the concrete and moisture. - Saturated state during which water will start boiling as soon as the temperature of the moisture reaches 100 ~ The boiling heat transfer coefficient is estimated using an empirical relation proposed by Jakob and Hawkins (1957)

h w = 7.9

(ATx)3 W/m 2 oC

(6)

where p, p l and ATx are the system pressure, standard atmospheric pressure and temperature difference between the surface and saturated liquid, respectively. Estimates of the free water content were made in this study on the basis of water-to-concrete ratio of 0.5 and a 100% hydration level. The free water concentration is assumed to increase linearly from a value of

650 zero at the centre of the column to a maximum value at the steel-concrete interface during the heating process. This is based on observation of unloaded test specimens heated in an electric furnace and on temperature-time histories of large columns heated under standard fire test conditions. In the former, moisture seemed to diffuse radially towards the surface forming a ring of moisture at the interface, which eventually evaporated. In the latter, the temperaturetime histories at locations close to the interface exhibit larger constant-temperature plateaus, which decrease towards the centre, indicating the evaporation of larger amounts of water toward the interface. - Superheated state for temperatures above 100 ~ The moisture in this state is completely in vapour form at a pressure of 1 atmosphere with the specific heat of the superheated steam being calculated using a correlation by Rivkin (1988) 2.4 Contact resistance

There is always resistance to heat conduction across solid-solid interfaces caused by the relatively few points of contact between the surfaces (surfaces can have some degree of roughness) and the presence of entrapped gases in the void spaces present at the interface. The second factor is normally dominant because the thermal conductivity of the gas is normally lower than that of the solids. In steel-concrete composite elements under high temperature conditions the void spaces will increase as a result of differing thermal properties and both air and water vapour can be present at the interface. The dominance of the gaseous interface can be gauged by considering the fact that the thermal conductivities of concrete and steel, for example, are respectively 16 and 550 times greater than that of steam. Since little is known about the exact geometry of the contact area and the presence or otherwise of gases in the void spaces between the steel casing and the concrete core at the interface, the problem is best treated as an Inverse Heat Conduction Problem (IHCP). Inverse modeling estimates the surface temperature and heat flux fields from the temperature-time histories at any number of locations inside the solid. These boundary conditions can be obtained by coupling the temperature measurements with a heat transfer numerical model and minimization of differences between the calculated and measured temperatures at given locations and times. Contact resistance can then be estimated from the temperature gradient across the interface and the flux densities determined by the IHCP analysis. To this end, composite steel-concrete samples were prepared and heated in a modified electric furnace to temperatures up to 900 ~ Both ends of each specimen were insulated to simulate a 2-D heat transfer problem as closely as possible and temperature-time histories obtained at ten locations in a cross-section of the specimen. INTEMP (TRUCOMP CO, Fountain Valley, California, USA), which solves linear or nonlinear inverse heat conduction problems, was used to solves for the unknown flux-time histories at the specified locations. Figure 2 shows the average estimated contact conductance hc as a function of temperature t (~ for both a circular and square steel columns over the test temperature range 30-600 ~ Curve fitting yields the following two expressions: c

h c = a + bt + - i T

W/m 2 oc

(circular column)

(7)

a = 403.00105, b = -0.25589772, c = 880801.49 hc = at b

W/m 2 ~

a =1377.4142, b = -0.34447549

(square column)

(8)

651 The thermal contact resistance at the interface is written as R c = 1/hcAA i KlW.

1500

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.~ |0 C

round ,~ 1000

~

'~

500

C 0

0

0

......

,...............

0

,.................... , ...............

200

400

600

800

temperature (t), ~ Figure 2. Thermal contact conductance as functions of steel surface temperature for circular and square columns 3. RESULTS To test the capability of the model the predicted temperature-time histories were compared with measured values for a circular (Lie, 1994) and square (Lie and Irwin, 1995) steel columns filled with bar-reinforced concrete and exposed to standard fire. Figure 3 compares predicted and measured temperature response of a circular column at three different locations: column surface, at a depth of 65 mm and at the column core. Figure 4 shows comparison results for a square column at two locations: column surface and a depth of 71 mm. There is generally good agreement between measured and predicted results in both cases. Discrepancies are evident for the temperature profile for the column surface. This could be attributed to the radiation exchange between the heating elements and furnace walls and the 1200 oo 1000 .9. = tl)

S

u

r

f

a

~

=

,

800

65 mm depth

600

x.

400 E (1) i.-- 200 !

0

20

"1

40

60

'

!

80

!

........

!

'

....

1

100 120 140 160 180

Time, min

thermocouple. Figure 3. Predicted (solid lines) and measured (markers) temperatures in circular steel column filled with concrete (373 mm outside diameter, 6.35 mm steel thickness)

652

1200 o0 1000 ~_6 800 (~

600

Q- 400 E 200 0

surf -

e

a

~

9

-

0

i

i

20

40

......

l

60

....

i

80

'

'

1'

i

i

100 120 140 160

Time, minute Figure 4. Predicted (solid lines) and measured (markers) temperatures in square steel column filled with concrete (203 x 203 mm, steel thickness 6.35 mm) CONCLUSIONS 9 The proposed model is suitable for the calculation of temperature-time histories in composite concrete columns exposed to fires; 9 The obtained results compare well with the published experimental data; 9 The moisture in concrete has stronger effect on the temperature profile at locations close to the surface; 9 The accuracy of the model can be further improved by accounting for the mass transfer of moisture in concrete. REFERENCES Barthelemy, B.(1976) Heating calculations of Structural Steel Members, J. St. Div., ASCE, 102(8), 1549-1558. Edwards, D.K., Matavosian, R (1984) "Scaling rules for total absorptivity and emissivity of gases", Transactions of ASME, Journal of Heat Transfer, Vol. 106, pp. 684-689, 1984. Ghojel, J.I. (1998) A New Approach to Modelling Heat Transfer in Compartment Fires, Fire Safety Journal, 31,227-237. Jacob, M., Hawkins, G. (1957) Elements of Heat Transfer, 3'd ed., John Wiley & Sons, New York. Lie, T.T., Celikkol, B. (1991) Method to Calculate the Fire Resistance of Reinforced Concrete Columns, ACI Materials Journal, V 88, No. 1, January-February. Lie, T.T., Irwin, R.J. (1993) Method to Calculate the Fire Resistance of Reinforced Concrete Columns with Rectangular Cross Section, ACI Structural Journal, V. 90, No. 1, January-February. Lie, T.T. (1994) Fire Resistance of Circular Steel Columns Filled with Bar-Reinforced Concrete, Journal of Structural Engineering, Vol. 120, No. 5. Rivkin, S. L (1988) Thermodynamic Properties of Gases, Fourth Edition, Hemisphere Publishing Corporation, New York (text translatexl from the Russian by J.I. Ghojel) Smith, C.I., Stirland, C. (1983) Analytical Methods and the Design of Steel Framed Buildings, International Seminar on Three Decades of Structural Fire safety, Fire Research Station, Herts, UK, pp. 155-200. Wong, M.B., Ghojel, J.I., Crozier, D.A. (1998) Temperature-time analysis for steel structures under fire conditions, Int. J of Structural Engineering and Mechanics, Volume 6, Number 3, April, 275-289.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

653

Non-elastic load capacity of compressed steel truss member during fire G.Ginda a and W.Skowrofiski a aFaculty of Civil Engineering, Technical University ofOpole, ul.Katowicka 48, 45-061 Opole, Poland

Results of numerical analysis of steel truss member subjected to elevated temperature caused by fire are discussed in the paper. Steel strain is described using non-linear BaileyNorton formula. Conditions of heating of steel and creep steel behaviour are included when estimating values of steel model parameters. Influence of buckling is taken into account. Balance of the member is described by modified equations ofVlasov's theory of thin-walled members. Numerical analysis is based on the finite difference method. Finally discussion on influence of various factors including slenderness ratio, load ratio and intensity of heating of steel on load capacity of the member is presented. Obtained results make it also possible to assess effect of steel creep on load capacity of analysed members and its interaction with other parameters.

1. INTRODUCTION Rheological phenomena appearing due to steel creep can cause large quantitative and qualitative changes of steel strain at elevated temperature. So non-linear relation between stress and strain is required to include activation of theological phenomena appearing in heated steel. Because of fire safety reasons it is necessary to assess influence of these phenomena on process of structural collapse. So it is intended to present results of analysis of bearing capacity of steel truss member subjected to elevated temperature caused by fire and to assess influence of rheological phenomena and its dependence on geometrical and load parameters of the member, conditions (rate) of steel heating and level of steel temperature.

2. TENTATIVE ASSUMPTIONS 2.1. Heating of steel Temperature of steel is function of time t (1) that depends on constant value of steel heating rate T and initial value of steel temperature To: T(t) = 7'.t + To The member is equally heated along its length.

(1)

654 2.2. Load and geometry of the member

The member of initial length of L0 is initially compressed by force of constant value No. Imperfection of straight shape of member's longitudinal axis is included considering eccentrics of compressive force. Conditions of member's support are described using coefficients of compliance: Cr (direction along member's longitudinal axis), Cr C , Co (adequate directions of generalised displacements of member's cross-sections) (Fig. 1). b C C,q,O /

C {,rl,O p

o

/

]"!

No

i

]~oj l

,~

';I'

Figure 1. Scheme of the member 2.3. Strain of steel

It is assumed that basic parameters of steel depend on steel temperature T and conform to isotropic nature of material. Influence of only normal components of stress (~) and strain (6) on collapse of the member is taken into account. Current value of strain 6 is described through the sum of thermal expansion and mechanical (elastic-non-elastic) components (2):

The mechanical component is given by formula (3) [1] (A, n are functions of parameters)"

,~_,., = A(r,~,).,,.I,,t

"(~'')-'

2.4. Description of member's equilibrium

It is assumed that equilibrium of the member at time t is described by modified balance equations (4) of Vlasov's theory of thin-walled members [2]):

[~,(~,,,,,,~,).~,],,-[~.(~+,c.~)],+(M..o),,+s ~ =o [~7.(~,,,,~,~). ~,i,,-[N.(~,-Xc.~)].+(~,.o)..+s d~-o E-7~,(T,A,n,o')-0"1"-(G-7,.0')'-[N .yc ~"]'+[N 9 .xc r/'l'+M:, 9 .~"+My. r/"-

+ o.[q,.(,o-xc)_ q,-(e=-,~)] + s

~ -0

(4)

655 w

~

m

where E l , G I values of cross-sectional stiffness, evaluated applying integration over member's cross-section F according to steel model (3), M, s - functions resulting from including steel model non-linearity. Because of interdependence between displacements of cross-sections and cross-sectional parameters, application of step-wise method for resolving of set (4) is required. -

3. NUMERICAL ANALYSIS 3.1. Components of numerical model Application of the finite difference method makes it possible to transform set (4) into equivalent set of linear algebraic equations. Influence of restriction of member's elongation caused by thermal expansion on changes of member's length and value of compressing force (AN) is given using formulas (5,6) where Cr = Cr ~+ Cr p (Fig.l):

(5) L(T)= Lo.(I+C,)-eT

(6)

[In]

Influence of support conditions changes on member's behaviour is taken into account thanks to relations (7), where g denote values of adequate curvatures:

Iml ,

o'=co.eo

Iml

(7)

Criteria of member's collapse are based on limit value of mechanical strain component of steel Elim.(Usually equal to 1%). General form of algorithm estimating value of critical temperature Tcr and verification of proposed numerical model is contained in [3]. 3.2. Results of numerical analysis Numerical analysis deals with members made from typical American structural steel ASTM A36. Functions of temperature describing values of basic steel parameters: ratio of thermal expansion (XT,Young modulus E, limit of plasticity cry, coefficient of steel strengthening/l and theological parameters (limited to range of temperature from 20 to 600 ~ are presented in [4]. Poisson's ratio is considered constant (v=0,30). The load case only covers eccentrically put compressive force. Considered values of force eccentrics are equal to e~~'p =e/'p =0,001m. The lack of transversal force and considered

unlimited warping of member's terminal cross-sections (IC0t'Pl= oo) cause that bimoment B vanishes when considering cross-sectional forces. Two different values of heating rate of steel are considered: 4 ~ (structure efficiently protected against quick temperature rise) and 20 ~ (structure protected less efficiently)

656 600 -

9

i

i

i

550 500

e~L=0,876

~,

450

~--2,921 "~'

350

'~ ~

300

~=-1,287

250 200 . . . . . . 0 O,10

".... 0,30

0,20

' 0,40

'

' 0,50

-0,60

/k 0 , -

Figure 2. Influence of slenderness ratio and initial load ratio on critical temperature 100 90

'

-

1

'

|

"

i

.....

i

1

80

70 6O so

<~ 40 30 20

§

10 0 L..---

0,125

0,15

0,175

0,20 A 0, -

0,225

0,25

0,275

Figure 3. Results of ignoring of steel creep geometrical parameters and load of the member e.g. decreases with increasing of slenderness and decreasing of load. Difference of critical temperature resulting from ignoring of steel creep reaches tens of ~ Above difference testifies to necessity of taking into account

657 when including influence of time on relation between stress and mechanical part of strain. Influence of thermal expansion of steel on changes of compressing force value changes (AN) is not included in numerical analysis thus Cr = 1. Analysed members include typical polish steel U-iron (C65E), I-bar (I80), T-bar (1/21 120) and angle section (L60x60x6) of initial length that range from 1,Sm. to 2,25m. Various conditions of member's support are considered. Most representative results for conducted analysis are obtained for three members: slender one - I80 - pinned on both ends, regarding directions of displacements ~, r/(L0=2,25m), typical one - Lt0x60x6 - pinned on one end and fixed on the other end (L0=l,875m.) and stocky one - C65E - fixed at both ends (L0= 1,Sm). Initial value of compressive force No ranges from 10 to 140 kN. Influence of following parameters is included in analysis: 1. slenderness ratio A, 2. load ratio A0 at initial temperature (8):

No

[_1

(8)

3. steel heating rate 7". Comparative computations without including influence of time on value of mechanical part of steel strain are also conducted. Influence of slenderness ratio and load ratio on critical temperature (for 7" = 4 C/min) is illustrated in Figure 2. Difference in critical temperature ATkz' (9)

ATkx'-'[Tkr]E-P--[TIx]E'p't

loci

(9)

where [Tkr]E'p''t denotes critical temperature evaluated without including steel creep, and

[Tkr]E'p - value obtained including steel creep at elevated temperature, that results from ignoring influence of steel creep on steel strain for slender member is presented in Figure 3. Figure 4 deals with members that have reserve of load capacity (considering load ratio A0=0,375) and shows influence of steel heating ratio on difference in critical temperature ATkr (caused by change steel heating ratio value from 20 to 4 ~ relative to slenderness ratio.

4. CONCLUSIONS Solution of load capacity problem of steel truss member subjected to elevated temperature requires application of relations expressing unfavourable changes appearing due to heating of steel structure by fire. In case of compressed members the relations should include ratio of steel heating, geometrical and load parameters of the member. Obtained results show that increase of slenderness ratio and initial value of load ratio leads to decreasing of load capacity of themember. Influence of steel heating ratio and related activation of rheological phenomena is important at steel temperature exceeding 350~ Degree of this influence

658 70

|

'

!

.....

!

"

!

i

u

|

!

=

|

60

50 o

o 40 30 20 10 0

0,8

1,0

1,2

1,4

1,6

1,8 2,0 2,2

2,4

2,6

2,8

3,0

Figure 4. Influence of slenderness ratio on effects related to steel creep depends on additional steel weakening appearing due to activation of rheological phenomena in case of design of compressed steel truss members collapsing at temperature exceeding 350 ~ To avoid complicated calculations based on non-linear theory, new concept of design ratio formulation should be considered, that would allow correction of design calculations of structure subjected to fire considered as exceptional load case. Presented results reveal differences in effects of rheological phenomena that result from differences in member's slenderness, steel heating ratio and load ratio, thus showing possibility of such concept.

REFERENCES 1. P. Fedczuk, W. Skowrohski The Analysis of Norton Model Parameters For Structural Steel Under Heating, Proceedings of the First International Conference "Fire Safety of Building', Spata (Poland), November 1995, ITB Warsaw, 1995, pp.217-225 (in Polish) 2. W.Z. Vlasov, Thin-Walled Elastic Members, Izd. Gos. Mat.-Fiz. Lit., Moskva 1959 (in Russian) 3. G.Ginda, W.Skowrotski Elasto-plastic Creep Behaviour and Load Capacity of Steel Columns During Fire, Journal of Constructional Steel Research, Vol.46,No.l-3, 1998, pp.206-216 4. W.Skowrotski, Buckling Fire Endurance of Steel Columns, Journal of Structural Engineering ASCE, vol.119, No.6, 1993, pp.1712-1732

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Seience Ltd. All rights reserved.

659

Fire r e s i s t a n c e o f c o n c r e t e filled steel tubular b e a m - c o l u m n s in C h i n a - s t a t e o f the art Lin-Hai Han" and Xiao-Ling Zhao b ~'Harbin University of Civil Engineering and Architecture, Haihe Road 202, P.O.BOX 689, Harbin, 150090, P.R. China bDepartment of Civil Engineering, Monash University, Clayton, VIC 3168, Australia

Abstract: Finite element method is applied for the calculations of temperature fields of concrete filled steel tubes under fire condition. A theoretical model that calculates deformations and strength of beam-columns at elevated temperatures is described in this paper. The predicted results show good agreement with test results. Based on the theoretical model, influence of the change in material strength, sectional dimensions, steel ratio, load eccentricity and slenderness ratio on the fire resistance is discussed. 1. I N T R O D U C T I O N Concrete filled steel tubular beam-columns have been used extensively in China as well as in other countries. They have been proved to be economical leading to rapid construction and thus additional cost savings. An important criterion for the design of concrete filled steel tubes, besides the serviceability and critical load bearing capacity, is fire resistance. Research to determine the fire resistance of concrete filled steel tubular column has been carried out in several countries L~H81. In recent years, Harbin University of Civil Engineering and Architectures (HUCEA) has been engaged in research to calculate the fire resistance of concrete filled steel tubular columns with the support of the Chinese Natural Science Foundation. Both theoretical and experimental studies were carried out. Most of the work has been carried out on columns with both circular and square cross-sections, and the columns have been subjected to concentric or eccentric compression loads. The composite action between the steel and concrete has been considered, which was often neglected by other researchers in the theoretical analysis. A theoretical model that calculates the strength and the fire resistance of the columns is described in this paper. The influence of the changing parameters of the beam-columns on the fire resistance is analyzed. 2. S T R E N G T H OF COLUMNS DURING FIRE E X P O S U R E 2.1 Division of Cross-Sections The cross sectional area of a concrete filled steel tubular column is divided into a number of elements as show in Figure 1 in order to calculate the temperature, deformation, stresses and strength of the column.

660

Y-AXIS

l

Y-AXIS

Concreteelement / ~Steelelement I'/ I i,

!" Ii' ,

II

II II

, ,

1 (

i

i

i

,

,

,

|-

1

X-AXIS

II 1

1,"

0

Ii "

"

X-AXIS

(b) square sections (a) circular sections Figure 1 Arrangement of elements

2.2 Temperature of Columns during Fire The column temperatures are calculated by using a finite element program f91. The method for deriving the heat transfer equations and calculating temperatures, with the thermal properties is described in detail in Reference [9]. The temperature at the center of an element is defined as the temperature of that element in Figure I. The temperature in building structures under fire increases as the time increases. The commonly used temperature versus time curve is given by ISO-834: Fire resistance t e s t s - Elements of building construction. 2.3 Stress-Strain Relations of the Concrete Based on the test results of concrete filled steel tubular stub columns under constant high temperature, the relations between longitudinal stress oc and longitudinal strain E of the core concrete has been derived tt~ i.e. E

E

Oc = o o - [ A - - - -

l-(--)-]

Eo

for e_< Eo

(la)

for e > eoand ~ > ~ o

(lb)

for e > eoand ~, < ~,o

(Ic)

Eo

o~. = O o . ( I - q ) + O o . q . ( ~ e

)~ Eo

E

Oo " ( ~ ) Eo

Gc "-

E

'~ . ( ~g)

E,,

Eo

p.(---l)

Composite action between the steel and the concrete can be taken into account f~~ by the fy "As ratio ~ = ~ where fy and As are the yield stress and cross-section of steel tubes, fck and fck "Ac Ac are the compression strength and cross-section area of filled concrete. Details for determining the parameters Oo, 4 , [3, ~, q and ~ can be found in Reference [10].

2.4 Stress-Strain Relations of the Steel Figure 2 shows typical stress-strain curves of steel tubes (fy = 345 MPa at temperature of 20 degrees) at high temperatures. Details of the model were described in Reference [3].

661

500 I

400

it

' T=20 c I . . . . . o

'

1

3ooO t

!

T= 2oo~c! 300

i

-

200 IOO 0 0

i

I

10000

20 000 30 000 40 000 50 000 e (la~) Figure 2 Stress-strain curves of steel tubes 3. C A L C U L A T I O N O F S T R E N G T H D U R I N G F I R E The calculation of the fire resistance of a column involves the calculation of the fire temperatures, to" which the column is exposed, the temperature in the column and its deformation and strength during the exposure to fire. The strength of the beam-column decreases with the duration of exposure to fire. A numerical model was developed to analyze the ultimate load. The model considers both material and geometric non-linearity. In this method, for the calculation of column strength, the following assumptions were made: (1) The cross sections remain plane in the fire exposure, (2) The shape of the deformed member is regarded as semi-sine curve, (3) Concrete has no tensile strength and (4) The stress-stain curves of the steel and concrete under high temperature can be determined using equation (1) and Figure 2 respectively. Based on assumption ( 1), the strain of the steel elements can be expressed as: (~, = ~. x,, + ~:,, - ~:.,T and that for the concrete element is given by: ~ = q). x ~ + (~,, - E~T

(2) (.3)

where e,, is the axial strain of the column; Esv and ~cv are strains of the steel and concrete due to thermal expansion [~~ xsi and x~i are horizontal distances from the center of the steel element and concrete element to a vertical plane through X-AXIS of the column section respectively. ~ is the curvature at mid-height of the column, which can be derived from assumption(2) as: 71;2

, = (--~-.). u,,,

~4)

where L is the column length and Um is the deflection at mid-height of the column. Axial force within the column section at mid-height is: l!

Ni,, = 2 Z ( ~ i=l

"dAsi +oci "dAci)

(5)

662 where o~. and Gci are longitudinal stresses of the steel and concrete elements respectively. Internal bending moment of the column section at mid-height is: I|

M,,, = 2.~__,(o~,- x~i-dA~ i +o~.u" xr "dAci)

(6)

I=1

For any given curvature ~, and thus for any given deflection Um at mid-height of the column, the axial strain Co is varied until the internal moment at the midsection is in equilibrium with applied moment, i.e." M~, = N , , - ( e o + Urn) (7) where, e,, is the initial load eccentricity. Based on the method introduced above, the strength of beam-column during exposure to fire was calculated. Figure 3 and Figure 4 show the calculated N-urn and N-M relations respectively, where D is the diameter of the circular section and B is the overall width of the square section, t is the thickness of the steel tube, 3'~.k=0.67 .fc,,, .1~-. is the cube strength of the concrete.

T~

12 O(X)

=I

()m~"

~

time = 0 rnin.

t

12(x~ ~ _ . - - - - - - ~ - - - - - - - T

(xx)~ 4

zs~

(XX)

4 (XXI

i

() ()

t"

2(1

i

I,

I.

40

60

0

80

0

J' 4(')

20

~ 60

tqO

ttm(mm)

ttm(mm)

(b) Square section

(a) Circular section [ D x t x L = 400 x 12 x 4000mm;~ \ [ y = 345 MPa;fck= 41 MPa //

['B x t x L = 400 x 12 x 4 0 0 0 m m ~ k,.ty= 345

MPa;[ck=41 MPa

J

Figure 3 Calculated N-um relations 12 ( I ) 0

0~176 U----T

T

,i,,;~ = o,,~in.

I~

i (XX)

0110

"

;

!

!

!

i

4 (XX)

'i ........\ 0 ~ 0

i

n

9

.

21X)

4(')0

600

800

I000

0

2(')0

41111

(a) Circular section [ D x t x L = 400 x 12 x 4000mm;~ lx f y = 345 MPa;fc k= 41 MPa / |

6011

81111

!(X)O

2011

141111

M (kN.m)

M (kN.m) |

(b) Square section /' B x t x L = 400 x i 2 x 4000mm;'~ \ .l'y = 345 MPa;./ck = 4 ! MPa ]

Figure 4 Calculated N-M relations

663 The failure criterion for the column proposed in ISO-834 standard has been adopted. A comparison of the calculated results with the test results is shown in Figure 5. 50

9

I

'

,

2oV 7

,0-do;,

o

,o

Test (min.)

'

'

l)

/

,,

/ c

;o ..... .,'o..........

.,o

",Y-

,.,, Test (rain.)

Test (rain.)

(a) (b) (c) Figure 5 Comparison between calculated and experimental fire resistance: (a) columns with circular sections, (b) beam-columns with circular sections, (c) column with square sections By the use of the analytical method, influence of the change in section dimensions (D or B). steel ratio (or = AdAm), slenderness ratio (X = 4L/D for circular columns; X = 2 if-3 L/B for square columns), load eccentricity ratio (= e/ro where ro=D/2 or B/2), strength of the concrete 0%k) and the steel 0'y) on the fire resistance (&)were analyzed [~~ in the calculations ,the applied load of the beam-column was taken as the design load given in Code DL/T5085-1999 for circular sections and GJB-2000 for square sections in China. It was found that only the dimensions and the slenderness ratios have significantly influence on the fire resistance of the columns under the load condition, as shown in Figure 6 and Figure 7. I()C) 80

(~ = O. 12; e/ro= 0" I y= 345 MPa:I ok= 41 MPa

c t = O. 12; e/ro= O: I y= 345 MPa: tck= 41 MPa

-

-. 4 0 .~

60

"r

40-

2(1 -

20 C

I 400

0

, 1.... 800

I !200

I 1600

I 400

21)00

I 800

1 1200

I 161)11 2000

B (ram)

D (ram)

(b) square sections (a) circular sections Figure 6 Influence of the dimensions on the fire resistance 40 B = 4C~)mm: cz = 0.12; e h i ) = 0:

D = 4 0 0 r a m ; cz = 0.12; ebb = 0; 30 _

30 -

/'y= 345 M P a ; f c k = 4 1 MPa

ty = 345 MPa: f c k = 41 MPa

._= E

~ 2o 10

,._.,

20

i

0 0

I

I

30

60

I....

90 A

I

! 20

0

150

0

I

I

I

I

30

6()

90

120

A

(a) circular sections (b) square sections Figure 7 Influence of the slenderness ratio on the fire resistance

150

664

4. CONCLUSIONS 1. 2.

3.

The analytical method introduced in this paper is capable of predicting the fire resistance of concrete filled steel tubular beam-columns. There is a composite action between the steel and concrete under fire. The composite action has been considered in the model for calculating the fire resistance of concrete filled steel tubular beam-columns. Section dimensions and slenderness ratios have significant influence on fire resistance of concrete filled steel tubular columns.

REFERENCES

1.

.

.

o

10. II.

12. 13.

Klingsch, W. (1985), New Developments in Fire Resistance of Hollow Section Structures, Symposium on Hollow Structural Sections in Building Construction, ASCE, Chicago Illinois Falke,J. (1992), Comparison of Simple Calculation Methods for the Fire Design of Composite Columns and Beams, Proc. of an Engineering Foundation Co~!['er on SteelConcrete Composite Structure, ASCE, New York, pp.226-241 Lie,T.T. (1994), Fire Resistance of Circular Steel Columns Filled with Bar-Reinforced Concrete, J. of Structural Engineering, ASCE, Vol. 120, pp. 1489-1509 Lie,T.T. and Chabot,M. (1993), Evaluation of the Fire Resistance of Compression Members Using Mathematical Models, Fire Safety Journal ,20, pp. 135-149 O'Meagher, A.J., Bennetts,I.D., Hutchinson, G.L. and Stevens, L.K. (1991), Modeling of Concrete -Filled Tubular Columns in Fire, BHPR/ENGIR/91/031/PS69, Melbourne British Steel, (1990), SHS Design Manual for Concrete Filled Columns, Part 2, Fire Resistant Design, London British Steel Tubes and Pipes (1990), Design for SHS Fire Resistance to BS5950:Part 8, London, 1990. ECCS-Technicai Committee 3 (1988), Fire Safety of Steel Structures, Technical Note, Calculation of the Fire Resistance of Centrally Loaded Composite Steel-Concrete Columns Exposed to the Standard Fire. Xu, L. and Han ,L.H. (1999), Nonlinear Finite Element Analysis on the Temperature Field of Concrete Filled Steel Tubes, J. of Harbin Universit), of Civil Engineering and Architecture, Vol.32, No.5, pp.34-38. Han, L.H. (2000), Concrete Filled Steel Tubes-From Theory to Practice, Science Press, Beijing (in Chinese). Tao, Z., Han, L. H. and Zhao, X.L. (1998), Behavior of Square Concrete Filled Steel Tubes Subjected to Axial Compression", Proc. 5th Int. Symposium on Structural Engineering for Young Experts, ShengYang, P. R. China, pp. 61-67 Lie, T.T. and Chabot, M. (1992), Experimental Studies on the Fire Resistance of Hollow Steel Columns Filled with Plain Concrete, NRC-CNRC Internal Report, No.611 Hass, R. (1991), On Realistic Testing of the Fire Protection Technology of Steel and Cement Supports. Translation ofBHPR/NIJT/1444, Melbourne

Earthquake Loading

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

667

Expe~mental study on steel bridge piers with inner cruciform plates subjected to cyclic lateral loads K. Iwatsuboa, T. Yamaob, T. Yamamuro ~ and M. Ogushi c a Department of Civil and Engineering, Yatsushiro National College of Technology, Hirayamashinmachi 2627, Yatsushiro City, Kumamoto Ken, 866-8501, Japan b Department of Civil Engineering and Architectural, Kumamoto University, Kurokami 2"39" 1, Kumamoto, 860-8555, Japan c Shimizu Construction Co. Ltd., Sibaura 1-2-3, minato-ku, Tokyo, 105-07, Japan The strength and ductiliW of steel bridge piers with inner cruciform plates were studied experimentally. A total of 4 specimens with inner cruciform plates and compact sized sections were tested under constant compressive axial and cyclic horizontal loadings. The effects of inner cruciform plates on the ductility and strength of bridge piers were investigated. It was shown that piers with inner cruciform plates had increased ductility and energy absorption capacity.

1. INTRODUCTION The Hyogo-ken Nanbu Earthquake caused severe damage in a great number of buildings, highway bridges and railway facilities. Damage of bridge piers resulting in a local buckling of steel plate was observed in some bridge piers of the highway and railway viaducts. Following the earthquake, investigation of plastic ductility of steel bridge piers has been carried out experimentally and theoretically by a number of research organizations. It has become evident that it is necessary to develop steel bridge piers possessing higher ductility at a lower cost 1)2). This paper presents the experimental results of tests on steel bridge piers with inner cruciform plates. For the investigation of the effect of inner cruciform plates on the ductility and the strength of piers, test specimens designed according to the current design practice in Japan with unstiffened and stiffened columns and columns with inner cruciform plates were tested. The results are discussed in the light of improvement of ductility and energy absorption of bridge piers

668 2. OUTLINE OF E X P E R I M E N T The dimensions of box-section test specimens are shown in Table 1 and Figure 1 respectively. Type N is an unstiffened box-section column and Type S is a stiffened one. Types C-20 and C-40 are box-section columns with inner cruciform plates where the heights of inner plates are 20% and 40% of the height of the specimen (h, see Figure 2) . All test specimens were composed of SM490Y steel plate of 9 mm thickness. Here Rf is the slenderness parameter of plate panels between longitudinal stiffeners, and is defined as, b /% i2(1-v ~) Rf = t ~ - ~ - 4 n 2 n

(1)

2

where E is Young's modulus, ay is yield stress, n is a panel number and v is Poisson's ratio, y/~* in Table 1 is the rigidity of longitudinal stiffeners compared to Ver6~ a plate panel. Hmmntal

-'•'L_

tT

I

b V Type N Type S Type C Figure 1 Cross-sections of specimens Figure 2 ~est specimen Table 1 Dimensions of test s)ecimens b t h Height of Specimen (cm) (cm) (cm) cruciform plate Type N 19.85 Type S 19.93 .-'"'~'-'---~. 0.87 94.0 Type C-40 19.85 40% T y p e C-20 20.00 20% ,=

bs (cm)

t~ (cm)

Rf

y/y*

o.51

5.2

0.87

"<-._

0.87 0.87

0.25

3.36

Vmm2) 50(

Table 2 Material properties Thickness t 8.7 mm E 205.93 kN/mm 2 ay 408.91 N/mm2 e~ 0.002273 v 0.29

40( 30( 20( lO( C

' 10~00.... 20600' 3 0 ~ 0 0 ' 4 0 ~ ) 0 strata (tl)

Figure 3 S t r e s s - strain curve

669

8

Load

35~'- ............... ~ 4~y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28y

-A

I

......

18:~ ~

/~

"28,~.......... ....... JlJr

/

Cycle

~

Vk /

k

-

V

Figure 4 Loading steps

Straingage ~ Dial gage /

~i "

FlangePlate

F

,

+ ,

v

gePlate

--,,,.,i , .,.. T,o:,,, ,,...I. .~ ,

TM

RWebPlate

l

5cm !

' ~ ~

I

",,~!'

......

+".., '"'......-"

Figure 5 Location of strain gages and dial gages Figure 2 shows a test specimen with inner cruciform plates. The mechanical properties of material obtained from tensile coupon tests are shown in Table 2. Figure 3 is a stress-strain curve of SM490Y steel. The axial compressive load and the cyclic lateral load were applied by a hydraulic jack as shown in Figure 2. The magnitude of the constant axial load was 0.15 x ~y x A (the cross-sectional area of a specimen). The lateral load was controlled by the tip displacement of the specimen and the cyclic loading was given with the loading step as shown in Figure 4. At each cycle, the tip displacement increases the magnitude of 5y defined by.

H+ My = h = 3EI (2) where My is yield moment and I is moment of inertia of a cross-section. Figure 5 shows the location of strain gages and dial gages. Strain gages were attached to the flange panel surfaces of the specimen.

3. RESULTS AND DISCUSSIONS 3.1 Hysteretic behavior Horizontal load versus horizontal displacement hysteretic curves for all column types are shown in Figure 6. Figure 7 shows envelope curves of horizontal l o a d horizontal displacement hysteretic curves. From these figures, it was found that the behavior and the strength degradation of test specimen Type C-20 is similar to those of stiffened specimen Type S. In the case of Type C-20, however, the load suddenly dropped due to local buckling deformations in unstiffened panels where there are cruciform no plates. Though specimen Type C-40 shows that the behavior is very stable and only slight local buckling deformations were observed in the column base panels, the strength degradation was not improved because only local buckling occurred in the panels of junctioned with of cruciform plates due to insufficient welding. The hysterisis loop area, namely the energy absorption capacity, of each specimen at every load steps is shown in Fig~are 8. It can be found that all

670 specimens have almost the same energy absorption capacity until 5=50mm and that it is increased by raising the ductility. It can be recognized from these results that the column with inner cruciform plates and height larger than 0.2 h(h: height of specimen) is effective for improving the ductility and energy absorption capacity. ! '

~,0"5

- " -"" ~ - "

,,.!, " - " """ ~ -

-"-4-~-'-~-~-,-lr~- -~-i

l"

-

~~-~-!-,-~-

,

cloln~

, :

:

cracx,

, ,~,,

,

r

:

~ o.o,-!-',-!-~?//.,~,//~-~~

~ -i -!- h .i,//l~, ,, !t/lL ~- i --i -

.o.<

"

' [wo~m~,,r~kl-:--:-:--:-:-:--:--:-~

o., :-!-!-!-i!~'Yl~ ~ i-

~o.~ ,

!~~/It/, ~/tl~ ~//;

.o.,. i,,,llt-,f,i,f,lt ~ll/i,~---~ .,.<-~~~~~: ~-::-~ ~-

)61~y

-

'

(a) Type N 9

,

,

,

,

,

,

,LVVC4omgcraczcl:

-~- - r - , - ~ - r - - ~ - r - ~ - ~

o.,:

,-1

r ,_

. . . . .

'

,'

1'8/6y

'

,

~o.~ '-~li.'l-,lll~ 'l, lll~~ ~ I,, .o.,~ .//,:/,,/-/,,,ttf~,~tl1~f ., -2 ,

. . . . . . . .

(b) Type S

-i-i-,>,~~-',/~N ~-

_,7/,I_H,'

'

~::.,_

l.C

~-ii~~! '~gll"l.,~ '- ~ - i ~ ~ ~ ~ ~ 0.0 _//,,./.~'~~, :. -0.5 '~t'/-~,.~.i!.-i-!-!-i.' -1.C 0.5

(c) Type C-20 (d) Type C-40 Figure 6 Horizontal load- horizontal displacement hysteretic curve Energy absorptioncapacity(kN- ram)

HorimntalLoad H

46

200t ::

!

-~ o -

"I . . . . .

; l i - z ~ TypeN Type S

Type C-20 30- ---IF-TypeC-40

,, ; : .....

/o

,

~r ~

i

,ooI i

0

40 60 Displacement 6 (ram) Figure 7 Envelope of horizontal loadhorizontal displacement curve

20

20

Figure 8

.

40 60 Displacement5 (ram) Energy absorbing capacity

671 3.2 Collapse modes Figure 9 shows collapse modes of four specimens. In the unstiffened and stiffened specimens local plate buckling was first observed in the plate close to the column base immediately after the peak horizontal load. During the cyclic loading, eventually the specimen lost its lateral resistance after either vertical cracking in welds of flange-web junctions or horizontal cracking m welds between the web and the base plate. In contrast, in test specimens with inner cruciform plates slight local deformations were observed in the column base panels and then in unstiffened panels where the cruciform plates are absent. The latter buckling deformations progressively grew and the lateral resistance was lost.

(1) Type N During the loading cycle with 5=+35y, yield lines could be seen on the plate close to the column base, and during the loading cycle with 5=+55y, local buckling deformations occurred in the same place. (see Figure 9(a)). When the load was increased to 5=+85y, the vertical cracking in welds of flange-web junctions occurred and the strength suddenly dropped. (2) Type S Specimen Type S showed the same behavior as specimen Type N. During the loading cycle with 5=-75y it sounded like a fracture of welds in the stiffened plate. When the load was increased to 5--+115y, horizontal cracking occurred in welds between the web and the base plate, and the lateral resistance was suddenly lost. The local buckling modes were depicted by sinusoidal half-waves in the panel of the column base (see Figure 9(b)).

672 (3) Type C-20 In the case of specimen Type C-20, during the loading cycle with 5--+38y, yield lines could be seen not only on the plate of the column base but also in unstiffened panels where the cruciform plates are absent. When the load was increased to with 5--+78y, local buclding deformations took place in unstiffened panels where the cruciform plates are absent and the vertical cracking in welds of flange-web junctions occurred in this part at 5--+11~ (see Figure 9(c)). (4) Type C-40 Specimen Type C-40 showed the same behavior of as specimen Type C-20 until 5--+38y. However, when the load was increased to with ~---+95y, a big cracking sound occurred and local buclding occurred in the panels of the junction of cruciform plates due to insufficient welding (see Figure 9(d)) . Specimen Type C-40 showed very stable behavior and only slight local buckling deformations were observed in the column base panels. It was found that welding conditions are very important for getting rich ductility and attention must be paid to welding when making the box-section column. Type C can be expected to show the same behavior of the concrete filled columns because inner cruciform plates work to prevent occurrence of local buckling at the column base panels. However, it is necessary to determine the suitable height for inner cruciform plates in order to increase the ductility and energy absorption capacity. To see a point of view of cost, volume of inner cruciform plates on Type C-20 is a three less than one of stiffener on Type S and it'll be designed on low cost 4. CONCLUSION A total of 4 specimens with inner cruciform plates and compact sized sections were tested under constant compressive axial and cyclic horizontal loadings. From this study the following conclusions may be drawn: 1) The column with inner cruciform plates with a height is larger than 0.2 h (h: height of specimen) is effective for improving the ductility and energy absorption capacity. 2) Welding method is very important factor for getting high ductility and much attention must be paid to welding for fabrication of the box-section column.

REFERENCES 1.T. Tominaga and H. Yasun~mi, An experimental study on ductility of steel bridge piers, which has the thick and less stiffend cross section, Journal of Structural Engineering, VoI.40A, (1994), 189. 2.T. Tominaga and H. Yasun~m~, Evaluation of costs and seismic capacity on the thick-walled and less-stiffened steel bridge piers, Steel Construction Engineering, Vol.2, No.5, (1995), 37. 3.Japan Road Association, Specifications for highway bridges, 1996.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

673

Evaluation of steel roof diaphragm side-lap connections subjected to seismic loading C.A. Rogers a and R. Tremblayb aDepartment of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, H3A 2K6, Canada. bEPICENTER Research Group, Department of Civil, Geological and Mining Engineering, l~cole Polytechnique, P.O. Box 6079, Station Centre-ville, Montreal, Quebec, H3C 3A7, Canada.

Single-storey steel structures represent the vast majority of buildings that are constructed for light industrial, commercial and recreational uses in North America. For these buildings, it is often more cost effective (comparing cost to safety increases) to construct a non-ductile structural system, despite the fact that this type of structure and its occupants would be more vulnerable to seismic ground motions. The overall objective of this research is to investigate the possibility of allowing the metal roof deck diaphragm to absorb earthquake induced energy through plastic deformation along with the vertical bracing system. This paper provides preliminary information on the inelastic cyclic response, including load vs. displacement hysteresis and energy absorption capacity, of 39 screwed, button-punched and welded deck side-lap connections. The results of monotonic, cyclic and quasi-static tests revealed that the type of connection influences the ultimate capacity and ability to dissipate energy.

I. INTRODUCTION This paper addresses the seismic performance of the side-lap connections that are typically found in steel roof decks of single-storey buildings. Structures of this type are used for light industrial, commercial and recreational buildings in North America. In Canada, a large proportion of these structures are located in the St-Lawrence and Ottawa River valleys, as well as along the Pacific coast, the most active seismic regions in the country. Seismic provisions have recently been included in Canadian design standards to ensure that an adequate level of seismic performance for steel structures exists. However, these required changes have increased construction costs, thus making it more attractive (comparing cost to safety increases) to use a non ductile structural system; despite the fact that this type of structure and its occupants would be more vulnerable to seismic ground motions. The aim of this research project is to develop alternative solutions to increase the cost efficiency of the bracing system while improving the seismic behaviour of the structure. One possible solution is to account for the inelastic response of the metal roof deck diaphragm in energy dissipation calculations. In Canada, the seismic design of steel structures must conform to the National Building Code (NBCC) [1], which refers to the CSA-S16.1 [2] and CSA-S136 [3] Standards for steel design related issues. In design it is possible to use NBCC specified lateral seismic loads that are

674

Figure 1. Typical low-rise steel building significantly lower than the maximum forces that would be expected under the design level earthquake, provided that the lateral load resisting system exhibits a stable and ductile cyclic inelastic response. Under seismic ground motion, lateral inertia forces develop at the roof level due to the horizontal acceleration of the roof mass. To transfer and resist these lateral loads, the structure generally includes a metal roof deck diaphragm and vertical steel bracing (Fig. l). The roof diaphragm is made of steel deck units that are fastened to the supporting steel roof framing to form a deep horizontal girder capable of transferring lateral loads to the vertical bracing elements. The vertical bracing then transfers these loads from the roof level to the foundations.

2. CONNECTION TESTS The main objective of this phase of thc investigation was to measure the performance of various side-lap connections, i.e. the attachments between two adjacent deck sections, with regards to: stiffness, capaci~', ductilit3' and ener~' dissipation capability under different types of loading. A total of 39 side-lap specimens, with screwed (10-14x7/8"). button-punched (10 mm diameter) and welded (25 mrn length using a 410-10 MPa welding rod for 2-3 sec at 200 V) connections, were tested in the structures laboratory at l~cole Polytechnique using the test set-up shown in Figure 2. The test apparatus, modelled after the AISI Specification [4] recommendations, was bolted to the floor of the shake table (used to displace the specimen), with one edge of the side-lap connection attached to the shake table connection plate. The other half of the test specimen was secured to a rigid support connection plate, which in turn was connected to a load cell and a rigid support mounted on the strong floor of the laboratou,. Initial monotonic tests were completed, followed by quasi-static tests using the ATC 24 [5] seismic testing guidelines, and finally cyclic tests at 0.5 and 3.0 Hz. The protocol defined for the quasi-static and cyclic tests required displacements that ranged from +1.0 to +15 ram, vfith 5 increments of 3 cycles at the same amplitude and 3 decrements with 2 cycles, as shown in Figure 3. The most common deck profile found in Canada. i.e. 38 mm in depth • 914 mm in width, which requires the use of button-punched or welded side-lap connections, as well as a modified version of this deck section, which allows for the use of screwed side-lap connections, were included in the research program. Sheet steels with a thickness of 0.76 and 0.91 mm meeting ASTM A653 [6] Grade 230 specifications with minimum specified yield and ultimate strengths of 230 and 3 l0 MPa, respectively, were used. Load and relative longitudinal displacement measurements were recorded for all tests using a data acquisition computer system capable of sampling at 200 Hz.

675 /--Threaded

Shake Table - ~ . Connection . a Plates [-'-"

r

S~dl~

"L

Connection

25~n dia Pin

Section B

Rigid ~r-ll I011 0 I Support N FII*II I

9|

!I lii'' i .

.

Guide-~ I Plate ~

Inverted t

SteelSpecimen/ Deck---a Test

Rigid Support \T Connect,on ..... ~ r ' t a t e / - - lenon Shake Table--~. ~t .~r / Sheet Connection ~ 2 ] ' v ~ / ( . ~ Guide Plate .. ~ ~ . Plate 12.7mm.*Allen-~._x " ""-,,~,',,~

.

-~.

Key Bolt (typ) TeflonX X-Gr%~ed Sheet ~urtace

" li

P-3615 Button Punch and Weld Side-Lap Connection

Load#Y Rigid Support --~ Connection Plate Cell ~ B Guide Plate

~ ~. r ' ~

Shake Table Direction of Movement

.

~

':' ~ ,

P-3615 B Screw Side-Lap Connection Section A

Figure 2. Deck-to-Deck Side-Lap Connection Test Set-Up 20 -i

20 -

.i ;'vvvv vv _

v

'

vv"

-15 -20

0

-

'

i~ -10 ~

500 1000 1500 2000 2500 3000 Time (see)

~ .

-15 1 -20 J 0

5 10 15 20 25 30 35 40 45 Time (see)

Figure 3. Test Specimen Quasi-Static and 0.5 Hz Displacement Protocol

3. TEST RESULTS

Information regarding the displacement versus load behaviour for typical 0.76 mm welded, screwed and button-punched side-lap connections, cyclically tested at 0.5 Hz, is provided in Figures 4-6, with specific data for all load types listed in Table 1. (Note: each specimen consisted of 2 connectors). The complete load-displacement hysteresis is shown, as well as graphs for the 1, 2, and 5 mm displacement cycles. These figures illustrate that the ultimate capacity, Pu, depends on the type of connection, with the welds providing the highest resistance and the button punches the least. The ultimate resistance of the screwed and button-punched connections did not vary significantly with load type (see Table 1), although the Pu results for the welded side-laps ranged in value from 4.83-8.05 kN, a characteristic most likely caused by difficulties in fabricating two connections exactly alike. Typically, the welded connections failed by sheet tearing during the 10 mm cycles, and thus provided only minimal resistance in the remaining cycles, as indicated by the flattened end portion of the energy curves in Figures 5 and 6. In some instances inadequate welding of the two adjoining sheets caused failure to occur soon after loading, e.g. specimen II4b. The performance of a welded connection of this type is highly dependent on the skill of the welder, as well as the voltage and time settings of the equipment. The screwed and button-punched

676 WELD

SCREW

B U77"ONP UNCH

Overall

Overall

Overall

. . . . . . . . . . . . .

A

9 ~=

z

, i

A

_J

-=~.,

:

g

,

2 1

1 ~rr,.. _

-

- 2 0 -t. 5 -t. 0

--I

-

. r -,.,

~t0--1-5-20

5 I

o,

-

5 20 ---

Disp (mm)

l m m Cycles

l m m Cycles

l m m Cycles

....

-2

Disp (mm)

Disp (ram)

g l

A

z. . ~

~

v

"O

.... z"

"I~

lo ~5-~o

g-2o-ls-~~~-~

5 I,~

z

A.~-")

" I. Z # L J /

~

,

o -3 _1

- '1- ~~'

-2

1

I

I

2

3

..

,1.~r

.....

5

9

-3

"

9

-2

-, -

,

~,~

_1

gr

,

!--

. . . . . . .

9__,','_.. . . . . . . . . . .

.-

...

"~m

'

~ P ' ~ ~ ' 4

I

-,,a

~

r-

T

,

~' . . ",~ ~" A -" '"5

Dlep (ram)

Disp (mm)

t

~

---,

. . . .

t

Disp (mm)

=:

o, . ~ ~ :~: ~ _ - ~ - - ' ; T ~

I

3 r

,

,

5ram Cycles

5ram Cycles

2

2. I

2 ---3

Disp (mm)

(ram)

i

2mm Cycles

. . . . . . . . . .

o.

4

Disp (mm)

s

--

6..............

n -o

'

2mm Cycles

2mm Cycles

Disp

T

Disp (mm)

Disp (mm)

. . . . . . . . . . .

_

i

Smm Cycles

i

,2

I

......

i

-2

I

i

Disp (mm)

J

,

Figure 4.0.76 mm Test Specimen 0.5 Hz Overall, 1 mm, 2 mm and 5 mm Load vs. Displacement Cycles 500 I"400

500~

....................................

i

200

We/d

,

:u 100

................................................

~ A . -_...'r - ' r -~------'--~'" -f ~_..~,."r~~onP ......h

0 1 . ,

09

.....

400 i

Scrr

~ .............

i

"~

.

5

.

.

10 15 20

.

.

25

.

30

Time (see)

--,

35

40 45

o 0

100

200 Cumulative

300

400

500

Disp (mm)

Figure 5.0.76 mm Test Specimen 0.5 Hz Energy Dissipation History connections exhibited a pinched displacement versus load curve for different reasons. The screws tilted significantly when subjected to shear loading, which caused a reduction in load due to the

677

500

,oo

5~176

Weld

/,.

_, ,~176

_

E=200

'

i

,

"~200

~oo - ~ - ~ 0-!, 0

.,,

I

Scre~

_

~' 1 2

' 3 4 5 Time (set)

6

7

0

8

100

200

t

300

400

Cumulative Disp (ram)

500

t

t

Figure 6.0.76 mm Test Specimen 3 Hz Energy Dissipation History Table 1.0.76 mm Test Specimen Pu and Energy Dissipation Results

Test Specimen

Connection Type

Load Type

la lb 3a 3b 5a 5b 7a 9a 9b lla lib 13a 13b 15a Ilia lllb lI2a lI3a l13b l14a lI4b

Button-punch Button-punch Button-punch Button-punch Button-punch Button-punch Button-punch Screw Screw Screw Screw Screw Screw Screw Weld Weld Weld Weld Weld Weld Weld

Mono Mono 0.5 Hz 0.5 Hz 3 Hz 3 Hz Quasi-static Mono Mono 0.5 Hz 0.5 Hz 3 Hz 3 Hz Quasi-static Mono Mono Quasi-static 0.5 Hz 0.5 Hz 3 Hz 3 Hz

i

,

P. ,(kN) 1.64 1.49 1.94 1.44 1.43 1.48 1.59 4.91 4.56 4.48 4.68 4.62 4.79 4.32 6.13 7.16 6.23 8.05 7.91 6.27 4.83 i

z Energy ~Energy / Pu

,

(k N mm) ll9 112 115 104 137 144 131 145 140 152 274 608 381 453 101

(kN mm / kN) 61.7 78.0 80.3 70.3 85.7 32.1 28.0 31.4 29.3 35.3 44.0 75.5 48.1 72.2 20.9 i

,,,

reduced resistance of the fastener in the pull-out mode as compared to the bearing mode. In cases where large displacements occur, i.e. when 6 >screw length, it is possible for the screw to be completely pulled out of the sheet steel. The button-punched connections exhibited a pinched behaviour because of elastic relaxation between the punch and the die portions of the two joined sheets after deformation of the material in the punching process. Typically, after a button punch connection had displaced in excess of 2 mm, the resistance of the side lap resulted from the friction between the two adjoining sheets and not from the bearing between the punch and die. T h e e n e r g y dissipation c u r v e s illustrated in Figures 5 and 6, for the 0.76 m m s p e c i m e n s tested

at 0.5 and 3.0 Hz, provide comparisons of the total absorbed energy with time and with the cumulative displacement of the connections. The results indicate that the screwed connections had a slightly increased capacity to absorb energy (131-152 kN mm) in comparison with the buttonpunched connectiom (104-137 kN mm), In contrast, there is a marked increase in energy dissipation level for side lap connections that are joined with welds, although the measured values are inconsistent (101-608 kN ram) due to the difficulty in forming welds of this type. It is also noted that

678 the button punch connections exhibited the most stable energy dissipation capacity over the duration of the cumulative inelastic demand (see Figures 5 and 6), and compared with the maximum load level reached, were able on average to absorb more energy per kN of load than the screwed and welded connections (see Table 1).

4. CONCLUSIONS The type of loading had no discernible effect on the ultimate capacity of the screwed and button-punched side-lap connections, although the process used for welding resulted in inconsistent connections and hence test values. The welded connections carried the highest ultimate loads and absorb the most total energy, whereas the button-punch connections yielded the lowest values. However, when comparing the energy absorption per unit of force the buttonpunched connections outperformed their counterparts. Further full-scale seismic cantilever tests of steel roof deck assemblies would be beneficial to better understand the relative performance of these side-lap fasteners when subjected to earthquake loading. In parallel, analytical studies on typical buildings should be undertaken to assess the ductility demand in the various deck fasteners and to compare calculated values with those measured in tests. The analytical models used in these studies should be capable of reproducing the measured inelastic response of the fasteners. Analytical results detailing an allowable ductility level for the design of steel diaphragms are as of yet inconclusive. However, further research is currently being carried out to isolate the influence of local fastener ductility on the overall structmal ductility of single storey steel buildings. ACKNOWLEDGEMENTS The authors would like to thank the Natural Sciences and Engineering Research Council of Canada, the Canadian Institute of Steel Construction, the Canadian Sheet Steel Building Institute, the Canam Manac Group, Hilti Limited, ITW Buildex and the Steel Deck Institute for their support. The authors would also like to acknowledge the assistance of the laboratory technicians at l~cole Polytechnique, G. Degrange, P. BSlanger, and D. Fortier. REFERENCES

1. National Research Council of Canada. (1995). "National Building Code of Canada" 11th Edition, Ottawa, Ont., Canada. 2. Canadian Standards Association, S 16.1. (1994). "Limit States Design of Steel Structures", Etobicoke, Ont., Canada. 3. Canadian Standards Association, S 136. (1994). "Cold Formed Steel Structural Members", Etobicoke, Ont., Canada. 4. American Iron and Steel Institute. (1997). "1996 Edition of the Specification for the Design of Cold-Formed Steel Structtaal Members", Washington, DC, USA. 5. Applied Technology Council. (1992). "ATC24 - Guidelines for Cyclic Seismic Testing of Components of Steel Structures", Redwood City, CA, USA. 6. American Society for Testing and Materials, A653. (1994). "Standard Specification for Steel Sheet, Zinc-Coated (Galvanized) or Zinc-Iron Alloy-Coated (Galvannealed) by the Hot-Dip Process", Philadelphia, PA, USA.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

679

L o w cycle fatigue o f concrete filled steel robe m e m b e r s K. Tateishi a , T. Saitoha and K. Muramta b a Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo, Japan b Railway Technical Research Institute, 2-8-38, Hikarimachi, Kokubunji-shi, Tokyo, Japan Low cycle fatigue, which is one of the final failure modes of Concrete Filled steel Tube (CFT) members, was investigated experimentally. In order to estimate the low cycle fatigue strength, strain behaviors in the steel tube must be investigated in detail. In this study, a new strain measuring system based on the photogrammetry technique was applied. This method made it possible to measure the large strains around the buckling portion and reveal the unique characteristics of the strain field. Based on the measured strain data, low cycle fatigue assessment was performed. As a result, it was shown that this assessment was effective to deal with the crack problem in CFT members. 1

INTRODUCTION Concrete Filled steel Tube (CFT) members have high mechanical performance. For

example, deterioration of load carrying capacity after the yielding is gradual, while steel members suddenly lose the restoring force after a certain displacement level. However, some reports have pointed out that a crack was sometimes formed in the steel tube of CFT member, and after that, the restoring force decreased suddenly. Therefore, the occurrence of cracks, as well as the local buckling failure, should be considered as one of the final failure modes of CFT members [ 1]. In this study, the applicability of low cycle fatigue approach to the crack problem in CFT members is investigated. For low cycle fatigue assessment, plastic strain developed in the material must be known. Particularly, for CFT members, plastic strain fields in the steel tube around the buckling portion must be investigated because cracks always occurred around the

680 Table 1. Mechanical properties and chemical composition of the steel Y.S. T.S. El. C Si Mn P S (MPa) (MPa) (%) (%) (%) (%) (%) (%) STK400 373 451 28 0.11 0.10 0.48 0.021 0.005 Gmax

(ram) 20

Table W/C (%) 60

2. Mix proportion of the concrete s/a Unit Weight (kg/m~) (%) W C i S G 50 195 325 ] 907 921

Slump (cm) 6.0

Air (%) 2

buckling portion. However, conventional strain measuring methods, for example, strain gauge method, can not be applied to measure the strain in an object with three dimensional deformations like the local buckling deformations. In this study, a new strain measuring method based on the photogrammetry technique was applied to measure the strain fields. 2

STRAIN MEASURING SYSTEM Here, only the outline of the system is shown because more detail information was

already given in RefI4]. The system consists of digital cameras and personal computers as shown in Fig. 1. From the stereo image taken by the digital cameras, the coordinates of some target points placed on the specimen are determined by using photogrammetry technique. If the coordinates of a point before and after the deformation are known, the displacement vector for the point can be easily calculated, and then, finally, the strain field in the region can be quantified. 3

SPECIMEN AND LOADING METHOD

Specimen is shown in Fig.2. The steel tube had a circular section and was made from STK400 steel of which chemical compositions and mechanical properties were shown in Table 1. The steel tube was welded to the base plate. The compressive strength of concrete cast in the tube was 37MPa. Two specimens were connected by high-tension bolts and loaded

681

Fig.4 Target Points

Fig.5 Loading Patterns

at the center as shown in Fig.3. Axial force was not applied. Six specimens (three pairs of specimens) were prepared. One pair of them was steel tube (ST) specimen in which concrete was not filled. Another two pairs were CFT specimens (CFT-1, CFT-2). The mix proportions of concrete are shown in Table 2. On the surface of the steel tube near the base, target marks with 5mm intervals were drawn by the paint (Fig.4). These target marks were traced during the loading test and used as the points for calculating the displacement vector. The loading sequence is shown in Fig.5. For ST specimen and CFT-1 specimen, the displacement level was increased, while the displacement amplitudes were kept constant in 7 6 y for CFT-2 specimen. 4

EXPERIMENTAL RESULTS

4.1. Load-Displacement Relationship Load-displacement relationships are shown in Fig.6. For ST specimen, the restoring forces are suddenly reduced with the displacement. Local buckling deformation was observed near the base, and it grew with the displacement level. However, no crack was observed in ST specimen. For CFT-1 specimen, the hysteresis loops are stable, and the decrease of the restoring force is gentler than that of ST specimen. At the displacement level of 25.2mm(7 ~ y), a crack was 3O

0

"

2O

~10 0

0

-I0 -20 -3(3 -40

{ I lll

t -1(

j -2c 1

-20 0 20 Displacement(mm) (a) ST Specimen

-3(: 40 -40

,

,

t

i

,

-20 0 20 40 -30 -20 -10 0 10 20 30 Displacement(mm) Displacement(mm) (c) CFT-2 Specimen (b) CFT-1 Specimen Fig. 6 Load-Displacement Relationships

682

"-

=.,

:!:::ili:!iiii::

-5 -" -6

~ \

:~ 0.10[ "~

-" + 3 ~ 0.10 ~ + 5

/

~

i

!

B -o2o

)

~. _ ~(r/lm) --

-

-.-

i

B

,; t

-0.30

I'

-( (a) under compressive loading

(b) under tensile loading

Fig.7 Strain Distribution of ST Specimen 0.20 ---~y .1 + + 3 x +4 0.15 I --4~+5 + + 6

0.20 0.15

=:

i -*-+7

=9o 0.10

?_.9~o o.lo

.0.05 "UI

~9 0.05

.~ 0.00

.~ 0 . 0 0 ~ 0 ,~ .~.-0.05 ~

0

o

-0.05

A

-+-+8

201"

...

40

1' 6

d~(mTm)

B

.E -0 10

~9 -0.10 -0.15

....

B

)

x m m

-~

+4 +6

A

~;0

C ! ]

m -0.15

-0.20 (a) under compressive loading

-0.20

(b) under tensile loading

Fig.8 Strain Distribution of CFT Specimen detected at the location 5mm away from the base. This crack propagated with loading cycles and leaded to sudden deterioration in load carrying capacity at the displacement of 10 6 y. For CFT-2 specimen, the restoring forces gradually decreased with displacement cycles, and crack was detected near the base at seven displacement cycles.

4.2. Strain Behaviors The comparison of the measured strain in ST specimen and CFT-1 specimen, which were tested under the same loading pattern, is shown here. Fig. 7 shows the distribution of strain in x direction (see Fig.4) for ST specimen at each displacement level. Though three displacement cycles were repeated in each displacement level, the result for only one cycle in them is shown in the figure, because the differences among them were very small. When the compressive displacement are loaded, the tensile strain increases with the displacement level at the top of the buckling portion (point B), while the compressive strain becomes larger at the both loots of the buckling portion (point A and C). Even under tensile loading, residual plastic strain remains in compression when the displacement level exceeds 5 6 y.

683 0.25 0.20 I _'~"axial directionI 0.15! ,.'~'-hoov t~t direction 0.10 .~0.05 ~o.~ -0.0~ -0.t~ -0.15 -0.21 -0.2~ !

i

r

|

t

I

|

I

i

I

0.25 0.20 [ 4 - axial ~ (x/ direction [ 1 0.15 0.10 S005 ~o.o~ -0.05 -O.1C -0.15 -0.2C -0.25

!

0.20 [ _-~-axial {x) directionl -U,hoopiv) directlonl 0.10

~o.00 -0.1( -0.2(

,x ,,x ~ ,~q,.'b • k, x~,S xS ,~ xb dy

-0.3(

.x x\ .q,xq,.'~,,'~.~ ~, h ,,5 .~,do dy (c) Lower foot of the buckling

(b) Top of the buckling Fig.9 Strain History of ST Specimen 0.251 . . . . . . . . ~ 0.25 ........ n,~n[l-c--axial (x)direetion[ [iiii!iill 1 [ ,~,,~ ]--o--axial(x)directiorl[ o ~ 0.251"]+axial (x)direetionlI!i:ili:ii] .... / [-o-hoop (y) direetior~ Ji!i~:~!!i!.~.~.[ "'~ 0.20t I..o_hoop (V)direetion~. (a) Upper fogtYf the buckling

o.10[

I 0.1c

0.00

O.OC

oo )

oo

0.1q .......

.

I

-0.15. .~.•215215 . . . . . •. ~• . . •. .~ -0.1~ . . .~. •215215 . . . . . . • .~ •. .• . .-0.t~ . ..~•215 . . . •. . .

9

9

o

~Y (a) Upper foot of the buckling

s

.

.

.

.

.

.

.

.

.

.

.

.

.

9

.

.

_

9

~iv (b) Top of thi~ buckling

s

• ~•215 ~

9

9

(c) Lower foot of Yhe buckling

Fig.10 Strain History of CFT-1 Specimen Fig.8 shows the strain distributions for CFT specimen. The same characteristics to ST specimen can be observed under the compressive loading. However, under the tensile loading, the strain distributions are relatively flat and almost strains are tensile, which means the local buckling deformation becomes small. This characteristic is considered to be caused by the fact that the inner concrete resists the vertical shrinkage due to the buckling deformation of the steel tube and delays the progress of the buckling deformation. Fig.9 and 10 shows the strain histories at the three points in ST specimen and CFT specimen where large strains were observed in Fig.7 and 8. At the top of the buckling, large strain in hoop direction (y direction, see Fig.4) is observed. Particularly, for CFT specimen, hoop strain is much larger than axial strain. This large hoop strain in CFT specimen is considered to be the result of the restriction of the deformation of the steel tube by the inner concrete. At the upper and the lower foot of the buckling portion, the strain behaviors are similar, and the large fluctuations can be observed for the strain in x direction. Particularly, the strain fluctuation at the lower foot of the buckling portion in CFT specimen was largest, which was coincident with the fact that a crack was formed at the position.

684

5

LOW CYCLE FATIGUE ASSESSMENT

(D 1 CX0 ..... . t~ I-4

Based on the measured strain data at the foot of

~ 9 0.1

buckling portion where crack was formed, the plastic strain range, A%.,, was calculated, and

~0.01

summed up according to the linear damage rule. The equivalent strain range, Ae~,, was calculated by the following equation. E m,~eq _.

j~r = E

!i'~'~...... O-i- ~_~.

i It

'

, , I Ilia i i! I]H

, CFT-1 ,~-~..] 0.00r

L Illl[~

10 100 Number of Cycles Fig. 11 Low Cy'cle Fatigue Assessment 1

kp A6p e "ni _~,kp9

il i

_

ni

(1)

where, n; is the number of cycles, and of kp is material parameter. Here, kp was taken as 0.54 according to the former experimental study [3]. Fig.ll shows the relationship between the number of cycles when the crack was detected and the equivalent strain range. The line in the figure shows the low cycle fatigue strength given in Ref.[3] which was the result of material tests. The obtained data in this experiment are located near the line, that means the low cycle fatigue assessment is effective to estimate the occurrence of cracks in CFT members. 6

CONCLUSIONS

As a fundamental study on the low cycle fatigue of CFT member, strain measurements of the steel tube was carried out by a newly developed strain measuring system based on the photogrammetry. By using the system, the unique characteristics on deformation and strain field in CFT members under cyclic loading were clarified. The fatigue strength of the specimen used in this study was coincident with the strength curve obtained by material test, which showed the low cycle fatigue assessment is effective for considering the crack problem in CFT members. REFERENCES 1. Murata, K., Watanabe, T., Nishikawa, Y. and Kinoshita,M., Proc. of the 50th annual conference of JSCE, I-A, pp.222-223.(1998) 2. Murai, S., Okuda, T. and Nakamura, H., Report of the Institute of Industrial Science, The University of Tokyo, 29(6),pp. 1-15. 3. Nishimura,T. and Miki, C., Proc. of JSCE, No.279, pp.29-44 (1978). 4. Tateishi K. and Murata K., Proc of the EASEC 7, pp.949-954 (1999)

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

685

The importance o f further studies on the capacity evaluation o f concrete-filled steel tubes under large deformation cyclic loading Chen Lee, Raphael H. Grzebieta, and Xiao-Ling Zhao Department of Civil Engineering, Monash University, Clayton, Victoria 3800, Australia This paper reviews recent studies concerned with evaluating the capacity of concrete-filled steel tubes under different cyclic loading paths. The review has been divided into two sections. The first section discusses cyclic loading paths including both partial and full cycle oscillations. The second discusses the capacity analysis of conerete-fiUed steel tubes and how the stiffness, strength, ductility and energy dissipation are determined in the plastic region for different cyclic loading histories. This paper suggests that further investigation of the effect of cyclic loading paths on the behaviour and capacity evaluation of concrete-filled steel tubes is necessary to establish a standardised means in assessing a member's suitability for seismic design. 1. INTRODUCTION In a seismic design, the performance of a member's plastic deformation depends on the material properties that will enable it to maintain stability, deform inelastically, and absorb the imposed seismic energy. Overall evaluations of stiffness, strength, ductility and energy dissipation need to ensure adequate balance of these properties. However, the cyclic response associated with a loading history and the failure mechanisms of concrete-filled tubes has not been fully understood and quantified particularly in regard to plastic deformation. The material responses including the yield plateau, the BauscbJnger effect, cyclic strain hardening and softening vary with diverse cyclic loading paths. These different characteristics in the plastic region can lead to the different values of member capacities being predicted. 2. CYCLIC LOADING PATHS Cyclic loading paths are mainly categorised into partial and full cyclic oscillations. In a partial cyclic loading path, a specimen is unevenly loaded in either tension or compression over different displacement ranges from a starting position or zero load. By contrast, in a full cyclic loading path, a specimen is evenly loaded in tension and in compression over the same displacement range oscillating about a starting position or load. There have been only a few experimental investigations into the relationship between the capacity of test specimens and various cyclic loading histories. These loading paths are now reviewed in the following sections. 2.1. Partial Cyclic Loading Path Liu and Goel [ 1] conducted an experiment on tubular braces. Both hollow and concrete-filled rectangular tubes were subjected to axial cyclic loading, which were mainly compression

686 loads. Figure l(a) shows the layout of the hysteresis load quadrants. Figure l(b) shows a typical test where the specimen is loaded in all four quadrants. Liu and Goel's tests were predominantly in quadrant three. The concrete-filled tubes with a larger width-to-thickness ratio had the better capacity improvement. Cyclic peak compression forces for the concretefilled tubes decreased at a lower rate than that of hollow tubes due to the delay of local buckling. Ge and Usami [2] investigated the strength and ductility of concrete-filled steel box columns with large width-to-thickness ratios under axial cyclic compression loading (Fig 2). Some of the cohmms were reinforced by longitudinal stiffeners to increase the stiffness. In this experiment, all columns were subjected to three consecutive cycles before the displacement was increased. Ge and Usami's tests were restricted to quadrant three. They concluded that the concrete filling was damaged before the peak strength was reached. Zhao et al [3] also examined concrete-filled cold-formed RHS columns under axial cyclic compression and tension loading. The steel tubular braces were subjected to cyclic-direct or cyclic-incremental loading (Fig 3). The load versus displacement curves appeared in quadrants three and four as expected. There was little difference in residual strength between the cyclic load and the monotonic load for the thicker tubes. Using welded rectangular hollow sections, Fukumoto and Kusama [4] conducted an investigation on the local instability of plate elements under uniaxial cyclic compression and tension loading. A typical cyclic loading path (Fig 4) was applied to the test specimens. The maximum compression loads were recorded in every cycle, whereas the tension loads were maintained below yield. The limited tension load restricted the hysteresis curves to the third and fourth quadrants as expected. It was observed that the cyclic envelope curves agreed well with the curves from the monotonic compression tests.

P/Py 1.5 ~ Typical envelope 1~ j~.~ ~ curve Hysteresis loops ,

,~P Force

ITelsio. I Quadrant 4

.

/ Quadrant1

Qu'ad;ant3 / Quadrant2

Icom.. l,- on]

0 ~ ~ ~

~~

Fig l(a). Layout of Hysteresis Load Quadrants

-1.5 -~ Fig l(b). NormalisedAxial Load-Displacement Hysteresis Behaviour in All Four Quadrants

Maison and Popov [5] tested circular bracing pipes under axial cyclic loading (Fig 5). The bracing pipes were subjected to uneven axial displacements in compression as well as in tension. The load displacement curves mainly appeared in quadrants three and four.

=-0.2 "--" i

~/k h 10 ;v V ~ k ~

15 20 N~ ~ Cycles

ffl

~ -0.6 ~ -O.8

~i

< Fig 2. Cyclic Loading Path for Box Columns (C-~ and t.~mi, 1992)

8/8y 3~ 2~, o

0

..~5, ~ ~ . ~

10

20

p

No. of Cycles

30

40

50

Fig 3. Cyclic-Incremental Loading Path for RHS Braces(Zhao et al, 1999)

687 e/ey 2 ~,

Tension No. of Cycles

_~

E

20

-s j

Compression

of Cycles

~ -2o "~

-6 ~,

No.

.

-60

_

-100 -'

Fig 4. Cyclic Loading Path for Box Columns (Fukumoto and Kusarna, 1985)

Fig 5. Cyclic Loading Path for Braces (Maison and Popov, 1980)

2.2. Full Cyclic Loading Path For all of the tests cited in this section, the load-displacement curves appeared in all four quadrants due to the full reversal cyclic loading. Usami and Ge [6] investigated the strength and ductility of steel box columns partially filled with concrete, which were subjected to a constant axial load P and lateral cyclic loading H (Fig 6). The investigation showed that the partially concrete-filled columns significantly improve earthquake-resistance capability. In a further study, Ge and Usami [7] examined similar partially concrete-filled steel box columns over a larger displacement. The reversal displacement was gradually increased to 68y, where 8y is the displacement at the top of the column when first yield occurs. Nakanishi. et al [8] conducted an experiment on double skin tubes filled with low strength concrete. The outer tube was steel, whereas the inner tube was either steel or plastic. Hollow and single concrete-filled tubes were also tested for comparison. The tubes underwent dynamic loading and were then reloaded with constant axial load and lateral cyclic loading (Fig 7). Substantial strength decrease was observed in the reloaded concrete-filled steel tubes. However, the strength of only-cyclic-loaded specimens showed no significant difference to the strength of dynamic preloaded specimens. Popov and Black [9] conducted an experiment on steel struts of square hollow sections subjected to full axial cyclic loading. This investigation established that the subsequent compression capacity decreased as soon as buckling occurred in the test specimens. Due to the cumulated strain, the tangent modulus decreased significantly with each cycle.

8/~y

H

No. of Cycles

IL^AAAAAAA yoVV

.10-.. x~V V4V v6V ~V -3

~y

No. of Cycles

!AAAAA_#,AAAA

H -~~-~ ~

-

Fig 6. Cyclic Loading Path for Concrete-Filled Steel Box Columns (Usami and Ge, 1994)

Fig 7. Cylic Loading Path for Concrete-filled Steel Columns (Nakanishi. et al, 1999)

3. CAPACITY EVALUATION Various methods have been used to calculate the properties of stiffness, strength, ductility and energy dissipation. In practical seismic designs, the ideal cyclic behaviour of a member is where a sufficient level of strength is guaranteed no or little deterioration over a minimum number of cycles.

688 3.1. S t i f f n e s s a n d Strength In modelling the plastic behaviour under complex loading, Dafalias and Popov [ 10] proposed the following formula to calculate the plasticity modulus: 1

E'

1 = ~+

1

E~

(1)

Ep

where E', E ~, and E p are the tangent, the elastic and the plastic moduli respectively. Hajjar and Gourley [11 ] also used a similar approach for their nonlinear cyclic model. Popov and Black [9] investigated bracing struts under severe axial cyclic loading. The tangent modulus and reduced modulus were obtained from the hysteresis envelope curves to calculate stiffness. From these curves, the stiffness in compression showed a significant deterioration, while the stiffness in the tension was steadily decreasing. They suggested that a reducing modulus was more suitable than a tangent modulus in determining the cyclic buckling load. ECCS [12] recommend that a maximum rigidity ratio ~:+ and a minimum rigidity ratio ~:- are defined as follow: + = tga7 ~Ji tga,.

+

'

and ~- = tga. tga~r

(2)

where, tga~. and tgct; are the initial tangent modulus at first yield, tga 7 is the tensile tangent modulus in quadrant one (Fig 9), while tga7 is the compressive tangent modulus in quadrant three (Fig 9). Resistance ratios were also suggested for estimating the strength capacity. The ratios, r and e [ , are used such that: + = F / + ands/-

c,

&+,

F~-

= F;

(3)

where, F~+ and F~-, are the forces corresponded to the maximum tensile and compressive displacement in each cycle (Fig 9) and, Fy+ and Fy- are the forces when first yield occurs. The values can easily be obtained from test data and used to evaluate the safe load limit in design. However, the strength between two increment cycles may not be accurate for a large displacement increase. ASCE 7 [13] proposes that an effective stiffness, k,#, for each loading cycle can be used such that: k:,# = F~ - F 7

(4)

where F 7 and F 7 are the maximum tensile and compressive forces corresponding to maximum tensile and compressive displacement, A~, and A~, respectively. The formula is based on a tangent modulus approach. However, this may not accurately illustrate the relationship between load and displacement in the inelastic region as the tangent modulus varies during plastic deformation. Usami and Ge [6] presented the concept of a normalised strength ratio derived from loaddisplacement envelope curves. The ratio is expressed as the peak load from a cyclic test relative to the load at yield for a monotonic test.

689

3.2. Ductility and Energy Dissipation Ductility and energy dissipation of a member are highly sensitive to displacement history. For a seismic design, ductility requires that a member is able to sustain deformation beyond the yield point without significant loss of strength. Ductility does not take into account the number of cycles in estimating the deformation. However, energy dissipation can provide a good indication of cyclic history. Usami and Ge [6] used the collapse point to evaluate ductility and dissipated energy in the study of concrete-filled steel box column. In a load-displacement envelope curve, the collapse point was defined as the post-peak softened load that is equivalent to the elastic yield load Hy as shown in Fig 8. Therefore, the ductility (~) was defined as the ratio of the ultimate displacement at the collapse point (Su) to the displacement (Sy) at which the first yield occurs. The energy dissipation was also defined as the summation of all of the areas enclosed by the hysteresis loops up to the collapse point. In a further study, Ge and Usami [7] used the concept of a 95% maximum load. In the loaddisplacement envelope curve, the failure point was defined as a point where the post-peak load softened to 95% of the maximum load. Ductility and energy dissipation were then calculated up to this point. ECCS [12] proposed that ductility (Fig 9) is represented as the ratio between the absolute value of maximum displacement in the tensile or compressive force range (Ae~ or Ae[ ) and the corresponding tensile or compressive yield displacement (ey or e~ ) in each cycle such that: + Ae; Ae, /z; = ' - 7 - , and/z[ =

ey

(5)

ey

The energy dissipation ratios, r/; and r/;, were also defined as the ratio of the real energy dissipated and the energy dissipated at yield in a half-cycle as follow: +

7?;

A; F v (e+ + e . - e,+, - e; )

and r/7 =

A;

(6)

F)7 (e+ + e . - e+_v- e; )

where ,4;+ and A, are the area as shown in Fig 9.

4. CONCLUSIONS From the above review, it is clear that there is considerable variation in the different studies. These differences lead to different results in regards to seismic capacity of a member. In order

690 to standardise the properties of a member, innovative concepts and criteria are needed for inelastic analysis, which is increasingly used in seismic design. Thus, further research has the following aims: 1. To compare the capacities of concrete-filled circular, square and rectangular section tubes for different cyclic loading paths; 2. To re-evaluate the definition of stiffness, strength, ductility and energy dissipation in inelastic and plastic regions in the case of cyclic loading; 3. To find the most representative loading histories associated with capacity evaluation; 4. To investigate the effect of different failure mechanisms associated with capacity evaluation. ACKNOWLEDGEMENTS

The authors are grateful to the Australian Research Council for financial support. Thanks are also given to Jane Moodie, for her advice on the written expression in this paper. REFERENCES

1. Liu, Z.Y. and Goel, S. (1988). "Cyclic Load Behaviour of Concrete-filled Tubular Braces." J. Struct. Engrg., ASCE, 114(7), 1488-1506. 2. Ge, H.B. and Usami, T. (1992). "Strength of Concrete Filled Thin-Walled Steel Box Columns: Experiment" J. Struct. Engrg., ASCE, 118(11), 3036-3054. 3. Zhao, X.L., Grzebieta, R.H., Wong, P. and Lee, C. (1999). "Concrete Filled Cold-Formed C450 RHS Columns Subjected to Cyclic Axial Loading." Proc., 2"d Int. Conf. Advances in Steel Structures, Hong Kong, China. 429-436. 4. Fukumoto, Y. and Kusama, H. (1985). "Local Instability Tests of Plate Elements under Cyclic Uniaxial Loading." J. Struct. Engrg., ASCE, 111(5), 1051-1067. 5. Maison, B.F. and Popov, E.P. (1980). "Cyclic Response Prediction for Braced Steel Frames." ,Z. Struct. Engrg., ASCE, 106(ST7), 1401-1416. 6. Usami, T. and Ge, H.B. (1994). "Ductility of Concrete-Filled Steel Box Columns under Cyclic Loading." J. Struct. Engrg., ASCE, 120(7), 2021-2040. 7. Ge, H.B. and Usami, T. (1996). "Cyclic Tests of Concrete Filled Steel Box Columns." J. Struct. Engrg., ASCE, 122(10), 1169-1177. 8. Nakanishi, K., Kitada, T. and Nakai, H. (1999). "Experimental Study on Ultimate Strength and Ductility of Concrete Filled Steel Columns under Strong Earthquake." J. Construct. Steel Research, 51,297-319. 9. Popov, E.P. and Black, R.G. (1981). "Steel Struts under Severe Cyclic Loadings." J. Struct. Engrg. Division, ASCE, 107(ST9), 1857-1881. 10.Dafalias, Y.F. and Popov, E.P. (1975). "A Model of Nonlinear Hardening Materials for Complex Loading." Acta Mechanica, 21, 173-192 11. Haijar, J.F. and Gourley, B.C. (1997). "A Cyclic Nonlinear Model for Concrete-Filled Tubes- I: Formulation." J. Struct. Engrg., ASCE, 123(6), 736-744. 12. ECCS-CECM-EKS. (1986). Study on Design of Steel Building in Earthquake Zones, ECCS, Brussels, Belgium. 13. American Society of Civil Engineers, 1995, ANSI/SCE 7-95 Minimum Design Loads for Buildings and Other Structures, ASCE, Reson, VA.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

691

Design of Large Bridge Over the M a t c h e s t a River m Seismic Zone A. Likvermana, G. Shestoperov b, V. Seliverstova "Giprotransmost J.S. Co 2 Pavia Kortchagina Str, Moscow 129278, Russian Federation b TSNIIS J.S.Co 1 Kolskaya, Moscow 129329, Russian Federation

This paper deals with the major seismic design features of the 900 m bridge over the fiver Matchesta near the Sochi city, South Russia. Efficient structural measures to mitigate earthquake action are discussed. Detailed local site conditions assessment resulted in elaborating of detailed mapping that allowed for modification of the value of seismicity.

1. INTRODUCTION The new 900 m viaduct crosses a valley of the river Matchesta and forms a part of the Peripheral Motorway around the city of Sochi, South Russia. The bridge deck accommodates two traffic lanes of 11.5 m wide and two sidewalks of 1.5 m wide each. The bridge is designed with three expansion joints, resulting in two sections of 800 m and 100 m long. Some piers in the middle part of the bridge reach a height of more than 40 m. For typical structural details of one of the central intermediate piers see Fig. 1. The piers and abutments are on bored piles, being 1.5 m in diameter. The choice of span arrangement was governed by geologic and geophysical conditions taking into account the layout of existing communication lines (cables). The configuration of spans is 80+85+91+ 126+ 114+2x68+2x63+2x53+46 m. The main part of the superstructure is a continuous ten-span structure straight in plan. The latter two spans are on a curve, being 250 m in radius. The superstructure comprises a single steel box girder permanent in depth, being 3.6 m. The total design mass of the deck main part exceeds 10000 t.

2. DESIGN BASIS

The design was carried out in accordance with the requirements of the Russian Bridge code (CHHII 2.05.03-84*). The bridge had to withstand seismic forces corresponding to a ground acceleration of 0.4g. Earthquake forces in particular were based on the requirements in the Seismic code (CHHI111-7-81"). A concept of maximum seismic accelerations forms a basis for the Russian seismic standard. Normally the design seismic intensity is obtained from maps developed for the territory of the former USSR.

692

~,

k,j

10000

_

k

l ~ ~' - ~ 2 1

I.

-i" i

~

~

!

9

~

!

I

[

Fig. 1. Configuration of the central intermediate pier

3. SEISMIC MAPPING AND SOIL CONDITIONS According to the data of seismic and tectonic researches, the bridge site is characterized by an earthquake intensity of 9 on the MSK-64 scale (seismic forces corresponding to a ground acceleration of 0.4 g) with a return period of 1000 years. However taking into account local geological conditions at bridge site and results of detailed mapping, the value of seismicity was modified. In the Russian practice the seismicity of each particular construction site is determined by the table in the Seismic code and using data of geological surveys. However the seismicity of construction site determined by this method is typically considered overestimated and may be qualified as preliminary only. Therefore for bridges, length of which exceed 500 m, special seismological investigations are required by the Seismic code. To consider the influence of local conditions on seismicity of each particular pier site, a method based on seismic rigidity of ground layers was used for design. The adoption of this method call for the data on velocities of seismic waves in ground layers under investigation. Values of these velocities may be obtained from field geophysical surveys. The other approach, which is more preferential in some cases, is to use correlation equations, which provide a relationship between velocity of seismic wave, soil properties and conditions of their layering. A basis of seismic detailed mapping of each pier site is justified by a special analysis. A design scheme of ground base for pier no. 5 is shown in Fig. 2.

693 Results of this analysis comprises velocities of seismic waves, dynamic modulus of elasticity, Poisson ratio, dynamic coefficient of soil stiffness, coefficient of soil conditions for a layer, coefficient of soil conditions for the particular pier location. E.g. velocities of transverse wave and coefficient of soil conditions for each layer at pier no. 5 are given in Table 1. Finally the coefficient of soil conditions for pier no. 5 location was calculated as 0.9.

1! c=y

[_

!

Gravel

11m 32m

Loose ;Irgi, Ilite

6m

Weathered ',Zrgi Ilite

7rn

Fig. 2. Design scheme of ground base for pier no.5

Table 1 Velocities of transverse wave and coefficient of soil conditions Soil layer Velocity of transverse wave, (m/s) Clay 192 Gravel 464 Loose argillite 579 Weathered argillite 690

Coefficient of soil conditions for each laver 1.37 0.82 0.71 0.65

Review of values of seismic waves velocities at pier bases lthrough 5 within a depth of 32 m have shown that seismicity of the bridge site varied from the seismicity adopted for the Sochi region using the Seismic code. To calculate the real ground acceleration at location of the pier base under investigation, a nominal acceleration of ground (for an average layer by seismic properties) is multiplied by the coefficient of soil conditions determined for each particular site. Based on the analysis of results e.g. the modified value of ground acceleration for location of pier no.5 is 0.36g.

4. SEISMIC ANALYSIS AND STRUCTURAL MEASURES Because the bridge could be subject to seismic loading, the superstructure of the main section is installed on the fixed bearings at seven central intermediate piers and on movable bearings at

694 four extreme piers. To decrease seismic loads on anchor piers, their structures are designed as reinforced concrete frames with low rigidity in the direction of bridge axis. Piers nos 3-5 in the central part of the bridge are 42.25, 45.8, 44.95 m high respectively. These are the highest piers. Period of self vibration for these piers with account for superstructure mass is 1.8 s, and this allows adopting a minimum value of dynamic coefficient and therefore reducing seismic load on piers. A relationship between dynamic coefficient and period of self vibration is shown in Fig. 3. Due to the large flexibility of pier columns, the range of superstructure horizontal displacement in the elastic stage reaches about 120 mm. It is expected that with account for cracks in concrete the horizontal displacement of superstructure may reach at least 25 cm under the seismic action. When large displacement occurs, movable bearings at piers nos 1, 2, 1O, 11 may not function. Also the superstructure may cause damage to wall of abutment no 1 and edge of adjacent superstructure over pier no 11. Dynamic

Coefficient 2.50

9

2.00

9

1.50

1.00

:

0.50

:

9

! . . . . . . .

o.0o

0.~i0

'~

1.00

;

1.50

2.00

Period of self vibration

Fig. 3. Dynamic coefficient To eliminate the abovediscussed effects the following measures were adopted. Support length of movable bearings were designed to accommodate superstructure displacement of 250 mm. To prevent damage to abutment no 1 and edge of adjacent superstructure over pier no 11, relevant parameters of expansion joints were chosen. For the bridge frame system under the design seismic action with account for cracks formation in the pier columns and reduced decrement of vibration of flexible piers, a range of displacements along bridge axis is estimated as 31-32 cm. To satisfy the requirement of column strength of pier No. 9, the range of superstructure displacements due to vibration during seismic event is recommended to be limited to 23-24 era. To reach this goal, the following measures have been recommended. In the first design effort it was recommended to restrict the superstructure motion using reinforced concrete curbs to be constructed at piers Nos. 1, 2, 10, 11 and steel stoppers to be attached to the superstructure. The curbs and stoppers were designed to static load of 104 t. To alleviate an effect of blow, buffers are installed between curbs and stoppers. Each buffer corn-

695 prised five steel/resin elements. Totally there were six buffers, they were recommended to be installed at piers Nos 1 and 11 (one per each pier) and at piers 2 and 10 (two per each pier). Besides buffers should not limit the superstructure displacement due to temperature effects which reach 16.8 cm. Therefore buffers are installed in such manner that a distance between surfaces of buffer and stopper is 17.0 cm. However later on the more efficient measure has been adopted. In the final design concept the superstructure motion is restricted by means of Maurer hydraulic dampers installed at piers 1 and 10. These dampers are designed to concentrated horizontal load of 150 t. The antiseismic measures were selected on the basis of experience. Also general principles and structural requirements of the Russian standards for construction in seismic areas were considered. 5. CONCLUSIONS The adopted method of assessment of local site conditions allowed modification of the value of seismicity. Considering real soil conditions parameters of each particular site in the analysis, efficient structural measures to minimize damage produced by seismic motions were elaborated. The bridge is currently under construction and will be opened to traffic in 2000. REFERENCES 1. CHnI'I 2.05.03-84*. (Building norms and regulations). Bridges and culverts. Minstroy (Ministry of Construction) of Russia, M, 1996 (in Russian). 2. CHnl-111-7-81". (Building norms and regulations). Construction in seismic regions. Minstroy (Ministry of Construction) of Russia, M, 1996. (in Russian):

This Page Intentionally Left Blank

Fracture/Fatigue

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

699

Tensile fracture b e h a v i o u r o f thin G 5 5 0 sheet steels C.A. Rogers a and G.J. Hancock b a Department

of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada.

bDepartment of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia.

This paper reports on the fracture of G550 sheet steels subjected to uniaxial tension. The fracture resistance of 30 different notch test specimens with initial fatigue cracks was measured over a range of temperatures, and a numerical study on the effect of cracks in the elastic load range was completed using the FRANC2D finite element computer program. The main objective of this investigation was to determine the critical Mode I, i.e. crack opening, stress intensity factors, Kc, for 0.42 mm G550 and 0.60 mm G550 sheet steels. Tests were completed to measure the magnitude of the crack tip stress field where ultimate failure was caused by trustable fracture of the notch specimens in the elastic deformation range.

1. INTRODUCTION The Australian / N e w Zealand design standard for cold formed steel structures (AS/NZS 4600[1 ]) allows for the use of thin (t < 0.9 mm), high strength (fy = 550 MPa) sheet steels in all structural sections. However, due to the lack of ductility exhibited by sheet steels that are cold reduced in thickness, engineers are required by standards and specifications to use a yield stress and an ultimate strength reduced to 75% of the minimum specified values. Test results of G550 sheet steels by Rogers and Hancock [2] have shown that, in some instances, it is possible for thin G550 sheet steels to fracture soon after the exiting the elastic range of deformation. The fracture resistance of two different G550 sheet steels was measured at different temperatures, and a numerical study detailing the effect of cracks on structural performance in the elastic load range was completed using the FRANC2D [3] fmite element computer program. Fracture resistance properties were determined by testing notch test specimens that contained initial fatigue cracks. Finite element analyses were performed on notch, as well as bolted connection structural models. 1.1. Basic Fracture Mechanics Stress distribution in a loaded member is greatly affected by the presence of cracks or discontinuities. The classical structural mechanics approach deals with these matters through the use of a numerical multiplier referred to as a stress concentration factor, i.e. the increase in stress caused by a change in geometry such as a notch. Fracture mechanics, however, recognises that the stress intensity at the tip of the crack can be expressed as a stress intensity factor, K, as follows,

700

r = o-,,,.~

(1)

where crappis the nominal stress applied to the member and a is the size of the crack. As the stress intensity factor at the tip of the crack, K, increases with increased loading, it may reach the value of Kc, when the balance of elastic energy release from the loaded body exceeds the energy requirement for crack extension. At this point a nmning crack that is known as unstable fracture takes place. The stress intensity factor, K, at the tip of the crack should be kept at a value less than the characteristic Kc of the material under investigation if unstable fracture of the structure is to be avoided. This is analogous to the requirement that the cross-sectional stress must lie below fy if one does not want yielding to occur. There are a number of factors that influence the value of Kc, one of which relates to the thickness of the loaded member. In thick sections generally plane-strain conditions emerge, where it is more difficult for plastic deformation to occur beyond the crack tip. This lowers the value of the material toughness and consequently lowers the plane-strain critical stress intensity factor, which is known as Kic. In thin sections, where plane-stress conditions prevail, crack extension requires more energy in the form of plastic work, and thus the fracture toughness of the material is higher.

2. M E A S U R E M E N T OF THE CRITICAL STRESS INTENSITY FACTORS, Kc A total of 30 notch ~ i m e n s were tested in the J.W. Roderick Laboratory for Materials and Structures at the University of Sydney. The main objective of this phase of the investigation was to detemaine the critical Mode I, i.e. crack opening as opposed to crack sliding (Mode 1/) or crack tearing (Mode m) [4], stress intensity factors, Kc, for 0.42 mm G550 and 0.60 mm G550 sheet steels. Tests were completed to measure the magnitude of the crack tip stress field where ultimate failure was caused by unstable fracture of the notch specimens. The material properties of cold reduced sheet steels have been shown to be anisotropic [2], hence, ~ i m e m were cut from three directiom within the sheet; longitudinal, transverse and diagonal, with respect to the rolling direction.

2.1. Critical Stress Intensity Factor, Kc, Measurement Test Procedure The test specimens were milled to size, as shown in Figure 1, with a notch placed in one edge using two circular cutting blades. A fatigue crack was then initiated and allowed to extend by cyclically loading the specimen in tension (from 4000 to 9400 cycles at 10 Hz) between 8% and 39% of the yield strength, calculated at the net section. Test specimens were milled with only one notch because of the difficulty in accurately machining a notch of identical dimensions on either side of the specimen, and in developing symmetric fatigue cracks. Each test specimen was then loaded to failure under stroke control with a cross-head speed of 0.02 era/rain. The load vs. deflection graph of each specimen was observed to ensure that deformation remained elastic prior to failure. The maximum load, recorded when fast fracture of the sgecimen commenced, was used to determine a Kc value following the method documented in the Compendium of Stress Intensity Factors [5]. The basic test procedure can be found in ASTM E338 [6] and E399 [7], although plain stress conditions occurred, not plain strain, due to the thinness of the sheet steels that were tested. Hence, the calculated critical stress intensity values that were obtained are characteristic of the steels that were tested and not of G550 sheet steels in general.

701

Si~im~

",, ..~y_i

9 I

..................

9

oripLength ,,!

o

.~149 !~ '

'

o

Fatigue--->~." ..,~ Cr~k r---__... ..

-

: (kip t~4th

t~.; 5O

240

Figure 1. Schematic Drawing of Notch Specimen Test Set-Up

2.2. Critical Stress Intensity Factor, Kc, Calculation and Test Results The general stress intensity factor, K~ for a sheet of width b and height 2h with an isolated crack of length 2a, which is subjected to a uniform tensile stress, 3~ can be related to the stress intensity fagtor of an edge cracked sheet where the ends of the test specimen are free to rotate. This relationship can be estimated with the use of the following equations (Brown and Srawley [8]).

/co = f,.F

(2)

Kc = 1.12- 0.23(a/b) + l O.6(a/b) 2 - 21.7(a/b)' + 30.4(a/b f

(3)

ro

These equations are valid in the following range; h/b > 1.0 and a/b _<0.6. The load was assmned to originate at the edge of the grip, thus h = 75ram and h/b = 1.5 for the notch specimem that were tested. The values ofa/b = 0.232 to 0.294 for the m ~ range of fatigue crock lengths. Critical stress intensity factors were c~culated for all of the notch test specim~_s using Eqs. 2 and 3. Of the 30 notch tests, 18 were completed at a t e m ~ of 21.5~ (room t ~ m ~ ) and the remaining at temperatures that varied from 1~ to -21~ The room temperature mean value test results are provided in Table 1, and detailed information for each individual notch slx~imen can be found in Rogers and Hancock [9]. All of the sheet steel types that were tested have critical intensity factors that exceed 3000 MNrn"3a. Ashby [10] associates the failure of materials that are found to have Kc values in the range measmed with the plastic rupttae ductile ~ failure mode. A significant decrease in the ~ toughness of the (3550 sheet steels is not evident for the tramverse direction in comparison with the longimdinfl and diagonal directions. However, the transverse Kc values do fall below the longitudinal and diagonal values for both the 0.42 mm G550 and 0.60 mm G550 sheet steels. The meamaed Kc values are atypically high partially ~ of the thinness of the (3550 sheet steels, which did not allow for plane strain conditions to occur during testing; hence, these values are only valid for the thicknesses tested.

702

Table 1 Mean Measured Kc Values at Room Temperature (21.5~ Material'Type ' Kc' Material'Type Kc & Direction (MNm "3a) & Direction (MNm "3n)...... 0.42 mm G550 Long. 0.42 mm G550 Tran. 0.42 mm G550 Diag. i

3767 3182 3748

0.60 mm G550 Long. 0.60 mm G550 Trail. 0.60 mm G550 Diag.

3551 3260 3743

i

No significant variation in the measured critical stress intensity factors of the G550 sheet steels was observed for the range of temperatures used in testing, i.e. 21.5~ to -21~ [9]. This is an indication that the transition tempemtme from ductile to brittle fiactme behaviour of the G550 sheet steels that were tested lies below the range of temperatures used.

2.3. An Evaluation of Stress Intensity Factors, Kc, Using FRANC2D An analytical study of cracked specimens fabricated from G550 sheet steels was completed to determine the design implications of possible failure by unstable fracture in the elastic load range. Critical stress intensity factors were computed using a finite element model, and then compared with the measured critical stress intensity factors obtained from tests. The FRANC2D finite element computer program, distributed and written by the Comell University Fracture Group, was used because it has been specifically developed for the analysis of crack behaviour. In this analytical study it was assumed that, for the modelled test specimens, rapid unstable fracture resulting in ultimate failure would occur when the stress intensity at the prescribed crack tip reached the critical measured Kc value. This is a conservative assumption that is dependent on the following; 1) once the specimen has reached its ultimate load canying capacity the maximum load does not decrease as the crack size increases, hence rapid fracture of the specimen is not abated, and 2) that loading occurs over a short period of time so that stable crack growth does not occur prior to ultimate failure, i.e. the length of the fatigue crack is not extended by any further crack growth except at ultimate failure. All of the elements that were used in the finite element models had elastic-isotropic material properties, which were defined with the results obtained from coupon tests of G550 sheet steels [2]. A linear analysis method that incorporates a direct stiffness Gaussian elimination solver was used with isopammetric quarter-point elements at the crack tip, as well as quadrilateral 8 node and triangular 6 node second order shell elements for the body of the structure. The crack type was defined as non-cohesive and, when necessary, crack length was propagated manually in the direction given by the maximum circumferential tensile stress around the crack tip.

2.4. Comparison of the FRANC2D and Measured Kc Values for Notch Specimens Models of the notch slxx:imens tested at room temperature were analysed using the FRANC2D finite element computer program to obtain Mode I, i.e. crack opening, critical stress intensity factors, Kc. These numerically calculated critical stress intemity factors were then compared with the mean measured results obtained from the notch specimen tests. The FRANC2D program contains three numerical methods with which Kc values can be calculated; 1) a displacement correlation stress intensity factor method, 2) an equivalent area formulation of the J-Integral, and 3) a modified crack closure integral method. The values of the mean measured critical room temperature stress intensity factors that were obtained using the Brown and Srawley equations (Eqs. 2-3) and the FRANC2D calculated Mode I factors were found to be consistent, as shown in Table 2. No significant loss in the accuracy of the predicted Kc values occurred with the use of any of the three available

703 Table 2 Comparis.on.0 f FRANC2D with Measured R0om..Temp. Kc Va!ues Material Type & Direction

0.42mm G550 Long. 0.42mm G550 Tran. 0.42mm G550 Diag. 0.60mm G550 Long. 0.60mm G550 Tran. 0.60mm G550 Diag.

Elastic FRANC/Meas.

J-Integral FRANC/~eas.

CC-Inter FRANC/Meas.

0.996 0.998 0.998 0.998 0.998 0.997

0.998 0.997 0.997 0.997 0.998 0.997

0.992 0.992 0.992 0.991 0.991 0.991

computational methods. However, the J-Integral method was used for the numerical analysis of all other finite element test specimens that are contained in this paper. Individual numerically calculated Kc values for each of the modelled tests specimens are provided in Rogers and Hancock [9].

2.5. Triple Bolt Connection Specimens Triple bolt G550 sheet steel connections that failed through rupture of the net section (see Rogers and Hancock [ 11]) were also analysed using the FRANC2D computer program. Additional 0.5 mm long non-cohesive cracks were placed on either side of the perforation of the innermost bolt hole at the position of the highest stress concentration, perpendicular to the direction of load, as shown in Figure 2. Quadratic shaped distributed loads were applied to all of the bolt holes along the edge where the bolt and sheet steel were in contact. The ultimate load that was reached during actual testing was used to calculate the necessary load applied to the cross-section. The results in Figure 2 refer to various bolted connection specimens, e.g. 0.60-G550-IT, a 0.60 mm thick sheet steel specimen cut l}om the transverse direction, T, and connected with integral bolts, I. Other connections were composed of a 0.42 mm sheet steel, connected ,afth conventional bolts, C, and cut from the longitudinal direction, L [11 ].

Figure 2.Triple Bolt Model Finite Element Mesh for FRANC2D and Kappvs. Kc Ratios Analysis of these bolted connection specimens using the FRANC2D computer program reveals that the applied stress intensity factors, K~, do not reach the measured critical level, Kc (see Figure 2). These results indicate that failure of the bolted connection G550 sheet steel specimens can be attributed to yielding and ultimate rupture of the material at the net section and not unstable fracture

704 in the elastic load range. The applied stress intensity factors fall short of reaching the critical level for the Iransverse 0.42 mm G550 test specimens. Further increases in the crack length would result in elevated applied stress intensity factors and ultimately, unstable fi-acture of the test specimens in the elastic load range. The high stress intensity levels can be attributed to the greater width of the test piece, greater crack length and more localised load distribution.

3. CONCLUSIONS The Mode I, i.e. crack opening, critical stress intensity factors, Kc, for the 0.42 mm G550 and 0.60 mm G550 sheet steels were deterrrfined for three directions in the plane of the sheet. Single notch test specimens with fatigue cracks were loaded in tension to determine the resistance of G550 sheet steels to failure by unstable fracture in the elastic deformation range. The measured critical stress intensity factors were then used in a finite element study to determine the risk of unstable ~acture for different G550 sheet steel structm~ models. It was determined that the previously tested bolted connection specimens were not at risk of failure by unstable fracture in the elastic zone. Notch specimens were tested over a range of temperatures (21.5~ to -29~ to determine an approximate ductile-brittle lransition temperature for G550 sheet steels. The measured critical stress intensity factors showed that crack resistance did not vary significantly between the notch test specimens that were completed at different temtmatures. These results indicate that the ductilebrittle transition temperature lies below the range of temperatures that was used in testing. ACKNOWLEDGEMEN'I~ The authors would like to thank the Australian Research Council and BHP Coated Steel Division for their financial support. The first author was supported by a joint Commonwealth of Australia and Centre for Advanced Structta-al Engineering Scholarship. The FRANC2D fracture analysis program and CASCA pre-processor were provided by the Comell University Fracture Group.

REFERENCES 1. StandardsAustralia / Standards New Zealand, Cold-formed steel structures- AS/NZS 4600, Sydney, NSW, Australia (1996). 2. C.A. Rogers and G.J. Hancock, J. Strut. Eng, ASCE, Vol 123 No 12, (1997) 1586-1594. 3. FRANC2D, Tutorial and user's guide, Version 2.7, Comell UniversityFracture Group, Ithaca, NY, USA. 4. B. Dodd, Y. Bai, Ductile Fracture and Ductilitywith Applications to Metalworking,Academic Press Inc. Ltd., London, England, UK (1987). 5. D.P. Rooke, D.J. Cartwright, Compendiumof Stress IntensityFactors, Procurement Executive, Ministry of Defence, Her Majesty's StationeryOffice, London, U.K. (1976). 6. American Society for Testing and Materials A 338, Standard Method of Sharp-Notch Tension Testing of High-Strength Sheet Materials, Philadelphia, PA, USA (1986). 7. American Society for Testing and Materials A 399, Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials, Philadelphia, PA, USA (1983). 8. W.F.Brown, J.E. Smwley, STP 410, American Society for Testing and Materials (1966). 9. C.A. Rogers and G.J. Hancock,Tensile fracture behaviour of thin G550 sheet steels, Research Rept. No. R773, Dept. of Civ. Eng. Universityof Sydney, Sydney,NSW, Australia (1998). 10. M.F. Ashby, Prog. Mat. Sci., Chalmers AnniversaryVolume, (1981) 1-25. 11. C.A. Rogers and G.J. Hancock, J. Struc. Eng, ASCE, Vol. 124,No. 7, (1998) 798-808.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

705

Fatigue Strength Properties of Stainless Clad Steel T. Mori

Department of Civil Engineering, Hosei University 3-7-2 Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan

This study aims at making clear the fatigue crack propagation rate and fatigue strength properties of stainless clad steel composed of steel and stainless steel, which possess different mechanical properties. For this purpose, fatigue crack propagation tests, fatigue tests of cruciform welded joints, stress intensity factor analyses and fatigue crack propagation analyses are carried out.

1. INTRODUCTION A steel struc~e has the advantages of high strength, lightweight and so on compared with a concrete structure, but it is said to be apt to suffer corrosion and fatigue. The use of stainless clad steel is one of useful methods to reduce the corrosion problem of a steel structure. Stainless clad steel is composed of two materials; they are steel and stainless steel that possess different tensile strength, Young's modulus, and other mechanical properties. It is therefore impossible to apply directly the strength properties of steel or stainless steel to stainless clad steel. In this study, fatigue crack propagation tests, fatigue tests of cruciform welded joints, analyses of stress intensity factor and analyses of fatigue crack propagation are carried out in order to make clear the fatigue strength properties of stainless clad steel or clad steel composed of other material to use it as a structural material.

2. MATERIALS AND SPECIMENS Three types of materials were utilized, which were structural steel JIS SS400 of 9ram thick, stainless steel JIS SUS316L of 9mm thick and stainless-clad steel of 10.5mm thick (SUS316L:2mm + SS400:8.5mm). Mechanical properties and chemical compositions of those are shown in Table I. Specimens used for examining the crack propagation rates were rectangular plates with single sharp notches as shown in Figure I. In these specimens, TIG-dressing was done along the centerline of the specimen in order to introduce the tensile residual stress. The residual stress measured in the center of the specimen was about 350MPa in the SS400 specimen and 200MPa in the SUS316L specimen. In the clad-steel specimen, tensile residual stress was also observed, whose value was about 400MPa on the surface of SS400 and about 10OMPa on the surface of SUS316L. Two types of cruciform joint specimens were fabricated as shown in Figure 2. The non-

706 Table 1 Chemical comlxmitions and mechanical properties material

~,

SS400 SUS316L

CUd

Steel

Mateiml

ss400 SUS3 i6L

C 0.12 0.02

o.14 0.02

Si 0.20 0.57

Mn 0.65 1.01

o.ts

0.76

0.57

1.01

Ova,s)

SUS316L

292 252

417 551

0.271 0.293

Clad ~eel

297

sls

0.248

SS400

Cr 0.04

}do

Cu 0.01 -

Ni 0.02 12.20

17.5o" 2.~

0.03 "'

-

12.20

17.50

~)

M~

'o.oi

ratio

Ooa)

P 0.01 0.03

16.5 29.4 21.7

-

-

,,

2.09

Young's

(OPa) 206 186 202

l . l x l O "~

" 'l.6x ld"

'

load-cam3~g type of joint was fabricated by fillet weld and the load-c,arrying type of joint was done by groove weld with full penetration. The welding was done using CO2 arc process.

3. F A T I G U E C R A C K P R O P A G A T I O N R A T E S

Fatigue crack pcopagafion tests were carried out mskr constant amplitude loading and decreasing load as a crack extension in order to measure a wide range of crack propagation rates. A traveling microscope of 50 magnifications with an accmacv of 0.01 mm was used for the measurement of crack length. In these tests, minimmn load was set at lkN. In the load decreasing tests, the load ranges were reduced stepwise by about 5% every crack extension of about 0.25ram In these tests, fatigue crack opening and closing behaviors were measured, and the crack did not close in all the tests because of tensile residual stress induc~ by

TiG-dre~tng

Results of the crack propagation tests are shown in Figures 3, 4 and 5, in which the relationships between fatigue crack p m ~ o n rates da/dN and stress intensity factor ranges

707

~

SS400

1o_6

SUS316L

.

G

"4~

'"

l

~o-~ .

.

m

~

:

~

"

10-l~

JSSC (steel)

.i i

10 100 stress intensityfactorrange AK(MPafm) Figure 3. da/dN- A K relationship

"l

10-6" "

~o - s .

Clad steel o SUS316L ~ SS400

!

F" .

.

'

9

~

JSSC

(steed

-

10 100 stress intensityfactorrange AK(MPa4-m) Figure 4. da/dN- ~ K relationship

1

lo4!

' :

'

JSSC (~eel)

9 1

i

"f n SS400 a SUS316L 10-6 o clad steel(SUS316L) , s clad steel(SS400)

"

~ 10-1~

'

~ 10_10'

-

1

~

.

10

stress immmityfactorrange AK~ f - l n ) Figure 5. da/dN-A K relationship

"

10-lo JSSC

"

9 100

l~

10~

strmnintensityfactorrmge AK/E(~m) Figure 6. da/dN~A K/E relationship

AK are indicated. In Figure 3, da/dN-AK relationships of the structm~ steel (SS400) are indicated. Solid line in this figure is da/dN-AK relationship of steel specified in "Fatigue Design Recommendations of Steel Stmc~es" by the Japanese Society of Steel Construction (JSSC Recommendations)[1]. The relationship in the JSSC Recommendations was derived from a large number of test results on steel. The da/dN-AK relationships obtained here are plotted on the line of JSSC Recommendations. The relationships of the stainless steel shown in Figure 4 is almost the same as that of the steel in the region of comparably small AK, but the da/dN of the former is slightly higher than the latter in large AK region. As shown in Figure 5, the da/dN-AK relationship of the clad steel is almost the same as that of the steel. As one of the reasons why the da/dN of the stainless steel is high compared with the structural steel, it is considered that Young's modulus of the stainless steel is lower than that of the structural steel. That is, the da/dN is depend on the strain range at the crack tip, so it

708 might be arranged by the strain intensity factor range (AK/E, E : Young's modulus). For example, it was indicated that da/dN-AK relationships of aluminum alloy, stainless steel and carbon steel which had different Young~s modulus were scattered dependent on the material, but da/dN-AK/E relationships of these materials are almost the same as each other [2]. All the test results ananged by strain intensity factor range AK/E are plotted on Figure 6. The da/dN-AK relationship for steel ~ i f i e d in the JSSC Recommendations is expressed as equation (1). da/dN = 1.5 x 10"II(AK 2"75 --AKth2"75) A Kth = 2.9 MPa~fi~,

(1) da/dN" m/cycle,

AK" MPa

This equation can be transformed into equation (2) in consideration that Young's modulus of steel is 2.06 x 105 MPa. da/dN = 6.15 x 103[(AK/E)2.75 __(~)th2.75 ] (AK/E)th = 1.41 x 1 0 s ~ ,

(2) da/dN:m/cycle,

~ d E : .fro

Solid line in Figure 6 is the da/dN-AK/E relationship expressed by equation (2). All the test results obtained here are plotted on the solid line regardless of the material.

4. FATIGUE STRENGTH OF CRUCIFORM JOINTS Fatigue tests were performed on 2 types of cruciform joints (see Figure 2) using

Electro-hydraulic servo system testing machine with dynam/c capacity of 500kN. Cyclic loads were pulsating tension of 10kN in the minimum load. Figure 7 shows an example of failure staface of non-load carrying type joints. Fatigue crack originated from the weld toe on the stainless steel side of joint. The crack form is semi-elliptical and it does not change at the boundary between the stainless steel and stmcUnal steel. The fatigue crack in the load carrying type of joint initiated and propagated from the stainless steel side of the joint The test results are shown in Figure 8. The ordinate is stress range Ao and abscissa is fatigue life N. The two broken lines in the figure show regression lines for non-load carrying and load carrying type of joints. The fatigue strength of both of the joints is almost the same. In Figure 8, Ao-N relationship specified for these types of joints by JSSC Recommendations is also drawn as a solid line. All the fatigue test results obtained here satisfy the specified relationship.

5. STRESS INTENSITY FACTOR ANALYSES AND FATIGUE CRACK PROPAGATION ANALYSES The finite element analyses were carried out for a non-load carrying type of cruciform fillet welded joint shown in the Figure 2. A Yotmg's modulus of steel was set at 2.06xl05N/mm 2, and a Young's modulus of clad material was set at 0.3, 0.6, 0.9, 1.0 and 1.5 times of the steel.

709 Hereafter, these ratios will be called as the ratio of Young's modulus. The ratio of 0.9 corresponds with the stainloss clad ster The elom~nt size was made to be 0.1mm, and a Poisson's ratio of steel and clad material was set at 0.3, then the analyses were performed under the plane strain condition. The fatigue crack from the weld toe was simulated by setting a gap of 0.01mm width there. Stress intensity factor K was calculated on the basis of the energy release rate g, which was obtained by calculating the load point displacement at each 0.1mm of crack extension. The g-value can be transformed to the K-value using following equation.

K-4E'.gl(I.-v2 )

(3)

E, v : Young's modulus and Poison's ratio of the material where the crack tip exists Since the Young's modulus of clad material and steel is different, the stress intensity factor of clad steel is also different from that of unclad steel. It was considered that this difference could be expressed using correction factor Fc. That is to say, the stress intensity factor of the clad steel can be obtained by multiplying the Fc-vahe to the stress intensity factor of the unclad steel. The Fc-values obtained here are shown in Figure 9. In this figure, Es and Ec is a Young modulus of steel and clad material, and Ec/Es is the ratio of Young's modulus. When a ratio of Young's modulus is smaller than 1.0, the correction factor Fc is also smaller than 1.0, and its minimum value occurs at the boundary between the steel and the clad material, and it becomes almost equal to the ratio of Young's modulus. The value of Fc gradually increases afar the crack enters the steel, and it becomes 1.O at about 5mm of the crack length. The tendency in the case of the ratio of Young's modulus being bigger than 1.0 is the reverse of the above case. The analytical result of the correction factor Fc in generating fatigue crack from the steel side is shown in Figure 10. The value of Fc increases linearly as crack becomes large when

710

,

,

.

,

9

.

.

.

.

.

.

I

1.2

I

1.2

.]

~.

m

/a

I

m

"~, "=.--Z.'._.~ o

,/" /

o

o Ec/Ee, O.3

0.8

0

o Ec/Es=0.6 a EcJEs=0.9 A EcJE$= 1.5 2

4

crack length a (ram)

6

Figure 10. Fo-values for crack from the steel

~

1[

/ 0

.

[

/

8

]fatigue crack from

./~"

[ ~ clad surface I ~ steel surface

/-

9

=O.e

I

i

! ~

9

05.

t

I

I

I

!

. i1

i

i

9

1

ratio of Young's modulus

!

1.5

Figure 11. Fatigue strength analyzed

the ratio of Young's modulus is larger than 1.0, and decreases when the ratio is less than 1.0. The Fc-value is 1.0 at 4.0mm of crack length regardless of the ratio of Young's modulus. For the case in which the fatigue crack was generated from clad material side or steel side of cruciform welded joint, fatigue crack propagation analyses was carried out in consideration of the correction factor Fc and the expression of fatigue crack propagation rates indicated in equation (2). Then, the relationship between fatigue strength at 2 million stress cycles and the ratio of Young's modulus was obtained. The result is shown in Figure 11. The vertical axis of the figure is the fatigue strength normalized by the fatigue strength of unclad steel. The fatigue strength in case of generating fatigue crack from the clad material side lowers as a ratio of Young's modulus is smaller, and the fatigue strength in case of originating the fatigue crack from the steel product side has the reverse tendency.

6. CONCLUSIONS (1) Fatigue crack propagation rates of stainless clad steel are the same as those of conventional steel if the rates are arranged by strain intensity factor ranges. (2) Fatigue strength of welded clad steel is almost the same as that of welded conventional steel. (3) When Young's modulus of clad material is smaller than that of conventional steel, fatigue strength of weld clad steel becomes low as the modulus becomes low. REFERENCES

1. Japanese Society of Steel Construction : Fatigue Design Recommendations for Welded Structures, 1995.12 2. Ohta, A., et.al. : An influence of Young's Modulus on Fatigue Crack Propagation Rates, Proceeding of the National Meeting of Japan Welding Society, No.61, pp.362-363, 1990.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

711

Testing of Welded T-Joint with Fatigue Cracks and Comparison with Failure Assessment Diagram T. lwashita a, Y. Makino b, K. Azuma c and Y. Kurobane c aDepartment of Architecture, Ariake National College of Technology HigashiHagio-Machi 150, Omuta, Fukuoka, JAPAN bDepartment of Architecture and Civil Engineering, Kumamoto University Kurokami 2-39-1, Kumamoto, JAPAN CDepartment of Architecture, Kumamoto Institute of Technology Ikeda 4-22-1, Kumamoto, JAPAN Brittle fracture was reported for many welded oonnections in the Kobe Earthquake. Post earthquake investigations revealed that failure of connections were caused also by crack growing from weld defects. The plastic deformation capacity of connections was found to be decreased by occurrences of brittle fracture from these defects. Therefore it is necessary to evaluate quantitatively the effect of weld defects on the performance of the joint and determine acceptance standard of weld defects. It is important to establish rational criterion for evaluation of weld defects because the inspection of welded structure is greatly dependent on ultrasonic testing at present. In our study, the final goal is to establish reject criteria of weld defects to prevent brittle fracture. The welded T-joints with fatigue cracks at weld toes were tested under two loading methods as shown in Fig. 1 (a), (b). Welded T-joints 1) showed large-scale yielding around the weld defect, and sustained brittle fracture and/or ductile tear after showing significant ductile crack growth. Test results for cracked welded T-joints were evaluated on the basis of the failure assessment diagram (FAD) according to BSI PD64932). 1. SUMMARY OF FAILURE ASSESSMENT

1.1. Procedure of Failure Assessment Assessment procedures of weld defects adopted here are as follows: 1) Determination of assessment curve 2) Calculation of plastic collapse parameters 3) Calculation of fracture parameters The defect is assessed by evaluating the fracture and plastic collapse parameters and plotting the corresponding points on the FAD. Ductile crack growth was ignored in the calculations shown below. If a plotted point exists within the assessment curve, the defect is acceptable. If a plotted point is outside the assessment curve, the defect is unacceptable.

1.2. FAD Based Assessment Curves The failure assessment criterion curve in BSI PD6493 is given by the follow equations:

s~

-Jr -K2

(1)

712

cot

s~pmnnlo~k

e, ,ne

~:~"~Rib

//i".-

Main P k z ~

".

Plate

/IV"

Plate

Marl e Block

Def~ Angle 0

@Po (a)

Po "Jr

DeformationAngle 0

Loading Method I (b) Loading Method II Fig. 1 Loading scheme and mechanical model for welded T-joints

Two parameters, Sr and Kr, are used to assess the susceptibility of a crack to brittle fracture. The fracture parameter Kr is the square root of Jr, and Jr-~JJ is the ratio of the crack tip driving force to the fracture toughness. Jc is Elastic J-integral. The plastic collapse parameter Sr is the ratio of the applied load P and the plastic collapse load Pu and taken as Sr=P/Pe. The assessment curve is obtained by two parameters, Sr and K~. The following equations are obtained by Level2 approach in BSI PD6493.

o/oy

s~

-

-

o" O'y -

(3)

In the assessment curve of Level2 in BSI PD6493, the plastic collapse parameter, a / a y, is in the range of 0-1, which is inappropriate on material with high work-hardening. The modified failure assessment curve according to Level3 in BSI PD6493 considered the effect of high work-hardening of material, which is based on the ratio of the ultimate tensile strength of the material, a u, divided by the yield strength of the material, a y. Therefore, Kr for Level3 is redefined by following equation: K r -(1 + 0.14S 2){0.3 + 0.7 exp(--0.65S~)}

(4)

where the maximum value of a for Sr is % (i.e. ( a u+ a y)/2). The assessment of the FAD is attempted using these two assessment curves. 2. FAILURE ASSESSMENT USING FAD 2.1. FAD compared with Test Results The elastic J-integral, Je, was calculated by using a nonlinear FE analysis because it was impossible to calculate the stress intensity factor at defect tips in the case of welded T-joints under combined M and N loads. The FE analysis was performed by using ABAQUS3) on models constructed from 8-node biquadratic plane strain 2D elements. The plasticity of materials was defined by the yon Mises yield criteria. The stress-strain curve used for FE

713 analysis was determined by tensile coupon test of base material used. The isotropic hardening parameter was used. K, is the ratio of the crack tip driving force to the fracture toughness and it is obtained to calculate the ratio of Je to Jic obtained by Single Edge Notched Bend (SENB) test according to the JSME standard 4). Sr is the ratio of the maximum load Hf to the plastic collapse load Hu of cracked T-joints. Kr and Sr are calculated by the following equations:

K,- JI-i~,~

(5)

S, = Hf Hu

(6)

The plastic collapse load is calculated by means of simple plastic analysis, incorporating P-8 effects. Stresses on the cross section of the main member are assumed to be uniformly distributed on both tension and compression sides and equal to • (rectangular stress block assumption). Fig. l(a) and (b) represent mechanical models for Loading Method I and II respectively. The plastic collapse load is given by: n , = 2(N sin 0 + M 1 +M2 cos 0) l

(7)

In Eq.7, N and M signify the axial and bending loads in the main plate, respectively. The subscripts 1 and 2 denote the positions on the main plate as shown in Fig. 1. Note that fatigue cracks exist at the position 2. From the rectangular stress block assumption, N = 2xB Oy

(8)

M Bt2-~yo_4x2 " 4

-~-)

(9)

where x is the distance from the central axis of member to the neutral axis and shown in Fig. 2. When Loading Method I is adopted, x can be calculated from equilibrium equations because Po is known. According to the lower bound theorem of simple plastic analysis, the maximum value of Hu at the ultimate limit state for Loading Method II is given when neutral axis moves to the outside of the cross section of the main member. Therefore, H,, = 2N sin 0 = 2BT 9' try sin 0 crack

9'=1 gross

area

sectional

(10) (11)

area

V is the area reduction factor. Where the crack depth was defined as the maximum depth of m

% Fig.2 Stress Distribution across the thickness

714 fatigue cracks. Table 1 shows test results and the depth of fatigue cracks. Fatigue cracks were measured using a low power magnifying microscope after testing was completed. "Brittle & Ductile" in Table I means they occurred on each side at the same time. Fig. 3 compares test results with the FAD. All the data points except one fall outside of the assessment curve, far above the curve. In fact, these specimens sustained sufficient plastic deformation, and then reached the ultimate strength. This means that ductile welded joints can stand the load rising well into the unsafe region. One specimen that are plotted close to the assessment curves sustained brittle fracture following plastic deformation smaller than the deformation sustained by the rest of the specimens (sin0=0.245). This may suggest that FAD is applicable to welded joints that show brittle behavior. When plotting test results, the value of J~c used to calculate the K+ parameter was that determined by SENB test. SENB specimens may be subjected to much grea~r plastic constraint at the notch root as compared with notch roots of surface cracks in welded T-joint specimens. The literature 5) reported the value that the fracture toughness measured by SENB test is five times the fracture toughness observed by wide plate test. Thus, it is possible that evaluated K~ values become too large, over-emphasizing effects of brittle fracture on ultimate collapse. The effect of the plastic constraint on welded T-joints was not considered because the precise degree of the plastic constraint is not clear. Ductile crack was observed in the test for welded T-joints, but effects of ductile crack growth were not considered for the FAD in this paper yet.

2.2. FAD compared with IrE analysis results The failure assessment using FE analyses was conducted at the load when the value of elastic-plastic J-integral by FE analysis reached J~c. Kr is calculated by Eq.5, where elastic J-integral Jc was determined by elastic IrE analysis at a load when J according to the elastic-plastic FE analysis reached hc. Sr was evaluated as the ratio of the load at which J reached JIc to the plastic collapse load on an M-N interaction diagram (See Fig. 4). Namely, S~ is defined as the ratio of two radiuses from origin. The one radius reached the point on the curve given by FE analysis where J= J~c while the other radius reached the points at which the former radius crosses M-N interaction curve. M-N interaction curve was constructed on the assumption that the distribution of stresses on the cross section of main member follows the rectangular stress block assumption as described previously. 1.2

Table I Specimen Material

.

C22ssl SS400 C22ss2 SS400 C22ss3 SS400 C22ss4 SS400 C22ss5 SS400 C26ssl SS400 C26ss2 SS400 C26ss3 SS400 C26ss4 SS400 B26snl SN490B B26sn2 SN490B B26ssl SS400 B26ss2 SS400 B26ss3 SS400 .

.

.

.

.

Test results

t Depth (ram) (ram) 21.87 1.2 21.92 0.7 21.92 0.5 21.93 0.8 21.93 1.8 25.92 1.9 25.98 0.6 25.93 0.5 25.97 0.6 25.99 13.0 25.97 11_3 26.01 14.0 26.02 9.5 26.05 10.4 .

.

.

.

.

H (kN) 518.0 492.9 527.5 502.8 502.6 612.9 621.0 616.9 618.6 361.8 482.9 389.3 493.3 472.2 .

sinO

Failure Type

0.52"7 0.458 0.437 0.487 0.481 0.479 0.478 0.477 0.470 0.245 0.360 0.337 0.377 0.367

Brittle & Duc~'ile Brittle & Ductile Brittle & Ductile Brittle & Ductile Brittle & Ductile Brittle Brittle & Ductile Brittle & Ductile Brittle & Ductile Brittle Ductile Ductile Ductile Ductile

0.8

.

.

0.6

I

Test1

|

A Test3

0.4

L

0.2 0

--

0

'~'U 1 I~ t

l

,t.t4 i

x

,

0.5

i,

1

1.5

2

Sr

Fig. 3 FAD compared with test results Level2 & Level3: Assessment Curves Testl:C22ssl-5 Test2:C26ssl--4 Test3:B26sn 1,2 Test4:B26ss 1~3

715 Fig. 5 (a) and (b) show the FAD compared with test and FE analysis results for the two loading - - M-N Curvel methods respectively. It is seen that FE analysis results agree with the assessment curve. Table 2 shows the load H according to FE analysis plotted in Fig. 5. The notations of 1.5ram ~6.0mm in Table 2 denote the fatigue crack depths assumed in the FE analysis. Test results are inconsistent with FE results on the FAD because test results are apart from FE results in the maximum values of the load M H as shown in Fig. 6. It was assumed in the failure Fig. 4 M-N interaction diagram assessment of test results that the crack driving force reached J~c when specimens reached the maximum values of H. In the failure assessment of FE analysis results, the load H was assumed to have reached the maximum values when the elastic-plastic fracture toughness J according to FE analyses reached J1c. These maximum loads are significantly lower than those from tests. Both of the tests and the FE analyses used Jr the elastic J-integral from FE analysis results for the failure assessment. Accordingly, J~ of the test was high and J, of the FE analyses was low. Even if fatigue crack depth is deeper than 6.0mm in FE analysis, results on the FAD are not so different, though the deeper fatigue crack depth is, the lower the load when J reaches J1c is. This is another reason why test results did not agree with FE analysis results. It is necessary to consider two problems to apply the FAD. First, FE analysis with crack growth analysis was not performed in the failure assessment. Second, in the case of the failure assessment of test results, the value of J~c used to calculate the Kr parameter was determined by a SENB test. SENB specimens may be subjected to much greater plastic constraint at the notch root as compared with notch roots of surface cracks in welded T-joint specimens 5). If that value, factor of 5, as described previously, was multiplied to Jtc, the test and FE analysis results would get closer. However, the effect of smaller plastic constraint is still not clear in the case of welded T-joints. Further experimental verifications are required to enhance the reliability of the FAD approach.

1.2

1.2

"

A&

0.8

0~ Levi2 --Levi3 [] Testl II Test2 X l_~mm X 3.0ram O 4.~mm + 6.0ram

0.6 0A 0.2

0

0.6

I i 0.4 t

I

0.2

0.5

1

Sr

(a)

1.5

2

L

0

L~ Test3 t A Test4 X 1-Smm

]\x I \

o

]

x

3.omm

I

4~mm

0.5

1

1.5

Sr

(b) Loading Method 11 Loading Method I Fig. 5 FAD compared with test and analysis results

2

716 Table 2

H load of plotting by FE analysis

Fatigue Cracks

lo5mm 3.0ram 4_~mm 6.0ram

H load (Method I ) 171(kN) 131(kN) 103(kN) 89(kN) H load (Method lI) 189(kN) 166(kN) 152(kN) 140(kN)

7 0 0 . 600 500 ~400 " m 300

I '..

Fig. 6

.

.~1 ~-~ '

200 . z " ~" 100 / ./-m q ~ J I c 0

'

..

,, ~,o*

....

iTest ~ ,-' t~"' FIEA n a l y s i s ~ ." ,,. ~ Ap0 ~ t~ - Jlc for Test

0.I

....

for FE Analysis 0.2

0.3

0.4

0.5

sin 0 Comparison of FE with test

3. CONCLUSIONS The conclusions drawn from this research can be summarized as follows: 1) Test results for welded T-joints with fatigue cracks showed that the failure assessment curve is overly conservative when the joints failed after extensive plastic deformation. A test result for only one specimen, which failed in a less ductile manner, was found to be close to the assessment curve. 2) SENB specimens may be subjected to much greater plastic constraint at the notch root as compared with notch roots of surface cracks in welded T-joint specimens. Thus, it is possible that evaluated Kr values become too large, over-emphasizing effects of brittle fracture on ultimate collapse. 3) FE analysis results agree with the assessment curve of the FAD, but FE analysis results were inconsistent the test results on the FAD. It is necessary to consider effects of the plastic constraint on welded T-joints and also to include effects of ductile crack growth. Further experimental verifications are required to enhance the reliability of the FAD approach. ACKNOWLEDGEMENTS This project was partly supported by the Japanese Ministry if Education Grant-in Aid for Scientific Research under the Numbers A07555178, C10650581. REFERENCES 1. Iwashita T., Azuma K., Makino Y., Kurobane Y., (1999). "Fracture from Fatigue Cracks at Weld Toes of Plate-to-Plate T-joints," International Society of Offshore and Polar Engineers, Vol.lV 1999, pp.90-96 2. BSI. BSI PD6493, (1991). "Guidance on Methods for the Assessing the Acceptability of flaws in Fusion Welded structures" 3. ABAQUS, (1999). ABAQUS v5.8 Manuals (User's Manuals I andlll), Hibbit, Karlsson and Sorensen, Inc. 4. JIME S 001, (1981). "Standard Method of Test for Elastic-Plastic Fracture Toughness Jlc" 5. Minami F., Ohata M., Toyoda M., Afimochi IC, (1997). "Determination of Required Fracture Toughness of Materials Considering Transferability to Fracture Performance Evaluation for Structural Components, -Application of Local Approach to Fracture Control Design-," J1. Naval. Arcit. Japan, No.182, pp.647-657

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

717

Crack surface contact under alternating plasticity C. H. Wang and L. R. F. Rose Aeronautical and Maritime Research Laboratory, DSTO, 506 Lorimer Street, Fishermans Bend, VIV 3207, Australia The opening and closing of a crack subjected to alternating plasticity is analysed, with a view to characterising the closure behaviour of short fatigue cracks under cyclic plasticity. Elasticplastic finite element analysis has been conducted where a non-linear kinematic hardening constitutive law is used to model the material's cyclic plasticity. Both crack-centre opening displacement and crack-tip opening displacement have been determined under fully reversed cyclic loading, with the maximum applied strain being up to ten times the material's yield strain. Under cyclic loading conditions, the crack opening displacement at the maximum stress has been found to increase with the number of cycles, approaching an asymptotic limit that is more than twice that corresponding to the first half cycle. An analytical model has been developed to characterise this transient ratchetting in the crack opening displacement.

1. INTRODUCTION Short fatigue cracks have been found to grow faster than long cracks for the same applied stress intensity factor (Miller,1987; Shin, 1994; Wang and Rose, 1999). The lack of or low level of plasticity-induced crack closure has been considered as the main cause for this anomaly, as supported by some experimental observations. The objective of the present study is to quantify the opening and closing of fatigue cracks under cyclic plasticity. For simplicity, an ideal crack without plastic wake is considered first; the effect of plasticity-induced crack closure of a growing crack will be addressed separately. 2. PROBLEM DESCRIPTION

The problem to be considered is a centre crack of half length a in an infinite plate subjected to a remote cyclic load, as illustrated in Fig.1 (a). The material is assumed to obey the non-linear kinematic hardening constitutive relationship by Armstrong and Frederick (1966). It is noted that the conventional power-law hardening model, which has been widely employed in the study of crack problems, is unable to simulate the cyclic deformation behaviour under nonproportional, multiaxial loading. Furthermore, the power-law hardening relationship predicts an indefinite strain hardening: stress continues to increase monotonically as the strain increases. By contrast, the non-linear kinematic hardening model overcomes both these deficiencies. More detailed description of the theoretical framework of this non-linear kinematic hardening model can be found in (Chaboche, 1986). In the special case of uniaxial loading, the centre of the yield surface evolves according to the following the equation, = (C-),x)p (1)

718

where the parameters C and 7 denote a material's strain hardening constants, and the parameter p denotes the plastic strain. The variable X is the back-stress, representing the centre of the yield surface, i.e., a = o"o + X (2) where o"o denotes the yield stress of the material. The relationship between the stress and the strain can be obtained by integrating equation (1),

o;[

e=--+

c 1

In

e

c-

(,,-,,o)r

(O'>O" 0)

,,

(3)

9 9 I

. . . . .

~yy -- O'**

tt

tt

tt

tt

tt

o

y

~

/

/

]

o-

l

h

" [:C=83MPa -231

i-

~ "0"5~

.v x

/

/

;

1/

//,=2.45

TI--'3"76

1

1

2.

(a)

-1.s

17

-4

'

"-~, '

....

-2

i . . . , , .(b.)]

0

2

4

Fig. 1 A centre-cracked plate subjected to cyclic stress; (a) crack geometry and coordinate, and (b) cyclic deformation behaviour of the material. It is apparent that as the strain increases the stress asymptotically approaches but never exceeds a maximum saturation stress, i.e., lim o" = a 0 +CI7. (4)

e---~ne

The strain given by equation (3) can be normalised,

k=~+~-l~ t~o

cro

r/

I 1-(or/or o-1)71

9

(or >

cro)

(5)

where tzo = cr0 / E, ~ = E / C, 71 = cr07/C. An example of the non-linear kinematic hardening law (5) is shown in Fig.1 (b). 3. FINITE ELEMENT ANALYSIS Finite element analysis of the centre crack problem shown in Fig.l(a) was performed using ABAQUS, with the material obeying the above described non-linear kinematic hardening model. The applied load was fully reversed with the maximum strain being 10 times the material's yield strain. Crack face inter-penetration was avoided through the use of compression-only springs positioned between the crack faces. Emphasis will be placed on two crack opening displacements: crack-centre opening displacement (COD) and crack-tip opening displacement (CTOD); definitions of these two quantities are depicted in Fig.2.

719 Y

-a

a

(a) Co) Fig.2 Definitions of (a) crack-centre opening displacement and (b) crack-tip opening displacement. The results of the opening displacement at the centre of the crack are shown in Fig.3(a). The arrows indicate the direction of loading. It is seen that, upon the first unloading, the crack surfaces would come into contact at a strain level just below zero. However, a significant increase in the maximum crack opening displacement is observed when the crack is reloaded to the maximum strain. Furthermore, crack surface contact then occurs at a much lower strain level. The progressive increase in COD seems to asymptote after three cycles, when the crack closure level approaches the minimum stress in the cycle, as shown in Fig.3(a). The solid curve shown in the figure represents the predictions of a theoretical model presented later. The asymptotic value of the COD seems to be approximately twice the COD corresponding to the first half cycle. A similar trend can also be observed for the crack-tip opening displacement as depicted in Fig.3(b). The CTOD seems to asymptote after about three cycles. Unlike the COD, the saturation value of the CTOD appears to be two and half times that of the first half cycle. Predictions of a theoretical model to be presented later are represented by the solid curve. 100 . orj o

E

9

.

.

,

!

.

. .

.

.

.

!

.

.

.

.

!

.

.

.

.

Symbols:IrE results

.~"0

t

80

o u

"~

(b)

60

.E o

~" 0

41) 20

"~ o

z

~

D

f

w'"

.o... ~

.

20

0 -1(

-5

0

5

Normalised applied strain r

10

..... ; ..... x Applied strain

l0

de 0

Fig.3 Crack opening displacement under cyclic plasticity (a) crack-centre and (b) crack-tip.

720 4. O R T H O T R O P I C ELASTIC SOLUTION 4.1 Monotonic Loading Under monotonic loading, an analytical solution of the problem shown in Fig.l(a) can be obtained by extending the perturbation solution by He and Hutchinson (1981) developed for power-law hardening materials. The essence of the perturbation method is to treat the problem of a fully plastic crack in an isotropic solid as an elastic crack in an orthotropic solid subjected to an infinitesimal stress increment; the elastic properties of the orthotropic material can be determined from the incremental stress-strain relation. Assuming that the deformation throughout the cracked body is proportional, the deformation theories of plasticity would apply. In this case, the increments in deviatoric stress and strain are related via the incremental modulus L, ku = L#eg,j (ca = 0) (6)

where the incremental moduli LUktfor non-linear kinematic hardening material are, after some algebra manipulations, Lu,, = 2 .

(t~,,t~,,+t~ut~s,)-3~u~ ~ + 2-= 1 - n euene :2} 3 n

with

n--

e,,,, dtr,,~

=

oe,~ [1E+

(7)

1]

(8)

C - (G ,q - Cro ) 7

l Creq # = -----

(9)

3 eeq

The parameters tr.q and e.q denote the equivalent stress and equivalent strain, respectively, which are related through equation (3). The value of exponent n for a particular hardening material ( C = 0.1E, 7' = C / cro ) is shown in Fig.4(a).

50

,

|

C/E=0.1

~

C/~o=1

=

10

I~

/ / A i t r e s s exp~

o

"~ 20

n

//C

B

~

5

m

0

l= O M

10

.=

9

\

\/

fC/~~

9

1.0

,I

i

1.2

I

1.4

J

I

1.6

I

I

1.8

Equivalent Stress O=q/a~

2.0

~ -10 I "=~ <

\/

Applied strain

Time

(a) (b) Fig.4 Stress exponent for a non-linear kinematic hardening material, (a) under monotonic loading and (b) cyclic loading.

721 It is clear that the parameter n is not a constant but varies with the applied stress, contrary to the case of power-law hardening material where n is a material constant. Nevertheless, the crack-centre opening displacement and the J-integral can be obtained using the perturbation solution method of He and Hutchinson (1981), under plane strain conditions, o'** C O D = ~ n 4~t =2~ne~q.pa (10) J =~a~n (cr*'y= tra~ncr~qe~.p 4/.t

(11)

where tr~q and e~q.p denote the applied equivalent stress and plastic strain. The contributions of the elastic strain can be readily included by means of the superposition principle,

COD=4e**a + 2~ne~.pa

(12)

J= (l-v2)~a (or**) 2 + ~ra~rn(r~qtZ~q.p (13) E From which the crack-tipopening displacementcan be determined (Wang and Goldstraw, 1999), J CTOD . . . . o~o-r

(14)

where ct = 2 / ~ and tr r = tr 0 + C / y . The above solutions for the first half cycle are also shown in Figs.3, indicating a reasonably good correlation.

4.2 Cyclic Loading and Crack-Surface Contact The solution developed in the previous section can be extended to the case of cyclic loading, provided the crack surfaces do not contact during unloading. The influence of crack surface contact will be addressed later. To facilitate the following analysis, relative stress and relative strain are defined as the absolute values of the difference between the instantaneous stress or strain and their values at the previous turning point (Wang and Brown, 1994),

<.

"

,

* =lei, - e,; I

(15)

By integrating equation (1), the following relation between the equivalent relative strain and equivalent relative stress can be obtained (Wang and Rose, 1998),

e',q = ~cr"q + _l ln C + y( X R,q + tr,q" -2O'o)

(16)

where the positive sign "+" corresponds to unloading and the negative sign "--" corresponds to re-loading. Solutions pertaining to the cyclic loading can now be obtained by re-casting the solutions derived in the previous section in terms of the relative stress and relative strain. It is interesting to note that the maximum stress exponents corresponding to subsequent turning points, B and C in Fig.4(b), are greater than the stress exponent for the first half cycle, point A. Recalling the expressions for COD and CTOD, it is therefore clear that the observed increase shown in Figs.3 is partly due to this increase in the stress exponent. For fully reversed loading, crack surface contact would occur. For example, when a crack in an elastic material is subjected to a fully reversed cyclic loading, crack surface contact would

722 occur at zero stress. In this case, the cyclic crack opening displacement and the cyclic Jintegral can be obtained by equating the perturbation stress to the effective stress range, ACOD = 4----7--At:r'U"+ ~n Acr'U`,41.1 = ~Ae'a + 2~nAe~,va]U`,

(17)

AJ =--~

(18)

4#

where U`, denotes the effective stress ratio,

U`, =

(19) Act Here the crack opening stress is taken to be the same as the contact stress during previous unloading. As can be seen from the comparison shown in Figs.3, the present analytical model correlates well with the numerical solutions. r

max - - r

op

5. CONCLUSIONS The opening and closing of a crack subjected to alternating plasticity has been analysed using the finite element method, using a non-linear kinematic hardening constitutive law is to model the material's cyclic plasticity. Under cyclic loading conditions with the maximum applied strain being ten times the material's yield strain, the crack opening displacement at the maximum stress has been found to increase initially with the number of cycles, approaching an asymptotic limit that is more twice that corresponding to the first half cycle. An analytical model has been presented to characterise this transient ratchetting in the crack opening displacement.

REFERENCES Armstrong, P. J. and Frederick, C. O (1966) A mathematical representation of the multiaxial Bauschinger effect, C.E.G.B. Report, RD/B/NN 73. Chaboche, J. L. (1986) Time independent constitutive theories for cyclic plasticity, International Journal of Plasticity, Vol.2, 149-188. He, M. Y. and Hutchinson, J. W. (1981) The penny-shaped crack and the plane strain crack in an infinite body of power-law material, J. Applied Mechanics, Vol.48, 830-40 Miller, K. J. (1987) The behaviour of short fatigue cracks and their initiations, Part II-a general summary, Fat. Fract. Engng. Mater Struct., Vol. 10, 93-113. Shin, C. S. (1994) Fatigue crack growth from stress concentrations and fatigue life prediction in notched components, Handbook of Fatigue Crack Propagation in Metallic Structures, Andrea Carpinteri (Editor), Elsevier Science B.V., 613-652. Wang, C. H. and M. W. Brown (1994) A study of the deformation behaviour under multiaxial loading, European Journal of Mechanics, A/Solids, Vol.13, pp. 175-188. Wang, C. H. and Rose, L. R. F. (1998) Transient and steady-state deformation at notch root under cyclic loading, Mechanics of Materials, Vol. 30, 229-241. Wang, C. H. and Rose, L. R. F. (1999) Crack-tip plastic blunting under gross-section yielding and implications for short crack growth, Fatigue and Fracture of Engineering Materials and Structures, Vol.22, 761-773. Wang, C. H. and Goldstraw, M. W.(1999) Plastic deformation at the tip of a tensile crack in a non-linear kinematic hardening material, International Journal of Fracture. (in press)

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

723

M o d e l l i n g of t h e cyclic r a t c h e t t i n g a n d m e a n s t r e s s r e l a x a t i o n b e h a v i o u r of m a t e r i a l s e x h i b i t i n g t r a n s i e n t cyclic s o f t e n i n g W. Hu and C.H. Wang Aeronautical and Maritime Research Laboratory, Defence Science and Technology Organisation, Melbourne, Australia.

From experimental studies of 7050 aluminium, it has been observed that this material exhibits a much higher yield stress during initial monotonic loading than during subsequent cyclic loading. This cyclic soi~ening plays an important role in the accurate prediction of cyclic stresses, and it has been noted that existing constitutive models cannot adequately address this issue. To characterise this softening behaviour, a method is proposed in which the Armstrong-Frederick type of constitutive model is associated with two sets of material constants, one for the initial monotonic loading and the other for the subsequent cyclic loading. The transition from the constants for monotonic loading to the constants for cyclic loading takes place upon the first reverse yielding. A comparison between experimental results and the model predictions shows that the softening model proposed here can accurately reproduce the experimental results for the initial half cycle, which, together with a modification of the dynamic recovery term in the non-linear kinematic hardening model, leads to significantly improved prediction of the cyclic response of the material. 1. INTRODUCTION For structures subjected to time-var~ng loads in service, determination of the cyclic stress/strain at critical locations is essential for the design and analysis of the structures to determine their fatigue life and residual strength. While the current engineering practice in design is still to use the steady-state cyclic stressstrain curve, more elaborate models are being developed to account for such transient behaviour, to make it possible to perform more sophisticated analysis using crack growth models. Among the various non-linear kinematic hardening models developed for cyclic plasticity, the extended Armstrong-Frederick (Armstrong and Frederick, 1966) rule has emerged as the most promising constitutive law that adequately models cyclic ratchetting and mean stress relaxation. Based on yield surface plasticity, this model introduces a back stress to represent the effect of micromechanisms such as the pile-up of dislocations at grain boundaries. The evolution of the back stress depends on two terms: a hardening term and a

724 dynamic recovery term. Chaboche (1986) suggested that multiple back stresses could be introduced to extend the capability of the original Armstrong-Frederick model in capturing strain hardening and the rate of ratchetting under fixed stress limits. The model has been further improved by modifying the dynamic recovery term, to improve the correlation between the predictions of the model and experimental observations (Jiang and Sehitoglu, 1996; Otmo, 1997; Wang et al. 1999). For many engineering materials, significant differences exist between the initial monotonic behaviour and subsequent cyclic behaviour in terms of the yield stress and hardening characteristics. One such example is a common high strength material, 7050 alumininm alloy, which exhibits a much higher yield stress during initial monotonic loading than during the subsequent cycles, as illustrated in Figure 1. This transient softening plays a significant role in the accurate prediction of the subsequent cyclic behaviour. Using the material constants determined from the steady-state cyclic stress-strain curve, the nonlinear kinematic hardening model could not predict the sudden softening observed in experiments (Hu et al., 1999), and this one-off error significantly affects the performance of the model in its prediction of subsequent cycles. In this paper, a constant-switching method is proposed to characterise this transient softening behaviour, aiming at improving the overall prediction of the rate of mean stress relaxation. 2. THE CONSTITUTIVE MODEL AND TRANSIENT S O F T E N I N G This section summarises the theory of the non-linear kinematic hardening model to highlight the parameters affecting the transient cyclic softening behaviour. Discussion is restricted to time-independent plasticity, which is appropriate for modelling the cyclic deformation behaviour of most metallic materials at ambient temperature. According to the small strain theory, the total strain e can be decomposed into an elastic part and a plastic part, e = e ~+ e P, and the stresses are related to the elastic strains via Hooke's lawcx = E: ( e - e P). Here, and in the following, bold symbols represent tensorial quantities and ":" signifies a tensor product. The tensor E is a rank 2 fourth order tensor of elasticity. Let cry be the uniaxial yield stress and R a scalar function of the equivalent plastic strain representing the effect of isotropic hardening, then the yield surface for materials obeying the von Mises yield criterion can be described by, f

= J2 - R - o ' y

-- 0

(1)

where J~ is the second invariant of the stress tensor. For the original kinematic hardening model developed by Armstrong and Frederick (1966), the back stress obeys this evolutionary law

725 X=kle p -k~Xp

(2)

where kl and k 2 are material constants. From the normality rule, the plastic strain increment can be expressed as e p = ,~N, where ~ is the plastic multiplier to be determined using the consistency condition 3~ ---0 during plastic loading, and N is the unit exterior vector normal to the yield surface at the loading point. Noting Of/O~ =-Of/OX, and denoting OR/Op= R', the plastic multiplier can be determined as 1

=<-N'O > (3) h where < > denotes the MacCauley bracket (i.e., <x>=(x+H)/2), and the plastic hardening modulus h is given by h = kI - k2X: N + R'

(4)

Together with

(2) the above equation represents the nonlinear relationship between the back stress and the plastic strain. Following Chaboche (1986) the isotropic hardening parameter R can be defined through R'=b(Rs-R)p, with p=[~t~ p : t r ] ~2, and b and R~ denote the rate of isotropic hardening and the maximum isotropic hardening respectively. In the multiple-back-stress model proposed by Chaboche (1986), the centre of the yield surface in deviatoric stress space is given by the sum of a number of back stresses, each evolving according to a similar law as equation (2) x = ~x(')

,

:~('> = k~')e~ - . -~(,)y(,) 2 -- P,

( i - 1 , 2 ..... M )

(5)

t=l

In the above model of Chaboche and that of Armstrong and Frederick, kl and k2 characterise kinematic hardening and dynamic recovery, respectively, and Rs and b represent isotropic hardening behaviour. For 7050 aluminium these constants have been determined by Hu et al. (1999). Figure 2 shows the numerical results from the above model together with the experimental data. Whereas the model provides a fairly accurate description of the shape of the hysteresis loops there is an obvious downward shift. Two factors contribute to this apparent inaccuracy. Firstly the yield stress used in the model, which is determined from the cyclic loading curve, is significantly lower than that shown by the monotonic loading curve, and secondly the hardening parameters are different in the initial loading and the subsequent cycles. The latter is apparent from the difference in the evanescence in the hardening segments of the monotonic and the cyclic loading curve.

726 500

600

...................

400

450

"~ 2 0 0 400 |

wl

350

9 Cyclicdata

0

m

Solid curves: best fit

-200

3O0

-400

ZS~ ................... 0.005 0.010 0.015

ol020

Plastic strain Figure 1 Transient cyclic softening

0.000

0.005

0.010

0.015

0.020

Strain Figure 2 The initial monotonic and steady-state cyclic stress-strain curve

the

3. M O D E ! J J N G OF T R A N S I E N T CYCLIC S O F T E N I N G From the simple tensile-compressive test data, Figure 1, it is natural to suggest t h a t the monotonic data should be used to model the initial monotonic loading branch, and as soon as the first unloading is detected, the cyclic data should be used to model the subsequent cyclic behaviour. This concept can be extended to three-dimensional loading. At any material point, after the first yield, the first unloading needs to be monitored, by checking the following condition: f <0 or f = 0 and f <0 (6) In the following, discussion is restricted to uniaxial loading. Implementation of the analysis in multiaxial finite element codes will be dealt with separately. Based on the above discussion, two sets of material constants will be defined for the constitutive model, one for the initial monotonic loading and the other for subsequent cyclic loading. Let the number of back stresses in the model be M , then the range of cyclic stress at steady state is related to the range of plastic strain through (Hu et al., 1999), A~2 = ~" 3 N~' k~ kf) Ae p _/~,k~tanh

+o'y +R,

(7)

A similar relationship exists between the stress and plastic strain during the initial monotonic loading

727 500 400 300 200 "~

100

0

~, -100

40

-31111

-500

Prediction (ml=0.2, m2=0.8, m3=1.2) B. -m Experimental 0

0.005

0.010

Su~in

0.015

O.02G

Figure 3 Comparison of experimental and numerical hysteresis loops

0 10 20 30 40 50 60 70 80 9010ff110 Number of cycles Figure 4 Comparison of the rate of mean stress relaxation

REFERENCES

Armstrong, P. J. and Frederick, C. O (1966) A mathematical representation of the multiaxial Bauschinger effect, C.E.G.B. Report, RD/B~N 73. Chaboche, J. L. (1986) Time independent constitutive theories for cyclic plasticity, International Journal of Plasticity, Vol.2, 149-188. Hu, W., Wang, C. H. and Barter, S. (1999) Analysis of cyclic mean stress relaxation and strain ratchetting behaviour of aluminhlm 7050, DSTO-RR0153, Aeronautical and Maritime Research Laboratory, Melbourne, Australia. Ohno, N. (1997) Current state of the art in constitutive modelling for ratchetting, Transactions of the 14th International Conference on Structural Mechanics in Reactor Technology, Lyon, France, 201-212. Jiang, Y. and Sehitoglu, H. (1996) Modelling of cyclic ratchetting plasticity, part I: Development of constitutive relations, Journal of Applied Mechanics, Vol.63, 720725. Ohno, N. (1997) Current state of the art in constitutive modelling for ratchetting, Transactions of the 14th International Conference on Structural Mechanics in Reactor Technology, Lyon, France, 201-212. Wang, C. H., Hu, W. and Sawyer, J. (1999) Explicit Numerical Integration algorithm for a Class of Non-linear Kinematic Hardening Model, International Journal of Computational Mechanics (accepted).

728 Table 1 Material properties of 7050 Aluminium alloy under monotonic loading

k(~') (MPa) k~i)

k~') = 45025 k~" : 736

k~2) = 28894 k~2) = 731

m <~ m ~ -'- 0 m (2) = 0 E = 69 GPa, Cry = 276 MPa, R, = 32 MPa, b = 12

k[ 3) = 6698 k ~3) = 74 m O) = 0

Table 2 Material properties of 7050 Aluminium alloy under steady-state cyclic loading

k~') (MPa) k~') m (i) E=69GPa, r

k~') = 49260 k~(') : 739

k[ 2) = 3241 k~2' :86.8

m ~ =0.2 m (2) =0.8 = 276MPa, R, =32MPa, b = 1 2

M kl(i)

k~3) = 1334.7 k~3) : 30.1 m ~ = 1.2

(8)

The corresponding material constants are obtained (Wang et al., 1999) (see Tables 1 and 2) by fitting these curves to the experimental data. 4. NU1VIERICAL R E S I S T S Using the above material constants and the switching procedure, both the initial monotonic behaviour and subsequent cyclic behaviour can be predicted. Figure 3 shows the predicted hysteresis loops and the experimental data. Comparing to earlier results in Figure 2 the new predictions clearly demonstrate significant improvement over the whole loading history. The initial monotonic stress-strain curve has now been reproduced accurately, and as a result of this the prediction of subsequent hysteresis loops is also improved. The mean stress is shown in Figure 4 as a function of the number of cycles. It is evident that the new model gives a much better prediction of the rate of relaxation of mean stress than does the old model, especially in the first 60 cycles. 5. C O N C L U S I O N S The transient cyclic soi~ening of aluminium 7050 alloy has been modelled by introducing two sets of material constants, one for the initial monotonic loading and the other for subsequent cyclic loading, into the nonlinear kinematic h a r d e n i n g constitutive model. The switching of the two sets of constants is performed at the first unloading after initial yield. Comparison with the experimental data shows t h a t the modified model, coupled with a modified representation of dynamic recovery and a switching of material constants, provides an accurate prediction of the hysteresis loops and the rate of relaxation of m e a n stress.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

729

Influence of Specimen and Maximum Aggregate Size on Concrete Brittle Fracture Mohsen A. Issa", Md. S. Islama, Mahmoud A. Issab and A. Chudnovskya 'Department of Civil and Materials Engineering The University of Illinois at Chicago 2095 ERF, 842 West Taylor Street Chicago, IL 60607, USA bT.y. Lin International Bascor 5960 North Milwaukee Avenue Chicago, IL 60646, USA

ABSTRACT This paper presents an experimental investigation on the influence of specimen and maximum aggregate size on concrete brittle fracture. Approximately 250 concrete specimens with various dimensions of (width x total depth x thickness) 105 x 105 x 12.5 mm to 1680 x 1680 x 200 mm made with various maximum aggregate sizes of 4.75, 9.5, 19, 38, and 76 mm were tested. The quasi-static loading-unloading tests were performed in a closed loop displacement control mode using a servo-hydraulic Instron machine equipped with two dynamic extensometers. Fraetal dimensional analysis was done on the fracture surfaces of specimens using Image analyzing technique. A significant size effect was observed. For same size specimens, as the aggregate size increased, fracture parameters, i.e., critical energy release rate, G~c, and fracture energy, GF increased. For same maximum aggregate size specimens, fracture parameters increased with increase in specimen sizes. Bridging and other forms of crack face interactions that are the most probable causes of high toughness, were more pronounced in the specimens with larger maximum size aggregates. Higher values of fractal dimension were reported for the specimens with bigger aggregates.

1. INTRODUCTION Fracture mechanics deals with the mechanical response of flawed or cracked members subjected to the application of forces or stresses. In actual stractures, sub-critical crack extension frequently leads to a critical crack extension resulting in a catastrophic failure. Brittle fracture is often observed in service while the applied stresses are substantially lower than the yield strength of the material. Fracture behavior of concrete

730 has been the subject of extensive research for many years, and while much has been learned, concrete still remains among the engineering materials with insufficiently understood fracture behavior. Concretes are highly heterogeneous by design and thus the microdefects are inevitably present. The crack propagation in concrete is accompanied by a process zone of variable size. It may result in a size effect different from the conventional fracture mechanics prediction. Many researchers reported controversy with respect to size effect in concrete fracture. In this study wedge splitting specimens of various sizes with various maximum aggregate sizes were tested under well controlled experimental conditions and the results were compared in terms of energy release rate, fracture energy, and fractal dimension.

2. BACKGROUND Extensive literature related to the influence of specimen and maximum aggregate size on fracture behavior of concrete have been reviewed. Many authors reported that the fracture toughness increases with an increase in maximum aggregate size used in the specimen[ 1-7], while some authors reported that they did not find any significant change in fracture toughness with the increase of aggregate size [8, 9]. Some researchers reported that fracture energy increases with an increase in maximum aggregate size [3-7, 10], while some reported that no significant difference was found [11, 12]. A significant number of experimental study was conducted to observe the influence of specimen size on fracture. The reported results were found to be inconsistence. Some researchers reported that fracture toughness increases with specimen size [4-7, 13], whereas others noticed no significant influence of the specimen size [12, 14, 15]. Many authors reported that the fracture energy increases with specimen size [ 14, 16], while some authors reported that the specimen size has no significant effect on fracture [10, 17]. Different results were also reported in terms of fractal dimension [12, 18, 19]. Thus, it is well understood that size effect and controversy are evident in concrete fracture.

3. MATERIAL, SPECIMEN GEOMETRY AND TESTING PROCEDURE Concretes of compressive strength ranging from 44 to 49 MPa with various maximum aggregate size of 9.5, 19, 38 and 76 mm were used. Wedge splitting specimens of sizes (width x total depth x thickness) varying from 105 x 105 x 12.5 mm to 1680 x 1680 x 200 mm were prepared. Cylinders, 150 x 300 mm and 100 x 200 mm with different mix proportions, were prepared and tested according to ASTM C39 standards. Experimental setup of a typical wedge splitting specimen is shown in figure 1. A downward vertical load was applied to the rollers through a wedge with an angle of 8.75 ~ The applied vertical load was translated as the splitting load to the specimen. The testing machine was equipped with displacement, load and strain channels. The fracture process was monitored by optical and acoustic imaging systems. The tests were conduced in a closed loop displacement control mode at a contact speed of 0.125 mm/min. The specimens were loaded until the crack advance was noticed and at this time the machine was switched to unloading mode and crack length was measured. The specimens was

731 again loaded and unloaded. The step was repeated until the specimen broke into two halves. _~.~ad from actuator

---Wodgo Bearing miler

__

Test specimen

L W/2 ~ , W/2 .._1 F" "1 Figure 1. Schematic diagram of test setup . v

r~,.

4. RESULTS AND DISCUSSIONS The influence of microstructural parameters, such as aggregate size, and macroscopic parameters, such as specimen dimensions, on brittle fracture has been experimentally investigated. The specimen geometry was scaled and similarity chart was prepared to address the size effect problem in brittle fracture. The significance of the study is evident through three basic forms of comparisons with respect to size effect. The first corresponds to the various specimen sizes east with the same maximum aggregate size. The second criteria presents geometrically identical specimens east with various maximum aggregate sizes. The third and final form of comparison entails equally scaling down both geometry and maximum aggregate size, i.e., having identical t/d~x. The energy release rate for geometrically identical specimens increased as the maximum aggregate size increased as shown in Figure 2. The increase in fraet~are toughness with increasing maximum aggregate size is associated with the increasing resistance encountered by the propagating crack. Similarly, for different size specimens made with the same maximum aggregate size, G~o increased as the specimen size increased as showninFigure3. For geometrically similar specimens (i.e., the shape and all dimensionless parameters are the same), the R-curve for larger specimenswas noticeably higher than that for the smaller specimens. The fracture energy, GF dependence on specimen dimension and maximum aggregate size is well pronounced as shown in Figure 4. The variation of fractal dimensions with maximum aggregate sizes are presented in Figure 5. For the same maximum aggregate size specimens, the fractal dimension also increased with specimen size.

732

3.0

500

3.0

2.5 400 2.0

~

"

300 ~

I I

J,~ q

//-~

400

[

300

1.5

~

1.0

200 r~-

0.5

100

0.0

500

2.5 :1 -o-.. s3 (s~o~o~ ,oo,=) :1 -*--s4(4zo,,4zox 5O,.m) $5 (210 x 2.10 x 25 ram) 2.0 " ~ -'-I . . . . . . . . . . . . 1.5

~, ~-~ 0

200 olO0

0.5

0 0.20

0.30

0.40

0.50

0.60

0.0

Observbed crack length/Width

Figure 2. Variation o f G:c with aggregate size in $4 specimens.

0.30 0.40 0.50 0.60 Observed crack length/Width

Figure 3. Variation o f G~c with specimen size ( m a x i m u m aggregate size = 9.5 mm).

300

2.30

S1 specimens -"- $3 specimens *, S4specime__._.._~m

r

0.20

S 1 specimens (1680 x 1680 x 200 ram)

250

2.25

u_

#

0 /./

.~ 2.20

~ 200

~3 o

i~4 I.LLN/'~*" /.:~i/"~'''")I!

150

o o.~

100

.... 0

a. 2.15

o.olo 0.020 0.030 o.11411 CMOD (in.)

, .... t ..... i . 20 40 60 Maximum aggregatesize, d ~ (ram)

Figure 4. Variation o f GF with specimen and aggregate size.

80

2.10

I

10

'

I

20

'

I

30

'

I

40

'

I

I

50

60

'

I

70

Maximum aggregate size (ram)

Figure 5. Fractal d i m e n s i o n dependency on aggregate size.

'

I

80

733 5. CONCLUSIONS Approximately 250 wedge splitting concrete specimens of various sizes with different maximum aggregate sizes were tested. As a result of this study, the following conclusions can be drawn:

.

~

~

For fixed specimen size, both toughness G~c and fracture energy G~ increase with an increase in the maximum aggregate size. For the same maximum aggregate size specimens, the toughness and fracture energy increase with specimen size. Bridging that contributed to toughness is more pronounced in the specimens with larger maximum size aggregates. For the same size specimens, the fractal dimension increases with an increase in maximum aggregate size and for the same maximum aggregate size specimens, the fractal dimension increases with the specimen size. The value of the fractal dimension is directly proportional to the surface roughness. The rougher the surface, the higher the fractal dimension.

ACKNOWLEDGMENTS This study was funded by a contract awarded to the University of Illinois at Chicago by the National Science Foundation (Grant No. CMS 9522306). Their financial support is gratefully acknowledged.

REFERENCES [1]

[2] [3] [4]

[5] [63

Shah S.P. and Chandara S., Critical stress, volume change, and microcracking of concrete. ACI J., 65/57 (1968):770-781. Walsh P., Fracture Of plain concrete. Indian Cone. Journal, 46,11 (1972): 469-476. Nallthambi P., Karihaloo B. and Heaton B., Effect of specimen and crack sizes, water/cement ratio and coarse aggregate texture upon fracture toughness of concrete. Mag. Conc. Res., 36,129 (1984):227-236. Issa M. A. and Chudnovsky A., Reliability prediction for brittle materials based on small scale testing. Final report submitted to National Science Foundation, Virginia, USA, (1999): 1-215. Issa Mohsen. A., Issa Mahmoud. A., Islam Md. S. and Chudnovsky A., Size effects in fracture of concrete: part I, experimental setup and observations. International Journal of Fracture, in press, 1999. Issa Mohsen. A., Issa Mahmoud. A., Islam Md. S. and Chudnovsky A., Size effects in concrete fracture: part II, analysis of test results. International Journal of Fracture, in press, 1999.

734 [7]

[8] [9]

[10] [11] [121 [13] [14]

[16] [17] [18]

[19]

Issa Mohsen A., Issa Mahmoud A., Islam Md. S., Shulkin Y., Chudnovslcy A., Shlyapobcrsky J., and Dudley J. W., Scale effects on fracture resistance of brittle materials in the presence of compression along the fracture plane, Proceedings of the Sixth Pan-American Congress of Applied Mechanics, PACAM VI, Brazil, (January 1999): 1059-1062. Carpinteri A., Experimental determination of fracture toughness K~c and J~c for aggregate materials. Proe. 5th Int. Conf. Frae., France, 4 (1981): 1491-1498. Bazant Z., Size effect in blunt fraeture: eonerete, rock metal. J. Eng. Mech., ASCE, 110,4 (1984):518-535. Hillerborg A., The theoretical basis of a method to determine the fracture, energy GF of concrete. Materiaux et Constructions, 18,106 (1985):291-206. Petersson P., Fracture energy of concrete: Practical performance and experimental results. Cem. Cone. Res., 10 (1980):9 l- 101. Saouma V. and Barton C., Fraetals, fractures and size effects in concrete. J. Eng. Mech., 120,4 (1994):835-854. Boseo C., Carpinteri A. and Debemardi P.G., Fracture of reinforced concrete: Scale effects and snap-back instability. Eng. Frae. Mech., 35,4/5 (1990):665-677. Hilsdorf H. and Brameshuber W., Size effects in the experimental determination of fracture mechanics parameters. In Application of Fracture Mechanics to Cementitious Composites, Northwestern University, ed. S. P. Shah (1984):361-397. Bazant Z., Scaling laws in mechanics of failure. J. Eng. Mech., 119,9 (1993):18281844. Bazant Z. and Kazemi M., Size dependence on concrete fracture determined by RILEM work-of-fracture method. Int. J. Frae., 51 (1991):121o128. Liang R. and Li Y., Study of size effect in concrete using fictitious crack model. J. Eng. Mech., 117,7 (1991):1631-1651. Issa M. and Hammad A., Assessment and evaluation of fraetal dimension of concrete fracture surface digitized images. Cement and Concrete Research Journal, 24,2 (Mar-Apr 1994): 325-334. Issa M., Hammad A. and Chudnovsky A., A correlation between crack tortuosity and fracture toughness in cementitious materials. Int. J. Frar 60 (1993):97-105.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

Fatigue design o f welded classification m e t h o d

very thin-walled

735

tube-to-plate joints using the

F.R. Mashiri, X.L. Zhao and P. Grundy Department of Civil Engineering, Monash University, Clayton, VIC.3800, Australia

Tube-to-plate T-joints, made up by welding a square hollow section tube to a plate, are tested under fatigue loading. Constant stress-amplitude cyclic loading is applied to these connections as in-plane bending load. The effects of in-line galvanizing, steel grade, stress ratio and tube wall thickness on the fatigue strength of these joints are discussed, using S-N data expressed in terms of nominal stress ranges. Analysis of the fatigue test data using least squares method is carried out to determine the design S-N curves of the tube-to-plate T-joints under in-plane bending, for the classification method. Two methods are used in this analysis, assuming that either log N or log S is the dependent variable. Different design S-N curves are obtained using the two methods. The design S-N curves derived from the analysis show that the S-N curves for the tube-to-plate joints lie above the design existing S-N curves in AS4100-1998 (SAA 1998) and CAN/CSA-S16.1-M89 (CSA 1989).

1. INTRODUCTION In recent years, there has been a trend towards using thin-walled tubular sections in the construction industry. Thin-walled hollow sections result in lighter structural connections that are cheaper but at the same time providing enough static strength (Zhao et al 1996). Thinwalled welded tubular sections have also found use in construction of structures under fatigue loading. The undercarriages and support systems of agricultural and road equipment, such as trailers, haymakers, graders and swing-ploughs are made from thin-walled hollow sections (Mashiri et al 1999). This equipment is subjected to fatigue loading under service conditions and some fatigue failures have been observed, especially for tube wall thicknesses less than 4mm. Current design codes contain fatigue design rules for wall thicknesses including and above 4mm only (EC3 1992; SAA 1998; CSA 1989; Zhao et al 1999). As a result there is a lack of fatigue design guidelines for wall thicknesses less than 4mm. Fatigue tests of tube-to-plate T-joints with square-hollow section tubes of wall thicknesses less than 4 mm have been carried out. Specimens with tube wall thicknesses of 1.6 mm, 2 mm and 3 mm have been fatigue tested under constant stress- amplitude loading. The specimens were tested under cyclic bending stress. This paper uses statistical analysis to determine S-N curves, for the classification method, from the resulting fatigue data. The least-squares method of analysis is used. Traditionally the S-N curves are determined from the assumption that the number of cycles is the dependent variable (Nakazawa & Kodama 1987; ASTM 1980). This is because most fatigue tests are controlled through the monitoring of stress or strain. Stress or strain is therefore considered to

736 be the independent variable. In this paper however, the same basic equations of the leastsquares method will also be used to determine the S-N curves, but with the stress as the dependent variable. The stress level and the number of cycles to failure of each specimen are interrelated. It can also be argued that the stress level applied to the specimen before failure depends on the number of cycles the specimen is subjected to before failure. Tests of the tube-to-plate T joints under in-plane bending have been carried out for in-line galvanized steel known as DuraGal and non-galvanized steel. The effects of galvanizing, steel grade, stress ratio and wall thickness are discussed. The resulting S-N curves will be compared to S-N curves of existing tube-to-plate connections from the Canadian Standard, CAN/CSA-S 16.1-M89 (CSA 1989) and the Australian Standard AS4100-1998 (SAA 1998).

2. DETERMINATION OF S-N CURVES FROM FATIGUE DATA The common statistical method of analyzing S-N data is the least-squares method. Only failed data and high cycle runouts can be used. The method assumes implicitly that the data follows a constant lognormal distribution. Little & Jebe (1975) derive the equations used in estimating the slope, intercept and random measurement error of a model for simple linear regression by the least-squares. Similar equations for calculating the slope and intercept of a straight line S-N curve passing through fatigue data are given in guidelines for statistical analysis of fatigue data by the Japan Society of Mechanical Engineers, JSME (Nakazawa & Kodama 1987) and the American Society for Testing and Materials (ASTM 1980). The definition of design S-N curves given in the Department of Energy guidelines was adopted in this analysis. The Department of Energy Guidelines define the design S-N curve as the meanminus-two-standards-deviation curve of the relevant experimental data (Department of Energy, 1990). For a normally distributed population of given mean, ~t and standard deviation, a, the/a+2cr contains about 95% of the population (Little and Jebe, 1975). The design S-N curves for the two cases are therefore given by equations defining the upper and lower bounds of the scatter of the S-N data as follows; (i) when log Nis the dependent variable, logN = A + BlogS + 2cqogN and (ii) when log S is the dependent variable, logS = a + blogN + 2crb,gs.

3. FATIGUE TEST RESULTS Tube-to-plate T-joints were tested for three different thicknesses. Square hollow section tubes of thicknesses 3mm, 2mm and 1.6mm were used to make the test specimens. For each tube wall thickness, the welded joint specimens were tested at 3 or more different stress levels to allow the determination of the slope of the S-N curves resulting from this fatigue data. Replication was also employed, allowing more than one specimen to be tested at any given nominal stress range, for most of the stress ranges considered. This improves the reliability of the results by defining a scatter band for the S-N data. Specimens were also made up from different steels as given in Table 1. Cracks initiated on the weld toe at the comer of the brace on the brace-plate interface and were noticed as surface cracks. Failure was defined as a through thickness crack along the entire width of the brace. The through-thickness crack occurred on the weld toe in the brace

737

on the side under tension. This failure mode corresponds to complete failure of the connection. Fatigue test results for the 3ram, 2ram and 1.6ram tubes are shown in Figure 1. The tubeto-plate T-joints made up of 3ram square hollow section tubes were tested at two different stress ratios of 0.5 and 0.1. The rest of the tests were carried out at a stress ratio of 0.1. Table 1" Steels used in fatigue tests

Steel Type

Origin

Minimum Yield Stress (MPa)

C450,'non-galvanized C450LO, galvanized (DuraGal) C350, non-galvanized S355JOH, non-galvanized

Australia Australia Australia Europe

450 450 350 355

'

Minimum Ultimate Tensile Strength (MPa) "

'

500 500 430 " 490-630

' ......

4. DISCUSSION OF FATIGUE TEST RESULTS All the tubes used to make the specimens for fatigue test are cold-formed. However Grade C450LO steel, also known as DuraGai is a high strength structural product with a light galvanized exterior coating. The other steel sections from grades C450 are non-galvanized. Grade C450 and C450LO have the same yield stress and ultimate tensile strengths. The galvanized tubes are welded in their natural state, without removing the galvanizing, by increasing the voltage by 0.5volt to l volt. Compared to black steel, a 0.5volt to l volt is enough to maintain a given arc length if the speed is kept constant. In order to be able to determine the effect of in-line galvanizing on fatigue strength of these joints, S-N data for specimens with the same stress grade, stress ratio and thickness, but with different surface coatings can be compared (for example the solid square box symbols versus solid triangle symbols or empty square box symbols versus empty triangle symbols, in Figure 1). It can be deduced that there is no visible trend of the effect of in-line galvanizing on the tube-to-plate welded connections, for each stress ratio considered. To demonstrate the influence of stress grade, specimens of the same thickness and stress ratio, but with different stress grades are compared. Two different steel grades, C450 (or C450LO) and S355JOH, were used for the 3ram thick specimens and for the 2ram thick specimens as well. The S-N data for the different grades of steel, for the 3ram thick square hollow section tubes tested at a stress ratio of 0.1, are shown in Figure 1. The 3mm thick specimens tested at a stress ratio 0.5 are also shown in Figure 1. It can be seen (solid square box and solid triangle symbols versus solid round symbols in Figure 1) that there is no noticeable trend in the distribution of number of cycles at a given stress relating to steel grade. There is also no noticeable influence of steel grade on fatigue strength for the 2ram thick tube connections tested at a stress ratio of 0.1, as shown in Figure 1 (solid diamond symbols versus empty diamond symbols). The tube-to-plate T-joints, made from 3ram thick square hollow section robes were tested at two different stress ratios of 0.1 and 0.5. The specimens tested at a stress ratio of 0.5 have reduced fatigue life compared to those tested at a stress ratio of 0.1. The damaging effect of a fully tensile cyclic stress range tends to increase as the mean stress or stress ratio increases (Maddox 1991).

738 A plot of the S-N data for the 3ram, 2ram and 1.6ram tube-to-plate T-joints is shown in Figure 1. Because it has been shown that there is an influence on fatigue strength caused by stress ratio, results corresponding to a stress ratio of 0.1 only are discussed here. Using a deterministic approach it can be concluded that for lower nominal stress ranges the lower bound of the scatter is determined by the thinner sections of 2mm and 1.6mm. This may be a result of defects such as undercut. In those connections where these defects occur, they may have a greater influence on fatigue strength on the thinner sections of 2mm and 1.6ram compared to the 3mm thick sections. Other researchers have found an increase in fatigue strength as the wall thickness of the tubes failing under fatigue decreases (van Wingerde 1992; van Wingerde et al 1996, 1997; Zhao et al 1999; IIW Subeommission XV-E 1999. In this plot the increase in fatigue life, which is normally observed as the tube wall thickness decreases, is not realized. Instead, the smaller wall thickness tubes have produced fatigue data defining the lower bound of the scatter, especially at lower nominal stress ranges.

5. DESIGN S-N CURVES FOR TUBE-TO-PLATE T-JOINTS UNDER IN-PLANE BENDING All the S-N data obtained from the experimental investigation is going to be analyzed using the statistical methods of least squares to determine the design S-N curves in terms of the classification method. The scatter of the data tested at different steel grades, for different surface coatings, stress ratio and thiekness do not warrant any separation and analysis of obtained data accordingly, when expressed in terms of the nominal stress range. A plot of all the S-N data obtained from the tube-to-plate T-joints is shown in Figure 1. The data eonsists of the following specimens, 25 specimens made from 3ram thick tubes, 13 specimens made from 2ram thick tubes and 5 specimens made from 1.6ram thick tubes. 1000

t

Jl

I

t1[ I

A

m Q,, v

'--4.

)

m o c

[ t

I I ] I

II

1

t [ I / I i-i :

I )il ]l II![

-ijj- i.i F--::i..il ! i !II

m 100 h,

13 DuraGal;50xS0x3SHS-Plate;R=0.5 9 C450;50xS0x3SHS-Plate; R=0.1

[1 El

9 0 X 9

fl -

[ , A C4S0;r,0xS0x3SHS..Plate;R"-.0.S

[1 II II

l i

|i J J~

9

"",,,

[~'-~,

7

S355JOH;50xS0x3SnS-Plate;l~q).1 S355JOH'50xS0x3SHS-Plate'R=0.5 C350;50x,'r:~l.6SHS'Plate; R ~ - I S355JOH'40x40x2SHS-Plate;R=0.1

] t o DuraOai;~4~Tfl~x~SHS-PZate;R=0.1 il I t - " AS4100-1998 ii t I - - - CAN/CSA-S16.1-M89 FI I I - .. - M e a n S-N curve II [ [ - . - M e a n + 2 S t a n d a r d Deviations Curve {J --~-Mean-2Standard Deviations Curve 1 -I--,1_ I t I I1

I/)

I

E

0 Z

10 1 .E+04

11ttl II! 1 .E+05

_

1 .E+06

N u m b e r of Cycles, N

Figure 1" Plot of S-N Data for Design Curves

I

[

i

II

I

I 191q4

i

e~

t ....

1 .E !+07

iTiii

1 .E+08

739 Four sets of analysis are proposed using the least-squares method of analysis as follows; (a) Assuming that log N is the dependent variable, such that the linear model, log N = A + Blog S is determined (ASTM 1980; Nakazawa & Kodama 1987). In this ease the following two analyses are carried out, (i) when both parameters A and B are estimated and (ii) when the parameter B is assumed to be 3 and parameter A is estimated. (b) Assuming that log S is the dependent variable, such that the linear model, log S = a + blog S is determined. In this ease the following two analyses are carried out, (i) when both the parameters a and b are estimated and (ii) when the parameter b is assumed to be 1/3 and the parameter a is determined. A summary of the results from the four analyses is shown in Table 2. Table 2 gives the parameters determined from the least-squares method, including the standard deviation and hence, the lower bound design S-N curve equation from each of the analyses considered. Figure 1 shows the mean, lower and upper bound design curves for the case when log N is the dependent variable, B=3 and A is determined. Classes, that is the nominal stress ranges corresponding to two million cycles, based on mean-minus-two-standard deviations curves are listed in Table 2. Table 2: Parameters, standard deviation and lower bound design S-N curve equations Class Design S-N Curve Standard Parameters Method (N=2E6 Equation Deviation, A or a and from Borb a~Nor mean-2SD ~/og S curve) (a) log N, dependent variable 43 log N=I 1.1488-2.95631og S A=11.7532 (i) A, B r B=-2.9563 determined 44 A=11.8286 (ii) A determined, O'1o8~=0.3022 log N=11.2242 -3log S B=-3 B=3 (b) log S~ dependent variable 54 log N=12.9742-3.84461og S (i) a, b determined a=3.5536 alog s'-O. 0895 b=-0.2601 44 log N=11.2253-3log S a=3.9428 (ii) a determined, Crlogs=O.100 7 b=-l/3 b=1/3 i

6. CONCLUSIONS Tube-to-plate T-joints made up of square hollow section tubes welded to a plate were fatigue tested under in-plane bending. (1) In-line galvanizing and stress grade of cold-formed tubes do not have any noticeable influence on fatigue strength. (2) Stress ratio has shown an influence on fatigue strength of the connections. Tests carried out using a stress ratio of 0.5 show reduced fatigue strength of the connections especially at lower nominal stress ranges, compared to those tested at a stress ratio of 0.1.

740 (3) The T-joints made from thinner tube wall thicknesses of 2ram and 1.6mm have been found to define the lower bound of the scatter of the S-N data points. The trend of increase in fatigue strength with a decrease in wall thickness is not realised in these test results. There is a possibility that the effect of weld defects such as undercut may have a greater influence on fatigue strength of thinner connections, causing them to have re.dueed fatigue strength. Thickness effect is the subject of further work. (4) The design S-N curves from the two different approaches both give lower bound design S-N curves which are above the existing design S-N curves from the Canadian Standard, CAN/CSA-S16.1-M89 (Class 40) and the Australian Standard, AS4100-1998 (Class 36). The equations derived for the ease when the inverse of the slope of the S-N curve is assumed to be 3 may be adopted for design in order to be consistent with design guidelines such as AS4100-1998 (SAA 1998) and CAN/CSA-S16.1-M89 (CSA 1989). The design S-N curve in terms of the classification method for tube-to-plate T-joints, made from cold-formed square hollow sections of thicknesses less than 4mm, is given by the equation log N = 11.2242- 3log S. This gives a detail category or class of 44.

REFERENCES

ASTM 1980, Statistical Analysis of Fatigue Data, ASTM STP 744, pp. 129-137 CSA. 1989, CAN/CSA-S 16.1-M89, Canadian Standards Association, Ontario, Canada. EC3 1992, Eurocode 3: Part 1.1, ENV 1993-1-1, European Committee for Standardisation Department of Energy, 1990, "Offshore Installations: Guidance on design, construction and certification", Fourth Edition, London, HMSO, Great Britain Little R.E. and Jebe E.H. 1975, "Statistical Design of Fatigue Experiments", Applied Science Publishers Ltd., Essex, England, 1975 IIW Subcommission XV-E 1999, Part 1, IIW Doe. XV-E99-251, Zhao & Packer (eds) Maddox S.J. 1991. Fatigue strength of welded structures. 2"d Edition, Abington Publishing, Cambridge, England Mashiri F.R, Zhao. X.L., Grundy P. 1999, Advances in Steel Structures, Vol. 11, pp. 983-990 Nakazawa H. and Kodama S. 1987, Current Japanese Materials Research, Vol. 2, Tanaka, Nishijima & Ichikawa (Eds), Elsevier Applied Science, 1987, pp. 59-68 SAA 1998, Steel Structures, Australian Standard AS4100-1998, Sydney, Australia van Wingerde A.M. 1992. Heron. Vol. 37, No. 2. van Wingerde, A.M., J.A. Packer & J. Wardenier 1996. Journal of Structural Engineering, ASCE, Vol. 122, No. 2, pp.125-132 van Wingerde A.M., D.R.V. van Delft, J. Wardenier & J.P. Packer 1997. Scale Effects on the Fatigue Behaviour of Tubular Structures. IIW International Conference on Performance of Dynamically Loaded Welded Structures, July 14-15, San Francisco, U.S.A Zhao X.L., G.J. Hancock & R. Sully 1996. Journal of the Australian Institute of Steel Construction, Vol. 30, No. 4, pp. 2-15 Zhao X.L., Herion S., Packer J.A., Puthli R., Sedlacek G., Wardenier J., Weynand K., van Wingerde A., and Yeomans N. (1999), "Design Guide for Circular and Rectangular Hollow Section Welded Joints under Fatigue Loading", Verlag TOW Rheinland, K61n, Germany (to be published)

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

741

Cosserat and non-local continuum models for problems o f wave propagation in fractured materials E. Pastemak 1 and H.-B. M0hlhaus 2 Department of Civil and Resource Engineering, University of Western Australia, Nedlands, WA 6907 Australia. E-mail" [email protected] 2 CSIRO Division &Exploration and Mining, Nedlands, WA 6009 Australia. High loading through mechanical impact or blasting leads to fragmentation of initially cohesive brittle solids. The subsequent wave propagation takes place in a discrete material the microstructural elements of which can rotate independently. To investigate the post-blast state of the material it is necessary to study wave motion in granular media with rotational internal degrees of freedom. In principle, such a medium can be modelled as a discrete system using, for example, the discrete element method. However in many cases this direct approach is computationally ineffective. In this paper we use a homogemzation approach instead which leads to the Cosserat continuum or even to non-locality.

1. INTRODUCTION It has long been recognised that internal degrees of freedom play an important role in the continuum mechanical modelling of material behaviour. What perhaps has been less widely appreciated is that these same degrees of freedom, and in particular the microstructural rotation or spin, may need to be recognised explicitly in the kinematical description of flow and deformation processes. We will demonstrate how such additional rotational degrees of freedom arise naturally by mathematical homogenisation of a discrete system. The aim of the present study is to investigate wave propagation in granular materials with internal rotational degrees of freedom in order to establish how accurately the Cosserat and non-local models describe their behaviour. For this purpose, a simple periodic discrete model of spheres connected to each other by both rotational and translational springs will be considered. This model allows the analytical derivation of a general closed form solution. In particular, the dispersion relations will be obtained. Two homogenization techniques will be investigated. One of them produces an anisotropic Cosserat continuum and the other one produces a Kunin-type pseudo-continuum. The wave propagation in the case of a higher-order gradient continuum for granular materials has been analysed by MOhlhaus (1996).

2. SIMPLE I-D STRUCTURES. PHYSICAL MODEL

Consider a material consisting of one dimensional, parallel chains of identical, spherical grains. The grains are connected by translational springs of stiffness k and rotational springs of stiffness k, (Figure 1). The potential energy density reads: = (2TIG3) -! {k((ld3i- ~3i_!) + F((D2 i -]- f~O2i_!)) 2 4- k,o(q~2i- (D2i_l) 2 }.

(1)

742 Here a designates the spacing of the mass centres of neighbouring spheres, r is the sphere radius, and a-zrl -~ is the number of chains per unit area of cross-section. The equations of motion assume the following form" m//s~ - k(lg3i+l - 2u~ + lg3i_! ) - kl'(q~2i+l - q~2i-i ) : q3i,

(20 (22)

J~i, + kr(ll/3i+! - H3i-i ) q- kr2 ((D2,+I -t- Zq)2, + cpi,_!) - k, (qiz,+l - 2q)2, + Ip2i_I) - ME,,

where u3i t~)2i are the independent Lagrange coordinates, and M2, are applied load and moment at ith-sphere respectively, is the moment ql t inertia of the ball. The equations (2) can be written in a homogenised (continuous) form by introducing continuous functions u3(x), tO2(x)which coincide with u3i and ~2, at discrete points

J=2mr"/5

x=ai:

mii3(xt)-k[u3(x , +a)- 2u3(x,)+u3(x I -a)]-kr[ql2(x ,

J~2(xl)+kr[u3(xl+a)-u3(x! -a)]+kr'[~,(x, - k , [r

(X ! +

a ) - 2r

(xl) + q)2 (Xi

" a)]

+a)-qJ2(x , -a)]= q3(x,),

+ a ) + 2qi2(xl) + qii(xl - a ) ] -

(31)

(32)

= M 2 (x I )

3. H O M O G E N I S A T I O N 3.1.

Homogenisation

by differential e x p a n s i o n ( C o s s e r a t c o n t i n u u m )

We replace the finite difference expressions in (1) by corresponding differential expressions Truncation of the Taylor expansions after the second order terms gives the following approximation W(XI) = (2T~a3)-' /

a2 +4k? .0~3

+ 4kr2r

+

k(Oq>2~a 2

"t

J

.

(4)

Differentiation of the energy density with respect to the deformation measures Ou3 YI3 =

0r 2 Y31 =-q~2, KI2 - OxI

+r

(5)

gives 1[~i3 -'- k(~a) -I [~13 + 0 - 2r/a)Y31 ]. 031 -- k(na)-' 0 -

2r/a)[y 13 + 0 -- 2r/a)Y31 ].

i~12 -" k, (ill/)-' K,2 (6)

Introduction of volume forces and moments and consideration of momentum and angular momentum equilibrium yields the equations of motion 0c~'3 O~Xl + Pf3 = P//3,

01xi2 ~

~!

-

0 13 4"t~31 4-Pro2

J "-

..

(7)

a33q

~

r

Formally, equations (5-7) represent a 1D Cosserat continuum (eg, Nowacki, 1970). After substituting the constitutive equations (6) into the equations of motion (7) we obtain the Cosserat equivalent ofLame's equations:

llkO2U3+ 2rkOqJ21+ Pf3 = P//3

~-~

0x, ~

a

0x,

1 1 k 02q)2 2r 0u3 ---k '~'~ ' 0x,' a Ox,

4r2 ] J k~2 + Pro2 = - - @ 2 a~ a3rl

(8)

Comparison of these equations with the homogenised form of Lagrange equations (2) shows that the Lame equations in the Cosserat approximation gives the same leading terms as the approximation of finite difference by the extraction of the Taylor series with the relative

743 difference in the term of order o(a3).

...

.

.

kJ .

.

Xl

U3i Figure 1. 1D chain of spherical grains connected by translational and rotational springs. 3.2. Homogenisation by integral transformation: Kunin's method ( Kunin pseudo-

continuum) Kunin's (1982) homogenisation procedure for discrete periodical structures is based on trigonometrical interpolation of discrete functions. For independent periodical chains of grains we have:

(~(x,)j

u3(xl))=a~_~(u3i)~)(xl-ia),Iu3i)=~)(xi-ia)(U3(Xl)~dxl , t,~, t.% t.~(x,)J

8(x) = (ax)-~ sin ~ '

(9)

a

This type of homogenisation produces so-called non-local constitutive relationships. The origin of this particular type of non-locality follows from the fact that the interpolation function for a given set of u3~, @2~is unique, hence the alteration of any local value leads to the change of the whole function. Using (9) the non-local equations of motion can be obtained in the following form: k .[ [ 2 8 ( x , - y, ) - 8 ( x , - Yl - a ) - 8 ( x , - y, + 0)][4 3 ( y , ) d y ,

+

-oo

+ kr j'[i~(x I - Y l - a ) - ~ ( x l

- Y i +a)]q)2(Yl)dYl -

qs(xi)-miis(xi),

(101)

,r .l'[IS(x,-y, + a ) - I S ( ~ , - S , - a)i,,(S,)~, + -at)

+kr 2 J'[8(x,-y,

+a)+ 28(x,-y,)

+8(x,-y,-

a)~2(y,)dy , +

(102)

+ k , J' [28(x, - y , ) - 8(x, - y, + a) - 8(x, - y, - a ) ~ 2 (y,)dy, = M 2 (x,) - a~2 ( x , ) .

It should be noted that insertion of (9) into (1) yields the expression for potential energy which after differentiation with respect to the deformation measures '~13-G'~//3/o~xl+(P2, ~r'~12-~l)2/G%Xl gives non-local constitutive law.

4. WAVE PROPAGATION. DISPERSION RELATIONSHIPS For a particular case of q3(xO=M2(xl)=O,let us consider the propagation of harmonic waves

u =Aexp(i~(x-v,t)~

tp=Bexp(i~(x-v,t)),

(11)

744 where ~ is the wave number, vp is the phase velocity. For the sake of simplicity let us write x instead ofxt, u instead of u3 and q~instead of q~2. Propagation of these waves will be studied for the original physical model (3) and for the Cosserat and Kunin approximations. 4.1. Wave propagation in the physical model By substituting (11) into (3) we obtain the following system:

_m~2vv2A+4ksin 2(~lA-2ikrsin(~a)B=O

(121)

-j~2vp2B+2ikrsin(~a)A + 4kr 2 c o s Z ( ~ ) B

(122)

+ 4k, sin 2(-~)B = 0

The system has non-trivial solution if its determinant vanishes, which leads to the biquadratic equation with respect to the phase velocity nOrvp4 - 4(Jk sin 2(_~)+ mkr 2 c0s2(_~]

+mk, sin2

-~-+vP16kk, sin 4

1 =0

13)

The discriminant is positive, hence there always exist two real solutions for the square of the phase velocity. Both solutions for vp2 can be shown to be positive. Let r=a/2, a=l. Then

....... .o,i

i 9

j

..g._Jsin (14)

Because the configuration is symmetric, it is sufficient to consider wave propagation in only one direction, ie the wave velocities will be assumed positive. Two types of waves have emerged with velocities determined by the sign plus and minus before the radical respectively. Since the determinant of the system (12) is zero and the solutions for displacement and rotation are linearly dependent it means that both displacement and rotation components exist simultaneously for each type of the wave. One can find the ratio of their amplitudes, for example from equation (121):

A/B = 2ikr sin(~a)[4k sin 2(/~a/2)- m~2vp2]-'

(15)

The first type of wave (positive sign before the radical in (14)) and the second (negative sign) have the following long wave asymptotics vp2---~,~2 as ~---}0. The corresponding asymptotics for the ratio of amplitudes is:

A

~0 ~ A -- 0 (rotational wave) ~0[oo ~ ~ B-- 0 (shear wave)

( 1 6)

Thus, we have two types of waves. The first one becomes the rotational wave in the long wave limit (~---}0), while the second one is shear wave. Otherwise both components are present, however asymptotically one type dominates. For that reason we will call these waves rotational-shear and shear-rotational. When ~--+oo (short length wave) this case can not be described properly. The ratio of amplitudes for different ratios of spring stiffnesses is shown in Figure 2. The phase velocity for rotational-shear and shear-rotational waves for different ratios of stiffnesses is shown in Figure 3. All the plots are given for the physically reasonable wave lengths, since in the considered system the wave length cannot be shorter than the ball size. Moreover, in the homogeneous models the wave length should be much greater than the ball size.

745

Figure 3. The phase velocity for rotational-shear (a) and shear-rotational wave (b). The circles show velocities corresponding to integer values of the normalised wave number.

4.2. Kunin's pseudo-continuum Substituting (11) into the non-local (integral) equations of motion (10) and calculating the corresponding integrals one can get the same system as obtained for the exact equation of motion (12), but with a restriction: ~<~/awhich does not appear mathematically for the exact solution This restriction reflects the fact that wave lengths must be larger than the microstructure size. 4.3. Cosserat approximation By assumingj~=m2=0, r=a/2 and substituting (11) into equations of motion (8) one gets:

ma-2~2v,2A -k~2A +i~kB =0,

.1a-2~2vp2B-k,p~2B-i~kA-kB

=0

(17)

746 leading, after setting r=a/2 and the normalisation a=l, to

[I ~~-*1 li~-13

k

1

1

+ + 2 1 ~ __J7 ~ 5 _ (

(18)

~_~..~)]

This expression is an asymptotics of (14) as ~---~0 with the accuracy o(~).. This is not surprising since the Cosserat model is a long wave (small wave number) apprommation. Figures 4 shows the phase velocity of rotational-shear and shear-rotational waves for the exact and the Cosserat solutions. They are in a quite good agreement for the small wave numbers, ie in the range where the Cosserat model approximates properly the exact solution. I

vp2(~a) 2 5

-

i

~1

i

i

1

vp2(~,3a)

(a)

(b) 1.5

Cosserat 0.5 \ exact L

, d.5 i .... ~'.~ ~ i5 ; ~o 0 0.5 1 1.5 2 2.5 3 ~a Figure 4. Cosserat phase velocities for rotational-shear (a) and shear-rotational (b) waves. .i

~

1.

i

i

5. CONCLUSION We .homogenised the governing equations of a 1D discrete system first by differential expansion and then by integral transformation leading to a Cosserat and Kunin-type non-local continua respectively. The Cosserat equations of motion are the long wave asymptotic approximation to the exact model. Two types of waves exist simultaneously: shear-rotational and rotational-shear waves. As the wave number tends to zero, ie for long waves the shear component dominates the shearrotational wave, while the rotational component dominates the rotational-shear wave. The non-local dispersion relations coincide completely with the exact ones given a specific restriction on the wave number ~<~/a. This means that for larger wave numbers the model should be changed for the ball microstructure should be taken into account.

REFERENCES M0hlhaus, H-B. and Oka, F.(1996). Dispersion and wave propagation in discrete and continuous models for granular materials. Int. d. Solids Structures. 33, 2841-2858. Nowacki, W. (1970). Theory of Micropolar Elasticity. Springer, Wien. Kunin, I. A. (1982). Elastic media with microstructure 1. One-dimensional models. SpringerVerlag, Berlin, Heidelberg, New-York

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

747

Dynamic tensile deformation and fracture o f metal cylinders at high strain rate

Manjit Singh~, H..R. Suneja a, M.S.Bolaa and S. Prakash b a Terminal Ballistics Research Laboratory, Sector 30, Chandigarh-160020, India b Department of Physics, Panjab University, Chandigarh-160014, India

An experimental study is presented on the radial expansion of metal cylinders of internal diameter 52mm and wall thickness 1-6mm, internally loaded with high explosives. A rotating mirror streak camera monitors the distance time (x-t) history of the cylinder wall, expanding under strain rate of 104-105sl, till it is ruptured. Aluminium cylinder is found to rupture at strain (e) of 75-150 %. For a fixed wall thickness, the rupture strain is found to increase with the strain rate. However, when the wall thickness is changed, a maxima is observed in the graph showing the rupture strain and strain rate relationship. Probably, aluminium follows the Ivanov rupture criteria. In deformation of copper cylinder the cracks initiation at strain of e=30-50 % is followed by rupture at very high strain up to 300 %. Recovered fragments show wall thinning by 50-60% and also exhibit that the shear fracture dominates the radial fractures.

1. INTRODUCTION The one-dimensional geometry of the radially expanding ring [ 1] is perhaps the simplest in considering the fundamental aspects of the fracture and fragmentation process. A ductile metal ring subjected to an outward radial impulse will decelerate due to circumfrential tensile force. The magnitude of this force will vary due to the work hardening in the metal and geometric softening caused by the thinning of the ring. Initially hardening dominates but eventually a maximum in the tensile force is achieved at which the flow changes from stable to unstable. The unstable flow is manifested by the onset of plastic necking and fragmentation. In a ductile metal, fracture proceeds through the multiple nucleation and growth [2] of necking regions. Concepts, which are found to govern the fracture and fragmentation in the expanding ring are generalized for the cylindrical and spherical shells. Researchers have used different techniques of driving the metal such as gun technique [3], electromagnetic force [4] and explosives [5,6]. Under a small impulse, by electromagnetic loading, cylinder wall will attain terminal velocity and then decelerate. The retardation history of the cylinder wall gives the stress-strain [4] behaviour of the metal. When the explosives are used as the driving force, the impulse generated is large enough to rupture the cylinder in the accelerating phase. Present studies are on the tensile deformation of metal cylinder when loaded internally with high explosives. Detonation of high explosives gives an outward impulse to the cylinder wall that expands under strain rate of 104-105 s-1, till the rupture occurs. The rupture strains of aluminium and copper cylinders have been found out under different strain rates.

748 2. FRACTURE HYPOTHESES Taylor [7] proposed a model to predict the radius associated with fracture based upon the assumptions that the fractures are radial. Compressive hoop stresses exist over the inner portion of the tube wall and are governed predominately by explosive pressure. The radial fracture cannot propagate into this compressive zone. Taylor concluded that the fractures would initiate at the outer surface of the tube wall where hoop stresses are tensile and penetrate up to the depth of (o T/P) which defines the boundary between the tensile and compressive hoop stresses. Here or, T and P are tensile strength, wall thickness and internal explosive pressure, respectivdy. Since P is a function of r/r0, ratio of instantaneous to initial internal tube radius, the fracture radius is defined as the radius associated with P= ~. This theory is essentially correct and is well supported by experimental observations. However, observations of fracture mode do not support, in general, the assumption of radial fracture. Hoggat and Recht [8] further developed this model for a cylinder. This model is based on the early appearance of small radial cracks in the tensile hoop stress region near the outer surface of the tube. With in the compressive hoop stress zone the tube expansion is accomplished by extrusion, which activates shear planes, rotated approximately 450 from the radial direction. When the component of stress normal to the shear directions changes from compressive to tensile, the preferentially weakened material in the shear zone, fractures. However, the material in the shear zone has been thermally softened by the heat of plastic deformation, resulting in the fracture along shear planes and the appearance of shear lips rather than the radial fractures.

2.1. Compressive Radial Stress in the Cylinder Wall Let P is internal explosive pressure acting on the wall of internal radius r. Then equation of motion for cylinder wall earl be written as [8]

d2a (1) where ro is initial internal radius of the tube, Ro is initial external radius of the tube, and p is density of the tube material. The pressure acting to produce the radial acceleration, d2a/dt", at any arbitrary radius 'a' within the wall, is defined by the equation of motion for the portion of the wall external to this radius, which is written in the same form as Equation (1) i.e. p,(2na) = p~(R2 - a2) d2a dt 2

(2)

From Equations (1) and (2) one gets p = .

9

]

.p

ro

(3)

Neglecting elastic strains and considering axial strain, ez. to be zero, equations for 'a' and R are written a = (r2-1-~3~2-1"o2) 1/2

and

R=

(1"2-1- Ro 2- ro2) 1/2

(4)

749 Above equations are written on the basis of conservation of mass of the expanding ring and also assuming that no appreciable change in density of the ring material occurs during expansion. Substituting Eqn. (4) into Eqn. (3) provides the required expression for the radial pressure in the wall at any radius 'a' as a function of the internal explosive pressure P

Pa=l(r2+a2_r2

;" R2o r2J

(5)

The internal explosive pressure can be represented by an isentropic expansion

P=

Po r

(6)

where Po is the effective detonation pressure acting on the wall when r = ro and y is the expansion exponent. The gas expansion exponent y actually varies during expansion, typically approaching a value of five during the early stages, and decreases as the expansion proceeds. A constant value of about y=3 [9] is generally taken for the calculations purposes. During expansion the radial pressure decreases and when it becomes equal to the tensile strength of the metal the cylinder fractures. 3.

EXPERIMENTAL PROCEDURES

Rotating mirror framing and" streak cameras were used to record the cylinder expansion and fracture. Framing camera views the whole cylinder, whereas, the streak camera views only a small annular ring through 0. l mm wide camera slit. An argon gas in a cardboard container was explosively shocked to produce the back light, thus to record the optical shadowgraph of the expanding metal cylinder. As the cylinder expands its wall thickness decreases till the onset of fracture and rupture.

Figure 1: Sequential framing camera photograph of an expanding aluminium cylinder initiated from both the ends. The rupture in the wall was identified with the outburst of detonation gaseous products through the cracks in cylinder wall. Figure 1 shows the framing camera photograph of an expanding aluminum cylinder detonated from both the ends. The expansion velocity of the

750 cylinder wall, V, is calculated by measuring the angle made by the expanding wall with the cylinder axis along with the detonation velocity. Figure 2a shows the streak camera record of an expanding outer wall of the metal cylinder. The initial outer diameter of cylinder is shown as the masked patch on the argon flash. The distance time plot of an expanding wall is shown in figure 2b. After some finite expansion the cylinder ruptures.

Figure 2: (a) Streak camera record and (b) distance time history, of an expanding aluminium cylinder of wall thickness 7mm and internal diameter 52mm, under TNT loading. From the measurements of cylinder wall velocity, V, and radius at fracture, rf. the strain and strain rate have been calculated from the relations [3]

e~ -

rr-r o 1"o

-

Ar ro

and

V I dr e. . . . . ro ro dt

(7)

The wall thickness of the cylinders and explosive compositions were changed to deform the metal under different strain rates. 4. RESULTS & DISCUSSION

4.1. Radial Deformation of Aluminium Cylinder The results of the rupture strain of aluminium cylinder under different loading conditions are plotted in figure 3(a). The results indicate that the aluminium cylinder ruptures at the strain of 70-150%, depending upon the wall thickness and the strain rate. Three explosive compositions, Baratol, TNT and Octol were used to deform the metal at different velocities. For each explosive composition, the wall thickness of the cylinder was changed to alter the strain rate. The results exhibit a maxima in the rupture strain for a wall thickness of 3-5rnm. This type of rupture criteria, showing a maximum strain at some value of strain rate, was put forward by Ivanov [I0]. By applying Ivanov criteria and assuming a visco-elastic relation, t~=t~0+rle, for flow stress, we have calculated macroscopic viscosity coefficient,rl, for alurninium to be 0.55-0.87xI() 3 Pas, in the strain rate region 3-7 x 104sI. However, when the wall thickness of the cylinder was kept constant and the explosive composition was changed to produce the different strain rate, then the rupture strain was found to increase with the strain rate, as shown in figure 3(b). This trend of increasing failure strain with strain rate was also shown by Slate et al [I I]. He expanded thin walled spherical shell of various metals by detonating a sphere of explosive located at the center of shell. He found out rupture strain in the range of 30-90%, under strain rate of 6-45xi03 st . Higher rupture strains observed in the present studies are may be due to higher strain rates.

751

Figure 3. Variation of rupture strain with strain rate for aluminium tube with (a) changing wall thickness (b) fixed wall thickness, for Baratol, TNT and Octol explosives.

4.2. Radial Deformation of Copper Cylinder The experimental value of rupture strain found out in copper cylinder of different wall thickness, are given in Table 1. The process of rupture in copper cylinder has been found to be different than that of aluminium. Aluminium cylinder expands uniformly till it ruptures and the point of rupture is clearly identified in most of the experiments. In copper cylinder it is observed that the crack is initiated at the outer wall and then it propagates inward through the expanding cylinder wall, till the rupture is complete. From the measurements of the time between crack initiation and rupture, the crack propagation velocity through copper cylinder wall has been found to be 270-290m/s. Wall thinning up to 60% has been assumed for calculating this velocity. This phenomenon could be observed only in few experiments as the crack observation requires a very high optical resolution of the streak camera. Fracture strain, the strain at which the crack initiates, has been observed to be 30-70%. Rupture strain up to 300% has been observed in copper cylinder when loaded with powerful octol explosive. Table 1 Experimental values of fracture and rapture Inner Wall Diameter Thickness Explosive ( mm ) ( mm ) 25 1 RDX/IqqT 52 2 52 1 TNT 52 2 52 3 52 1 Octol 52 3 52 4 52 1 Baratol 52 2

strains of copper cylinder Expansion Strain Rate Fracture Velocity Strain (mm/ms) (x 10 4s1) 1.45 5.80 --2.5 4.80 67 2.49 4.79 50 1.93 3.72 30 1.75 3.36 40 3.1 5.96 --2.5 4.80 102 2.2 4.23 --1.82 .... 3.50 ' 29 1.34 2.57 57

Rupture Strain (%) 160 ------200 257 300 328 158 190

The observed high rupture strain indicates that the copper metal is undergoing rupture by total necking. Very large rupture strain at high strain rates has also been observed by Taylor [12]. The rupture preceded by a complete necking was confirmed by Stelly[ 13] who performed the metallographic examination of the recovered copper fragments. A very pronounced grain

752 elongation with twining was also observed. Some defects, which will produce early instability at low strain rate, will appear to be more stable at the strain rates exceeding 10as~ .

4.3. Recovery Figure 11 shows the recovered fragments when a 45mm internal diameter aluminium tube was deformed by RDX/TNT (60:40) explosive. The thickness of the recovered fragments was 0.9 +_ 0.05mm, for a 2mm initial wall thickness. This indicates that the phenomena of metal extrusion takes place in the expanding wall to produce wall thinning up to 60%.

Figure 4: Recovered fragments after rupturing of 2mm thick and 45mm internal diameter aluminium tube. Close examination of the fragments reveals that the shear fracture dominates the radial fracture in high strain rate deformation of aluminium. Under explosive loading the outer layers of the cylinder are in tension as a result of the circumfrential stretching or hoop strains. However, in the inner parts of the cylinder wall, a state of hydrostatic pressure is produced which is sufficient to prevent the development of tensile stresses. Tensile crack can not propagate inward during that phase and the strains are accommodated by the propagation of 45 ~ shear failure. In general, the proportion of shear (45 ~ to tensile (90 ~ failure will increase with the increase in expansion velocity. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

D.E. Grady and D.A. Benson, Experimental Mechanics, 23, pp 393-400, 1983. L. Davison, A.L. Steveans and M.E. Kipp, J. Mech. Phys. Solids, 25, p 11, 1976. R.E. Winter et al, Proc. Mechanical Properties at High Rates of Strain, ed. J. Harding, Institute of Physics, London, pp 242-51, 1979 W.H. Gourdin, Proc. Mechanical Properties at High Rates of Strain, ed. J. Harding, Institute of Physics, London, 1989. I.R. Lambom, A.J. Bedford and B.E. Walsh, Proc. Mechanical Properties at High Rates of Strain, ed. J. Harding, Institute of Physics, London, pp 251-261, 1974. M.S. Bola, A.K. Madan, Manjit Singh and S.K. Vasudeva, Defence Science Journal, 42, pp 157163, 1992. G.I.Taylor, Scientific Papers of G.I. Taylor, Cambridge Univ. Press, England, 3, 1963. C.R. Hoggatt and R.F. Recht, J. Appl. Phys., 39, p 1856, 1968. F.E. Allison and J.T. Schriemp, J. Appl. Phys., 31, p 846, 1960. A.G. Ivanov, Strength of Materials, 8, p 1303, 1976. P.M.B. Slate, M.J.W. Billing, and P.J.A. Fuller, J1. Inst. Metals, 95, pp 244-51, 1967. J.W. Taylor, F.H. Harlow and A.A. Amsden, J1. Appl. Mechanics, 45, p 105, 1978. M. Stelly, Proc. Mechanical Properties at High Strain Rates, ed. J. Harding, Institute of Physics. London, 1979.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

753

Energy balance in dynamic brittle rock failure B. G. Tarasov Department of Civil and Resource Engineering the University of Western Australia Nedlands, WA 6907, Australia

This paper presents the experimental study of the dynamic brittle rock failure after the loss of stability by the "loading machine - rock sample" system. Violent brittle failure of laboratory samples is analogous to mckburst in mining pillars. For a proper understanding of this mechanism, it is important to consider the energy balance of the -whole system. The interrelation between all components of the energy balance needed to predict the violent nature of dynamic events is established. Two types of rocks are identified: A-type rocks that show an increase in post-failure energy consumption for dynamic failure compared to static failure, and B-type that show a decrease in the energy consumption. The latter type is more prone to rockburst.

1. INTRODUCTION N. G. W. Cook in his laboratory investigation of the progressive disintegration and loss of strength for rock samples loaded beyond their peak strength, established the mechanism of stability loss by the "loading machine - rock sample" system [1]. He recognised that the inadequate stiff loading system released more energy than could be absorbed by the slow deformation of the sample. The excess energy was used to accelerate the disintegration process. This mechanism was seen by Cook [2] and other researches to be analogous to mechanisms of mining rockbursts. Control of the energy release by manipulating the "local mine stiffness" was seen as a possible way to restrain rockburst. In practice, for a more successful control and operation of the dynamic process in mines, not only the cause of the stability loss need to be known, but also, the distribution of the released excess energy in the system. It is necessary to know, for instance, what part of the released elastic energy transforms into kinetic energy of flying rock fragments or to seismic energy, and what part of energy is absorbed by the dynamically disintegrated rock material. In order to address these questions, special experimental investigations have been undertaken. The author has carried out the laboratory studies in St. Petersburg Mining Institute, Russia, after developing the experimental technique in collaboration with A. N. Stavrogin, [3].

2. EXPERIMENTAL TECHNIQUE The loading setup [3] consists of a rigid frame 1 and a rigid wedge drive (2, 3; see Figure 1). This system stores a negligible amount of elastic energy when loading a sample.

754

Figure 1. System for study energy balance.

Figure 2. Components of energy balance.

The elastic element (4), fixed onto the wedge (2), is the source of the strictly controlled elastic energy accumulated in the loading system. By using such a set of elastic elements with varying stiffness values it is possible to change the elastic energy stored in the system by hhree orders of magnitude. Gauges (5) on the surface of the elastic element measure the strain in the element during both loading and fracturing of the sample. Variable inertial mass (6), supplied with a piezoelectric accelerometer (7), can be attached to the top of the element. In the experiments, the value of the inertial mass was varied by a factor of ten, from 0.5 to 5 kg. A rigid dynamometer (9) is attached to the stationary surface of the frame. A gauge (11) is used to measure the deformation of the sample. The experiment is conducted by placing the sample (8) on the inertial mass (6). The wedge drive generates a static load until the sample fails at either peak or post-peak strength. Unstable dynamic deformation is effected by the elastic energy stored in the elastic element. This stage of loading is not controlled. It is accompanied by flying sample fragments and by a vibration process generated in the elastic element after failure. The stiff dynamometer, gauges (5, 11) and the accelerometer provide data for both the static and dynamic loading cycles. Figure 2 shows a typical force P - displacement AI curve of a sample with line BG characterising a stiff loading system. Different areas of this diagram correspond to different energy kinds. The following notations are used: Wc = Wi + Wc + W~ is the total work of sample fracture (shaded area of the diagram); Wi is the work done in irreversible deformation before instability; We is the elastic energy stored in the sample before instability; Wp0 is the work used during the softening process; Wt = W~ + Wk + Wv + Wt is the elastic potential energy stored in the loading system before instability; Wk is kinetic energy of flying fragments of a fractured sample; W~ is the dynamic energy of the vibration process generated in the elastic element after failure of a sample; Wt is the thermal energy. The unstable deformation of rock occurs when the energy required for deformation is less than that available due to deformation. In laboratory tests instability occurs under the condition M > ML where M is the post-failure "stiffness" of the sample and ML is the stiffness of the loading system.

755 3. EXPERIMENTAL RESULTS Tests have been conducted on a variety of rocks: marble, granite, sulphide ore, two types of sandstone, brown coal, rock and potash salts [4, 5]. Except for marble, samples were extracted from rockburst-prone mine areas. All these rocks have shown absolutely stable postpeak deformation in the stiff loading system under condition M < ML. This means that the elastic energy stored in the rock samples at peak strength was completely absorbed by internal fracturing and was not transformed into dynamic energy. The only source of dynamic effect is the elastic energy stored in the loading system before instability. The next step of the investigation was to shed light on the energy consumption behaviour of the rocks in the stable (slow) and unstable (dynamic) deformation regimes. Typical experimental force-displacement curves indicating the post-failure behaviour of these rocks during unstable dynamic (curves 1) and stable static (curves 2) fracture are shown in Figure 3, with lines 3 characterising the stiffness of the loading system. The areas under the diagrams correspond to the energy consumption of the samples and the elastic energy stored in the elastic element before the loss of stability. The curves show that the energy of dynardc postfailure deformation Wpd differs fundamentally from the energy of static deformation. Two types of rocks were identified: (i) rocks that exhibit an increase in post-failure energy consumption for dynamic failure compared to static failure (A-type) (ii) rocks that exhibit a decrease in energy consumption for dynamic failure compared to static failure (B-type) The sandstone, shown in Figure 3, for example belongs to A-type behaviour; the rock salt belongs to B-type bvhaviour. Similar tests were conducted under varied amount of elastic energy WL stored in the loading system before Lnstability. This was achieved by using different elastic elements. The dependence of Wpd on the amount of elastic energy WL is shown in Figure 4 for the same rock types. We observe a fundamental change in post-failure energy consumption but this dependence attenuates with increase of WL. The explanation of this Wpd-WL behaviour, typical for different rocks, is given in [4, 5]. A seven-fold increase in energy consumption was observed in granite.

100: P, kN

25

L

Sandstone

20

80" 60"

~ 3

15

P, kN

9

~

Rock salt 31~~23~.~

40

20

x" d 2 . . ~ ~ 0.1

Al, mm 0.2

0.3

,

.

0.5

1

.|' m m 1.5

Figure 3. Post-failure portions of the dynamic (1) and static (2) force-deformation curves for sandstone and rock salt. Line (3) is the stiffness characteristic of the loading system.

756

20

Wpa, J

ne

Rock salt

10 10

5 I

I0

I

20

I

30

IW-~' J 40

W L, J 10

20

Figure 4. Post-failure energy consumption W~ vs. elastic energy of loading system WL.

The difference in energy consumption for various situations must be considered when estimating the energy balance in rockburs~. Particular emphases should be given to the existence of two rock types. For B-type rocks, small excess in the loading system energy leads to extremely violent dynamic fracture. Figure 5 illustrates the difference in the estimation obtained (a) without consideration, and (b) with consideration, of the energy consumption change during unstable deformation. The unshaded triangular areas under the graphs correspond to the energy consumption of the materials, the shaded areas correspond to the energy released from the loading system and transformed to dynamic effects. Thus, consideration of the energy consumption change leads to fundamentally different results.

Figure 5. Illustration of the energy release estimation for two cases: without consideration (a) and with consideration (b) of the energy consumption change during unstable deformation. Next, the transformation mechanism of the potential elastic to dynamic energy in the form of vibration energy of the loading system W,, kinetic energy of flying fragments Wk, and thermal energy Wt, was considered. The energy Wv was measured in experiments using an accelerometer (7). The accelerometer registered acceleration of the inertial mass (6) during and after the fracturing of the sample. The amplitude of the accelerometer signal was proportional (for a given inertial mass and stiffness of the elastic element) to the energy of the vibration process generated in the elastic element after failure. A special method was used for the calibration of the accelerometer. A thin-waUed glass tube was placed in the testing system instead of the rock sample. This glass sample was loaded up to a value equalling the strength of the rock sample. At this stage, the elastic energy stored in the elastic element through its load and deformation was determined. The glass sample was then broken by striking its side with a special device. The time required to fracture the glass (3 x 10.5 s) was about a hundred times less than the natural period of vibration of the elastic element. In this case, practically all the potential energy of the elastic

757

element was transformed into vibration energy. The vibration process was generated in the elastic element with the attached inertial mass. In this situation (with the glass sample) the amplitude of the accelerometer signal corresponded to the total energy stored in the elastic element before fracture. The same calibration tests were conducted under different loads and, consequently, the relationship between the released energy and the amplitude of the accelerometer signal was established. Using this calibration relationship, and knowing the amplitude of the accelerometer signal obtained in an experiment with a rock sample, the value of the vibration energy Wv, was determined. The remainder of the released energy transformed into kinetic energy Wk, in the form of flying sample fragments, and to thermal energy Wt. Experiments over a wide range of stiffness values, inertial mass ml, and sample mass ms have shown that the energy ratio Wv/Wk is determined by the mass ratio mCms: Wv/Wk = m~/ms

(I)

This equation implies that the fragments of the fractured sample gain velocity due to the dynamic movement of the inertial mass of the loading system. The velocity of the fractured mass is equal to the maximum velocity of the inertial mass motion in the first cycle of the vibration process. In other words, the mechanism of the energy transfer from the loading system to the fractured mass of the sample is based on the principle of a catapult operation. Analysis of the complete energy balance using the experimental results for the behaviour of WL,Wpd,Wv,Wk energies has shown that under uniaxial loading the amount of thermal energy Wt is negligible. Thus, the complete energy balance of dynamic brittle rock failure can be written as follows: WL = W ~ + Wv + W~,

(2)

The relationship between the different components of the energy balance is determined by equation (1) and the following equations: W, = ml (Wt.- W ~ ) / (ml + ms)

(3)

Wk = ms (WL- Wpa ) / (mi + ms )

(4)

20 W, J ~

10 W,J

Sandstone

Rock salt W

10

5 T

10

I

I

20

30

i

40

Wl,J ~'

10

20

Figure 6. Components of the energy balance in dependence on the elastic energy, stored in the loading system before instability.

758 Figure 6 represents the experimental results reflecting the relationship between all the mentioned energy kinds determined for sandstone and rock salt. The experiments were conducted using different elastic elements that provided different elastic energy WL values in the loading system before instability. 4. CONCLUSIONS The most important experimental results considered in this paper are: - The interrelation between all components of the energy balance of dynamic brittle rock failure. - Two types of rocks were identified: 1) rocks that showed an increase in post-failure energy consumption for dynamic failure compared to static failure (A-type). 2) rocks that revealed a decrease in energy consumption for dynamic failure compared to static failure (B-type). The latter type is more prone to rockburst.

ACKNOWLEDGEMENT The author is very grateful to Dr. A.V. Dyskin and Dr. E. Sahouryeh for reviewing the paper.

REFERENCES

1. Cook, N.G.W. The failure of rock. Int. J. Rock Mech. Min. Sc. V. 2, 389-403 (1965). 2. Cook, N.G.W., eL al. Rock mechanics applied to the study of rockbursts. J. So. Afr. Inst. Min. Metall. V. 66, 436-528 (1966). 3. Stavrogin, A.N., Tarasov B.G. A test machine to study energy balance of rock sample fracture. Authors Certificate No. 1024796 Inventions Review, No. 23 (1983) [in Russian]. 4. Tarasov B.G. Energy consumption in brittle fracture of rock. Cand. Theses, Leningrad, All Union Inst. For Rock Mech. & Surveying (VNIMI) [in Russian] (1983). 5. Stavrogin, A.N., Tarasov, B.G. Energy balance of rock fracune. FTPRPI, No. 1, 18-27 (1985). [in Russian].

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

759

Stress intensity factors for tubular T-joints with a curved surface crack Bo Wanga, Seng Tjhen Lieb and Zhihai Xiang a aDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China. bSchool of Civil and Structural Engineering, Nanyang Technological University, Singapore. A new finite element (FE) mesh generator has been developed for three-dimensional curved crack problem in T tubular joints. FE mesh for T tubular joints with a curved surface crack can be obtained by transferring a plain plate with a semi-elliptical crack. Stress intensity factors (SIFs) for tubular T-joints with a surface crack are calculated by using FE method. SIFs for the same model are also obtained from the mixed FE and boundary element (FE/BE) method. In FE/BE analyses, four subdomains, which include two BE subdomains for the region of high stress concentration and other two FE subdomains for remaining parts, are used to model flawed tubular joints. Comparison of numerical results from FE and FE/BE methods with experimental results indicates that these two numerical results are in good agreement with test results. This conclusion verifies the reliability of numerical simulations. I. INTRODUCTION In civil and offshore engineering, many tubular structures suffered from earthquake, wave loading and damaged generally in the form of fatigue cracking. These cracks develop and curve at the weld toes, which join the contoured ends of the brace members to the outside of the larger chord members. The formation of macro-cracks and the behavior of crack growth in tubular joints subjected to the load have been investigated[I-3]. Crack driving force, SIF is a very important fracture parameter to govern the crack initiation and growth. During recent years, FE method becomes a very useful tool to analyze the mechanical behavior of tubular joints. The available commercial FE packages have been widely applied in nonlinear analysis on uncracked and cracked tubular joints[4]. Shell elements were usually used to model members for different types of tubular joints. However, most work has been limited for through cracks in tubular joints, since the mesh generation is still the most time-consuming work for the FE analysis of joints. In the case of the three-dimensional surface crack, it is very difficult to carry out numerical analyses for tubular joints and the mesh generation is one of the key parts in FE analysis. Very fine meshes are required for modeling cracks and the transition between the fine mesh for a crack and the coarse mesh for the region outside the crack is very complicated[2,3,5]. In the present work, following the program ABACRACK [6], a new mesh generator has been developed for T tubular joints with three-dimensional crack. SIFs for tubular T-joints with a curved surface crack will be calculated. On the other hand, the mixed FE and BE method is introduced. The same geometry and loading cases will be simulated by using the mixed FE/BE method. Two numerical results will be compared with experimental results.

760 2. FE ANALYSES 2.1 Tubular T-joint model

470kN Y

Dimensions in (ram) Brace Chord Outside diameter 457 914 Wall thickness 16 32

1210 I

J

Z '~r

crack tip ----~X

3900 Figure 1. The geometry of T tubular joint The configuration of T tubular joint specimen is shown in Figure 1. Because of the symmetry in the x-y and y-z planes, only one quarter of the joint is modeled. The two ends of the chord are restrained rigidly in all directions. The joint is axially loaded with a force of 470 kN in the y-direction. The properties of elastic material include Young's modulus E = 2.1• and Poisson's ratio v = 0.3. When the joint was subjected to this fluctuating axial force, it was found in the test that fatigue crack initiated at the weld toe on the chord close to the saddle point, and propagated through the chord wall. Huijskens[7] measured crack profiles by using the beach-marking technique and the geometry of the surface crack is shown in Table 1. Table 1 Geometric parameters for T tubular joint with a surface crack Chord: Diameter ( D ) . 914mm Thinkness (t) 32mm Brace" Diameter (d) 457mm Thickness (t) 16mm Crack: Length ~2c) 123mm Depth (a) 18mm 2.2 FE Mesh Generation

Figure 2a. Mesh for plain plate with an elliptical crack

Figure 2b. T tubular joint with a surface crack

761

Figure 2c. Semi-elliptical crack

Figure 2d. Mesh with a surface crack at-weld toes

The tubular T-joint model with surface crack starts from a plain plate containing a semielliptical crack. Using the program ABACRACK[6] generates the plate mesh shown in Figure 2a. In this program, the width of plate b, the length of plate h, half width of crack c, and the depth of crack a, may all be specified as a multiple of the thickness of plate T, which is always unity. Further generation for tubular joints, a new program has been developed and can be used to transform the plane model into a tubular joint intersection. Firstly, a 3D T-butt mesh is produced, and then is mapped irto a tubular joint intersection. The modeling details are as follows: An attachment is added to the plain plate mesh by specifying the number of elements through the thickness of the attachment and adjusting the plain plate mesh such as that the attachment footprint will have the desired width. The attachment, with the required dimensions and mesh grading, is then mapped to provide a weld profile at its base. Geometric parameters like attachment thickness t, weld angle, and weld toe radius r can be easily specified. Since the problem is not symmetric about the crack plane, the next stage is to complete the main plate by simply reflecting the original plate about the crack plane. This results in a full 3-D T-butt mesh with a refined weld toe and crack mesh. A series of mapping on the T-butt mesh has been performed. First, the length of the main plate, which eventually becomes the chord radius (D/2), is adjusted to give the desired geometry of the final joint. The mesh is then curved round 900 so that the attachment becomes the brace. The main plate, now a quarter circle, is further mapped into a square, as shown in Figure 2b. Further mappings turn the main plate into the top section of the chord and bring the brace down to join the chord giving the model the familiar tubular joint intersection shape. The rest of the chord is then added to form a quarter tubular T-joint with a surface crack, shown in Figure 2b. This program will produce the input file, which is used for the general purpose FE package ABAQUS. Brick elements are used for chord, brace and weld toe. Full integration for 20node quadratic brick elements (C3D20) are used in the crack region and reduced integration for 20-node quadratic brick elements (C3D20R) are used elsewhere. 15-node quadratic triangular prism elements (C3D15) are used in the crack front. There are 3316 elements and 16703 nodes in this model. Semi-elliptical surface crack mesh and local mesh with a surface crack at weld toes are shown in Figure 2c and Figure 2d, respectively. 2.3 Stress intensity factor Elastic solutions for the displacements near the crack tip are used in the method of displacement extrapolation. In plane strain case, displacements can be expressed as follows. Mode I:

762

u, = ~

cos(

0)[1 - 2 v + sin 2 ( 0 ) ] ,

v, = -~-

sin(

0 ) [ 2 - 2 v - cos 2 ( 0)],

w t= 0

( l a , b,c)

Mode II: Kn~r 1 1 Ku~ ur=--G-- ~--nnsin( 0)[2-2v+cos2( 0)], Vn =--~--

1 1 COS( 0)[-l+2v+sin2(~0)], w,=0

(2a,b,c)

Mode III: ur =0,

v. =0,

w t =_K~~nsin(20 )

(3a,b,c)

where u, Vn, wt are the local radial, normal and tangential displacements, G is the shear modulus and v is the Poisson's ratio. The plane stress form of these equations is obtained by substituting v with v/(1 +v). At~er running ABAQUS program, stress, strain and displacement fields will be available for tubular joint model. Stress intensity factors can be evaluated at any point on the crack front from the asymptotic behaviour of the displacement near the crack front. For the quarter-point elements used on the crack front, stress intensity factors K1 and Kn can be calculated by using the following equations: K,

-

2/r2_.~ ~ 1 ~/L 2/'~"i [4(uS -- UC) _ (UO _ UE)] x:bt + 1 ~t., i [4(vS - vC)--(VD -- vE)]' K,, = ~--i'+

(4a,b)

where u and v are the shearing and opening displacements at the end-nodes D, E, and at the 88 B, C in the quarter-point element, Li is the length of the side of the element internally adjacent to the front, ~t = E/2(1 +v) is the shear modulus of elasticity and ~: = (3-4v) for plane strain and ~: = (3-v)/(1 +v) for plane stress. 3. FE/BE

ANALYSES

To use the mixed method of FE and BE, a region f~ in Figure 3a is divided into two parts: f~ for the FE method and f22 for the BE method. Ful, Ft I and Fu 2, Ft 2 are boundaries with known displacements and tractions for f2~ and Q2 respectively. F~ is an interface between subdomains f2~ and f22. Following the combined technique of FEM and BEM[5], the relation between nodal displacements and tractions on the interface from FE equations is introduced into boundary integral equations as a natural boundary condition. The same geometry tubular T-joints with a semi-elliptical surface crack is simulated by the combination of FEM and BEM, shown in Figure 3b. Two layers of BE are used to model the region of high stress concentration along the intersection between the brace and the chord. The thickness of the two layers is equal to 32 mm which is the main thickness of the chord. The BE region is then further divided into two subdomains BE1 and BE2 along the semi-elliptical crack surface. FE1 and FE2 are used to model other parts of tubular T-joints including chord and brace. The subdomain BE1 is then connected to FE1 and likewise BE2 to FE2. In this model, 8-20 nodded isoparametric elements are used for the two FE subdomains. There are 515 nodes and 210 elements in FE1,334 nodes and 116 elements in BE1,210 nodes and 75 elements in BE2, and 553 nodes and 226 elements in FE2. The stress intensity factors for the same crack profile

763 are obtained by using Eq.(4).

Figure 3b. FE and BE meshes 4. R E S U L T S AND D I S C U S S I O N S --"--

3DFEM

--o-4o

9

BE-FE Test

mrda ~9

;

-'-

40"

~,1

~____----m~|

~. ~"~ 1

./e''1"

~o

u)

|'------m--_~. n_\./om~m / ~-s

/

~A._.___..__A~A

; " ;o " ~ " 3; " 4'0 " ~o " /o " ;o Distance

along

the crack

front (mm)

Figure 4a. K~ distributions along crack front

, 10

',

, 20

Distance

-

, 30

.

, 40

. . . . A ~ h' -

,

-

50

, ...... 60

, 70

along the crack front (ram)

Figure 4b. Kx, Kn, Km distributions along crack front

SIF, K~, distributions along the surface crack front from FEM and FE/BE are shown in Figure 4a. These two numerical results for the same crack profiles are compared and the tendencies of two curves are very similar. From FE analyses, the maximum value, K ~ x takes place at the deepest point along the curved crack front. The two KI values at the deepest point are in good agreement with experimental results. It means that those FE and FE/BE meshes and simulation are reliable. However, there are some errors between two values at the surface point along the crack since there is difference between two meshes. This difference will not

764 affect surface crack growth at the deepest point. Stress intensity factors K I, K n and Km are plotted in Figure 4b. From these results, it can be concluded that SIFs of the mode I is the most important factor to affect the crack growth at the deepest point. Near the surface point, SIFs of mode II and mode III will affect the crack propagation as well. But crack growth at the deepest point is the most dangerous for T-joints with a surface crack. Therefore, it means that in the stress-sWain field around the crack tip, K~ can be taken as a fracture parameter to govern the crack initiation in cracked tubular structures. 5. CONCLUSIONS 1) A FE mesh generator has been developed for three-dimensional curved crack problem in T tubular joints based on the existing ABACRACK program. Comparing the two results from FE and FE/BE with test results has proved the reliability of this generated model. 2) Stress intensity factor distributions along the crack front have been obtained by using FE and FE/BE analyses, respectively. These two numerical results show to be in good agreement. The maximum SIFs take place at the deepest point along curved semi-elliptical crack front and agree very well with experimental result. The SIFs of mode I is the most important factor to affect the crack growth at the deepest point for T tubular joint with a surface crack. ACKNOWLEGEMENT The first author would like to be grateful to Tan Chin Tuan Fellowship at Nanyang Technological University in Singapore. The project is partly supported by Opening Laboratory from Educational Ministry of China- Failure Mechanics Laboratory in Tsinghua University and Base Science Foundation in Tsinghua University. REFERENCES 1. B. Wang, N. Hu, Y. Kurobane, Y. Makino and S. T. Lie, Damage criterion and safety assessment approach to tubular joints, Engineering Structures, in press, (1999). 2. B. Wang and K. C. Hwang, An engineering approach to safety assessment for non-Jcontrolled crack growth, Engineering Fracture Mechanics, Vol. 57, No. 6, (1997) 689-699,. 3. D. Bowness and M. M. K. Lee, The development of an accurate model for the fatigue assessment of doubly curved cracks in tubular joints, Int. J. of Fracture, 73, (1995) 129-147. 4. ABAQUS, ABAQUS v5.5 Manuals (Users' Manual I and II, Theory Manual, Example Manuals I and II, Verification Manual), Hibbitt, Karlsson and Sorenson Inc., (1995). 5. S. T. Lie, G. Li and Z. Z. Cen, Analysis of tubular joints using coupled finite and boundary element methods, Engineering Structures, (1998). 6. ABACRACK, Three-dimensional surface crack generator, v3.1, FRCR-003, (1989). 7. H. A. M. Huijskens, Fracture mechanics based predictions of the effect of size of tubular joint test specimens on their fatigue life, Master Thesis, TU Delft, The Netherlands, (1988).

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

765

Effect of the E n v i r o n m e n t and Corrosion on the Fatigue Life of a Simulated Aircraft Structural Joint S. Russo a, P.K. Sharp a, R. Dhamari b, T.B. Mills c, B.R.W. I-Iintona, K. Shankar b and G. Clarka aDefence Science and Technology Organisation, Aeronautical and Maritime Research Laboratory (AMRL), 506 Lorimer Street Fishermens Bend, Victoria 3207, Australia bSchool of Aerospace and Mechanical Engineering, University of New South Wales, Australian Defence Force Academy (ADFA), Northeott Drive Canberra ACT 2600, Australia eunited States Air Force Research Laboratory, Wright Patterson Air Force Base (VASE), Ohio 45433, United States of America.

Corrosion and fatigue are major factors in determining the structural integrity and life expectancy of ageing aircraft. The presence of corrosion on airframe structures can have a detrimental effect on the integrity of the aircraft structure by promoting fatigue crack initiation and accelerating crack growth. This paper describes the results of ongoing work aimed at (i) gaining a better understanding of when and to what extent the environment degrades the fatigue strength of a typical aircraft structural joint, and (ii) identifying the effect Corrosion Preventative Compounds (CPCs) and corrosion have on the fatigue life of the joint. Fatigue test results indicated that the application of a CPC at the faying surfaces decreased the fatigue life at 144MPa, whilst the effect was not statistically significant at the higher stress level (210MPa). A similar effect was also observed for tests under humid conditions. The presence of intergranular corrosion in the bore of the countersunk fastener holes reduced the fatigue life by at least one order of magnitude for both stress levels. The addition of a CPC also reduced the fretting corrosion at the faying surfaces of the test joint and shifted the fatigue initiation sites to within the bore of the fastener holes. The scatter in fatigue life for specimens under identical environmental conditions was found to be associated with the location of fatigue crack initiation sites. Cracks initiating at the edge of the fastener hole had lower fatigue lives compared to those that initiated away from the fastener hole. 1. INTRODUCTION Many military aircraft are operated well beyond their original design life, and age-related issues affecting aircraft structural integrity need to be addressed. These include the development of corrosion and fatigue cracks on ageing aircraft structures, which will have a significant effect on maintenance cost, aircraft availability and flight safety. The repair of corrosion and fatigue cracks in military aircraft is costly, both in economic terms and aircraft

766 operational availability. Thus, it is crucial that these issues be fully investigated to gain an insight into their effects on aircraft safety and to determine appropriate maintenance actions. 2. EXPERIMENTAL DETAILS The fatigue specimen, also used in a round robin testing program (1), consisted of an aluminium alloy 11/2 dog-bone coupon assembled using a pair of cadmium-plated steel HiLok interference fit fasteners. This single shear joint is meant to simulate the load transfer and secondary bending moments commonly experienced at stiffener runouts attached to the outer skin of airframe structures (2). The material was 3.2ram thick 7075-T6 bare aluminium alloy; specimens were machined in the longitudinal (rolling) direction. All specimens were protected with a chromate conversion coating, applied to Specification MIL-C-5541 "Chemical Conversion coatings on Aluminium and Aluminium Alloys", and an uninhibited epoxy enamel topcoat to Boeing Specification BMS 10/11 Type II "Chemical and Solvent Resistant Finish". The fastener holes were drilled following the application of the protection scheme, to ensure the bore of the holes unprotected. Fatigue testing was performed in either dry air (relative humidity<40%) or in air at 100% relative humidity, both at 295 K. A series of specimens was tested following the application of a CPC. The CPC used is a fast-acting and deep-penetrating oily lubricant containing a corrosion inhibitor (3). It was applied to the faying surface of the joint prior to fatigue testing. This CPC is currently used on several RAAF aircraft and meets specification MR~C-16173DGrade 3. Pre-corrosion was carded out in a constant immersion test using EXCO solution (4) which is widely used to assess the susceptibility of aluminium alloys to exfoliation and intergranular corrosion. To ensure corrosion was only localised in the bore of the hole, the countersink surfaces were masked, and a plastic cell with a circular contact area measuring 12ram in diameter was fitted over each fastener hole prior to assembly. Fatigue tests were performed using tension-tension constant amplitude sinusoidal loading with a stress ratio R = 0.1, and a frequency of 4 Hz, at stress levels of 144 MPa and 210 MPa. The four testing schedules included (i) fatigue in dry air, (ii) fatigue in humid air, (iii) the application of a CPC followed by fatigue in dry air and (iv) pre-exposure followed by fatigue in dry air. Further details are summarised in Reference (1). All fatigue testing was undertaken at ADFA in collaboration with AMRL. Optical analysis of the fracture surfaces and adjacent regions using a Zeiss stereo microscope at magnifications of 60X to 400X was used to characterise the types of degradation at the faying surfaces, and the origin, length and distribution of fatigue cracks. 3. RESULTS A comparison of log mean values for fatigue life as a function of environment is presented in Figure 1 for the two loads investigated. As expected, the life to failure under each environment was lower at the higher stress. The statistical package ANOVA (5) based on the Analysis of Variance was also used to determine any environmental effects. Both sets of results indicated that the application of a CPC at the faying surfaces decreased the fatigue life at 144MPa, whilst the effect was not statistically significant at the higher stress level (210MPa). A similar effect was also observed for tests under humid conditions. Pre-exposure

767 of joints in a corrosive solution prior to testing led to a reduction in fatigue life of over one order of magnitude at both stress levels. Io'

,

" " !

....

, ....

i

t"" ~""21 I - - e - - 210MPaH

! ....

i ....

i ....

t

10s

10 4

lo'

....

ENVIRONMENT Figure 1.

Comparison of the log-mean fatigue life data for the various environments.

Following fatigue testing the fracture surfaces from each specimen were closely analysed using both visual and optical techniques, and the number of fatigue cracks on each specimen determined. It was found that at 210MPa, the average number of fatigue cracks per specimen was approximately five, which was twice as many as at the lower stresslevel.Fatigue cracks were classified as primary, ie. those that initiated fn-st,and secondary. In most cases the primary fatigue crack was the crack with the greatest length and was easily identified by fractographic analysis. They ranged in length from 1.5mm to 3.5mm. The origins of each primary and secondary fatigue crack were classifiedas Fatigue Crack InitiationSites (FCIS) and labelled according to Figure 2. T'nese locations were divided into three distinctzones, the laying surface zone containing regions G or S, the hole zone containing locations B-E and M Q inclusively,and the edge zone containing locations F or R. All fatigue cracks originated from the upper plate of Figure 2. The majority of primary fatigue cracks initiated at the faying surface zone in the dry and humid environments. The application of a CPC changed the distribution of FCIS, so that there were more sites in the hole and fewer at the faying surfaces (figure 3). As expected, the presence of corrosion in the bore of the holes resulted in the initiation of all fatigue cracks in this zone.

Figure 2.

Key to locations of fatigue crack initiation sites.

Detailed observations of the fracture surface, especially at the faying surface, indicated the presence of regions containing a black corrosion product. Qualitative microanalytical techniques using Energy Dispersive Spectroscopy (EDS) identified the black product as

768 primarily aluminium, with oxygen and cadmium. The cadmium would be from the fasteners while the black product was probably the result of fretting corrosion. This fretting product was most pronounced on specimens exposed to a dry atmosphere and was observed both adjacent to and remote from the FCIS. The application of a CPC to the joint almost eliminated fretting at the faying surfaces. Humidity had a similar effect, reducing the amount of fretting at both stress levels. Interestingly, fretting did not seem to play a role in the formation of cracks but instead was a result of the extra displacement in the joint caused by fatigue cracks. These results are shown in Figure 4.

Figure 3. The dependence of the location of primary fatigue crack initiation sites on the environment.

Figure 4. The effect of the environment on the degree of fretting at the faying surfaces

In addition to characterising the FCIS into designated zones, a two coordinate axes system in the z-plane was also used to document the exact location of crack origins. Crack origins using this axis system for specimens tested at 210MPa under various environments are shown in Figure 5. Only the initiation site of the primary crack is plotted. It should be noted that no

Figure 5. Fatigue Crack Initiation Sites using X-Y the coordinate system of specimens tested at 210MPa under various environments. primary fatigue cracks initiated on the countersink or top surface from the upper plate. Hence, points shown remote from the inner circle indicate cracks initiating on the faying surface. It is

769 evident that cracks initiated at a number of locations, in and away from the minimum net section. A larger number of fatigue cracks initiated towards the free edge side, ie. at the left side of fastener hole 2 or the right side of 1 for all environments at both stress levels. Similar observations have been made by others using identical specimens (6). In over 90% of the cases, investigation using the Field Emission Electron Microscope (FESEM) indicated that both primary and secondary fatigue cracks fomaed at Fe and Cu rich intermetallics in the microstructure. The relationship between the location of primary FCIS and the fatigue life was investigated (Figure 6). Data for the pre-exposed environment are not included in this figure since all FCIS for this testing environment occurred at the one location. It is evident that for specimens tested dry or with the application of a CPC, primary fatigue cracks originating at either fastener holes or edges produced lower lives than those originating at faying surfaces. The relationship between location of primary FCIS (X coordinate) and fatigue life is shown in Figure 7. For primary FC1S located close to the minimum net section (X=0), the fatigue lives were lower than those for other failure sites. Of the many attempts to correlate fatigue life to some observed failure feature, the relationship between inclusion size and total fatigue life was the strongest (Figure 8).

Figure 8. Relationship between the Inclusion Size and Fatigue Life

770 4. DISCUSSION The results have shown that the test environment plays a major role in both the cycles to failure and locations of the FCIS for the 11/2 dog-bone specimens. While several investigators have tested this specimen under a variety of conditions, including dry and humid atmospheres (1-2,6), this is the fwst investigation to report on the effects of exposing the joint to CPCs or to a localised corrosive environment in the fastener holes. For the specimens where CPC was applied, the reduction in fatigue life at the lower stress level may be explained by the CPC reducing the fraction of load transmitted by friction, and concomitantly increasing the load transmitted by shear in the fasteners. In the higher stress tests, the frictional load transfer is overshadowed by high bearing loads and high mechanical deformation. Also, it may be explained by the movement into the fatigue cracks of CPCs containing fretting debris which could provide fulcrum points for leverage near the crack tip. This effect would be less pronounced at high stress due to the shorter times to failure. The role of corrosion damage as a stress raiser and its associated impact on fatigue life is well known (7). This may account for the moderate reduction in log-mean fatigue life for specimens exposed to the humid conditions at the low stress level, and also the large reduction in fatigue life at both stress levels for specimens containing pre-corrosion. The shorter testing times for specimens exposed to humid conditions and tested at 210MPa may explain why corrosion was not present at the FCIS for these joints. The appearance of fretting corrosion product on several specimens indicates that there was relative movement of the two surfaces. Fretting corrosion is most often associated with the liberation of metallic particles that become oxidised and thus more abrasive. The reduced amount of fretting corrosion product associated with the presence of CPC may be explained by (i) the lubricating action of the CPC resulting in a reduction in the coefficient of friction or (ii) the exclusion of oxygen from the environment. Moisture could also cause a reduction in the coefficient of friction at the laying surfaces, thus explaining the reduction in fretting damage on specimens tested in the humid environment. The lower fatigue lives of cracks originating in the bore of the hole or at the minimum net section (Figures 6 and 7) is likely associated with the high stress concentration at this location. Likewise, fatigue cracks that initiate at regions remote from the fastener hole experience a lower crack driving force and thus exhibit higher fatigue lives. Figure 6 shows the wide discrepancy in fatigue lives of cracks forming from the bore of the hole and those forming from the faying surface. Similarly, Figure 7 illustrates the trend of longer fatigue lives for cracks originating further from the line of minimum net section. 5. CONCLUSIONS The effect of treatment with CPC before testing in dry air and that of testing untreated specimens under humid conditions resulted in a reduction in fatigue life at the lower stress level (144MPa) whereas the differences in life were not significant at the higher stress level (210MPa). At the high stress levels, high bearing loads and high mechanical deformation are believed to have suppressed the effect of CPC and corrosion fatigue, respectively. At the lower stress, an increase in bearing load in the fasteners for the CPC cases, and the activation of corrosion fatigue in the humid cases, are believed to be the causes for life reduction.

771 Pre-exposure of joints to a localised corrosive environment at the fastener holes led to a significant reduction in fatigue life at both stress levels. The pitting/intergranular corrosion damage was quite severe, and the stress raiser effect drove the reduction in fatigue life. 9 Excluding the specimens that had been prior-exposed to generate corrosion damage in the hole bores, most fatigue crack origins were traced to faying surface Fe/Cu rich inclusions approximately 5-15 microns in size. In particular, this was most often the case for the specimens exposed to either dry or humid conditions. The application of a CPC showed a slight tendency to shift failure origins to the bores of the holes, while this was always the case for specimens exposed to prior corrosion. 9 The application of CPC to the specimen joints almost eliminated the existence of fretting corrosion at the faying surfaces for both stress levels. The effect was similar, though to a lesser degree, for specimens exposed to humid conditions. Still, fretting was not the cause of cracking in these cases. In general, primary fatigue cracks nucleating at either the faying surfaces or at a region remote to the minimum net section had higher fatigue lives. REFERENCES 1. R.J.H. Wanhill, J.J. De Luccia and M.T. Russo, "The Fatigue in Aircraft Corrosion Testing (FACT) Programme", North Atlantic Treaty Organisation, Advisory Group for Aerospace Research and Development, AGARD Report No. 713, (1989). 2. R.J.H. Wanhill and J.J. De Luccia, "An AGARD-Corrosion Fatigue Cooperative Testing Program", North Atlantic Treaty Organisation, Advisory Group for Aerospace Research and Development AGARD Report No. 695, (1982). 3. R.F. Mousley, "An Effect of corrosion preventative fluids on the fatigue of riveted joints",

The Influence on fatigue; Proceedings of the Conference, London, England, May 18-19, 1977, Institute of Mechanical Engineers, London, pp 131-136, (1977). 4. "Standard Test Method for Exfoliation Corrosion Susceptibility in 2XXX and 7XXXSeries Aluminium Alloys", G34-90, Annual Book of ASTM Standards, American Society for Testing and Materials, (1990). 5. J.L. Devore, "Probability and statistics for engineering and the sciences", Brooks/Cole Publishing. Company, Monterey, California, USA, (1982). 6. G. Segerfrojd, S. Zuccherini, G. Giovarmelli and L. Magnusson, "Fatigue Behaviour of Mechanical Joints - An Experimental Evaluation of Ten Different Fastener Systems and their Influence on Fatigue Life", FFA TN 1996-63, The Aeronautical Research Institute of Sweden, Stockholm, Sweden, (1996). 7. Metals Handbook, 9 th edition, Volume 13-Corrosion, ASM International, p1031, (1992).

This Page Intentionally Left Blank

Numerical Simulation

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

775

Plastic Instability Simulation of Steel in Tension S. Okazawa a and T. Usami b a Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA* b Department of Civil Engineering, Nagoya University, Chikusa, Nagoya 464-8603, Japan Ultimate plastic instability behavior of steel in tension is simulated by using a 3D elasto-plastic bifurcation analysis. In this study, after strictly detecting a bifurcation point and conducting a branch-switching procedure, localized deformations just before shear fracture of members with various width-thickness ratios are analyzed with the J2 flow hardening theory. Excellent correlation is demonstrated between the computational and the experimental results. 1. INTRODUCTION A steel member in plastic zone under pure tension has a limit point load. Beyond the limit point, necking occurs. This necking triggers a localization of plastic strain and leads to ultimate member fracture. Not limited to the steel, such behavior can be found for other metal materials and is known as a plastic instability phenomenon [ 1]. In most of conventional plastic instability analyses, 2D analyses assuming plane strain state have been employed [2]. It has come to be believed through such analyses that only diffuse necking can be simulated and that both localized necking and shear band could not be reproduced with the conventional J2 flow hardening theory. Therefore, deformation theories [3] or constitutive equations considering the occurrence of a micro void [4] has been used in plastic instability analyses. There is little research of 3D plastic instability analyses [5] in comparison with 2D analyses. A steel and other solids are 3D bodies fundamentally. Three-dimensional geometry gives great influence on instability behavior. Accordingly, a 3D analysis considering the geometrical shape of material is indispensable to clarify the plastic instability phenomenon. A study which examines whether or not it is possible to simulate the ultimate plastic instability behavior subsequent to diffuse necking by a 3D analysis with the J2 flow hardening theory is found only in the paper of Tvergaard [6] as long as the authors know. However, the analysis object in the study was only a thin member and a thick member was not treated. * formerly a postgraduate student at Nagoya University

776 Under such a background, we have conducted a 3D plastic instability analysis of tension members by using only the J2 flow hardening theory. It has been investigated whether or not simulating the localized deformation states after diffuse necking by a 3D plastic instability analysis in the members of various thickness is possible. Furthermore we compare the computed results with the deformation states just before fracture in tensile member experiments. 2. PLASTIC INSTABILITY PHENOMENON AS BIFURCATION BEHAVIOR The plastic instability phenomenon has come to be Limit~ n r c a t i o n Point ~ ~ Primary Path treated as bifurcation behavior after a limit point. Figure 1 is a concept of bifurcation behavior in plastic instability. A steel member continues to elongate just after a bifurcation point X Failure beyond the limit point along the .... Elongation primary path. On the other hand, diffuse necking occurs by The concept of bifurcation Figure 1. switching to the bifurcation Behavior in plastic instability path. We try t o simulate the localization phenomenon of the next stage by tracing the bifurcation path. In many of the existing plastic instability analyses, imitative bifurcation analyses introducing imperfection disturbances have been employed. In this study, we shall treat the plastic instability analysis as the bifurcation problem of a perfect system. The discussions with regard to the behavior of actual imperfect systems will become possible only after the bifurcation behavior of the perfect system is clarified [7].

/

3. BIFURCATION CRITERION The Hill's bifurcation criterion [8] has already been established as an elasto-plastic bifurcation criterion, and it is applied to structural plastic buckling and instability analyses. The Hill's bifurcation criterion of volume v is shown as follows with a rate of displacement vector u [9].

where S is the second Piola Kirchhoff stress tensor and E is the Green Lagrange strain tensor. The superscript 9 and the right superscript * are the rate and the difference in the solution of bifurcation and primary path direction respectively. The linear comparison solid in which yielding always continues is assumed in the Hill's bifurcation criterion. Equation (1) can be rewritten in the discrete form through the technique similar to the finite element method with the nodal displacement vector U:

777

The fictional tangent stiffness matrix K~in which the linear comparison solid has been assumed is defined as a sum of the linear part K / a n d the nonlinear part K~z" K c = K L + KNL

(3)

Equation (2) indicates that matrix K cis sing,Mar: dct K * = 0

(4)

Equation (4) is the same structure as the bifurcation criterion in elastic problems; i.e., the determinant of the tangent stiffness matrix K becomes zero. The difference between K c and K is only the part of the forth-order elasto-plastic constitutive tensor in the linear stiffness matrix, being induced by whether or not the possibility of unloading from yielding state is considered [9][10][11]. Therefore, by excluding all the possibility of unloading from the true tangent stiffness matrix, the same evaluation method as the method introducing the Hill's linear comparison solid is possible for the elasto-plastic bifurcation analysis. In conclusion, it is not necessary to introduce the linear comparison solid by HAll for the bifurcation criterion [ 12]. 4. BRANCH-SWITCHING We can evaluate the displacement predictor dU, for the branching direction on the bifurcation point with the linear injection of the critical eigenvector 0, corresponding to the zero eigenvalue of the tangent stiffness matrix and the displacement predictor dUx for the primary direction: dUll = C e s + dU1

(5)

Here C is the scaling factor of 0,. In the elasto-plastic bifurcation criterion of the previous section, the unloading possibility is excluded on the evaluated bifurcation point. Therefore, the bifurcation behavior does not match with the theory, if unloading occurs at the moment of branching instantaneously [13]. Hence, in the numerical procedure, the largest C can be determined so that the first unloading point remains still neutral. 5. COMPUTATIONAL RESULTS Ultimate plastic instability behavior is computed using a 3D bifurcation analysis. We use the J2 flow hardening theory for elasto-plastic constitutive equations. The relation between the following equivalent stress w and plastic strain ~P is assumed in the plastic zone.

Figure 2. The analysis model

778

~=oy 1+ ~-p

e y --

o~_ 1

(6) (7)

E 500 Here the subscript y denotes the yielding value. We set with Young's modulus E=200 (GPa) and Poisson's ratio v=0.3333. The hardening stress-strain relation is assumed in equation (6) by setting with n=0.0625. An analysis model is shown in Figure 2. Due to symmetry, the one-eighth of specimen is analyzed and 8-node isoparametric elements are employed with selective reduced integration to avoid volumetric locking. We apply a tensile load with boundary condition for uniform deformation. Three kinds of width-thickness ratio (W/t=lO,4,1) are computed. The numerical value in each model n_ame shows the value of width-thickness ratio. Figure 3 is the equilibrium path ofT-10. Where H is the nominal stress and u is the elongation in direction of the applied load. The bifurcation occurs immediately after the limit point. And other models T-4 and T-1 have bifurcation points after the limit points too. Also each bifurcation mode shows diffuse necking mode. The bifurcation occurs only once and a new bifurcation point has not been found on the bifurcation path. The localized plastic deformation becomes remarkable after diffuse necking. The localization phenomenon differs largely by the width-thickness ratio of the model. Figure 4 is the ultimate deformation state on the bifurcation path of the model T-10. The cross diagonal localized necking has occurred and shear deformation concentrates in the loca_lized necking area. The occurrence of this localized necking can be considered as the shear fracture. In Figure 5 that is the computational result of the model T-4, instead of shear type diagonal localized necking, the

Figure 6.

The two axial symmetric localized necking of T-1

779 narrow area sinking of thickness direction, so-called the concentratedlocalized necking occurs. In the model T-l, we can also observe two axial symmetric localized necking in the cross-sectional square in Figure 6. The reason that the localized necking behavior after the diffuse necking differs like this is the progress of unloading area on a bifurcation path. Unloading occurs at the edge of the specimen in all of the models firstly. However, the progress of unloading area of the next stage differs largely. Now, we adopt T-10 and T-4 to show progress of unloading. Figure 7 is the progress of unloading in model T-10. Unloading occurs once again from the central part, after unloading begins from edges and progressed to the central part. And deformation is localized into the diagonal plastic part left. However, in the model T-4, the unloading area from edges spreads furthermore and wraps a central plastic zone on the ellipse in the Figure 8. And, the concentrated localized necking occurs through deformation localized on the left plastic zone. 6. CORRELATION WITH EXPERIMENTS We carried out some experiments using tension members of structural steel to confirm the justification of the present 3D plastic instability analysis. The width-thickness ratios of steel members are 10,4,1 and these correspond to the models T-10, T-4, T-1 of previous section. Figure 9 is the ultimate deformation mode just before fracture in the tensile member of the width-thickness ratio 10. Shear type diagonal localized necking has occurred. This agrees well with the computational result of model I"-10 of Figure 4. Figure 10 is the tensile experimental result of the width-thickness

780 ratio 4.0 in which the localized concentrate necking similar to that of Figure 5 has been observed. Furthermore, in the case of the width-thickness ratio 1.0, the computed localized deformation of Figure 6 agrees well with the experiment of Figure 11. As we have seen, these experimental modes agree well with the 3D computational results without material instability from a qualitative standpoint. 7. CONCLUDING REMARKS Ultimate plastic instability phenomena just before fracture have been simulated by a 3D elasto-plastic bifurcation analysis with the J2 flow hardening theory. That t h e bifurcation of dii~se necking occurs after the limit point is the common behavior irrespective of member proportioning. By tracing the bifurcation path furthermore, the deformation moves to the diagonally localized shear type necking in the thinner members or to the concentrated localized necking in the th'cker members. After the bifurcation to diffuse necking occurs, no second bih~cation (i.e., in the meaning of the tangent stiffness matrix becoming singular) exists. In conclusion, behavior from the uniform state to ultimate deformation just before fracture, through diffuse necking, is a series of the geometrical instability. A 3D elasto-plastic bifurcation analysis for a perfect system is indispensable to elucidate the above phenomenon. It has been clearly demonstrated that there is no need to resort to other constitutive relations than the conventional J2 flow theory. REFERENCE

1. V. Tvergaard, Studies of Elastic-Plastic Instability, J. Appl. Mech., ASME, Vol.66 (1999) 3. 2. (for example,) R.M. McMeeking and J.R. Rice, Fimte Element Formulations for Problems of Large Elastic-Plastic Deformation, Int. J. Solids. Structures, Vol.ll (1975) 601. 3. (for example,) B. Budiansky, A Reassessment of Deformation Theories of Plasticity, J. Appl. Mech., ASME, Vol.26 (1959) 259. 4. (forexample,) V. Tvergaard and A. Needleman, Analysis of the Cup-Cone Fracture in a Round Tensile Bar, Acta Metallurgica, 32 (1984) 157. 5. (for example,) H.M. Zbib and J.S. Jubran, Dynamic Shear Banding : A 3D Analysis, Int. J. of Plasticity, Vol.8 (1992) 619. 6. V. Tvergaard, Necking in Tensile Bars with Rectangular Cross-Section, Comput. Methods Appl. Mech. Engrg, 103 (1993) 273. 7. F. Fujii and S. Okazawa, Pinpointing Bifurcation Points and Branch-Switching, J. of Engineering Mechanics, ASCE, No.3 (1997) 179. 8. R. Hill, A General Theory of Uniqueness and Stability in Elastic-Plastic Solids, J. Mech. Phys. Solids, Vol.6 (1958) 236. 9. Y. Tomita, Simulation of Plastic Instabilities in Solid Mechanics, Appl. Mech. Rev., ASME, Vol.47 (1994) 171. 10. R. de. Borst, Numerical Methods for Bifurcation Analysis in Geomechanics, Ingenieur-Archiv, Vol.59 (1989) 160. 11. J. P. Bardet, Finite Element Analysis of Plane Stram Bifurcation Within Compressible Solids, Comp. Struct., Vol.36 (1990) 993. 12. P. Wriggers and J. C. Simo, General Procedure for the Direct Computation of Turmng and Bifurcation Points, Int. J. Numer. Methods in Engrg, u (1990) 155. 13. J.W. Hutchinson, Post Bifurcation Behavior in the Plastic Range, J. Mech. Phys. Solids, Vol.21 (1973) 163.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta @ 2000 Elsevier Science Ltd. All rights reserved.

781

Several practical criteria for nonlinear dynamic stability of lattice structures* Z.-Y. Shen" ,Z.-X. Lib and C.-G. Deng" "College of Civil Engineering, Tongji University, Shanghai, 200092,China bDepartment of Civil Engineering, Zhejiang University, Hangzhou, 310027,China In this paper, nonlinear dynamic differential equations are transformed into general nonlinear equations without differential terms by the assumption of linear accelerations. The modified dynamic equations are similar to the nonlinear equations for static stability in expressing forms, so the general displacement controlling method for nonlinear static stability problems after improving can be used to trace the dynamic buckling and post-buckling equilibrium path of lattice structures. The conception of dynamic stability for nonlinear structures is interpreted according to the definition in the field of mathematics given by Lyapnov, and several practical criteria for dynamic stability are established. Finally, examples of elastic dynamic stability problems are analyzed, and the reliability and practicality of the proposed method and the criteria are demonstrated.

1. I N T R O D U C T I O N The phenomenon of dynamic instability exists in many fields t1'2]. Russian scientist Lyapnov had given a strict definition of dynamic stability and proposed two methods to analyze N The one is that by solving nonlinear dynamic differential equations, if the solution become divergent with time increasing, the equations can be regarded as dynamic instability. In this paper, such views are adopted for structural dynamic stability problems. The structural dynamic differential equations can be written as

/K(v,)}- -[M]{o, }- [el{o, }-

}+ {p}

(,)

where, {K(U,)} is nodal resistance vector, [M] , [C] are mass matrix and damping matrix,

~), }, ~, }, {Ut } are nodal acceleration vector, velocity vector and displacement vector respectively, {/ig} is earthquake acceleration vector, and {P} is static load vector. There are two cases that can lead the solution to divergence. The first one is that the equivalent load vector, the whole terms in the fight side of Eqn(1), is infinite. Periodic harmonic vibration and parameter excitation instability can be interpreted as this case. The second one is the structural tangent stiffness matrix becoming non-positive. Geometrical nonlinear dynamic instability can happen due to this case. In this paper, the elastic "This work is supported by National Natural Science Foundation and State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University

782 geometrical nonlinear dynamic stability of space frames under seismic actions is studied. It is difficult to define the structural dynamic instability strictly for a structure subjected seismic actions. In the past, there were many kinds of interpretations on structural dynamic stability. Bolotin got a series of achievements in the theory of parametrically excited dynamic stability under periodic loading t4]. Wang, Yang and others has been studying the theory of impact buckling tS]. The theory of dynamic buckling under step loading has also been studied by many authors [rj. But no one of the results can be adopted to analyze the dynamic stability problems of structures under seismic actions. Considering such situation, in the paper the elastic geometrical nonlinear dynamic instability is defined according to the response of lattice structures under the seismic actions as below: under seismic actions, the structural dynamic instability happens when the structural tangent stiffness matrix become non-positive, which leads to the resistance capacity decreasing or losing locally or globally and the dynamic response of nodal displacements and structural deformation increasing dramatically. When there are only local elements and nodes decreasing or losing resistance capacity, it is regarded as local dynamic instability. If the resistance capacity of the whole structure decreases or lose, it is called structural total dynamic instability. It is important to notice that a structure when it is unstable at certain time may be stable afterward due to the cycling behavior of the seismic action. In this case it is appropriate to say that the structural dynamic instability has occurred once. From this point of view, dynamic instability may occur several times for the same structures.

2. IMPROVEMENT AND APPLICATION OF THE GENERAL DISPLACE-MENT CONTROLLING METHOD By the linear acceleration assumption

{O,.,.,~}= ao({U,+~ }- {Ut }) - a2 ~l, I- a3~, }

}= {o, }+,,6 {o, }+,,, {c,,.,, }

(2) (3)

nonlinear dynamic differential equations can be transformed into general nonlinear equations without differential terms,

{K(U,+a,)}+ (ao [M] + a,[Cb{U,+~ }= {Pt+~}

(4)

where, ao, al, a2, a3, a6, a 7 are constants, {P,+A,} is equivalent dynamic load vector which can be determined at time t [71

[C]= Cl[M] + c2[KTg] ,c2= [Krg ]

2(co,g, -cojgj) 2

is tangent stiffness matrix corresponding to the global coordinate, to~ and coj are circle

frequencies vs i and j modes in initial time, ~, and ~j are critical damping coefficients of i and j modes.

783 For the initial iterating of the ith increment step, the incremental equations of Eqn(4) can be given a s I781 (5)

:

where , [K]i-' -- [KTg ]i-I + A~', 9(ao[M ] +

a,[C])

{P--,'+~,} = {P,+a,} -(aoiMl + a,[Cl){ uit+lAt } --{t F}, {'F} is the nodal resistance vector at time t. A21 is the load parameter increment in the initial step of ith increment and can be given as I

AA'~ = (- 1)" AA~IGSP}~

(6)

where, GSP is a general stiffness parameter, n is the number of the sign changing of GSP and A~ is a given load parameter increment. GSP can be calculated as follows

,T.{u,}'

GSP = {u, {u, }i_, }, r {u'} i I

i-I

(7)

1

i

mi_ !

where, {u, },, {u, }1 , {u, }, are the displacement increment vectors corresponding to {P,.a,} and tangent stiffness matrix at initial, (i - 2)th and (i - 1)th increment steps at time t + At respectively. At jth iteration of ith increment step, nonlinear incremental equations can be given as i

--i

i

(8)

For convenience, Eqn(8) can be rewritten as [KTg ]ij_, {U, } j

(9)

[K~, l~j_, (u: }'j " {R}'j_, {u}, j + {u~ }j where, {R} i

(~0) (11)

+/'e} - /

-i

F},

{,+AtF} is nodal resistance vector at j - l t h increment A)~'j =

J

{U, },_,T{u, }~ i j

( j >--2/

iteration, and A2'j is load parameter

(12/

At the jth iteration of the ith increment step, if the following condition arrives, the iteration in the ith increment terminates, and the next increment process begins.

784

{u}~jr(/~rL~{~--~t~}- {R};-' ) ~ oe1

(13)

In tracing the dynamic equilibrium path, the load parameter is at first given by the following equations for each increment step

(14)

'+~ F~(k)-'F(k)

Xo =

P:.~ (k)

where, k means the kth component of the resistance vector selected as the controlling parameter. In the process of iteration of each increment step, the load parameter increments can be accumulated directly. When the accumulated load parameter is approximate to 'l',and the controlling constraint Eqn(13) is satisfied, then the iteration process terminates at current time t and the next increment process at t + At begins. Since the equivalent dynamic load vector is related to ettrrent displacement vector, and it is still unknown, so the equilibrium path is traced dynamically as eqn(5). Where AX't is estimated approximately according to the condition in the last increment step, this may lead to k 0 > 1. If such ease occurs, the increment step must be decreased and reama to the initial state of current increment step.

3. E X A M P L E S O F E L A S T I C S T R U C T U R A L D Y N A M I C S T A B I L I T Y The example shown in Figure. 1 is a William plane frame t7..91 A mass of 5kg is exerted on the joint of the frame sustained the vertical component action of ELCENTRO earthquake with amplitude 1.5g. The calculated time-history curve for the nodal displacement is given in Figure.2, It is shown that there are several times that the nodal displacement responses are intense. The numerical results show that dymmaie instability occurs between 6s and 6.1s, 6.3s and 6.4s, 8.4s and 8.5s, 8.9s and 9.0s, 9.4s and 9.5s, and a 'snap-through' type dynamic instability happens in the upward and downward directions respectively. When instability happens, the structural tangent matrix becomes non-positive. When LDLT decomposition of the structural tangent matrix is carriedout, several components in the diagonal matrix D are negative or approximate to zero. This is a sign of dynamic instability occurring. h=6 172mm L--

tiT1

/St-

I

=9.804mm

L:657.5 mm Figure. 1. William's plane frame

j

785 0.016 0.012

%- O. 008 ~" O. 004 O. 000 -'0. 004 0.0

.

.

.

.

v'lv " ' - ' V '~}

.

i

2.0

4.0

6.0

8.0

I0.0

t(s) Figure.2. Time-history curve for nodal vertical displacement

--, Z u.

250

2.5

2OO

2.0 1.5 1.0 ~ 0.5 ~ 0.0 -0. 5 -t.0 0.015

150 I00 50

y

0 -50

6

-I00

-0. 005

O. 000 O. 005 O. Ol

- ' - F vs U ---GSP vs U

U(m) Figure.3. Curves for vertical resistance and general stiffness parameter vs displacement The partial curves for the vertical nodal resistance and general stiffness parameter vs the vertical nodal displacement of structures suffering dynamic instability are given in Figure.3 at the time around 8.9s. It is shown that the structural resistance decreases with displacement increasing in part of the curve duringthe load incrementing process. It dernonstmtes that the dynamic instability occurs. It is also shown that GSP decreases with structural tangent stiffness deteriorating, and its value is approximate to zero and becomes negative at the time near the critical point. Other examples have also been calculated and the same conclusions as the above example are obtained. The results are not given in the paper for page limitation.

4. C O N C L U S I O N By analyzing of the elastic examples, it is demonstrated that the theory and method in this paper are reasonable and valid in nonlinear dynamic stability analyses. It is also shown that the structural tangent stiffness matrix, the time-history curve for nodal displacement response, the equilibrium path curve and the magnitude of the general stiffness parameter can be used to

786 determine the dynamic instability conveniently. Therefore, several practical criteria for dynamic instability can be drawn as following 1. Criterion based on displacement. If the time-history curve for nodal displacement waves intensely in some time field, and the amplitude overpasses the critical value for static instability under the same loading form( for dynamic problems, it means equivalent dynamic force vector), it can be regarded that the dynamic instability occurs. This criterion can be used to estimate the structural dynamic instability approximately. 2. Criterion based on the tangent stiffness matrix. In tracing the equilibrium path, if the structural tangent stiffness matrix becomes non-positive, it can be regarded that the structures have gone into critical state. If tangent stiffness matrix is decomposed as LDL T and there are negative components in diagonal matrix D, then a conclusion can be drawn that the structural dynamic instability has happened. 3. Criterion based on general stiffness parameter (GSP). When structures go into critical state, GSP is approximate to zero. When the equilibrium path goes over the critical field, and the structural post-buckling path is unstable, GSP becomes negative and this is a sign of dynamic instability happening. 4. Criterion based on the dynamic equilibrium path. Under dynamic loading, if the structures go into post-buckling state and the post-buckling path is unstable, or the tangent line of equilibrium path appears horizontal or decline, it can also be regarded that dynamic instability occurs. REFERENCES 1. Z.X. Li, Nonlinear Dynamic Stability Analyses Of Steel Lattice Structures, Ph.D.Thesis Univ.of Tongji, 1998. 2. J.H. Ye, Dynamic Stability Analyses of Single-Layer Reticulated Shell. Ph.D.Thesis Univ.of Tongj i, 1995. 3. S.Q. Shu. Theory of Stability of Ordinary Differential Equations. Shanghai Science and Technology Press, 1962 4. V. V. Bolotin. The Dynamic Stability of Elastic Systems. Hadden-day. San Francsco, 1964. 5. R. Wang and at. el. Advances of Impact Dynamics. China University of Science and Technology Press, 1992. 6. A. T. Brewer and L. A. Godog. Dynamic Buckling of Discrete Structural Systems under combined Step and Static Loads. Nonlinear Dynamics, 1996, Vol.9, No.3, pp249-264. 7. Z.X. Li, Z.Y. Shen and C.G. Deng, Application of General Displacement Controlling Method in Dynamic Stability Analyses, J. Tongji Univ, Vol.26,No.6(1998),pp409-412. 8. Y.B. Yang and M.S. Shieh, Solution Method for Nonlinear Problems with Multiple Critical points. AIAAJournal, Vol.28,No.12" (1990) pp2110-2116. 9. G.Z. Voyiadjis, G.Y. Shi. Nonlinear Post-buckling Analysis of Plates and Shells by Four-Noded Strain Element, AIAA Journal,Vol.30,No.4(1992),ppl110-1116.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

787

Snap-through Analysis of Toggle Frame using the software package, NIDA, by I element per member Siu-Lai CHAN and Jian-Xin GU Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hong Kong, CHINA This paper discusses the development of the exact tangent stiffness matrix for an initially curved beam-column element. After incorporation into the analysis program, NIDA (Nonlinear integrated design and analysis), the large deflection analysis of the toggle flame with member initial imperfection is conducted. Only one element is sufficient to model a member even it is under a high axial force. It was further found that many researchers over-looked this point of analysing structures with members under small axial force to claim their element being capable of modeling a member by a single element. Conventional elements under high axial forces are inaccurate and resort must be made to the present development of high-precision element. 1

INTRODUCTION Second-order analysis for checking of the large deflection or the P-A effect has been common to date. This checking is superior to the uses of simplified design formulae in various national design codes in that it is simplier, more accurate and more general. However, the P-5 effect, referring to the second-order effect between axial force and deflection along an element, can only be included by using a refined element stiffness or by dividing a member into many elements. This makes the widely used cubic element deficient in second-order analysis since the stiffness matrix becomes incorrect when under high axial force. The computer software, NIDA (Nonlinear Integrated Design and Analysis) is developed using curved element for simulation of member initial imperfection which is mandatory in various design codes of practice. Further to this, the powerful Pointwise Equilibrating Element (PEP) and the exact stability function elements are used in place of the Hermite cubic element which is deficient in buckling analysis of columns and frames. NIDA has been used for design of a number of slender frames and trusses. This paper demonstrates the proficiency of the element and the robustness of the NIDA in the second-order analysis of a toggle flame. The method of stability function has been developed for decades. It differs from the fmite element method in that it derives the exact tangent stiffness matrix in stead of assuming a displacement function. Livesley and Chandler ~ presented a careful derivation of the stiffness matrix of a member under an axial force. Oran ~formulated the stiffness matrix allowing for the bowing effects, but no examples were given on the application of his element matrix Since the introduction of high grade steel, member buckling and frame instability become more important. The design of steel scaffolding by Chu et al. 3, Peng et al. 4 and of steel trusses by Chan, Zhou and Koon 5 are some of its application to practical structures.

788 2

INITIAL I M P E R F E C T I O N AND EQUILIBRIUM EQUATIONS In reality, the initial imperfection of a member is random and of arbitrary shape. Imperfections are, however, assumed to be in a half sine curve with amplitude at mid-span as follows (see also Figure 1). v0 = Vmosin ~zrx L

(1)

in which v0 is the lateral deflection, Vmois the magnitude of imperfection at the mid-span, x is the distance along the element longitudinal axis and L is the element length (Figure 1).

l /

Y

deformedcurvature e~.~~/// initialcurvature

M1 t

Lc .

.

.

.

.

L

.

.~~2

M2 LUL

'1

1

Figure 1 The Element with Initial Crookedness

The equilibrium equation along the elemem length can be expressed as, EI ~d2=vl dx 2

- P(vo + vt) + Ml- + M., x-Mr L

in which E is the Young's modulus of elasticity, I is the second moment of area, the nodal moments and v~ is the lateral displacement induced by loads.

(2)

M l

and

Making use of the boundary conditions that when x-0 and x-L, v~=0, we have,

v,

_ M r sin(#-kx) -(-~ sin~

L-x M2 sinkx x + vmoSm---L "-P- sin~ L L

(3)

M 2 are

789 Superimposing the deflection to the initial imperfection, we have the f'mal offset of the element centroidal axis from the axis joining the two ends of the element as,

V

"-

V

1

+

V

(4)

0

~,I -~'~sin0 . L~x].~I ~i~s~n0LXl+11-q-v. ~i.--*XL

'~'

k=~E ~

(6)

#=kL

(7)

in which,

q-

P PL 2 - ~ p~r 7r2El

(8)

2

Per is the buckling axial force parameter given by Per-

3

E1

FORCE VERSUS DISPLACEMENT EQUATION Differentiating Equation (3) with respect to x, and expressing the rotations at two ends d Vl

.

dVl I

as the nodal rotations as, ~xlX= o = 01, ~ x Ix=L= 02, we have,

790

El{

M~ =--~- c101+ c2 02 + co

M2 =

(v.:/1

(10)

c20~ + c~02 - c0

z_8o

&+u

L

L

(9)

(11)

L

in which cl, c2 and Coare stability functions. Axial strain can be expressed in terms of the nodal shortening and the bowing due to initial imperfection and deflection as,

in which u is the nodal shortening and 5o and imperfection and deflection given by,

1

J~

1i,; are the shortening due to bowing of initial

[dv<,l:dx.8o ,sr<,vl' ' =7 LLdx]dx

l dx J

(12)

Substituting Equations (1) and (4) into Equations (11) and (12), we have,

P

=EA

=EAz

(13)

-b,(O,+O2)2-b2(Ol-O212-bvs--~(O,-O2)-b,,~ Vmo

(14)

in which b~, b 2, bvs and b~, are curvature functions. A is the cross sectional area. These functions need to be derived for the cases of positive, zero and negative values of axial force parameter, q. The procedure for formulating the tangent and the secant stiffness matrices follows that by Chan and Zhou 6 to which interested readers should refer.

791 TANGENT STIFFNESS MATRIX

Defining [F] and [u] as the basic nodal variables at two ends of an elemem as,

[F] = [ MIz M2z Mly M2y Mt P ]r

(15)

(16)

In] = [Olz 02z Oly 02y Ot U ] T

The tangem stiffness equation for the incremental forces and displacements can then be written as,

(17)

[A F] = [L] T ({T]r [ke] [T] + [N])[L][A u]

in which [L] is the global to local transformation matrix, [T] is the transformation matrix relating the 6 local forces and moments to the 12 nodal forces and moments in local element coordinate system, [N] is the geometric matrix accounting for nodal translations and [1~] is the element tangent stiffness matrix in the local co-ordinate system given by,

r/~c==+ G2'

~2 H

77,c2~+ G,~G2~

~-'-2~I"

GiyGl~

[k

(o]=~El

2 H

rhc2, + G~.G2z

GlzGly /t-2H

GlzG2y ,r2H

0

~2H

G2~2 + zrH 2

G2zG,y 2 H

G2zG2y , r2 H

0

GlyG2~ rr2H

G~y tTycly+ ~.2H

GlyG2y

0

r/z Ctz

~TyC2y+ #2 H

Gl.~" LH G2z LH Gk.

--~LH (18)

G2yGl~ rr2H

G2yG2~ rr2H

r/yc2y+GlyG2y Ir2H

G2y r/yCjy+ rr2H

0

G2y ~LH

0

0

0

0

GJ+ Pro2 El

0

Glz LH

G2z L--'H

0

re2 L2H

G~y L--H

G2y L--H

792

in which

}7y =

L._Z.y; r/z = I~. I

I

A n important point should be noted here is the coupling bwteen the axial force and the bending moments, which is generally ignored in many other papers using the stabilityfunction for analysis.This coupling is important ifan accurate predictionof P4i effectis required. 5

NUMERICAL EXAMPLE-Wiiliam's Toggle frames with initial crookedness The William's toggle frame7 has been a benchmark example by researchers for testing of their techniques for second-order analysis. Most of the previous studies assume the members are perfectly straight and several cubic elements were used to model each member. Using the proposed element and the Minimum Residual Displacement method s, only a single element is needed to model each member with initial imperfection.

3t

2.5

0.00 No disturbing moment

~

~

~: E=,2.1e6,

-

......

L " 100ore b-lcm,

A-lcm

at top

-- -- -- 0.00 Disturbing moment =PUIO0

. =

2

0.001

-- -- -- 0.005 . . . .

0.01

,~

.

1.5

0.5

0

-1

-2 r

-3

-4

-5

v(cm)

Figure 2 Snap-through Buckling of William's Toggle Frame with initial crookedness Figure 2 shows the load versus deflection curve of the results for perfectly straight members with and without disturbing moment at loaded location. Also, the results of members with various magnitude of initial imperfections of anti-symmetric directions are also shown in the same figure. It can be seen that these results differ considerably, showing the stiffness and the snap-through loads are affected by the member crookedness. Whilst this crookedness is unavoidable in real structures, we cannot use straight elements for actual design. An interesting point worthy of notice here is that many researchers consider the error is due to large deflection or the non-vectorial property of rotation in three dimensional space. Their error is more commonly due to the result of inaccurate element stiffness when the member axial force is large. Also, some researchers demonstrate the accuracy of the element or method by the snap-through analysis of shallow dome, which is inappropriate since their element is under a small axial force and large discrepancy has not yet appeared in the loading range. The example described here should be used as a benchmark check of the accuracy of their elements.

793 6

CONCLUSIONS Modelling a member by a single element, NIDA has been applied to trace the equilibrium path of a toggle frame. The element stiffness is exact even under a high axial force whilst the widely used cubic element is deficient for buckling analysis. When used for analysis of large structures, the proposed method is more reliable, stable in tracing the equilibrium path and robust, making the "Advanced Analysis" practical and feasible in a design office to-date. 7

REFERENCES

1 Livesley, R.K. and Chandler, D.B., "Stability Functions for Structural Frameworks", Manchester University Press, Manchester, 1956. 2 Oran, C., "Tangent stiffness in space frames", Journal of Structural Division, ASCE, ST6, 1973, pp. 987-1001. 3 A.Y.T. Chu, Z.H. Zhou, C.M. Koon, S.L. Chan, J.L. Peng and A.D. Pan, Design of steel scaffolding using an integrated design and analysis approach, Proceedings of International Conference on "Advances in Steel Structures", Pergamon, 1996, pp. 245250. 4 Peng, J.L., Pan, A.D.E., Chen, W.F., Yen, T. and Chan, S.L., Structural modeling and analysis of modular falsework systems, dournal of Structural Engineering, ASCE, Vol. 123, No.9, September, 1997, pp. 1245-1251. 5 Chan, S.L., Zhou, Z.H. and Koon, C.M., Nonlinear integrated design and analysis by a single clement per member, keynote lecture, 6~ International Conference on STEEL & SPACE STRUCTURES, 1-3 September 1999, Singapore, organized by Singapore Structural Steel Society. 6 Chart, S.L. and Zhou, Z.H., Second order analysis of flame using a single imperfect element per member, dournal of Structural Engineering, ASCE, vol. 121, No. 6, June, 1995, pp.939-945. 7 Williams, F.W., "An aoproach to the nonlinear behaviour of the members of a rigid jointed plane framework with finite deflections", Quarterly Journal of Mechanics and Applied Mathematics, vol.17m 1964, pp.451-469. 8 Chan, S.L.,"Geometric and Material Nonlinear Analysis of Beam-Columns and Frames using the Minimum Residual Displacement Method", International dournalfor Numerical Methods in Engineering, vol. 26, 1988, pp.2657-2669. 8

ACKNOWLEDGEMENT

The authors are thankful to the support of the Research Grant Council of the Hong Kong SAP, Government on the project" Design and analysis of steel frames aaginst lateral-torsional buckling and warping effects" and the industrial users of the software NIDA.

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

Second-order members

inelastic

analysis o f steel gable

795

frames

comprising

tapered

Guo-Qiang Li and Jin-Jun Li Department of Building and Structural Engineering, Tongji University, Shanghai, 200092, P. R. China This paper investigates the inelastic behaviors of pitched-roof gable rigid-frames comprising tapered members. The accurate Timoshenko-Euler equilibrium differential equation for tapered members is developed and solved by employing Chebyshev polynomial approach. The elastic and inelastic stiffness equations accounting for simultaneous effects of axial force and shear deformation for tapered members are further obtained. The model of modified elastic-plastic hinges is used to analyze second-order inelastic behaviors of these frames. With this model, the effects of residual stresses, strain-hardening of material and shear deformation as well as those of geometry and material nonlinearities can be considered. Numerical investigations are carried out on inelastic behaviors of steel gable frames comprising both prismatic and tapered girders and columns. A number of useful conclusions are obtained at last. 1. INTRODUCTION It is common to use steel gable frames comprising tapered members for industrial buildings. These frames not only provide better distribution of strength, but also yield a lighter design on steel consumption. The approaches available for first or second-order elastic analysis of such frames are well established [1-~3]. However, a method of predicting the second-order inelastic behaviors of those structures is still an important topic for researchers. The purpose of this paper is to develop the inelastic incremental element stiffness equation for tapered members. For simplicity's sake, the model of modified elastic-plastic hinges is used. Employing the method presented, the effects of residual stresses, strain-hardening of material and shear deformation on the second-order inelastic behaviors of steel gable frames comprising tapered members could be considered. 2. THEORY AND FORMULA The cross-section of steel tapered members is usually an I-shape, formed by three plates. The height of the web is frequently linearly varied, while the flanges are symmetric and kept uniform in width along the length of a member, as shown in Fig. 1. For the tapered member described above, the applied forces and deformations could be modeled as shown in Fig. 2. Following the same procedure given by the first author of this paper [4] for dealing with uniform Timoshenko-Euler beams, the equilibrium differential equation of the tapered

796 t

.

L. . . . . . . . .

f -----3r--

N~

I

' . ' ~ . !~.

'i"

. . . . ~[

A-A

Fig. 1.

A steel tapered member

Timoshenko-Euler beam element can be written by or(z), y " - fl(z). N. y ' - N. y= fl(z).Q! -(M, -Q, . z)

Fig. 2. Applied forces and deformations of a element

(1) in which

ot(z)= E . l(z). ?'(z) ) fl(z)= E . l(z).

A" (z)

"'"7~_

)

In the above expressions, A, A, and I are respectively the area, web area and inertial moment of the cross-section at the location of distance z from the original point of the element. Elastic and shear modulus of material are represented by E and G. By nondimensionalizing equation (1) and using Chebyshev Polynomial to represent functions y, a, /~ for the solution of equation (1), the elastic stiffness equation of the tapered element could be obtained as

{i}

(2)

where

{8)-[a,0,,8 0,I {:)-[o, )

)

)

[k,l= )

)

~1

-~7 The expressions of ~, (i- 1,2.....9) are given in Appendix A and the detailed derivation of (2) could be found in the authors' another paper [5]. It is necessary to provide the initial and ultimate yielding surface for cross-sections for second-order inelastic analysis using the model of modified elastic-plastic hinges. A fully yielding limit surface equation for the maximum strength of a wide flange section was proposed in [6]. This equation for the case of uniaxial bending about the strong axis of the cross-section has the simple form as

797

N]'" + M = 1.o Ny )

(3)

Mr,

The initial yielding surface neglecting the effect of residual stresses may be defined as N + z.M =1.0 (4) Ny Mp To account for the effect of residual stresses, equation (4) is adjusted to the following form as

[7] N

z.M

~ + ~ = 1 . 0

(5)

0.8.Ny 0.9.Mp

In the above equations, N and M are the axial load and moment applied to the section, Ny - fyA and Mp are the axial squash load and limit plastic moment for axial load and moment applied respectively, fy is the yield stress of material and z is the shape factor of cross-section. Following the same way forming inelastic incremental stiffness equation for prismatic element [4,8], the similar equation for tapered element by using the model of modified elasticplastic hinges could also be obtained as [kp]. {AS}= {Af}

(6)

where {AS}and {af} are the vectors of incremental nodal displacements and applied loads, [kp] is the elastic-plastic tangent stiffness matrix, which could be calculated by (7)

[k, ]= [k. 1- [k. IGI~ILI~]~ [G]~[k.] in which

ILl= ([sy [Gy Ck, ]+ [aIk. NGIs])-' [A]- ,,,o~[~,,,~, ,~, ~l [El=

o

o0 o i l

[a]-~o~[ ~ '

LON,'

0, 0~,

0~

,,= , , : , ~

OM, 'ON,_,'

o,

] OM=J

In matrix [G], r, ( i= 1,2 ) is the yielding function at the end i of the element, which is defined as =

(N]"

M

(8)

In matrix [A], a, (i=1,2) is the parameter of elastic-plastic hinge representing the

798 degree of plasticity at the end i of the element, which is defined by R~ a i = ~I-R~

(9)

where

Rt

f =

1-

M1-Mm O-fl) Mj,,v -Mm fl

M<M,, v M m <M<Mv~. M > M~v

M, M,,v and M m stand respectively for applied moment, initial yielding moment and full plastic moment at the end i of the element in the presence of axial force N, and # is the strain-hardening parameter of material. Generally, # = 0.02 for low carbon steel. 3. NUMERICAL EXAMPLES Fig. 3 shows a typical gable frame. Dead, live, wind and snow loads are considered in the design of this structure, as illustrated in Fig. 3. The case with prismatic frame members had been studied and the limit load parameter obtained was 1.420 [9]. The same example is analyzed here to verify correcmess of the me~od proposed in this paper, which gives the limit load parameter to be 1.400. The gable frame as shown in Fig. 3 with tapered members was then analyzed. The limit load parameters of the tapered member frame and the prismatic member frame, are listed in Table 1, including eases either ignoring all effects (case 1), or considering effects of residual stresses (case 2), strain-hardening of material (ease 3), shear deformation (case 4) and effects of all the above (case 5). Fig. 4 shows the corresponding load-deflection curves of ease 1 and ease 5 for gable frames with prismatic and tapered members. In Fig. 4, A, and av represent horizontal deflection of the top of left column and vertical deflection of the top of pitched-roof respectively. The curves in Fig. 4(b) are different from those in Fig. 4(a). For the tapered member frame, the deflections increase more quickly and the effect of strain-hardening of material becomes less significant than in the prismatic member frame. 160 9

,

da J

8 7 ~ 18.3 8 / 8.7 I~I i 1.2 i i8.3 .- ~ l ~s-GJ-~~~~~s i ~ a is,3 3 5I ~ ~ p, _ . ~, d ~ ~ O 4 ~ ' - " - - - ti,d' ~ ~ I 2 , ~ 14.15

$ 171 J, 8@3000=240000

c~ J,

Fig. 3 A typical gable frame and its design loads (units: mm and kN)

~c~

799

Table 1. Limit load parameters of gable frames studied in this paper i

~Effects Case 1 ( O'cy = 0; Members ~ fl=0" G=+oo) Prismatic m e m b e r s (d, = d2 = d3 1.400 =d 4 =352)* Tapered members (d ! = 120 d 2 = 360 1.330

Case 2 (Cr~ys0) 1.400

Case 3 Case 4 Case 5 (cr~y~ 0 ; (fl=0.02) (G=g0Gpa) ,B=0.02;G=80Gpa) .............................. 1.610 1.340 1.530

1.330

1.450

1.310

1.430

d3=240 d 4=360) |

i

iii

*Note: All members are of A =

ii

5270mm

2,

I =

121x106

mm 4

i

i

i

ii

in prismatic member frame.

4. CONCLUSIONS Main conclusions could be drawn as follows, 1. Correctness of the method presented in this paper was verified by comparing the results of the load-bearing capacity of gable frame with prismatic members with those obtained by previous researchers. 2. The residual stresses have negligible influence on load-bearing capacity of steel gable frames, which is consistent with [9]. However, the effect of strain-hardening is significant in enhancing the load-bearing capacity of gable frames. The effect of shear deformation decreases the limit load parameter slightly. 3. The numerical results for gable frames with tapered columns and beams indicate that the appropriate variation in cross-sections of members along their lengths do not reduce the load-bearing capacity of frames but produce a design requiting less steel. It should be noted that the application of tapered members shorten the inelastic phase and reduce the influence of strain-hardening of material in gable frame, when compared with the prismatic member frame having approximately equal load-beating capacity.

o.8 -~ 0.4 [ / / 0

.

...............Case I Av

I00

...............Case 1 Case 5

-~

0.4 Deflection (ram)

Ire 0

o.8

200

300

(a) Prismatic members

400

Deflection (mm) 0

0

I00

200

300

400

500

600

(b) Tapered members

Fig. 4 Load-deflection curves of gable frames with (a) prismatic and (b) tapered members

800

REFERENCE AI-Gahtani, H. J.(1996). Exact stiffness for tapered members. 3. ofStruc. Engng.. 122(10):

1.

1234-

1239 2.

Banerjee, J. R.(1986), Exact Bemoulli-Euler static stiffness matrix for a range of tapered beamcolumns. Int. 3. Num. Meth. Engng..23(9): 1615-1628

3.

Just, D. J.(1977). Plane frameworks of tapering box and I-section. 3. ofStruc. Divi., 103(1): 71-863

4.

Li, G. Q. and Shen, Z. Y.(1998). Theory for analysis and calculation of elastic and elasto-plastic

5.

Li, G. Q. and Li, J. J. (1999). Effects of shear deformation on the effective length of tapered columns

behavior of steel frameworks. Shanghai Science and Technology Press.0n Chinese) for steel portal frames.. Struc Engng & Mech.. (submitted). Duart, L. and Chert, W. F. (1989). Design interaction equation for steel beam-columns. J.. of Struc.

6.

Engng., 115(5): 1225-1243 7.

King, W. S., White, D. W. and Chen, W. F. (1992). A modified plastic hinge method for second-order

8.

Li, G. Q. and Shen, Z. Y. (1995). A unified matrix approch for nonlinear analysis of steel frames

inelastic analysis of rigid frames. Structural Engineering Review, 4(1): 31-41 subjected to wind or earthquakes. Computers and structures, 54(2): 315-325 9.

Clarke, M. J., Bridge, R. Q., Hancock, G. J. and Trahair, N. S. (1992). Advanced analysis of steel building frames. 3. Construct. Steel Research, 23(1): 1-29

APPENDIX A ISl #1 _.

I]/'15

'

#2 "- ~r13~/15 --~br162

I~/'l l~r15 -- ~0r14[]/12 r

"-- ~fl3

--

~/11#2 ,

#6

,

#3 --"

~/! 1V15 -- ~r "- --

~/11#3

r

L

L

v,--G.A.(o)./o), v =rL.flo(N.9,', +L ) Vs =

2a ~

L2

~l/lO --"

2(L.N. "'

6ao

+ L.N , +

6a 0

#4

--"

1-q/,,#............_~ []/12

L

L.N.flo.V2

, V6 =-2-'~" ' ~ =

2a o

2(L.N.:o-,,,)r +L.(p, +LXu.v, +L) 9q =

,

, fb7 = # , L + N - # . , # s = # 2 L - # s , # 9 = # 3 L - # 6 ,

r L

~r162 ~r l~r15 -- []/14[]/12

2(L. N.Zo , ~u9 =

'

' 6ao

~/ll = Cl~rl "t- C2~r 5 -[- C3~/8 -{- C 4 ,

' ~/12 = C2~r6 "[" C3~lf9,

~u]3 = -(c~'2 + c2~'7 + c3~Uio), V~4 = csv/~ + c6~u5 + c7u + ca - ~u3, ~5 = c6~6 + c7~9, ~,,~ = - ( c ~ + c~'~ + c~,o)

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

801

A Parallel t h r e e - d i m e n s i o n a l elasto-plastic finite e l e m e n t a n a l y s i s in a w o r k s t a t i o n c l u s t e r e n v i r o n m e n t Z. Ding,t S. Kalyanasundaram,t L. Grosz,~t S. Roberts,:]: M. Cardew-Hallt tDepartment of Engineering, Faculty of Engineering and Information Technology :~School of Mathematical Sciences The Australian National University, Canberra, ACT 0200, Australia ABSTRACT This paper presents the solution schemes for the three-dimensional elasto-plastic problems using finite element method. Attention is focussed on their implementation in a parallel environment defined by a cluster of workstation connected by means of a network. A parallel substructure preconditioned conjugate gradient method combined with MR smoothing is employed. After the displacements are calculated, a substepping scheme is used to integrate the elasto-plastic stress-strain relations. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elasto-plastic problems whose size is much too large for a single work station becomes possible. 1

INTRODUCTION

The finite element analysis of elasto-plastic behaviour is a subject of great importance for fundamental and practical reasons. However, the use of complex three-dimensional models raises a number of questions in relation to accuracy and efficiency. Recent advances in parallel supercomputer technology provide the most promising way to reduce the computing time for large-scale elasto-plastic applications. In this paper, we will treat the problem as a whole rather than for the accuracy and efficiency separately. We will focus on the establishment of an algorithm which can be used to implement parallel computing for the solution of the linear system of equations. A parallel preconditioned conjugate gradient method will be employed. In the resulting parallel algorithm, the formation of the global system matrix is not performed, but the displacements for each substructure are computed directly. We will also introduce a substepping scheme for elasto-plastic stress integration process. This method is applicable to a general type of constitutive law and controls the error in the integration process by adjusting the size of each substep automatically. The utility of these algorithms in threedimensional elasto-plastic stress analysis and the performance of the resulting algorithms will be presented.

802

2

CONVENTIONAL INTEGRATION ALGORITHM

When analysing elasto-plastic problems using finite element method it is usual to integrate the stress as d~

d--T = D~,Ae,

T e [0, 1]

(1)

where Aa and Ae are the incremental stress and strain and D~p is the elasto-plastic matrix. In Eq. (1), Cr[T=0defines the stress state which already satisfies the yield criterion, and a[T=I defines the stress at an end of load increment or iteration. The solution method of initial value problem can be applied to this process. It is well-known that the method for integrating elasto-plastic stress-strain relations is to divide the integration process into several substeps and compute the stress-strain response over each substep. Traditionally, the number of substeps is determined from an empirical rule and each substep is assumed to be of the same size. Although this method has been used widely in the finite element codes, it has the following disadvantages: 1. If the correction-step is applied after each substep, the computational time will increase drastically. However, if it is done at the end of integration, it does not significantly affect the accuracy. 2. Since the number of substeps is usually determined by an empirical rule which is formulated by trial and error, the inappropriate choice of the number of the substeps usually lead to lose of either accuracy or efficiency. In the following section, we will introduce a substepping scheme which can be used to integrate the elasto-plastic stress-strain relation with an aim to control the error by adjusting the size of each substep automatically. 3

SUBSTEPPING SCHEME FOR INTEGRATION PROCESS

It is shown that methods of high order can be formulated for elasto-plastic problems, which are much more efficient than the first order algorithms used up to the present. Since the substepping scheme controls the error by decreasing the step size, it definitely involves a large number of substeps. As such, the cumulative effect of the per-step roundoff errors and their magnification in calculating subsequent substeps must be minimized. In this paper, we will employ Gill's fourth-order Runge-Kutta method which is known for its advantage of minimizing the roundoff error[4]. Similar to Sloan's the substepping schemes[i], the error estimate after a time step ATk is obtained by comparing the estimated stress increments which result from the third and Gill's fourth order Runge-Kutta method, respectively, r

-- a~.

+ (A~ 1 + 4Acz2 -F Act3)/6

(2)

and

(a)

803 in which ffl

a2

-

-

=

a3 =

(Tk

ak.+0.5Aal ak + 0.5(v/2- 1)Aal + 0 . 5 ( 2 - x/2lAcr2 (4)

Subtracting Eq.(2) from Eq.(3), we obtain an estimate of the local truncation error in ak+l according to Ek+l

[-(2 + v/2)Aa2 + (I + v/2)Aaa + Aa4)]/6

(5)

As an estimate for the local error in the substep from Tk to Tk+l -- Tk + ATk, we define the relative error for this substep as

R,.+, =11

II / II a-k+, II

(0)

Then RA:+I is compared with some prescribed tolerance TOL and the step is accepted if Rk.+l <_ TOL, and rejected otherwise. Furthermore, the value of Rk+l allows us to make an estimate for the asymptotically optimal stepsize: A T k + 1 "- A T k

(7)

~frOL/Rk+i

ATk+Ito

In case of rejection ATe.+1 is used instead of ATk; in case of acceptance we u s e continue the integration. In order to reduce the substeps rejected, we actually used AT~.+I - ATi:.min{2,max{O.l,O.9r

(8)

The constants 2 and 0.1 in this expression serve to prevent an abrupt change in the substep size, and the safety factor 0.9 is added to increase the probability that next substep will be accepted. As suggested by some researcher, for example, in [3], some form of stress correction must be used when the analysis involves strain (work) hardening. In this paper, a proportional scaling of stresses is used and the tolerance is set to be 10-4. 4

PARALLEL PRECONDITIONED CONJUGATE GRADIENT METHOD

Recently, Law[2] developed a parallel conjugate gradient algorithm by using transformation relationships between the displacement (as well as force) vectors local to each processor and the corresponding global vectors. In this paper, Law's element-by-element algorithm has been modified to a substructure-by-substructure algorithm and a diagonal preconditioner is used to accelerate the convergence rate. Throughout the process, the formation of global system is not performed. The storage space required for each processor includes an substructure matrix and vectors. A parallel substructure preconditioned conjugate gradient (PPCG) algorithm is described in Table 1.

804

Table 1. Parallel Substructure Preconditioned Conjugate Gradient Algorithm Step O:

Set {x (8)} = O, {r(')} = {f(')} Compute [C(')] and [C(')] (-') =

Ice's]

Go to Step 4 Step 1:

Step 2:

=

(Merge Sum) 1/a = ~a(S),s = 1,...,p

Step 3: {t(,)} =

Step 4:

Exchange {t (s)} with neighbour j

Step 5:

Step 6:

(Merge Sum) %e~ = ~ p('), s = 1, ...,p

Step 7:

If (%~,,,/7o < Tol) stop

Step 8:

Go To Step 1

5

PARALLEL ENVIRONMENT

The parallel computer used in this research is a Linux-Alpha workstation cluster. The Linux-Alpha Cluster consists of twelve 533MHz Alpha LX164s each with 256MB of memory and 5.3GBs of IDE disk connected by a HP fast Ethernet switch. To obtain optimal performance on the cluster, we use following techniques: 1. Since the PPCG algorithm involves the operations such as global reduction, we use Message Passing Interface (MPI) which has many useful features not available in the other parallel programming language. 2. Separate input and output files for each processor. These files are read from and written to by local copies of the program executable operating in parallel. This avoid communication at the phases of inputting and outputting data. 3. Horizontal strip-wise partitioning for load balance(Fig. 1).

805

(a) Cantilever beam

(b) Horizontal strip-wise partitioning on 4 CPUs

Figure 1: A three dimensional deep cantilever beam and the partitioning scheme (Young's Modulus = 1.47 x 1O5 MPa, Poisson's ratio = 0.3, Yield stress = 1.68 • 102 MPa) 4. Implementing the communication in such a way that the sends and receivers are ordered so that if one process is sending to another, the destination will do a receive that matches that send before doing a send of its own. 5. Diagonal storage scheme for equation solution. 6. Digital-Unix F90 compiler with full optimization. 6

NUMERICAL

RESULTS

The developed algorithms have been applied to the elasto-plasticfiniteelement analysis of a typical three dimensional deep cantilever beam (Fig. 1). As usual, the two important metrics, speedup and efficiency,are tested respectively. Speedup and efficiency are shown in Fig. 2. From the results shown in these figures, we can see that the speedup and the efficiencywill generally increase as the problem size increases. The efficiency of the parallel algorithm generally decreases as the number of processors increases. Without doubt, this decrease in efficiencyis due to the overhead required in the interprocessor communication. For this application, the speed-ups and efficiency are almost insensitivethe change of the number of processors when the problem sizes become larger than 7680 elements. In fact,for the problem with 30720 elements, the speed-up and efficiencyare almost perfect. This perfect performance can be attributed to that, a) the partitioning make every processor involved in the computation of the plastic regions, b) substepping scheme can integrate the strain-stressrelations accurately, so it uses less iterations, and thus makes the solution more efficient.It should be noted that, the cluster used is a multi-user system, and the elapsed clock time is dependent on the system loading which changes from time to time. This is the reason for the slightly oscillatory speedups and efficiency.

806 110

d.

.

.

.

.

.

.

.

,

!

:

i

9

f

l r~-

L _ _

. . . . . . . . . . . .

9x . . . . . . .

--

,.-~-

,-

_

9O

.- iiiili!.iii!,: "

a.

a

.

. . . . . . . . . .

. .

.

i

. . . . . . . . . . .9

~

_2_

~

" .................

~

3

4 S NUMBER OF PROCESSORS

(a) Speed-up

i ...............

i.............

.............

...............

9

i

1 ~ ~~

761~

..........

6

7

= ..............

1.

2'

,,,

~

: ...............

:

i

i ................................

i : ...............

: ................

:

: ................................

~'

1

2

:. . . . . . . . . . . . i

.

~

2i

g

i

- - -

Idul

~

80720 e~am~W lf~80 etuaenW

: .........

---0--

~ m

:l

~

960

4 5 NUMBER OF PROGESSORS

. . . . .

ekmNmw,,

8

7

(b) Efficiency

Figure 2: Performances of parallel algorithms 7

CONCLUSION

In this paper, a substepping scheme which can controls the error in the integration process by permitting the size of each substep to vary in accordance with the behaviour of the constitutive law and a parallel substructure preconditioned conjugate gradient method have been presented. This solution algorithm does not require the formation of the global system equations. Each processor in the parallel system is assigned a substructure and stores only the information relevant to the substructure that the processor represents. The combination of these two algorithms have been applied to a typical three dimensional elasto-plastic stress analysis. The result indicated that the combination of these algorithms shows a good speedup when increasing the number of processors. In summary, the combination of these two algorithms provides a powerful practical means for parallel finite element analysis of elasto-plastic problems. References [1] S. W. Sloan,"Substepping Schemes for the Numerical Integration of Elastoplastic Stress-strain Relations", Int. J. Numer. Methods Eng., Vol.24, pp.893-911 (1987) [2] K. H. Law, "A Parallel Finite Element Solution Method", Computers ~ Structures, Vol. 23, No. 6, pp.845-858, 1986 [3] A. Gens and D. M. Potts, "Critical State Models in Computational Geomechanics", Eng. Comput., Vol. 5, pp.178-197 (1988) [4] M. L. James, G. M. Smith and J. C. Wolford, Applied Numerical Methods for Digital Computation, Harper & Row, Publisher, New York, 1985

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

807

L i m i t A n a l y s i s of C y l i n d r i c a l S h e l l s S u b j e c t e d to R i n g L o a d A Comparative Study Between Analytical and Numerical Solutions J. R. Q. Franco a and F. B. Barros aDepto Eng. Est.-EEUFMG- Av. Contorno, 842-2o. andar- B.H./MG 30110-060-Brasil A kinematical approach associated to a 2D yield surface, capable of simulating bending and membrane behaviors, was used to obtain an analytical solution for limit analysis problem of a cylindrical shell subjected to a ring load. The displacement rate field during collapse defines simultaneously the strain rate and the rate of curvature. The solution for the differential equation, corresponding to the collapse mechanism, is trigonometric near the load and hyperbolic elsewhere. A more general linearized 3D yield surface was used to construct a Finite Element technique to solve the same class of problems. Finally, a comparative study of the numerical and analytical solutions is presented. 1

INTRODUCTION

In this work a new 3D yield surface proposed by Franco & Barros [1] is used to simulate bending and membrane behavior, describing the collapse mechanism of the discrete set of axisymmetrical cylindrical elements. The displacement rate field and generalized strain rate field associated to the discretized shell must be compatible with the new surface for the finite element analysis. Implementation of this formulation using the technique proposed by Franco et a1.[2,3] leads to improved mechanisms and to solutions closer to the exact one. An analytical solution based on this new yield law and on a kinematical approach was developed following similar procedures as those proposed by Druker [4]. 2

ANALYTICAL SOLUTION

The exact yield surface, Fig. l(a), for the case of an element subjected to membrane forces in the hoop direction and meridional bending in the absence of axial loads, was presented by Drucker [4]. The present analysis consider a simplified linearized yield surface represented by the inscribed hexagon in Fig. l(a). The problem of a long cylinder subjected to a ring load produces such a state of stress, where independent normal circumferential forces, No, and meridional bending moments, Me, appear.

2.1 Kinematical Analysis The hexagonal yield surface Fig. l(a) which simulates simultaneous membrane and bending behaviors, defines on DC and CB a relationship between the strain rate and the

808

N0

,n~ot

K:'~ ~ - / - - ~ - - - ~ 1 ~ a ~ ~

mr 2(l'ns)

El

{

}

oL,

I

/-

(

m-M,

)

3T llA- ~-~

,JH

-{ {o

P

I

(a) Drucker's Yield surface

(b) Collapse Mechanism

Fig. 1. Yield surfaces and Collapse Mechanism rate of curvature given by k ~ t / ~ o = -+-2. The mechanism proposed is shown in Fig. 1 (b). The states of stress and strain at x = 0 and x - ~ correspond respectively to points D and B of the hexagon. The differential equation governing the formation of the collapse mechanism can be written in terms of radial displacement rate as d2w

{

-+-2zb f o r 0 < x _ '7

rt~-x2 =

-2,~/or

,7 < x <

0

(1)

The solution for Equation 1 indicates a trigonometric variation for w in the neighborhood of P and hyperbolic further away as shown in Equation 2 :

=

I Wl = Cl COS ( ~ X )

-{- C28en

f or O < x <'7 -

-

(2)

Continuity and boundary conditions are used to determine the constants cz, c2, c3, c4. Upper bounds of the limit load for the inscribed hexagon, are then obtained by equating the rate of external work due the ring load to the dissipation of energy per unit area, which is determined by integrating the rate of dissipation per unit volume through the thickness, t. The dissipation function for the hexagon assume the following values :

1 0"o~2lkr

* for the stress point on vertical sides where Imp[ = 1, w i = ~

9 for the stress point on C where 1'7ol = 1, w i2 = aot I~ol 9 for the stress point on the slopping sides (me = 2(1 - '7o)) including the points line = 11 e {~o =

1/21,

3

Go'l;

~, = - y

{~o{ +

0"o~2

-(

I~,{

Using the symmetry of the collapse mechanism half of the internal dissipation energy

809 is given by

-

T

o

T

T-

(3)

where ~z-0 - ~dzbl Ix=0 and ~={ = -d~b2 ~ x [~={" The curvature and strain rates go and kr are given by -

go =

I

(v~ f o r O < _ x < _ ~ ;k~= r w___~2f or ~ <_ x <_ { r

d

2zbl f o r 0 ~ x ~ 71 r

d2~b2 f or ~? ~ x ~ ~ r

Half of the external work rate, per unit of circumferencial length of the ring load is We/41rr = P~/2. The kinematical theorem is now applied to obtain an expression for P = r ({, 77) in terms of the coordinates { and 7, since the parameter $ is canceled. Upper bound values for the ring load are then obtained and the minimum value is determined by solving the following system OP

OP

oT=0;N=0

The solution yields { = 1, 7 9 ~

and ~? = 0, 4 7 V ~ which give:

=

73

(a)

{70

for an optimum mechanism of size p = 2{ = 3, 58~7~. This solution is the same as that obtained by Drucker [4] using a lower bound approach. 3

NUMERICAL ANALYSIS The exact yield surface, Fig. 2(a) for a cylindrical element subjected to a complete set of stress resultants due to axial symmetric loading was determined by Onat [5]. The simplified linearized yield surface shown in Fig. 2(b) was proposed by Franco & Barros [1] and used for the present work. Application of the upper bound theorem to the ring load problem can be reduced to a minimization problem and solved by linear programming. The upper bound theorem was formulated in [1] as a minimization problem stated as: n = inf f(ac)Tgc(A)df~ X

n

where the infimum is taken for A satisfying /pTfl~. (A)dS -- 1. S

(5)

810

(a) Exact yield surface

(b) Linearized yield surface

Fig. 2. Yield surfaces for thin cylindrical shell The kinematically admissible collapse mechanism is defined by a displacement rate field d c and a strain rate field ec which is associated to a state of stress on the yield surface r ~ in equilibrium with the external load sp, where s is the load factor. Upper bounds on the limit load factor ~L < s can, then be found. The solution for this minimization problem requires the description of the yield conditions governing the material plastic behaviour based on the new yield surface, Fig (2(b)). 3.1

A N e w Finite E l e m e n t

A new axisymmetrical shell element was constructed by Franco & Barros [1], so that the displacement rate field and generalized plastic strain rate field associated to the discretized shell should be compatible with the new surface for the finite element analysis. A simple second order Lagrange interpolation polynomial is used to describe the total displacement field of the element. This shape function is capable of modeling the rate of curvature kr and the rates of membrane strain g~, go along the element. The discretized three nodes finite element is shown in Fig. 3. The Equation for the elastic-plastic strain field is given by e i1 3.2

= B~]

'

P l a s t i c S t r a i n R a t e Field

Definition of the plastic strain rate field is obtained from the flow law associated to the yield surface in Fig. 2(b). The associative rule relates the strain rate with eight plastic multiplier, shown in Fig. 2(b), and also establishes a relationship between the rate of curvature k~ and the following four plastic multipliers, ~1, ~4, ~5 e ~7. The latter allows the rate of curvature kr along the element, in the meridional direction, to be calculated. -2 The plastic strain rate field for an element i is then defined as: e~

= N;k i

3.3 Consistent Relation B e t w e e n IJ and Since the two strain rate fields were defined independently, a consistent relationship between the velocity field 0 , in terms of which the total strain rate Q.1 is described and

811 col

, !wi:s)

-

"~

9

/".~,

~

I

~ u ( s ~

Fig. 3. Discretized three nodes finite element Table 1 Cylindrical shell subjected to a ring load

PL/~o (m)

p (m)

Numerical

0.39096.10 -2

0.11250

Analytical- Equation(4)

0.38684.10 -2

0.11321

Limit Analysis

the rate of plastic multipliers A, which describe the material plastic behaviour Q92 had to be found. Such a relationship is obtained using the procedure, described in Franco & Ponter [2], [3], where the theory of conjugate approximations proposed by Brauchli & Oden[6] "' L'X was used to give U n -4

NUMERICAL

SOLUTION

A numerical solution is now presented for a cylindrical shell with the following geometry; r0 - 0, 20m, t - 0, 01m and L -- 0, 36m. The collapse mechanism and the limit load for a mesh with 64 elements are shown in Fig. 4. A comparison between the analytical solution, Equation (4), and the numerical results, Fig. 4 is presented in table (1) where the size of the mechanism and the limit load are compared. 5

CONCLUDING

REMARKS

An analytical kinematical solution for the problem of a cylindrical shell subjected to a ring loading has been presented and served to prove the reliability of the finite element technique developed to solve limit analysis problems of pressure vessels. The results from both approaches were practically coincident. Acknowledgement: This investigation was carried out in the Department of Structural Engineering of the Federal University of Minas Gerais. The authors wish to thank the

812 GEOMETRY AND COLLAPSE MECHANISM Load: Horizontal Ring Load at No 33 End Conditions: Node 1 : Fully constrained Node NN : Fully constrained Number of nodes : 65 Number of elements : 64 PL / S I G M A Y = .39096E-02 Thickness = .10000E-01 SIGMA(T) = .00000E+0( Weigth/m .0E+00

33

.04

................................. .30........................... Fig. 4. Numerical solution for a cylindrical shell subjected to a ring load Conselho Nacional de Desenvolvimento Cientffico-CNPq for the support of the research. REFERENCES [1] J.R.Q. Franco and F.B. Barros. An improved adaptive formulation for the computation of limit analysis problems on axisymmetrical shells. Proceedings of the V COMPLAS Barcelona, 1:625-632, 1997. [2] J.R.Q. Franco and A.R.S. Ponter. A general technique for the finite element shakedown and limit analysis of axisymmetrical shells- part 1 - theory and fundamental relations. International J. Num. Meth. in Eng., 40:3495-3514, 1997. [3] J.R.Q. Franco and A.R.S. Ponter. A general technique for the finite element shakedown and limit analysis of axisymmetrical shells - part 2 - numerical algorithm. International J. Num. Meth. in Eng., 40:3515-3536, 1997. [4] D. C. Drucker. Limit analysis of cylindrical shells under axially-symmetric loading. Proc. 1st Midwest Con/. Solids Mech, II1:158-163, 1953. [5] E.T. Onat.

The plastic collapse of cylindrical shells under axially symmetrical loading.

Quartely o] Applied Mathematics, 33:63-72, 1955. [6] H. Brauchli and J.T. Oden. Conjugate approximation function in finite-element analysis. Quarterly o] Applied Mathematics, 29:65-90, 1971.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

813

Finite element simulation of deep drawing of laminated steel Yee Foo Kwan and Monk Talda Department of Mechanical & Manufacturing Engineering, RMIT Bundoora VIC 3083, Australia

This paper presents the investigation results of using advanced finite-element techniques to simulate the forming process of vibration damping laminated steel sheets, which constitute a sandwiched steel-resin-steel structure. The research program" aimed at predicting potential manufacturing problems during the early design stage to reduce tooling cost and lead time. Experimental stretch tests using hemispherical punch were first performed on laminated steel specimens and the results were compared with those from numerical modelling.

1. INTRODUCTION The design and analysis of sheet metal forming processes and tools have been traditionally practised as an art, and implemented by a trial-and-error approach in the press shop. A typical stamping process development cycle involves an expensive and time consuming prototyping operation at the tooling design stage, in which the tryout dies are repeatedly machined and modified until the required stamping quality is expected to be achievable. This approach requires long lead time and large capital investment, and its relatively low reliability usually incurs substantial modification work to be carried out aider the tooling is made. The modem concept of concurrent engineering urges the need for a radical change in the design approach for sheet metal forming tooling and processes. An ideal approach should allow the process to be simulated in such a way that the potential production problems can be predicted and rectified during the design stage, and various plausible design alternatives can be experimented by conducting numerical sensitivity analysis. The ability of the Finite Element Method (FEM) to model sheet metal forming processes is still being explored by researchers world wide and examples of successful applications of FEM in solving practical stamping problems have also been reported [ 1-4]. Applying FEM in the analysis of sheet metal forming processes allows engineers to create a streamlined initial design that needs little or no modification. It provides the tooling engineers with design and analysis tools to accurately predict the 'end-product' at an early stage of the design cycle well before the tooling is made. With an ideal stamping process development cycle involving FEM, the prototyping process can be minirnised or even be entirely eliminated (Figure 1). " Thisworkhas beenmadepossibleby the AustralianResearchCouncilthroughan APA-Isupportedby Ford Motor CompanyAustralia.

814 ]~,oouc~ ,~.s,~N| { FINITE ELEMENTL SIMULATION__. I ~'"~[--"

i~IpRoCES~" DESIGN]

I )

[

I

I I

~176

.......

Figure I Stamping process development cycle involving FE simulations This paper presents the application of FEM in the simulation of sheet metal forming of laminated steel components. The explicit finite element code, ABAQUS/Explicit [5] was used to model the hemispherical punch stretching test on laminated steel specimens. 2. LAMINATED STEELS IN SHEET METAL FORMING The excellent performance of laminated steels in noise damping [6] has been recognised by the automotive industry world wide. The increasing breadth and complexity of customer demands for quality have been pushing the automotive engine oil pan to be re-engineered to reduce the noise resulting from the engine and improve acoustic conditions within the compartment. These requirements can be satisfied by using vibration damping materials, like laminated steel, but at the sacrifice of reduced material formability. The laminated steel constitutes a 0.05 mm thick middle layer of visco-elastic resin film sandwiched by two outer skins of 0.6 mm thick low carbon steel. It works on the principle that the energy resulting from vibrations transmitted from external sources will be converted to heat energy by the shear deformation of the middle layers (Figure 2). Compared with other low carbon steel sheets for stamping applications, which may show anisotropic properties, the sandwiched structure of laminated steel incurs not only variations in material properties on each single layer but also discontinuities across the whole cross section. (a)

~

SHEAR[----VISCO-ELASTICRESINLAYER

]

~

~

_

STEELSHEET

Figure 2" (a) laminated steel structure, (b) shear deformation of the resin layer 3. MODELLING OF LAMINATED STEEL AND PRODUCTION TOOLS Finite element simulation of sheet metal forming operations requires the material properties of the sheet metal blank to be clearly defined. Tensile tests were carried out using an Instron series IX automated materials testing system to obtain the non=linear material

815 properties required for the simulation. Specimens in three different orientations: 0 ~ 45 ~ and 90 ~ to the rolling direction were prepared and tested. The material properties of the resin film were obtained from literature as the corresponding values of thermosetting unsaturated polyesters; a common used resin matrix material for reinforced composites. Poisson's ratio of 0.45 was adopted to prescribe the nearincompressible condition, as using an exact value of 0.5 to describe the incompressible behaviour could result in numerical problems. It was also assumed that the low stiffness of the resin layer is negligible compared to that of the metal skin, and thus the tensile properties of the laminate were assumed to be the same as those of the steel layer (Table 1). The true stress-true strain relation of the outer steel skins in the plastic range was fitted to the power curve: o = 476.53 s ~ Table 1 Material Properties of the steel layer and the resin layer

Young's Modulus, E Poisson's Ratio, v Proportional Limit, Op 0.2% Proof Yield Stress, Oy Ultimate Tensile Stress, Outs Power Curve Constant, K Work Hardening Index, n Anisotropy, r Density, p

Steel Layer

Resin Layer

'164.35 G N / m 2

2 GN/m 2

0.33 93.177 MN/m 2 136.5 MN/m 2 337.696 MN/m z 476.53 MN/m 2 0.2452 ro = 1.566 r4s = 1.327 rgo = 1.728 7800 Kg/m 3

0.45

1120 Kg/m 3

In a typical sheet metal forming operation, the major components to be modelled can be divided into two categories: The blank which will be deformed during the operation, and the production tools, which are required to impart the required shapes to the blank. The production tools were assumed to be perfectly rigid and non-deformable during the analysis. They were modelled as rigid bodies, which were formed by bounding a number of Triangular R3D3 rigid elements together. The laminated steel blank was considered to be a three-dimensional continuum, which could be modelled by either solid elements or structural elements like membranes or shells. Both types of elements can be used for complex non-linear analyses involving contact, plasticity, and large deformations. In deep drawing operation, the primary forming mode of deformation is membrane stretching together with significant bending stresses, which can be well represented by using three-dimensional shell elements. Moreover, composite shell structure in which a shell composed of layers with different material properties can also be used to model the laminated structure. The S4R (4-node quadrilateral) element available in ABAQUS/Explicit was selected. The laminated steel was modelled with a 3-layer composite shell section in which 3 integration points were assigned to the resin layer while 5 integration points were assigned to the steel layers for higher accuracy (Figure 3).

816 5 integration points in each outer layer - ~ 3 integration points in the middle layer

F-g, O.6mm

/

\

shell element normal

ell

Layer 3 (steel)

10o _ 9

O.05mm ~

e 5

l

,.

,7

.,

Layer 2 (resin)

e6

4*

0.6mm

~

8

Layer 1 (steel)

,3

.

o 1

,..

Figure 3 Composite shell section definition for laminated steel In deep drawing operations, the total strain can be immense due to the large plastic strains which dominate the deformation of the blank. Except for the springback analysis, the elastic behaviour does not have a significant effect on the accuracy of the simulation. Thus the simplest elastic model, linear isotropic elasticity, was assumed for all layers for economical reasons. The ABAQUS/Explicit material library provides several models of inelastic behaviour. Classical metal plasticity is commonly used in metal forming and can only be used in conjunction with the linear elastic material model. It uses standard Mises or Hill yield surfaces with associated plastic flow to describe the isotropic or anisotropic yield respectively. The classical metal plasticity model together with the anisotropic yield model were adopted to model anisotropic plastic flow. Using the Hill yield surface to describe the anisotropic yielding requires a reference yield stress, o ~ to be defined. Then a set of yield stress ratios, Rij, are then defined with respect to o ~ so that the corresponding yield stress is Rij o ~ Thus, the r-values, the ratios of width strain to thickness strain, had to be converted to stress ratios. The reference yield stress, cr~ = measured initial yield stress (proportional limit) in rolling direction x = 93.177 MN/m 2 The strain ratio in x-direction, Rx = 1 The strain ratios in y-, z-, and xy-directions are given by V ro(rgo 71) thus, Ry = 1.018784,

R, =

(ro +

R, = 1.160215,

R~ =

+ O(ro + r o)

R,o, = 1.051271

4. HEMISPHERICAL PUNCH STRETCHING TEST Formability tests that stretch a specmnn of specimens of different widths over a hemispherical punch have been commonly used to determine the forming limit diagram of sheet metal [7, 8]. A finite element model of the tooling for hemispherical punch stretch test was developed (Figure 4) to verify the laminated steel model. In the simulation, a 25-mm specimen with its both ends fixed was subjected to a blankholding force of 15 tons, then the punch was displaced by 50 mm at a velocity of 20 m/s to deform the blank. As a common practice in sheet metal forming simulation using explicit dynamic analysis, the punch speed was artificially increased to accelerate the modelling process.

817

Hemispherical Punch 9 100mm x 150mm (height) 9 1234 R3D4 quadrilateral rigid elements

Biankhoider 9200mm x 200mm 9 140 R3D4 quadrilateral rigid elements

Blank 9 25mm x 175mm x 125mm 9 980 (10x98) S4R quadrilateral shell elements

Die 9 200mm x 200mm x 100mm 9 Punch-to-die clearance: 2mm 9 Die radius: 10 mm 9 1234 R3D4 quadrilateral rigid elements Figure 4 Finite element model of the tooling for hemispherical punch stretching test The experimental and numerical results were compared in 2 ways: 1. The shapes of the deformed blanks were compared (Figure 5). In the physical experiment, location of the necking region was about 59 mm from the pole and the necked width was reduced to 17.5 mm. In the simulation results, the predicted necking region and the necked width were 59.64 mm from the pole and 18.06 mm respectively.

0

The distributions of major and minor engineering strains, as well as the thickness along a line measured from the pole to the fixed ends were also compared (Figures 6 to 8). DISTRIBUTIONOF MAJORPRINCIPALSTRAINS J

-+ 75 .~

!i

A

V o

:2=; 0

20

40 60 80 100 120 DISTANCE FROM THE POLE (ram)

140

Figure 6 Comparison of major principal engineering strain distributions

818 DISTRIBUTION OF MINOR PRINCIPAL STRAINS o

~

\

, , 20

DISTANCE FROM THE POLE (nun) , 40

+-,O0

o 80

~ / 100

-15

-~,~: 1 2 0 J i

,.,.~

EXPI~RIM]ENTAL DA~i'A I "~- FEA DATA . . . . . . . . . . .

J i

Figure 7 Comparison of: minor principal engineering strain distributions DISTRIBUTION OF THICKNESSES 1.4

J

1.3 v = =

g

J

~o 1.2 1.1 1

O.9 0

i 20

I 40

4 80

- ; 80

I 100

I 120

I 140

[ i

DISTANCE FROM THE POLE (ram)

Figure 8 Comparison of: thickness distributions 5. CONCLUSION High degree of resemblance between the results of experimental testing and numerical simulations were achieved in terms of the prediction of deformed shape, strain distributions as well as thickness distributions. It can be concluded that the adopted material model was reasonably reliable and accurate. Further research work [9] was then carded out to simulate and optimise the deep drawing of automotive oil pan made of laminated steel. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

M.A. Ahmetoglu, G. Kinzeland T. Altan, J. Mater. Process. Tech., vol. 46 (1994) 421. A.A. Tseng, Journal of Metals, April (1988) 12. A. Makinouchi, J. Mater. Process .Tech., vol. 60 (1996) 19. E.V. Finckenstein, and E.J. Drewes, 19th IDDRG Biennial Congress, (1996) 215. ABAQUS/Explicit: User's Manual, HKS Inc., USA, 1998. G. Turner, MSc. Thesis, Cranfield Institute of Technology, U.K., 1984. K. Nakajima, T. Kikuma and K. Hasul~ Yawata Tech. Rep., No. 264 (1968) 141. S.S. Hecker, General Motors Research Publication GMR-1220, (1972). Y.F. Kwan, MEng Thesis, RMIT University, Australia, 1998.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

819

Analytical Solution for Semi-Infinite Body Subjected to 3D Moving Heat Source and Its Application in Weld Pool Simulation N. T. Nguyen, U2000 Research Fellow, Department of Mechanical & Mechatronics Engineering Center for Advanced Materials & Technology (CAMT), The University of Sydney, NSW 2006.

Abstract. The paper describes the analytical solution for double-ellipsoidal power density moving heat source in semi-infinite body and its application in weld pool simulation. The solution has been obtained by integrating the instant point heat source throughout the volume of the ellipsoidal heat source. Very good agreement between the predicted weld pool geometry and the measured ones has been obtained. This may pave the way for the future applications of this solution in various problems such as microstructure modeling, thermal stress analysis, residual stress/distortions and welding process simulation. I. INTRODUCTION Temperature history of the welded components has an significant influence on the residual stresses, distortion and hence the fatigue behavior of the welded structures. It is well known that classical solutions for the transient temperature such as Rosenthal's ones [1 ] which deal with the semi-infinite body subjected to instant point, line or surface heat source, can satisfactorily predict the temperature at a distance far enough from the heat source, but fail to do so at its vicinity. Eagar and Tsai [2] modified Rosenthal's theory to include a 2D surface Gaussian distributed heat source with constant distribution parameter and found an analytical solution for the temperature of a semi-infinite body subjected to this moving heat source. Their solution is an significant step forward for temperature prediction in the near heat source regions. Even though this 2D solution using the Gausian heat sources could predict the temperature at regions closer to the heat source, they are still limited by the shortcoming of the 2D heat source itself with no effect of penetration. This shortcoming can only be overcome if more general heat sources are implemented. Goldak et al. [3] first introduced the 3D Double Ellipsoidal moving heat source and used FEM to calculate temperature field of a bead-on-plate. They showed that this 3D heat source could overcome the shortcoming of the previous 2D Gaussian model to predict the temperature of the welded joints with much deeper penetration. However, up to now, analytical solution for this kind of 3D heat source still not yet available [4] and hence the researchers have to rely on the FEM for thermal history of the components and related simulation purposes. Therefore, if any analytical solution for 3D heat source is available a lot of CPU time could be saved and the thermal-stress analysis or related simulations could be carried out much more rapidly and conveniently.

820 In this study, an analytical solution for the transient temperature field of the semi-infinite body subjected to 3D moving heat sources [5] are described fLrst. Then, the simulated weld pool shape was obtained using the developed solution for various heat source parameters and compared with experimental results. 2. ANALYTICAL SOLUTION FOR DOUBLE-ELLIPSOIDAL HEAT SOURCE IN SEMI-INFINITE BODY

2.1. Goldak's double-eilipsoidal heat source Goldak et. al. [3] initially proposed a semi-ellipsoidal heat source in which heat flux is distributed in a Gaussian manner throughout heat source's volume. Later, the two different semiellipsoids were combined to give a new heat source called double ellipsoidal heat source. The heat flux within each semi-ellipsoids are described by two different equations. For a point (x,y,z) within the first semi-ellipsoid located in front of the welding arc the heat flux equation is described as

Q(x,y,z)=

6~rfQ ( 3x2 ahbhCu ~r~-~exp c~r

3y2

3z__]

a2 - b2

(1)

and for points (x,y,z) within second semi-ellipsoid covering the rear section of the arc as

(

/

Q(x,y,z)= ahbhChnXff-~exp -C~hb-- a2 - b2 (2) where a~bsCu = ellipsoidal heat source parameters as described in Fig. 1; x,y,z = movLng coordinates of the heat source; Q(x,y,z) = heat flux Q(x,y,z) at a point (x,y,z); Q = arc heat input (Q = r/IV); V,l = welding voltage and current; TI = arc efficiency; rs rb = proportion coefficient representing heat apportionment in front and back of the heat source, respectively and r/+ rb = 2. It must be noted here that due to the condition of continuity of the volumetric heat source, the values of Q(x,y,z) given by Eqs. (1) & (2) must be equal at the x = 0 plane. From that condition, another constraint is obtained for rf & rb as rJcu = r~/c~. Subsequently, the values for these two coefficients are determined as rI = 2cu cu c~; r~ = 2Chb/(Cu C~. It is worth noting that this double ellipsoidal distribution heat source is described by five unknown parameters: the arc efficiency r/, ellipsoidal axis: ah, b~ cyand c~.

2.2. Analytical solution The solution for the temperature field of single ellipsoidal heat source in a semi-infinite body is based on the solution for an instant point source, which satisfies the following differential equation of heat conduction in fixed coordinates [6]

_ dTt'

cYQdt ,)]3/2 . e x p pc[4~ra(t- t

(

(x

x )2 + (y

y )2 + (Z

4a(t- t )

t

Z') 2

(3)

where a is thermal diffusivity (a = k/cp); c is specific heat; k is thermal conductivity; p is mass density; t, t' are time; dTt' is transient temperature due to the point heat source 6Q at time t' and (x ',y'z ) = location of the instant point heat source 812 at time t'. Let us consider the solution of an instant double-ellipsoidal heat source as a result of superposition of a series of instant point heat source over the volume of the distributed Gaussian heat source. Substituting equation (1) & (2) for the heat flux at a point source into Eq. (3) and

821

take integration over the volume of the heat source, the analytical solution for double ellipsoidal moving heat source with a constant speed v from time t' = 0 to time t' = t was obtained as T - T~- 2 P : ~ t-~ !4(12a(t';i)+4)(i2a(t-t')+b~)~412a(;?t')+r

=+~12a(i--'t'i +

(4)

where

(3(x-vt') 2 3y 2 37,2 t /1'= rf .exp - 12a(t - tl) + c2f - 1 2 a ( t - t') + a 2 - 12a(t - t ' ) + b2 3(x - Vt') 2 3y 2 B'= rb.ex p - 1 2 - ~ - - t ~-~ ..~ - 1 ~ + ~2- ~ . ,

3Z2 .~ ,. - 1 2 a ( t - - - f ) + bh2 T is temperature at time t and To is initial temperature of a point (x,y,z).

)

Fig. 1. Double ellipsoidal power density distributed heat source It is worth noting that for ah=Chf=C~= ~ O" and bh=0 then Eq. (4) reduces to the same result as the solution published by Eagar and Tsai [2]. More details about the derivation of this solution are described elsewhere [5]. 3. RESULTS AND DISCUSION 3.1. Effect of heat source parameters on the predicted weld pool geometry In this study, a numerical procedure is applied to calculate the solution for the transient temperature field as described by Eq. (4) for double semi-ellipsoidal distributed heat source. A Fortran77 computer program is written in to facilitate the integral calculation in Eq. (4) and to allow for rapid calculation of geometry of the weld pool based on assumed melting temperature

822 of 1520~ for steel. Since the solution was obtained for a semi-infinite body, the mirror method which combines the temperature distribution in a plate of infinite thickness and its reflected images was adopted [ 1]. Using this program, the effect of various heat source parameters (a~ b~ Chl, Chh a n d 77) on the predicted shape of the weld pool were investigated. The following material properties were used for the calculation: heat capacity c = 600 J/kg/~ thermal conductivity k = 29 J/m/s/~ density p = 7820 kg/m 3. The arc parameters used are voltage, U = 26 V, current I = 230 A and welding speed v = 30 era/rain.

"f

.

/ 10

9

x (mm)

2

i

.

|

,

7.5

5

2.5

0

",

"',

~..

!

-2.5

-5

-7.5

-10

-12.5

-15

a) Top view of the weld pool 10 ) ,

7.5 ' ~

~

~

5

2.5 I

~.-~-'-.,...,~__

0 . n ~

,q.. 1.

-2.5 a h = 1 8 - 5 . ,

.

~

~

~

16 -7.5 . 14

-10 12

10

-12.5

7

5

-15

_ _ _ . ~ - - ~ ~ - ~ , . ~

-i,

b) Longitudinal cross-section Fig. 2 Effect of ah on the predicted weld pool geometry (b h = 2 m m , c u = 7 m m , 77 = 0.8 & Chb = 2Cu

Figures 2(a) and (b) show the effect of the heat source parameter ah on the top view of weld pool shape and its longitudinal cross-section, respectively, whilst other heat source parameters are kept unchanged (bh = 2 turn, cu = 7 rnm, r/= 0.8, c ~ / c v = 2). Fig. 2(a) shows that as ah increases from 5 to 14 the shape of the weld pool tends to be shorter and fatter i.e. its length decreases but its width increases. However, as ah increases beyond a certain value (ah > 14 ram), the weld pool becomes shorter and thinner. This behavior of the heat source can be explained by the nature of the distributed heat source. This means that the higher the value of ah the weaker the density of heat source becomes. At the lower values of ah (ah < 14 ram) when the corresponding heat density of the heat source is still high enough, the width of the weld pool increases as ah increases and weld pool length decreases for the same amount of heat input. However, at higher value of ah (ah > 14 ram) the heat density decreases substantially and the same heat input will result in lesser amount of melted metal i.e the smaller size of the weld pool.

823 i

.

....~. ~ y i m m , . . . . . . . . . . . . ~

.

4

~

3

!' ,

!

|

|

i

|

a

|

10 7.5 X (mm)

I

|

5

,

=

|

|

2.5

0

~, ir~

v

i

i

|

|

i

i

,

-2.5

.

! [

bh=2.0

/

b.=1.5 ! 1.0

i

.

|

i

I

!

-5

I

!

.

-7.5

.

.

!

.

.

|

~176 l

|

-10

-12.5

-15

-10

-12.5

-15

,

-17.5

a) Top view of the weld pool 10

7.5

5

=rx,mm

2.5

'

l

'

0 . . . .

-2.5

-5

=

'

. . . .

-7.5

.... '

. . . .

,

~

-17.5

. . . .

,_

_/

.

t

Oh=2.0 bh=l.5

"bh=lO

"

Z (mm)

b) Longitudinal cross-section Fig. 3 Effect of bh on the predicted weld pool geometry (ah = 5 m m ,

Chf = 7 rmn, rI = 0 . 8 & Chb = 2Chf)

t

10

..................

7.5 X (mm)

5

2.5

0

-2.5

-5

-7.5

-10

-12.5

-15

-lo

-12 ~

-1~

a) Top view of the weld pool lo

~5 ....

x (ram)

s

' ~ . " ~ ' " \ ' ' "

z 5 ....

o ....

Io"",

25

-s ....

,...'.'

-r s ....

'..'..':~...

"I

b) Longitudinal cross-section Fig. 4 Effect of cu the predicted weld pool geometry (ah = 5 m m , bh = 2 ram, rl = 0 . 8 & Ch~,= 2Cu

824 Figures 3(a) and (b) show the effect of the heat source parameter bh on the top view of weld pool shape and its longitudinal cross-section whilst other heat source parameters are kept unchanged (ah = 5 ram, chj.= 7 mm, r/= 0.8, Chu/Cht= 2). It Can be seen from Fig. 3(b) that there is an insignificant influence of bh to the pool depth whilst other parameters were kept unchanged. Its influence on the top view of the weld pool shape is minor. The weld pool length slightly decreases as bh increases from 0.5 to 2 mm but the width of weld pool is almost unchanged. Figures 4(a) and (b) show the effect of the heat source parameter chlon the top view of weld pool shape and its longitudinal cross-section whilst other heat source parameters are kept unchanged (a~ = 5 mm, bh = 2 mm, 7/= 0.8, cJcu = 2). It can be seen from this figure that as chlincreases, the weld pool width decreases but its length increases. The increase of weld pool length is more pronounced at its front half than that at its back half. The decrease of the weld pool width is at much lower magnitude. 3.2. Experimental result The geometry of the weld pool were obtained from a bead-on-plate specimen by means of the photos taken for the weld pool shape at the surface of the welded plate and at its transversal crosssection as shown in Fig. 5. The data for the weld pool profiles were measured directly from the photographs using the scales of Adobe Photoshop. The specimen was fabricated by using a welding robot and Gas Metal following welding parameters: voltage U = 26 V, the current 1 = 230 A, crn/min. Shielding gas of 80 % Ar plus 20 % CO2 is supplied at 20 l/min. high strength steel HT780 and filler material used was MIX-60B. Their and mechanical properties are given in Tables 1 and 2, respectively.

Arc Welding with the welding speed v = 30 The base material was chemical composition

Fig. 5 Geometry of the weld pool

Table 1. Chemical composition of the materials

L.T78o MaterII t L ~,IIX-60B

0.06

' "0.69

Mn 0.85 1.30

Pi ii s ! ca qNi ! tMot v f B l +ol

'0.005 0.001 0.23 0.008 0.009 0.20

1.25 -

0.44

0.5 0.3

0.03'5 0.0 i 1 -/'

0.09

825 Table 2. Mechanical properties of the materials Materials

Yield strength Ultimatetensilestrength

sy (MPa)

HT-780

.

.

.

Su (MP )

.

601

.

,_,

31

28

Q Exp. data

7 y (ram)

n

--------pred. data

6~ ~ 5

i'

....,

10 8 x (mm)

|

,

6

4

|

|

9 ,

Elongation (%)

....

859 662

821

,,,

MIX-60B

i

.

!:1

,

2

-2

-4

-6

-8

-10

-12

-14

-16

-18

a) Top view of the weld pool 8 t,~

. y (mm)

6

4

2

o

I

!

i

i

-2 /'~

i

-4 ....

1 _ ~ z

-6 ,

i

.

(mm)

.

.... ,,"

i

.

.

-8 ^

Exp. data . .

_,:-Cal.

data,

b) Longitudinal cross-section Fig. 6 The Weld Bead Geometry Simulation Figures 6(a) & (b) show comparisons between the measured and predicted data of the top view of the weld pool shape on the welded plate and its transversal cross-section. The predicted data were calculated using the following parameters of the heat source which provide the best fit with the measured data: ah = 10 m m , bh = 2mm, cu = 10 ram, r/= 0.85 and c~ = 2cu These parameters were selected based on the information of their effect on the weld pool geometry reported earlier. The heat transfer material properties used for the calculation was selected for HT-780 steel based on its ranges [7] and they were the same as previously reported. It is also seen from Fig. 6(a) that the present 3D heat source model can give very good agreements with the measured data given suitable parameters of the heat source are carefully selected. This means that the predicted model can be calibrated with the experimental data by selecting its heat

826 source parameters and can be used for various simulation purposes. However, Fig. 6(b) shows that the present model fails to predict the complex shape of the weld pool in the transversal crosssection. This is expected since many simplified assumptions have been used for the development of the present analytical solution. 4. CONCLUSIONS Analytical solution for the transient temperature field of a semi-infmite body subjected to double ellipsoidal moving heat source has been described and used for the simulation of the weld pool geometry. Both the numerical and experimental results from this study have showed that the present analytical solution could offer reasonably good prediction for the weld pool geometry by adjusting the various heat source parameters. This means that this solution has a great potential for use in various simulation purposes such as thermal stress analysis, residual stress calculations and microstructure modeling. ACKNOWLEDGEMENT This work has been sponsored by Science and Technology Agency (STA) Research Fellowship program which is ministered by JISTEC. The author would like to express his sincere thanks to his former boss Dr. A. Ohta and co-workers Mr. N. Zuzuki and Mr. Y. Maeda for their great support during the STA fellowship that the author took at NRIM. Special thanks to Drs. A. Okada, K. Hiraoka and T. Nakamura for their valuable support and for making their Daihen welding robot available for use in this project. REFERENCES

1. Rosenthal D. Mathematical Theory of Heat Distribution During Welding and Cutting. Welding Journal 20(5) (1941), 220-s to 234-s. 2. Eager, T.W. and Tsai, N.S. Temperature Fields Produced by Traveling Distributed Heat Sources. WeldingJournal 62(12) (1983), 346-s to 355-s. 3. Goldak J., Chakravarti A. and Bibby M.. A Double Ellipsoid Finite Element Model for Welding Heat Sources, IIW Doc. No. 212-603-85, (1985). 4. Painter M.J, Davies M.H, Battersby S., Jarvis L. and Wahab M.A. A literature review on Numerical Modelling the Gas Metal Arc Welding Process. Australian Welding Research, CRC. No. 15, Welding Technology Institute of Australia, (1996). 5. Nguyen, N.T., A. Ohta, K. Matsuoka, N. Suzuki and Y. Maeda. Analytical Solution for Transient Temperature in Semi-Infinite Body Subjected to 3D Moving Heat Sources, Weld. Res. Supp., WeldingJournal, August, 1998, pp 265s-274s. 6. Carslaw, H.S and Jaeger, J.C.. Conduction of heat in solids, Oxford University Press, (1967), pp. 255. 7. Radaj D. Heat Effects of Welding: Temperature Field Residual Stress, Distortion. SpringerVerlag, (1992), pp.28.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

827

Pseudorigidity m e t h o d (PRM) for solving the p r o b l e m of limit equilibrium of rigid-plastic constructions Yury Routman Design Bureau Of Special Agency Of Cosmos Research, Lesnoi Pr, 64, St.Petersburg, 194100, Russia Limit equilibrium calculations are a broad class of problems arising during investigation of strength of constructions. The solution of these problems gives the maximum level of loading, which the structure resists without failure. Today, most universal procedures used for solving such problems are methods of linear and non-linear programming [1] and [2]. They allow to examine all types of models both for linearized and non-linearized yield criteria. However, reduction of the limit equilibrium problem to standard relations of linear and non-linear programming is very complex and requires special skills of investigator. The Pseudorigidity Method (PRM) described hereinafter is a universal tool. At the same time, it can be easily automated with existing software for calculation of elastic constructions. The designer can easily and readily use such software.

1.

PROBLEM

DEFINITION

If the construction is in deep plastic deformation conditions, then its elastic deformation can be neglected. In this case, the model of the construction is a rigid-plastic body. Such a body is described by equilibrium equations and Drucker's postulate [3]. In the space of macro factors (generalized stress Ri and generalized transposition Ui) yield surface equation and Drucker's postulate are as follows [2] f (R, ) = l. (la) /)f ,~.>0, U = ~--~ -

if f = l ,

(Ib)

O, i f f < l . So, for example, iff(Ri) is a homogeneous quadratic form, then In (2) {0 ~ {R} are matrixes-columns, the elements of which are 0 i and Ri , [H] is a square matrix, the elements of which are defined according to (lb)if substituting {t.)} for {U}. Relations (2) are similar to dependences between deformation and force factors for elastic problem formulation. The further described PRM is based on this similarity.

828 Let us specify the mentioned similarity for rod models. We consider a rod in flat bend and torsion conditions. For this case of loading in [4], on the basis of Huber-Mizes criterion, the condition (lb) is obtained in the following form 2

~_~a,(S)M~(S)=I

(3)

i=l

In (3) M~(S) are the bending momentum 0=1) and torque (i=2) in cross sections of the rods; S is the coordinate of the cross section. Coefficients r

1

= ~ ,

where M~r is the

Mir(S) limit momentum of cross section at the "i" type of deformation. From (lb) and (3) we get

2a 2(S)

(4)

The elastic analog (2) in this case is the relation

{M where [D] =

[ E'It

0 ]

0

E212

is the rigidity matrix, I 1 and

I 2 are

axial and polar momentum of

inertia, E l = E the Young's modulus; E 2 = G the shift modulus. Using this analogy we can build an elastic scheme, the internal stresses in which under limit loading will be the same, as in the discussed rigid-plastic system. Let us demonstrate this. Let us assume that the rigid-plastic solution for the considered problem is known. We retain the geometrical dimensions, boundary conditions and system loading, and assume that its material has elastic properties. Let us assume the following distribution of rigidities of sections: 2

-

on sections of the rigid-plastic model, where ~ ct~M~ < 1 i=1

D i =

-

Eili = oo;

(6a)

in each deforming part, where ~a~M~ 2 = 1 1

1

D, = E,I, = 2t/q.(S)a, (S) = K ( S ) ~ a,. (S)

(6b)

In (6b) t is the coefficient with a time dimension, K(S) is the unknown function, which is alike for all types of deformation (bending, torsion). In the so formed elastic system we create deformation corresponding to the form of destruction of the elastoplastic system: Z = tZi. Then Mi(S)= D~Z~ =

zi(S) 2A(S)ct, (S)

(7)

If we compare (7) with (2) and (4), we can see that values M~(S) in deformed parts coincide in elastic and rigid-plastic systems. The values of internal momentum in places of formation of plastic joints completely define the limit loading and the values of internal

829 momentum on all sections of the system with nonzero velocity [2]. This is why formed elastic and initial rigid-plastic systems have a comparable loading and distribution of internal stresses. Thus, the considered rigid-plastic problem is reduced to a search for distribution of rigidities D i in an "equivalent" rigid system. Such values as Di shall be referred to as pseudorigidities. 2. I T E R A T I O N A L G O R I T H M OF T H E PSEUDORIGIDITY M E T H O D For finding K(S) we can propose an iteration algorithm. The result of (6b), (7) and (3) is" 1 K(S) = (8)

I zt + z__L ~ (S)

a2 (S)

Relation (8) is true for all sections of the rod system, including non-deformed sections (in this case Zi = 0 and K(S)= oo). Specifying in zero approximation K (~ ( S ) $ oo we can obtain from the solution of the elastic problem Z~~ (S), then using (8) K (~)(S), etc. Thus, the scheme of the iteration process is as follows (n is the number of iteration): K(.) (S) =

1

K("-~)(S)

_

1 [~,(n-,)(S)]

K ("-l)(S)

_

1 Iv(n_1)(S)~ 9(a)

K{"-l)(S) .

.

.

Ko'-l)(S) .

In other words, K <")(S) =

....

K ~"-1)(S) f (R/"-1) (S))

(9b)

where f (R i ) is the yield criterion reduced to a linear homogeneous function. At every iteration step the loading applied to the system changes according to the formula /~(") ( S ) = ,6(")P("-~) (S),

Bo o =

1 max 7(R~"-') (S))

( 10 )

s

The system destruction scheme is defined by values of: in plastic joints K ~ (S) are finite; in rigid zones Convergence of iterations (9) and (10) to a precise solution of the rigid-plastic problem is proved in [5]. 3. P S E U D O E L A S T I C D E P E N D E N C I E S F O R D I F F E R E N T M O D E L S It has been shown before, that with a quadratic yield criteria for realization of PRM (finding the distribution of rigidities in an "equivalent" elastic system) it is necessary to form a pseudoelastic model instead of (2):

830

{R}= K[FI]-~{U} (11) Next, the scalar multiplier K is determined using an iteration process similar to (9). The iteration procedure can be performed on the basis of existing software tools, if a certain standard elastic model with a rigidity matrix [1-I]-] will correspond to the matrix [D]=K[II]-~. For the main problems of limited equilibrium such a correspondence exists. For bending plates and beams-walls K[1-I]-1 =[D] v_0.5 , where [D] v=0.5 is a rigidity matrix for the elastic model with Poisson's ratio v = 0.5 and a variable modulus of elasticity E(x,y). For shells K[I-I]-~ = [H], where [H] is a rigidity matrix with variable E for an elastic multilayer shell at v = 0.5. For a 3D problem at v = 0.5 there is no matrix [D]. In this case to develop pseudoelastic dependences we should use models describing non-compressible materials. However, the PRM provides a high accuracy if we accept K[H~ ~= IDa=0.,9, where [D] is the rigidity matrix for a 3D stressed state. The previously described relations are based on the Huber-Mizes yield criterion. In [4] it is shown that the PRM can be applied also for linearized yield criteria. Let us designate the program performing elastic calculation with the rigidity matrix D=K[II]-~ as the basic program (BP). The iteration process is described by relations (9b) and

(10). Interaction between the iteration procedure and BP is realized as follows: 9 a strength model of the examined system is formed within the BP; 9 the elastic problem is solved for this model, whereas the level of the given type of loading can be selected arbitrary; 9 internal stresses of the model determined by elastic calculation enter the program realizing the PRM as an output file of the BP. PRM processes this file and changes the rigidity parameters of the examined model. After that the new rigidity parameters enter the BP as a new file of initial data for the problem.

4. TEST E X A M P L E S We have calculated a number of test models: beams, frames, plates, shells, 3D bodies. Results produced with the PRM differ from the known solutions no more than by 2% [4]. For example, arc calculation results (Hodgee's problem) are shown in Table 1 (where

Mr

k= 2~

PR

= 0.01, p = ' ~ r '

R is the arc radius, 2(/) is the aperture angle, 2P is the force in

the center of the arc.) Table 1 Limit loading of the arc accounting for the longitudinal force Limit !oading of the arc . . . . . ~0=10~ ..... ~0=20~ ~0=30~ p (according to [2]) 5.6 6.6 7.0 p (PRM) 5.75 6.6 7.0

~0__.40o' " ~0=50o. 6.6 5.9 6.55 5.9

~0=60o 5.2 5.2

831 Comparison of PRM with results given in [2] for a spherical segment is shown in Fig.1. ~. P l e r

O. 1 9

~|

I

0.17 O.15

1 L

I

0.13

9

0.11

~.

0.09

,~

0.07 0.05 0.03

'~" " 5

-

~ 10

~ 15

,j_ , 20

25

30

35

40

45

50 R.,,'h

Fig.1. Comparison of calculations using PRM with the results of [2]. PRM, ~ - results of [2]. The notations on Fig. 1 are as follows:P, uniformly distributed load on the segment; ~ s, yield limit of the segment material; R, radius of the sphere; h, thickness of the segment. *

-

5. APPLICATION OF PRM FOR SOLVING PRACTICAL PROBLEMS We have used PRM for solving of quite a number of practical problems. Below you can see the results of calculating a plastic damper and ring frame support for protection of a metalconcrete container. Plastic dampers are used to protect different technical objects from shock loading [6].The damper is constructed from curvilinear metallic rods with a round section. Each of the rods forms a semicircle. The calculation scheme of such a damper with load applied to it is shown in Fig.2. The results of calculation of the yield surface of the plastic damper are given instead of (2) in Table 2.

Fig.2. Plastic Damper (a) and its calculation scheme (b). Table 2 Yield surface of a three-dimensi~ plastic damper 50 60 70 ~0~~ 0 ......10 20 .30 40 Py/Pyo 1 0.881 0.731 0.587 0.462 0.344 0.250 0.156 Px/Pyo 0 0.156 0.262 0.337 0.387 0.412 0.431 0.437 In Table 2 Py, Px are components of limit loading; q~ is the angle between loading applied to the damper laying in the XY plane; Pyo is the value of q~=0, Pz=0

80 90 0.081 0 0.444 0.45 axis Y and the limit loading at

832

(a)

(b)

Fig.3. FE model of MCC (a) and calculation of stress in ring support (b). A metal-concrete container (MCC) has been developed for storing processed nuclear fuel. According to existing standards, MCC must preserve its durability after falling on a rigid basement from a height of 9 meters. For protecting MCC from shock loading plastic deforming ring frame supports are used. A finite element model of MCC and ring frame support is shown on Fig. 3.a. Results of the calculated the stress interaction between MCC and the foundation using PRM are shown on Fig. 3.b.

6. CONCLUSION The pseudorigidity method is a new technique for solving limit equilibrium problems. The technique is efficient (insignificant time losses for computer processing and problem preparation) and consise (requires little computer memory). Its important advantage is the ability to function on the basis of an elastic calculation program. Design and research companies usually always possess such software. At the same time the pseudorigidity method is highly versatilel.

REFERENCES 1. Cohn M.Z., Frzywiecki W. Nonlinear analysis system for concrete structure//Engineering Structures.- 1987.- Vol.9.- No.2.- P. 104-123. 2. Erkhov M.I. Theory of ideal-plastic bodies and constructions.- M.: Nauka, 1978. - 352 p. 3. Kamenyarzh Y.A. Limit analysis of plastic bodies and constructions. - M.: Nauka, 1997. 512p. 4. Mrazik A., Shkaloud M., Tohachek M. Calculation and design of steel constructions. - M.: Stroyizdat, 1986.- 455 p. 5. Routman Y.L. Pseudorigidity method for solving problems of limit equilibrium of rigidplastic constructions. - St.Petersburg, 1998. - 53 p. 6. Routman Y.L. Device for Protection of Buildings, Equipment, and Tubings//J. Const. Steel Res.- 1998.- Voi.46.- Nos. 1-3. - P.359-361.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

833

D a m a g e identification and restoration o f space frame using genetic algorithm Cheng-Wu Shen Xiao-Bing Tang and Hai-Hong Sun Civil Engineering Department, Wuhan Transportation University, Wuhan 430063, China ABSTRACT Structural damages usually result in decrease of natural frequencies. Resonance occurs when excitation frequency is equal to or close to the structural natural frequencies. This paper applies a genetic algorithm with real number direct operation to the problem of damage identification by the aids of the shift information of the structural first five frequencies due to damages. Numerical calculation shows that the proposed method is effective and reliable to identify damage with single connected domain and patching up the damages can recover the stiffness of the structure.

I.

INTRODUCTION

A Rotary machine mounted on a structure produces dynamic loads, which result in fatigue and damage of the structure. Damages in structures generally weaken the stiffness of the structure and reduce its natural frequencies. When excitation frequency is at or close to any of the structural natural frequencies, a resonance may happen and this may cause high stress in the structures if no sufficient damping is provided. So it is an important issue to detect cracks or damages in structures using a practical technology. According to the fact that the determination of structural natural frequencies is easy and reliable, P. G. Nikolakopoulos et al identified the structure with single crack by a diagrammatic method and gave the influencing curves of the first three frequencies vs. the two crack parameters of location and depth t~l. The structural damage identification is a reverse problem of structural dynamics, which can be transformed into multi-peak optimal one. If using standard optimization techniques, we may stop at a local minimum and can not detect the damage correctly. Genetic algorithm (GA) is now frequently applied to optimum design of structures t2-41, mad it is efficient to get a global optimal solution for a multi-peak optimal problem. A simple genetic algorithm is mainly based on three operators according to the genetics, which are reproduction, crossover and mutation. It usually characterizes design variables by introducing chromosomes with binary system coding, and is suitable to integer variables. This paper puts forwards a genetic algorithm that characterizes directly design variables with real numbers. The method takes the advantages of a simple genetic algorithm with binary "This work has been sponsored by National Natural Science Funds of China (Contract No. 19772038)

834 system coding, and applies this algorithm to an optimum problem with continuous variables. A numerical result shows the efficiency of the method.

2.

GENETIC ALGORITHM (GA) USING DIRECTLY REAL NUMBERS

The implementation of a simple GA includes five steps: (a) forming an initial seed population; (b) crossover which produces better feasible solution sets satisfying the constraints; (c) mutation that ensures genetic diversity by generating new, random individuals (chromosomes) (d) washing out the individuals whose fitness values are low and retaining the individuals with higher fitness values for mating. (e) repeating steps (b) to (d) until no any better individual is produced. The kernel of GA is the crossover and mutation. In the following sections, we will illustrate the GA proposed in this paper with emphasis on the difference of crossover and mutation with the simple GA. 2.1. Crossover A two-point crossover is illustrated below for two chromosomes. The crossover points are selected at random. A simple GA: the kth chromosome 001101001 crossover 000100101 the lth chromosome 100100110 101101010 The proposed GA: the jth variable of the kth chromosome akj crossover akj' = akj ( 1 - - , 8 ) + fl jao t thejth variable of the lth chromosome a 0 crossover a 0 = a~(1- flj)+ fljakj where ,8j (0< flj
j +ao):~(a;

+ab)

(1)

J=l

Besides the above condition to be satisfied, we should have akj+ a 0 = akj '+ a 0 '

That means the simple GA has done crossover of n times in order to make the exchange information diversity. 2.2. Mutation

The mutation operator preserves the diversity among the population, which makes the global optimum easy to f'md and avoid getting a particular local minimum.

835 A simtfle GA: Randomly select some genes in a chromosome and change 0 to 1 or vice versa in binary system coding. For example, 0110011 mutation 0010111 The proposed GA: Randomly select some variables (such as thejth variable a o of the ith chromosome) and give the two random parameters ~'j and q j ( ~ j , rlj ~(0,1)). The following transforms are carried out n

n

If 0 < ~j < 0.5, then a o = a o - ( a o - a j ) r l j and aj < a0. < a 0 < ~j If 0.5 < ~j < 1, then a o = a o. + ('6j - a o)rlj a n d a j < a o < a o < a j where a-, and a j = upper bound and lower bound ofthejth design variable respectively, and a,jtt = variable value after mutating, which is dependent on r/j.

3.

THE METHOD AND APPLICATION FOR THE DAMAGE IDENTIFICATION

Let's consider the flame structure shown in Fig. 1, in which a rotary machine with mass 4 0 k g is mounted on the frame top. The cross-sections of the horizontal and vertical beam of the frame are L40 • 3, and that of the inclined bars are round with d =10. We assume the mass of the rotary machine is distributed uniformly at the elements (~)-(~) and | The frame is divided into 21 elements and 20 nodes. The material properties are: E=210Gpa and v = 0.3. The natural frequencies of the undamaged frame can be found by FEM. The first five frequencies are 60.19Hz, 77.63Hz, 99.78Hz, 120.21Hz and 170.74Hz respectively. As the frame is damaged partly, the stiffness reduces and a severe vibration may occur because the 50 Hz frequency of rotary machine is almost equal to the fundamental frequency of the damaged structure. ll

i. i-"

looo _

Fig. 1 the frame structure

8t0 !

i

/2

m

~

n

JF

Fig.2 the element with damaged region

3.1. Dynamic analysis of the frame with dam~'.ged region Dynamic analysis of the damaged frame is made by FEM using beam elements. Each node of the element has 6 degrees of freedom {u, v, w , O , , , a y , o . } . Assume that there exists a damaged region in the element with the nodes i and j, and add

836 temporarily nodes m and n at the two ends of the damage region. In Fig.2, the i-m, m-n and n-j are l~, 12and 13 in length respectively. The state vector of generalized displacements (or coordinates) and generalized forces in each node is expressed as

{Vk}={uk,u,,wk,O,,k,Oy,,Oa,F,a, Fyk,Fa,,M,,k,Myk,Ma}

(2)

where k = i, m, n andj. Without any load between nodes i and j, we can get the transformation matrix [T] of the state vectors between the neighboring nodes,

{V.}=[T2]'{V.},

{Vm} = [T~1"{V,},

(3)

~"j }= [T3]" {V.}

where -1 o o o

~]-

o

o

-l,/A&

0

0

0

0

0

lk

0

0

1 0

-/,

0

0

ll/6EkI z 0

0 ll/6Ekly

0 0

0

0

1

0

0

0

0

0

-I,/G,I x

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0 o

-lk/Ejy o

0 -lk/ejz

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

O-

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

-1 k

0

-1

0

0

0

0

0

0

0

0

1,

0

0

0

-1

0

1 0

0 0

0 -l~/2E, ly l~/2e1,Iz o

0

0

0 -12/2Ejz l;/2E, ly 0

( k =1,2,3) (4) Ek. and (3, are the corresponding Young's and shear modulus respectively; A is the area of beam section; Ix, Ir and/z are the moments of inertia about the x-axis, y-axis and z-axis respectively. Assuming the damage is uniform in all directions and the scale damage factor is D, then we have E1=E3=E, E2=(1-D)E, GI=G3=G, G2=(1-D)G where E and G are the Young's and Shear modulus of undamaged structures respectively. If the transformation matrix of the damaged element is written as ITe ], then we get

L }-- ITs].{g,}

(5) [[A,] [A2]]

[r" ] = [r~l P'I" [r~ ] [rl It' l = L[A~] [A4IJ w er,

~

,is a6 6 unit m trix,

(6)

is a 6x6 suUmatrix From

837 equations (3)-(6), we can obtain easily the stiffness matrix [Ke ] of the damaged element

-p:]-'.[A,] [A:]-' ] [K.]: p, l- [A, ]. [a, ]-' . [.,l,] [A,].[A,]-'

(7)

Finally, we have

[u]+ [Kl){x} = {o}

(8)

3.2. The application of the proposed genetic algorithm The damage identification efficiency and accuracy are discussed by using GA based on the analysis of the frame in Fig.1. The dimensionless parameters aj (0
s (i.r, - zl/.r,) i=l

(9)

The 30 individuals with small values of 8 are kept by deletion, in which the rate of deletion is 2/3. Then new individuals are reproduced by crossover and mutation operators and a sum of 90 individuals are unchangeably maintained. In order to get the convergent result rapidly, the assumption that the individuals m and n are the same when [(a)m-(a),,[
838

listed in table 1 Table 1. Results of identifying the location and extent of the damage Experiment Analysis 16 runs " The best one

The damased element and extent element a2 a~ a4 (~) 0.15 0.81 0.95 (~) 0.075 0.796 0.912 "--0.153 "--0.926 "--0.942 (~) 0.115 0.839 0.936

Frequencies f, f, f~ f~ f5 50.05 68.58 82.20 109.60 130.24 49.81 68.33 82.14 109.39 130'.23 "---51.42 ~68.42 "--82.42 "--109.65 "--159.07 50.05 68.53 82.30 109.53 130.23

Depending on the initial population, it is possible to take more or fewer generations to converge. In this paper, the number of the generations is in the range of 39-175, and in most cases, about 100 generations are required. It can be seen that some errors exist about the identification of a2, a3 and a4 from table 1. This is mainly because that the solution of damaged extent and amplitude is not unique. In order to restore the stiffness of damaged frame to the original level, a steel plate can be patched at the element @. Based on the analysis of the GA, the repaired structure has the first five natural frequencies of 59.94Hz, 79.13Hz, 102.91Hz, l10.65Hz and 149.12Hz and the resonance can be avoided when a steel plate with 200mm • 40mm • 6mm is welded. 4.

CONCLUSIONS

The structural damage identification is a reverse problem, which can be transformed into multi-peak optimal one. GA is quite efficient to get a global optimal solution for a multi-peak optimal problem. Consequently, the damage location and severity can be identified using a GA. The modal parameters required are the test frequencies but not the mode shapes. Since it is often a difficult task to get mode shapes, the proposed method is convenient in use.

REFERENCES 1 2 3

4

P.G. Nikolakopoulos, D.E. Kalsareas, C.A. Papadopoulos. Crack identification in frame structures. Computers & Structures, 1997, 64(1-4), 389-406 A. Todoroki et al, Optimization of Composite Plate by Genetic Algorithms, Proceedings of Japanese Society of Mechanical Engineers(A), 61 (1995) 1453-1459 C.W. Shen, X.B. Tang and K.J. Yang, Genetic Algorithm and Fully Strength Design for Design Optimization of Space Truss Structures, Modern Mathematics and Mechanics, MMM-VII, 1997, 220-222, Shanghai University Publisher. M.I. Friswell, J.E.T. Penny and S.D. Garvey, A combined genetic and eigensensitivity algorithm for the location of damage in st'actures, CompUters & Structures 69 (1998) 547-556

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

839

Simulation of the hysteretic behavior of RC columns with footings Feifei Sun, Zuyan Shen and Xianglin Gu Department of Building Engineering, Tongji University, Shanghai, 200092, P.R. China

The critical sections of columns in a flamed structure subjected to earthquake loads are invariably adjacent to the beam-column joints or the footings. External restraint provided by the heavy elements alters the column behavior. The improvement of flexural strength in the critical column sections may not always prove conservative for the columns designed according to "strong-shear-weak-bend" seismic design philosophy [IJ Sheikh and Khoury reported t:l that a heavy footing provided additional confinement to the adjacent section and the failure thus moved to a section away from the footing where the restrain effect is minimal. However, no model is found to be able to simulate this phenomenon accurately. In this paper, a mechanic model which can simulate this phenomenon is proposed.

I. COMPOSITE SPRING MODEL

Composite spring model is first developed from fiber element model and multi-spring model to simulate the elasto-plastic biaxial behavior of steel members [31 and then developed to simulate the elasto-plastic biaxial behavior of RC memberst41. 1.1 (1) (2) (3)

Basic Assumptions Plane sections remain plane. The RC members are straight and prismatic. A member comprises two elasto-plasfir sub-elements (See Figure 1, elements A and C) at both ends and an elastic sub-element (See Figure 1, element B) in the middle. The sub-elements are connected with slave nodes (See Figure 1, nodes m and n). (4) An elasto-plasfir sub-element comprises several elasto-plastie axial springs (See Figure 2a), two elastic shear springs (See Figure 2b and 2r which are parallel to two main axes of the section and an elastic torsion spring(See Figure 2d). lit

Supported by National Key Projects on Basic Research and Applied Research and State Key Laboratory for Disaster Reduction in Civil Engineering, China.

840 B

sub-elements: A nodes: i

C 7/,

Figure 1. Composite spring model of RC members 0t

z

p

"]

(a)

\

(b)

(c)

(d)

Figure 2. Components of an elasto-plastic sub-element

1.2 Elasto-plastic sub-element The length of the elasto-plastir sub-element lp (See Figure 1) is determined as the length of the plastic hinge of an RC member at failure. Priestley and Park proposed the following equation according to test results: tSl lp = 0.08L + 6d b

(1)

where L is the length of the shear span and db is the diameter of the longitudinal steel bar. Because the shear springs and the torsion spring are elastic, the elsto-plastic behavior can only be simulated by the elasto-plastic axial springs. In order to introduce material characteristics into the axial springs directly, the section of an RC member is divided into several steel zones and several concrete zones. Each steel zone and each concrete zone is represented by a steel spring and a concrete spring at the centroid point of the zone, respectively. Therefore, the instantaneous rigidity Kt of the axial spring can be obtained as follows, K t = E t As

lp

(2)

where, A~ is the area of the axial spring, lp is the length of the elasto-plastic sub-element, and Et is the tangent modulus of material (steel or concrete).

841

Z

Z

t*/~,~

"

~'_,I/SJ ~ ~,-/\l' X/ '1 . .

_

Io'\1'/o ',/

"

.

I

Y_ .

~

~-l-~

. concrete spring 9steel spring

~

Figure 3. Setting of axial springs Thus the hysteretic behavior of the axial springs can be obtained from material constitutive models. The shift skeleton model t31 and m a x i m damage model ~41 were adopted for steel and concrete, respectively. According to the basic assumption 1 and the equilibrium condition at middle section, the stiffness matrix [I~p] of the elasto-plastic sub-element is obtained,

[[Ki~] [K~j11

[Kr = [[KTi] tK;lj T

0 ky

0 0 k:

[Kill= [K;]=

0 0 0 kt

sym.

[Krj]=[K;]~ =

9

0 0 0

0 -k;L,

Sy kyL 1 0

0

0

0

-k t

0

0

n

9

,

Sy =

0 0

-S z

p=l

(4b)

- I= + k~L 2 I~y I~y -Iyy +kyL 2

• kwhyp ,

I,. =

• k~h w ,

<40

p--I

11

9 2 * Iyy=~kxphy p , Iyz=Ekxphyphzp, p=l

(4a)

0 0 -k;

p--1 n

-Sy - kyL l 0 0 - Izy I~ + kyL 2

0 -ky 0

T ='~-~kxp , Sz = ~'kwhzp p-I

Sz 0 -k;L 1 0 I= + k~L2

-T 0 0

- Sz 0 k~L 1 Sy - k y L 1 0

n

(3)

Ll=0.51 p, L2=0.251p2

(4d)

p=l

where, kxp is the instantaneous rigidity of axial spring p; (hyk, h,.k) is the coordinate of axial spring p;

ky,

k z are the elastic rigidity of shear springs in y and z directions,

842 respectively; and k~ is the elastic rigidity of torsion spring. 1.3

Stiffness matrix of an RC member

According to compatibility conditions and equilibrium conditions at slave nodes, the contribution of all the sub-elements of an RC member can be condensed into the stiffness matrix of the whole member,

]

][f,m ][k i ]

where,

I [f'~ ] [f~ ]] - [[k~ ]+ [k~]

L[f ] [fnn]J [_

(Sa)

"

2. SIMULATION OF RC COLUMNS WITH FOOTINGS Two test specimens Unit 1 and Unit 4 [61,RC columns with footings, are selected to verify the composite spring model proposed above. The ratio of axial force to f o/As of Unit 1 and Unit 4 is 0.26 and 0.60, respectively, fo is the cylinder compressive strength of concrete and A s is the gross sectional area of a column. First, the sectional moment-curvature relationship of each specimen was computed using fiber element method with various confined concrete models including EA model, SR model, SU model, mKP model, MPP model and HKNT model t4]. It can be found that the theoretical section capacity is quite lower than the moment sustained by the sections near the footings(See Figure 4, in which the notations "testu" and "testp" indicate two envelope curves of the hysteretic curve of cyclic loading in opposite directions). This is consistent with the analytical results by other researchers t6, 7r, and it ought to be due to the external restraint by the footings. The hysteretic behavior of both specimens are simulated by the composite spring model. Analytical results coincide satisfactorily with test results (See Figures 5 and 6 ). The deformation and equilibrium conditions adopted in the composite spring model indicated that the curvature is calculated at the mid point of the elasto-plastic subelement and is assumed uniform along the elasto-plasfic sub-element, and the moment diagram remains linear along the whole member. This is consistent with real state in RC members. Moreover, in the model, the critical section changes from end section to the middle section of the elasto-plastic sub-element(See Figure 7), which reflects the actual condition of columns with footings [21. The moment of end section Mm~xAis always larger than the moment of the critical section M~rAat a quantity of 0.51pAV. Since the horizontal force of the test is computed according to MmaxArather than M~rg, it means that the composite spring model is able to reflect the effect of external restraint

843

by the footings automatically. 1000

1000

,-

,

,

"

I ~

-

,

,

'

-r

,

,

. . . . . . . . . . .

........... --L,~

=J ..... I !!

0

40

80 ~

e

120

160

,

,

,

r . . . . . . . .

l/.~_--i-+-._ ~ ~ .... ::i-------~-------~--- J .......

~ ~

t

,

li ....... i ......

I, '--~~"-"-"~'> , -,

O

~ 40

20

(xlO~lrnm)

li-',

60

Ciivi~e

i:

80

100

1213

(xlOelrnm)

Figure 4. Comparison between analytical results and test results of moment at the root section of each specimen

""'~"

;

"

":

~l

"

. . . .

I

""~"""

I~

A J

lkGliilONt~ll O I S P L A ~ N t ~r B~ $rU8 . i ~ M I

IlIN Unit

1:

a t mE4H S t U l . l W i l

kern, 0. 0 ~ . . i kN . O.t~$iqll

9i l ~ l t g l l

MMimmd

""<" .....

Horizontal

Loci.Deflection

H y x w m ~

Loopll

Unit

4:

Mnlmrml

HodzoMil

Loed.Defle4tion

Hystelreile

LOOlm

Figure 5. Test results of hysteretic curves [61 9

,

.

,,

9

,,

9

.

,,

9

,,

,

9

.....

15oo

t~o -. . . . . :---:-,~---:~, . . . . . / ,~,~. u . ~ (. , , . , ~

li

,ooo ; 1 - - . . . ~ . ,

::iIi

~o -1500

.....

, -, . . . . .

, ~ .....

9l . I -30 -20

, , .....

9 .l -10

~ .....

0

Displacement

Figure

, ,.....

, . -

9l . l 10 20

, _,_.

9l 30

9

.500

~

f.

~f

.4' .....

" ....

i

~~f/S

:

,

~o

v:

.... i ..... i---i .......

-

i

' ~ r - - ~

40

..... - -!. . . . :.....

!

.... ! i

..... i ..... i.... ..... I .... i ..... ! .... i----

i,~o"~~~,~o

of hysteretic

.

! ..... ! ..... i---:

Displacement (ram)

results

7

" . . . . . . . . . . .

...... ~ :;

'= .... ~

(mm)

6. Analytical

t ....

t .... ~ / _,/E_~~_/_:

i - ' [ -l l i ~ '

curves

A ~o

844

I

I

0.5tpA ~_~ l~r^

O.5h,Al/

l~rA

Figure 7. Transition of critical section in the composite spring model 3. CONCLUSION The composite spring model proposed in this paper reflects real deformation and equilibrium conditions of columns with footings. It not only simulates the hysteretic behavior of RC members, but also automatically accounts the external restraint provided by the footing. Ignorance of the external restraint will reduce the simulation accuracy of numerical models.

REFERENCES

1. T. Paulay, Deterministic Design Procedure for Ductile Frames in Seismic Areas, in Reinforced Concrete Structures Subjected to Wind and Earthquake Forces, ACI SP63, (1980) 357. 2. S.A. Sheikh and S.S. Khoury, Confined Concrete Columns with Stubs, ACI Struc. J., (1993) 414. 3. Y. Chen, Inelastic Behavior of Steel Frames Considering Varying Combined Stress in the Members (in Japanese), Ph.D. Thesis, the Univ. Of Tokyo, Japan, 1994. 4. F.F. Sun, Simulation of Seismic Response of Tall Reinforced Concrete Frame Structures, A dissertation for doctoral degree in engineering of Tongji University, CoUege of Civil Engineering, Tongji University, China, 1999. 5. M.J.N. Priesfley and R. Park, Strength and Ductility of Concrete Bridge Cohmms under Seismic Loading, ACI Struc. J., (1987) 61. 6. R. Park, M.J~N. Priestley and W.D. Gill, Ductility of Square-Confined Concrete Columns, J. Struc. Div., ASCE, No. 4 (1982): 929. 7. S.A. Sheikh and C.C. Yeh, Analytical Moment-Curvature Relations for Tied Concrete Columns, J. Struc. Engng., No. 2 (1992) 529.

Structural Failure and Plasticity (IMPLAST 2000)

Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All fights reserved.

845

An Analytical Method for Analysis of Curved Pair Members tied with Struts H. Ishihara', T. Yamao b and I. Hirai~ ' Kokudo-koei Consultant Corp., 4-3-5 Kitakyuhoji-chou Chuo-district, Osaka, Japan bDepartment of Civil Engineering and Architecture, Kumamoto University, 2-39-1 Kurokami, Kumamoto, Japan c Department of Civil Engineering, Kumamoto Institute of Technology, 4-22-1 Ikeda, Kumamoto, Japan This paper presents an analytical method for determination of the critical load of curved pair members tied with struts subjected to an axial compressive force. A numerical calculation was performed by using the proposed analytical method and frame-analysis based on the elastic finite element method by changing rise-to-span ratios to check the capabilities of the curved pair members. The buckling strength by the proposed analytical method showed good agreement with that of FEM. Validity and efficiency of the proposed analytical method was demonstrated. 1. INTRODUCTION New pairs of curved members tied with struts were proposed. It was shown that these pair structures subjected to axial compressive loads resisted higher buckling loads in comparison with usual two straight columns tied with struts. The critical loads of the proposed pair structures approach 4HE (H~: Euler's buckling load in Fig.l(b)) and their final deflections are represented by the first asymmetric mode. Especially, the new curved pair structures produce very small lateral deflections prior to buckling failure. This paper presents an analytical method for determination of the critical load of curved pair members tied with struts subjected to an axial compressive force. The proposed method based on the arch theory is obtained by considering the compatibility equation of the thrust

~H

H

stru

E I ~ E I axial direction

2E

E I ,--, E

l

curved member >--.r

~lateral \ direction ~ l /

/ H

(a)curved members (b)straightmember (c)straightmembers Figure 1. Curved pair members and straight members

846 displacement of the curved pair members. A numerical calculation was carried out by using the proposed analytical method and the frame-analysis method (called FEM) based on the elastic finite element method by changing rise-to-span ratios. Validity and efficiency of the proposed analytical method is discussed. 2. O U T L I N E OF P R O P O S E D ANALYTICAL M E T H O D The analytical method for determination of the critical load for the curved pair members is derived by using the compatibility equation, which determines the horizontal thrust produced in the arch due to the deformation of the system [2]. 2.1 Relationship between axial load and d e f o r m a t i o n of curved p a i r m e m b e r s Considering curved pair members subjected to the axial load Ho as shown in Fig.2.(a), the elongation A l 1and A l 2 at the end supports due to the axial loads AH1 and AH2 are given as follows [2]. : All =

16 }r

fl y, an "T . n

AHll fl fl 4 A. E { 1+.8( T )2+. 19.2 (-7-") }

"(1-1)

Al 2 =

-16

f2 E an l ~ n

AH21 f2 f2 PiE { 1 + 8 ( 7 - ) 2 § 19.2 (-7"-)4 }

(1-2)

Where Af~ and A f 2 are the rises of curved members (f~>f2), I is the span, h~ is the cross sectional area of the curved member and a~ is the coefficient in the series. The elongation of the two curved members must be equal. Thus, we have

in which

16 (fl +.f2) E ~ =- AH~ l Rl 7~ l n n A,E fl

fl 4

R1 "- 1+ 8( "7")2+ 919.2 ( T )

_

AH21

R2

(2)

A~E

f2 R2-" 1+ 8 (-T-)2+ 19.2 ( Tf2- )4

(3-1,2)

and Ho= AH~ + A H2 (4) As the lateral displacement w due to the axial load occurs, the distributed loads p.~ and P~2 must be applied to the curved pair members in order to maintain the stability of the given system as shown in Fig.2.(b). Hence, the unbalanced distributed load p, is found from the distributed loads P,1 and P.2P. = P.1 - P~2 (5,) in which 8f1 8f2 P.2 = 12 AH2 (6-1,2) P,I --- 12 AH~ , 2.2 R e l a t i o n s h i p between lateral d i s p l a c e m e n t and critical load o f c u r v e d pair members If the rise-to-span ratio of the curved pair members is small, it is possible to assume that the curved pair members are equivalent to the simple supported beam with flexural rigidity 2EI as shown in Fig.2(c). From this figure, we obtain the equation for the deflection curve in series form [31. 4 n~x 4 pgl Y. .~1 1 ( ) sin W'-" (7) 7r52EI -~ ns 1- ~ / r l 2 l n=1,3,5... in which Ho/2 2EI 7~2 (8-1,2) ~ = ~ Ho--~ 2EI 7r2 12 The equilibrium requires that; pg + p~ = 0 , pg = -p, = -p,~ + pa (9-1,2)

847 Ho (ext crnal 10ad)

Ho = AH, +AHz

aH, l r laH, qa3 ~a, a: / I I I

2El-"

.

E I (external 1

1

EI

~a i

:t~;~al oad)

axialdirection ~..

,

lateral direction

AH,AH,. (a) '

',(reaction force)

,

H0

(b)

~

I

=

(c)

Figure 2. Relationship between external load and deformation for curved pair members Substituting Eq.(9-2) into Eq.(7) yields the lateral displacement w for the curved pair members. The expression is w -_-

7r52E14/4' ( -8fll 2 AH1 +__78f2 A H 2 )

x E--~_ 1 ( "'l-~l/n ~''' )

Thus, we obtain an equation for determining the coefficient a." 4 14 -8fl 8f2 1

a~= ~r~2Ei ( ,12 a a x + - t 2

aH~)

•

sin nTrxi ....

(10)

)

01)

1

i-~/,?

Substituting Eq.(11) into Eq.(2) gives R2+- 512 f f2A, 1 1 7r621 E--~ ( 1 - ~ / n 2 ) AH~ . . . . . . . Ho R1+ R2"+ 512 taAa 1 1 7~'2I ..... ~ ' ~ - ( 1 - ~ / n 2 ) RI+

512 f flAa ~62I

AH2-

1

(12-1)

1

~'~

RI + R2 +~ 512 f~A~_

( 1-~/n 2 ) Ho 1 1 ~"~" ( 1.~'/n2 )

(12-2)

7r62i where f = fl + f2 (13) If the buckling of the curved pair members will occur with the first symmetric buckling mode, the axial loads AH1 and AH2 are equal to co. The dominators in Eqs.(12-1 and 12-2) equal zero. Thus,

in which

Rx+R~=R

R

F(~) =

- ___ a

a--

512 f2Aa 7r62I

(14) 1

F(;)= oZ7 ( ~

1

)

(15-1,2,3)

2 . 3 Analytical models

The analytical models of the curved members tied with struts that have hinged ends herein are shown in Fig.3. Each of the two curved members has the flexural rigidity EI(E:Young's modulus, I:moment inertia of a cross section) of a box cross section (see Fig.3(b)). Geometrical and structural properties of the models used in the numerical analysis are

848 Table 1. Properties of analytical models . . . . . /(cm) 207 [ 48 Span E(kN/mm2) 206 Young's modulus A~(cm2) 14.13" ' 1.66 , Sectional area of curved members Ia(Cm4) 15. 14 0. 095 . Sectional inertia of curved members .... /~ (cln 2) 1.41 O. 17 ,, Sectional area of struts fl/l 0.01 -~ 0.05 Rise-to-span ratio summarized in Table.1. Analytical models ~tI0 with the slenderness ratios//r,=200(r,: radius of gyration of cross section about weak axis) ..... ; ............. an are adopted to investigate the effect of the rises fl and f2 of the curved members upon the t/4 _, -I I_ 14~ J buckling load and behavior. Rise-to-span ratio (b-l) 1= 207cm fl/l of curved members are taken to be 0.01 to 0.05, and the rise difference A f (=fl-f2) I O.83cm between left and fight curved members which is equal to 1/500. The spans of these models are 207cm and 48cm each. The distance between left and fight hinged ends of the pair (b-2) != 48cm structures is a(=16, 6cm) as shown in Fig.3(a). The configuration of the curved member is a ( a ~ axes (b)Crosssectionmember parabola and all seven struts have the same Figure 3. Analyticalmodels cross section area(At=Affl0). Numerical analysis is carried out by using both the proposed method and FEM for the above models without initial crookedness in elastic region. The axial force at the ends is administered by controlling the incremental axial displacement in FEM analysis.

...,..,,.....

i

3.RESULTS AND DISCUSSIONS

3.1 Comparison of the proposed method and FEM Table 2 shows critical loads, axial forces and buckling modes for the curved pair members with/=207era and 48cm and fl//=0.01 ~'0.05 to check accuracy by comparison of results of the proposed method and FEM. The upper, middle, lower rows show results of the proposed method, results of FEM and errors of both results, respectively. The errors between the two methods are very small. From this, the results of the proposed method may be said to show good agreement with those of FEM. 3.2 Behavior of the curved pair members Fig.4 shows the relations between the axial loads (H0/Ha, H~: Euler's buckling load) and the lateral displacements w of the curved pair members with l=207cm and I=48cm, obtained by changing the rise-to-span ratios from 0.01 to 0.05 in the FEM analysis. The modes of the lateral displacements for the model of l=207cm with fl//=0.01 and 0.015 are shown in Fig.5. The critical loads Ho/HE for curved pair members with fl//_~0.015 approach 4 and their buckling shapes are represented by the first asymmetric mode as shown in Fig.5(b). However, it was not shown for the clear critical point of the model with f2/1~_0.013 and their deflection shapes approached the symmetry mode as shown in Fig.5(a). Hence, the

849 rise-to-span ratio of the m e m b e r may have a large influence on the critical load of the curved pair members. Table 2. Comparison of critical load between proposed method and FEM /-48cm Span l =207cm Rise-to-spanratio f~/l' 0.050 0.030 0.020 0.015 0.1Ji0 01050 0.030 0'.020' 0.015

....

0.010 574.609 390.332 67.257 67.257 67.257 67.257 45.785 563~887 378.143 i65.873 66.059 66.923 66.554-44.262 -2.06 -1.78 -0.50 -1.05 -3.33 -1.87 -3.12 32.899 32.236 301712 251008 co 212.945 c~ Left 213.613 -291.032 32.226 31.671 30'587 25.077 -33.921 Axial imember (AI-I,cAI-Itp)/AH,p (%) -2.43 -1.20 -1.03 0.31 -2.05 -1.76 -0.41 0.27 force of 34.358 35.021 36.544 42.249 co AH2p (kN) 293.535 299.210 312.275 361.664 members Right member AH2f (kN) 286.435!295.573 308.547 350.274 669.175 33.648 34.388 36.336 41.477 78.184 -2.07 -1.81 -0.57 -1.83 (AH2cAH2p)/AH2p (%) -2.42 -1.22 -1.19 -3.15 Buckling mode,, ' As~r , ' ' Symmetry, ,. ', ~ Asymmetry Symmetry H~'- 143.507kN(/=207cm), H~= 16.797kN(l =48cm) .~ upper: proposed method ( middle: F.E.M. (suffix "f') ' Load

I-I~ Hof (HorHw)/Hov A H~p AHu

Axial load Ho/H_ ,: ,3.91(4), ~ .9~ , 3 . 9 2 ( 4 ) 3.93(4) -~/ /~,//=0.05

3 ~

UIJ

/

(kN) 574.609 (ld~ 560.681 (%) -2.42 (kaN) 281.074 (kN) 274.246

574.609 567.671 -1.21 275.399 272.097

574.609 568.189 -1.12 262.334 259.643

Axial load ,

'

....... I ]2,,=u.u~-,

H0/HE. -I.,,,,~ .~.:,v 3.9,2~h

lower: error of both L89(4) /:~'9~8(4) '~3.96~4")

............................ -k~.IM) (4)~U / /fill=O.03. . . . . . . . h/l=O.O12. . . . . . . . . -13 45

"'""

-...................... .~3".56) ''' ."'''"?;;l'O"' fl/I-001. L--*-............. ." -~2.64

~Z,f ,,~,," . ...................... ,.-'"r e . o o -

.72)

1~ / -~datted line: symmetric mode at max. load ; full line: asymmetric mode at buckling od ( ~ )

'

~1/1=0.05 . .

''

'

.... '

'

. . . . . . . ~fl0..0_13 . . . . . . . . . . . .

-

/

2.72)

;i~;

,/'~dotted line: symmetric mode at max.load full line: asymmetric mode at buckling ( ):Buckling load by proposed method( ~ ) ,

I

0.1

,

I

0.2

Lateral displacemant(cm)

,

I .....

,,,,

I

0.3 0.4 Lateral displacement(cm)

(a)/=207cm (b)/=48cm Figure 4. Axial load vs. lateral displacement curves Axial direction (era) 2~'

Axial directi, )n (era) 200

lOC

1(3(

-----l-~a,~=l.o

~I-I~-I~--2.0

~

', ~1 k ~1

~

---'--HdHE--3.0 ~ \ ~! --'--~E--3"93 (max~ad) "-. ~li *- after collapse

l

i

i

3.90 ',4.05)

. ..-,'Zll:O~0~3 ......... fZ--0"0J'2" .......... 3.45 =i,'.." ;;;"'''" ;!! ./i: ~,.!~i102 ......................... ,1'I"001 . ___!',3.57)L64 (

t,

Lateral displacement(era)

-1

-0.3

u

I I /-I

Lateral displacement(era)

(b) fl//---0.015 (a) fl//=0.01 Figure 5. Modes of lateral displacement (1-207cm)

850 Axial load50_

(Ho/HE) Zg

I

,

,

symmetric " mode

,

r

9

/" 5=4

-

J

,

,,'"/'"'""

J

,

.... ""

.~.

-

Rise-to-span

at ~ = 4 fl/l

,,"

-

~

-I

'

I'

I

'

'

I

-

'

0.04 proposed ---'6"-- F E M

.."

ratio

I

0.05

,"

1( -

5

,

asymmetric mode -

method

0.03

_ ~~fl/l=2"4.~~(l~)etlSi~ide" ._ I

0.02

.

_ _a.ol3_ ........... 0.01 symmetric

| 0.C 1 2 9

I

0.02

,

I

003

Rise-to-span

|

ratiohll(1=207cm) ,

o.b4

o.b5

Figure6. Relationshipbetween maximum load and r i s e ~ ratio

'

i~0

'

2~o

'

mode

3~o

'

Slenderness

4oo' ratio l/r

Figure7. Relationshipbetween rise-to-span ratio and slendernessratio

Fig.6 shows the relationship between maximum load and rise-to-span ratio of the curved members by using the proposed method and FEM. The vertical axis represents the logarithmic maximum load ( ~ =H0/H~) and the horizontal axis shows the rise-to-span ratio fl/l. The maximum load of the curved members increases with an increase of fl/l. However, the maximum load remains with very small change beyond ~ --4 if the rise-to-span ratio increases in the FEM analysis: That is, it was found that the buckling mode of the curved members with fl//=0.0129 changes from a symmetric mode to an asymmetric one. Fig.7 shows the relationship between the rise-to-span ratio obtained by taking ~ =4 and slenderness ratio of the curved members. It can be recognized from this figure that the asymmetric buckling mode occurs at the upper part of this curve. Thus, this figure would be very useful for selecting the most suitable rise-to-span ratio of the given curved members in the elastic region. 4.CONCLUSIONS Numerical calculation were performed by using the proposed analytical method and FEM to investigate the behavior for the curved pair members tied with struts subjected to an axial compressive force. From this study the following conclusions may be drawn: 1)Both the elastic strength and deformation found from the proposed analytical method show good agreement with those of FEM. 2)The equations derived from the proposed analytical method are able to estimate the buckling strength with respect to the buckling modes from the value of ~. 3)A value of 0.015 for rise-to-span ratio is safe for curved members with a slenderness ratio of 200 without consideration of initial crookedness. Because the buckling load does not increase even if the curved members have fl/l larger than 0.015. REFERENCE 1. Ishihara,Y., Yamao,T., Hirai,I. and Mizuta,Y., Ultimate Strength and Behavior of Curved Compression Members tied with Struts, Proc. of the International Colloquium on Stability and Ductility of Steel Structures, Nagoya, Vo1.12, 1997, pp.615-622 2. Yoshimura,T. and Hirai,l., Dynamic Analysis of Langer Girder, Journal of the Japan Society of Civil Engineering, 1964, Vol.101(in Japanese) 3. Timoshenko,S., Strength of Materials Part II, Van Nostrand, pp.50-53

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

851

Numerical Analysis and Simulation for Cold Extrusion* Zhang Shaoxiong

Chen Binkang

and

Sun Haihong

Dept. Naval Architecture and Ocean Engineering, Wuhan Transportation University P.O. Box 430063 Wuhan, P. R. China Summary" In order to study the problems of large plastic deformation in axi-symmetric cold extrusion, this paper employs the penalty function approach to develop a calculation model of the rigid-plastic finite element method (FEM). Emphasis is laid on the numerical treatment for some key problems, such as determination of the initial velocity field, the converging factor and the decay coefficient, remeshing technique, the frictional loading on the boundary, the treatment of the boundary conditions, the singular points and the rigid region. Using a personal computer, the procedure described in this paper was implemented in a computer program with the visualization procedure, allowing us to analyze and simulate the whole process of different problems of axi-symmetric extrusion, as well as other types of metal forming.

1.

Introduction

Numerical simulation in metal forming is currently the object of intensive study. Many researchers have developed theoretical formulations of the material behaviour tll, and a large number of solution methods have been proposed t21. The rigid-plastic FEM has been recently applied to analyze many processes of metal forming as a powerful tool capable of simulating the material flow c31. As to the problems of stable flow, the rigid-plastic FEM is very mature up to now. But some problems associated with very large and severe deformation still remain, and it is commonly agreed that the treatment of these problems may affect directly the efficiency and accuracy of computation 141. To avoid the difficulties arising from the distortion of elements while severe deformation occurs, the mesh employed should be updated and re-generated continuously according to the deformed billet. Meanwhile, the boundary conditions should be adjusted. The more rationally is the initial velocity field obtained, the less steps of iteration are needed. The model of the frictional loading on the boundary, determination of the converging factor and the decay coefficient, and the treatment of the singular points and the rigid region, have also great influence on the rigid-plastic FEM simulation. "This work has been made possible by Hubei natural science funds (No. 94-52)

852 In this paper, the authors describe a calculation model of the rigid-plastic FEM for simulating large plastic deformation in axi-symmetric cold extrusion, based on the penalty function approach. Some problems such as determination of the initial velocity field, remeshing technique, frictional loading on the boundary and treatment of the singular points and the rigid region are discussed. The proposed method was tested and certified with the problems of upsetting and extrusion, and the final shapes of deforming bodies simulated are in general agreement with the practical metal forming ones. The procedure described in this paper was implemented in a computer programme using personal computer, including a post-processor for visualization of the computation of the velocity field and the strain-rates in the material. It allows us to study different problems of axi-symmetric extrusion and other types of metal forming. 2. T h e o r e t i c a l

Formulation

The rigid-plastic FEM formulation was derived from variational principle of the rigidplastic material, which consists of minimizing the functional of total energy considering the incompressibility constraint. Using the penalty function method, the problem can be expressed, without constraint, as follows I51

11=

I : c / V + - - ~av ( ~ ',

2

~ )2dV-

Isp T~vids

(1)

where ~ and ~ are the equivalent stress and equivalent strain-rate respectively; v~ is the velocity component; T, is the traction given; V is the volume of the deforming body; Sp is the surface on which the traction is prescribed; a is the penalty function; o~v is the volumetric strain-rate. In rigid-plastic finite element method, the equivalent stress is always regarded as the yield stress after hardening, which obeys the von Mises yield criterion and its associated flow rule. It is only required constant locally in each element and can be written as ~ = o",.(1 + a ~ ) b

(2)

where cr.,.is the initial yield stress, a and b are constant figures depending on the material properties and evaluated by experimental method. Supposing the deforming body is discretized by a finite number of nodes and elements, the functional in (1) is the summation of the corresponding ones in every individual element. The velocity field in an element can be expressed by its nodal velocities and shape function. From that we can derive the strain-rate distribution, the equivalent strain rate and the volumetric strain-rate. The minimum is found by deriving (1) with respect to the unknown nodal velocities, which gives a non-linear system of equations. The problem is solved by Newton-Raphson iterative method and the unknowns become the velocity perturbation {Au} (i.e. the first approximation correction to be applied to the nodal point velocity). This time the system of equations about {Au} is linear. In order to reach an optimum solution as near as possible to

853 the actual one in a relatively short time, it is important to estimate the initial velocity field by a rational approach. We can establish an approximate functional (3) which is similar with the original one both in the form and physical concept, in which the equivalent strain-rate ~ is in the form of square. So that it leads linear algebraic equations tS~.

2

)2

sp

)2

(3)

It will be positive when T~vj < 0 and negative on the contrary for the last term in the square root. From it we can get the initial velocity field immediately, and the initial velocity field calculated satisfies the boundary conditions of velocity as well as the incompressibility constraint. Starting from this initial velocity field, iterative calculation might be carried out until the corrective velocities converge to very small values, say 10.4 or less, compared to the nodal velocities. The criterion of convergence can be expressed as = U{

u}ll/ll{u}ll =

Au2

u2 < 60

(4)

In the iterative calculation, we should use the decay iteration instead of adding the velocity perturbation {Au} to the original velocity field directly. This is because that {Au} maybe not a small quantity in the early steps of iteration. The modifying relationship of the velocity field is written as {u},_, = {u},, + fl{Au},,

(0 < 13 < 1)

(5)

where the decay coefficient flis varying in the light of relative values of {Au}and{u}. Generally, the product offl and the largest component of{Au} should not be larger than 20% of the corresponding component in the {u} to be modified. Therefore,/3 is very small in the early step and approaches unity when the optimum solution is about to be reached. The whole plastic deformation process is divided into steps in which the velocity field can be considered constant. Stepwise calculation for incremental deformation is repeated from the stress-free state to the final state. The friction stress is treated as local constant in each element, which is represented by the friction factor m as r = m ~ / 3 . The use of friction coefficient allows us to identify the zone of adherence between die and workpiece. In this method, the calculation is somehow simple and the accuracy was proved to be enough trl. If the elements are fixed to the original mesh, the shapes of the elements will be severely distorted after large deformation, and the irregular shapes of elements would decrease remarkably the accuracy of computation. So it is necessary to update continuously the pattern of the deformed mesh according to the deformed configuration of billet, so the analysis can be continued from the new computation mesh which is not severely distorted. Only in this way could the simulation be continued until the final deformed shape. After remeshing, the boundary conditions of the new mesh should be adjusted in line with the relative position between the workpiece and the die or punch, and all of the history-dependent quantities of the

854 old elements must be properly transferred to the new ones, otherwise the calculation will become groundless. When remeshing has to be performed, the new velocity of each node can be evaluated approximately by the following tTl 4

1

{/,/new } ._ t~ 1 ~-(1 + ~:,~)(1 + T]iq){/d ~ }

(6)

where ~: and r/ are the local coordinates referred to the old element in which the new node is contained; u, is the velocity of i-th node; ~ and rh are node-dependent constants. Of course it is not economical and necessary to perform remeshing for every step. The redefinition o f the mesh is only needed in the following cases, (~) when the points of the workpiece surface leave the die, so modifying the boundary conditions,(~) when some points belonging to the lateral surface of the piece come into contact with the ram or the die wall thus changing the constraint conditions,| when one element reaches geometrical nonconformability or is over stretched. For case (~), the current mesh is analyzed by the degeneration index t71for each element E=(O/d) 2

(7)

where D and d are the large and small diagonal length respectively. Remeshing should be performed when E of one element exceeds the limited value (say 1.5). A mesh generator was developed by the author, which is based on the super element concept. It can divide the configuration of the workpiece into several super elements, and then generate the computation mesh by a few parameters automatically, while leaving the shape of the free boundary unchanged. The velocities change sharply near the comers of entry or exit section. The comers are socalled singular points. Finer elements can be used in the neighbourhood of the comer, but it will increase the CPU time remarkably. The better way to treat the singular points is considering them as bi-velocities points. That is to say, on every singular points, there are two nodes of the same coordinates, but they belong to different elements with different velocities in line with the boundary. After computation, the discrepancies of these points should be corrected in accordance with the total volume and the geometry of the tools. Another factor must be considered. Since the variational principles in plastic mechanics are only available when the whole body is in plastic deformation state. But acually, when reverse extrusion and combined extrusion are considered, there are rigid parts in the deforming body, and the stresses in the region cannot be determined. This difficulty can be handled by considering an offset of the equivalent strain-rate, which is several orders of magnitude smaller than the average strain rate in the deforming zone. The deviatoric stresses assumed to vary linearly from zero to the flow stress if the equivalent strain rate is smaller than this offset value tSj. By this means, the rigid region in the deforming body could be involved in computation.

855

3. N u m e r i c a l e x a m p l e s By using the computer program composed on the base of theoretical mathematics model of the present paper, we analyzed and simulated a practical metal forming process. The part of spring holder of steam valve (Fig.l-d) formed is illustrated schematically in Fig.1. An intermediate billet is formed by head upsetting from the original round billet firstly, then it is extruded by a taper ram within a stepped die, and finally, cutting the surplus parts gives the product. The part is made of aluminum alloy whose stress-strain characteristics are given by = 50.3(1 + 20~) ~ N/mm 2. A constant coefficient of friction m =0.2 is imposed on all the contact surface. -.

_

~12.5 I

I

.

I

I

31.

--

r

',

I

~

+ 27.5

I

'

!

;

ff29

_1

'

8

I ~14"9 1

! ~10 1

~,22

(b)

(c)

(d)

(a)

:

Fig. 1 The production of the spring holder for steam valve Fig.2 shows the configuration and computation mesh for the head forming. Fig.2 (a) shows the initial state, and (b) and (c) represent the situation when the punch pushes down with 10 and 14 mm respectively, and (d) gives the final shape. While analyzing the process of extrusion shown in Fig.l-c, finer mesh was employed because the flow of the material is more complicated. Fig.3 shows the states of original billet, when the ram pushing down with 3, 6, 9 mm respectively and the final product. I I 1 ' i ]

I II

I'

IIII 1111 111

]!11 lilt

1111 1

II

]!II 1111 i111 1 II 111

, l , ,

I

r

I

I I

I

l

i

l

i

i

iiiiii~

- : : :

iiii iiii ~iiii :4::: :-

::

:

iX!!:! i-!! ~

Jl

iiiii

IIII! II ! I I IIII!

l

I

l

I

I

I

U

I

i i[

i

. . . .

(a) (b) (e) (d) Fig.2 The mesh and configuration of the process of head forming The final shapes of deforming bodies simulated by the proposed method are in general agreement with the experimental or practical metal forming ones. It shown that the method is correct and valid. Fig.4 shows the distribution of the strain-rates in the material at some steps of calculation,

856

using the post-processor for visualization developed by the author.

..................

i!H',-l!'iil!![H! ::;;':;:::;: ;: :::

::::"iiiiiii ,:,:,,,,,,,,:

~

!!lllIllllllll:ll',

iii!!~"~:-':'i

.::::::::~'"

"q::: .............

H!iiil!ii iii :; : : : ............. i!:'.:~.:i~!!:"i~ ~Hiiii-H~i

ii'.-'i ~'~'-':':-'i ! .... iii!!!ii': ....

: ............ iiiii:~iiiii:,i

iiiiiiii]iiii lI-'ll I | ! I

Fig.3

The mesh and configuration of the process of extrusion

6~' I

1.2

0,:/, 0.8

Fig.4

t

0.4 0.8

The equivalent strain-rate distribution for some computation state

4. Conclusions The mathematical model and the corresponding computer programme were proven to be correct and practical, being capable of simulating and analyzing the problems of axisymmetric metal forming with high efficiency. The next step will be the detailed study of the flow and stress distribution in the deforming body in order to aid the design of the tools and the trouble shooting of the processes.

REFERENCES 1. Y.Tomita, Int. J Mech Sci. 24(1982) 711 2. S.I.Oh, Int. J.Mech Sci. 24(1982)479 3. J.J.Park, N.Rebelo and S. Kobayashi, Int. J. Tool Des. Res. 23(1982) 71 4. K.Mori, K.Osakada and M.Fukuda, Int. J Mech Sci. 25(1983) 775 5. Chen Ruxin, Hu Zhongmin: The plastic Finite Element Method and Its Application in Metal Forming, Chongqing University Press, Chongqing, China, 1989 6. G.Maccarini, C.Gladini and A.Bugini, J. Mater. Proc. Tech. 24(1990) 395 7. M.J.M.Barata Marques and P.A.F.Martins, J. Mater. Proc. Tech. 24(1990) 157 8. A.Alto, L.M.Galantucci and L.Tricarico, J. Mater. Proc. Tech. 31(1992) 335

General Structures

This Page Intentionally Left Blank

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

859

Experimental analysis on key components of steel storage pallet racking systems Nadia Baldassino a, Claudio Bemuzzi b, Riccardo Zandonini a a Department of Mechanical and Structural Engineering, University of Trento via Mesiano 77 - 38050 Povo, Trento, Italy b Department of Structural Engineering, Technical University of Milan Piazza Leonardo da Vinci, 32 - 20133 Milan, Italy

This paper summarises an experimental analysis on key components of steel storage pallet racking systems, which has been carried out since 1992 at the University of Trento (I) on behalf of 17 Italian Manufacturing Companies. Tests on both stub column and beam-end-connector are herein presented and the associated results are shortly commented. Furthermore, two new types of tests, i.e., base-plate connection and column in bending tests, currently in progress, are shortly introduced.

1. INTRODUCTION Design of steel storage rack systems is fairly complex, in comparison with the one associated with traditional steel buildings, owing to the particular geometry of the rack components. With reference to the European practice for pallet racks, beams are usually realised by means of boxed cross-sections and columns present perforated open section to accept the hooks of beam-end-connectors, which join beams and columns together, without the need of bolts or welds. Column behaviour is significantly affected by different buckling modes (i.e., local, distorsional and overall) as well as by their mutual interactions [1]. Moreover, the response of joints (both beam-to-column and column base) is typically non linear and, furthermore, the performance of base-plate connections depends strictly on the level of the axial load in the column. Due to the great number of types and to the different geometry of the key rack components, theoretical approaches for rack design are not currently available. Most important design standards for steel storage racks [2-5] require specific tests to evaluate the performance of members as well as of joints in order to understand and to quantify main factors affecting their response. A research project on steel storage pallet racking systems is currently in progress [6,7] with the goal of developing a procedure for the design of steel pallet racks. This paper deals with the experimental analysis of the study, which comprises of both stub column and beam-endconnector tests, the main results of which are herein summarised. Furthermore, base-plate connection and column in bending tests, which are currently on-going, are shortly introduced.

860 2. THE EXPERIMENTAL ACTIVITY Despite the fact that a large proportion of steel storage racks are manufactured from coldformed steel sections, the state of knowledge developed for cold-formed members, and, as a consequence, cold-formed design standards (such as [8-10]), can not be directly extended to racks. Experimental analyses are hence necessary to evaluate the behaviour of the key components of the rack system and to determine main parameters governing member/connection response. By means of these data, provided by re-analysis of tests results, design phase of rack systems is carried out by a suitable use of the approaches developed and codified for traditional cold-formed members. In this on-going study [6], attention has been up-to-now paid to the behaviour both of perforated stocky columns in compression and of beam-to-column joints in bending. Key features of these types of tests (i.e., stub column test and beam-end-connector test) are herein shortly introduced. However, additional types of tests are necessary to characterise the behaviour of other components influencing the overall behaviour of rack system. As a consequence, an experimental analysis on base-plate joints under axial eccentric loading (base-plate connection test) and on perforated columns in bending (column in bending test) has been planned and it is currently in progress to extend suitably the state of knowledge on racks. 2.1. Stub column tests Perforations in columns can produce significant reductions in their axial capacity, in addition to the ones associated with local and distortional buckling modes, which are typical of thin walled members [ 1,11]. Design rules for perforated members in compression are based on the definition of the form factor Q, which accounts globally for the effects of perforations as well as buckling effects. In accordance with the European approach for rack design [4], the value of Q can be evaluated on the basis of stub column tests, as follows:

q-- P~

f,.A

(1)

where Pu is the ultimate load of the stub specimen, fy the experimental yield stress of the material (obtained from tensile coupon tests) and A represents the gross area of the cross section. In total 339 stub cohmm tests have been performed up the collapse on 108 different column section types. On the basis of test results, it can be noted that: * only a small percentage of the tested specimens (approximately 1.8%) has an effective area (defined as Q'A) coincident with the gross area of the cross section. Owing to the hardening of the material in the comer zones of the section, in some cases Q resulted slightly greater then 1 and, hence, for practical design, Q=I has to be considered; . only a very limited number of specimens (9 in total) is characterised by a form factor less than 0.5. Moreover, with reference to the considered specimens, the mean value of Q results 0.73 and the associated standard deviation is 0.18. 2.2. Beam-end-connector tests The performance of pallet racking systems significantly depends upon the efficiency of the beam-end-connectors, which provide support to the beams and are, together with column bases,

861 the sources of stiffness for the down-aisle stability [12]. I~mowledge of the actual joint behaviour is hence of fundamental importance for a suitable definition of simplified moment-rotation (M9 ) joint relationships to use in the design analysis of semi-continuous frames. Experimental analysis on beam-end-connectors [4] comprised of 208 tests on 56 different types of connections. For each type of them, four tests were generally executed: three under hogging moments, to appraise the connection behaviour in the usual service conditions, and one under actions generating sagging moments to evaluate the performance in the presence of accidental upward action or of frame sway. Typical moment-rotation (M-O) joint curves obtained from these tests are reported in Figure 1. After the initial slippage due to looseness of the beam-end-connector, three branches can basically be identified under both hogging and sagging moments: elastic, inelastic (with a progressive deterioration of stiffness) and plastic (with a significant plateau and, in some cases, also a softening branch). Generally, tests were interrupted at a high level of connection rotation, out of the range of practical interest for the current usage of beam-end-connectors.

4

M

Ma

_

I

-100

I

-50

'- .

2

.

.

.

50

.

.

.

100.._TestI 150 --- T e s t II -'- T e s t m -~- T e s t R

Figure 1. Typical moment-rotation joint curves. The experimental M-O curves related to the response under hogging moments have been directly compared in non dimensional form, in accordance with the criteria of Eurocode 3 [13] for classification of joints in unbraced frames. In particular, from the original M-O curve, a non dimensional m - ~ relationship has been obtained. Terms m and ~ are defined, as: --

M

(2a)

m= M

p,b

-

~=~

EI b

(2b)

LbMp,b

where E is the Young modulus, Ib and Lb are the second moment of area and the length of the beam, respectively, and Mp,brepresents the beam plastic moment.

862 With reference to all the m - ~ joint curves [6], it should be remarked that: 9 for a great number of tests (approximately 30% of the tested specimens) joint response falls in the domain of flexible connection (as curve a in Fig. 2); 9 in some cases (in total 13% of the examined joint curves) joints can be considered semi rigid, owing to the value of the rotational stiffness (as curve b in Fig. 2); 9 in other cases (in total 10%) joints can be considered semi-rigid, on the basis of the value of the bending strength (as curve c in Fig. 2); 9 approximately half (47%) of the tested joints can be considered semi-rigid with reference to both stiffness and strength (as curve d in Fig. 2). From these results related to EC3 joint classification, simple frame model should be used in many cases for the design analysis. From a numerical study on the analysis models for steel buildings [ 14], it appears however that joint influence on fi~me behaviour also, in the case of flexible joints, results non negligible.

0.6

EC3-UPPER B O U N D

0.5

d

0.4

"

r

0.3 0.2

~ ...f" .....

0.1

...................................................................................b

/'"

a

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Figure 2. Non dimensional moment-rotation joint curves.

3. FUTURE EXPERIMENTAL ACTIVITY As previously mentioned, two other types of tests are planned in the experimental activity in order to provide a more complete set of basic information for rack design, in accordance with standards requirements. In particular, the following types of tests are considered: 9 test on columns in bending; 9 test on base-plate joint. Stub column tests, which define term Q to use for the design of axially loaded elements, don't provide complete information for rack design. Columns are, in fact, subjected to both bending and axial compression, owing to the geometry of the nodal zones. The influence of perforations as well as of different buckling modes affect remarkably also member response in bending and as a consequence, the definition of the effective geometrical bending properties, is required. Due to the absence of purely theoretical approaches, these terms can be determined directly by tests. The testing equipment and the specimen for this type of test are reported in Figure 3. The actual degree of restraints provided by base-plate connections influence remarkably the behaviour of the overall structural system, also if the degree of continuity is limited, as shown [12]. In absence of the possibility to define the experimental response of the base connection, the "ideal" restraint condition of hinged bases has to be considered for the structural analysis.

863 The testing equipment, which has been designed for base-plate joint test, is reported in Figure 4. The specimen is composed by two stub column connected to a concrete cube by means of the base-plate connection system. Hydraulic jack (1) axially loads columns in order to simulate their actual service conditions, while jack (2) applies a load to the centre of the cube to singled out connection flexural behaviour. /

it

COUNTER FRAME

i"

i"ii

!

HYDRAULIC JACK LOAD CELL I

----

|

"-"

////////~////////~/////////

• SPECIMEN

c__

I !1

~

'1 .......I ! ........i , l .........If

Figure 3. Testing equipment for column in bending test.

i

~HYDRAULIC JACK 2 BASE - PLATE CONNECTION

~,

LOAD..,.,,CELL /

~

~ /HAcD:IA U L.IC

Figure 4. Testing equipment for base-plate joint test.

4. CONCLUDING REMARKS This paper summarises an experimental research on steel storage pallet rack systems, which is currently in progress at the University of Trento. In accordance with recent standards for rack design, specific tests have been executed, owing to the absence of theoretical approaches capable of predicting the performance of the key components of racks. Up-to-now, attention has been focussed on the following types of tests: 9 stub column tests, which showed that the influence of local buckling as well as of perforations in columns, affect quite remarkably member resistance under compression;

864

beam-end-connector tests, which allowed to determine the moment-rotation curves of different types of beam-to-column joints under both hogging and sagging bending moments. As a general remark, it can be said that joints result very flexible, if classified in accordance with Eurocode 3 criteria. However, the actual response of beam-end-connectors provides a non negligible degree of lateral stiffness of the flame and, as a consequence, semi-continuous frame model is suggested for a more refined and "optimal" design analysis. As to the on-going experimental activity, both column in bending and base-plate connection tests are currently in progress in order to complete the set of basic data necessary to develop a safe and reliable rack design.

s

REFERENCES

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

[7]

[8] [9] [10]

[ 11] [12]

[13] [14]

G.J. Hancock, Design of Cold-Formed Steel Structures, 3rd Edition, Australian Institute of Steel Construction, 1998. RAL, Storage and Associated Equipment, RAL Deutsches Institut fur Gutersicherung und Kennzeichnung (German Institute for Quality Assurance and Marketing), 1990. RMI, Specification for the Design, Testing and Utilization of Industrial Steel Storage Racks, Rack Manufactures Institute, 1997. FEM, Recommendation for the Design of Steel Pallet Racking and Shelving, Section X of the Federation Europeenne de la Manutention, 1997. AS, Steel Storage Racking AS4084, Australian Standards, 1993. N. Baldassino, C. Bernuzzi, R. Zandonini and G. J. Hancock, Overall, local and distortional buckling in pallet racks, Proceedings of Structural Stability Research Council Conference (S.S.R.C.), Atlanta U.S.A., September, 1998. N. Baldassino and G.J. Hancock, Distorsional Buckling of Cold-Formed Steel Storage Rack Sections including Perforations, Proceedings of the Fourth International Conference on Steel and Aluminium Structures, Espoo, Finland, June, 1999. AISI, Specification for the Design of Steel Structural Members, American Iron and Steel Institute, 1996. AS/NZS 4600, Cold-formed Steel Structures, Australian/New Zealand Standard, 1996 CEN, Eurocode 3:EC3 Part 1-3, Design of Steel Structures - Goneral Rules. Supplementary Rules for Cold Formed Thin Gauge Members and Sheeting", European Commitee for Standardization, 1996. P. Dubas and E. Gehri, Behaviour and Design of Steel Plated Structures, ECCS-CECMEKS, Publication n~ 1996. N. Baldassino and C. Bernuzzi, Experimental Analysis of Beam-End Connectors for Steel Storage Pallet Rack Systems, in preparation to be submitted to Thin-Walled Structures. CEN, Eurocode 3: EC3, Design of Steel Structures - General Rules and Rules for Buildings, European Commitee for Standardization, 1994. C. Bernuzzi and R. Zandonini, Serviceability and Analysis Models of Steel Buildings, (IABSE) International Colloquium on Structural Serviceability of Buildings, Goteborg, Sweden, June, 1993, pp. 195-200.

Structural Failure and Plasticity (IMPLAST2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

865

Response of large space building floors to dynamic loads which suddenly move to a new position S. W. Alisjahbana Civil Engineering Department, Tarumanagara University J1. Let. Jen S. Parman No. 1. Jakarta 11440. Indonesia

The forced dynamic response of large building floors to dynamic moving transverse loads is investigated. The general analysis and solution of this problem are developed utilizing classical plate theory. The general solution of the force responses is derived in integral form, resulting in a series solution. The sudden change in load position requires the transient and the steady state deflection to be included at each of the constant load position, before and atter the sudden change. The resonance conditions may be said to occur if the deflection expression is maximised. 1. INTRODUCTION This paper presents a theoretical analysis of the response of a rectangular plate subjected to a concentrated transverse load of a harmonically varying amplitude P0eostot moving in the y direction at a constant velocity. The position in the x direction is constant until an instantaneous shift to a new x position. This kind of loading can be considered to represent human movements on ball-rooms or discotheque floors, performance stages and indoor sport facilities. 2. ANALYSIS A homogeneous, isotropic, thin rectangular plate of uniform thickness h is simply supported at all edges. The transverse surface load acting on the plate is taken in the general form p(x,y,t), representing all possible surface loadings that may be described as functions of position of the load and time. According to the classical plate theory, the governing partial differential equation of transverse motion for a damped rectangular plate subjected to a general transverse surface load is given by: DV 4 w(x, y, t) + ph -a-2w(x' y,t) + ),h o%v(x,y, t) _ p(x, y, t) &2 &

(I)

866 where D is the flexural rigidity, V 4 is the Laplacian operator, p is the mass density and 7 is the damping. 2.1. Homogeneous Solution The solution of the problem can be obtained by a method of separation of variables. This technique is particularly useful for the direct solution of boundary value problems, where the boundary conditions have a simple form. The procedure comprises the derivation of a sequence of solutions of a separable form, in such a way that superposition yields a solution satisfying the boundary conditions. According to this method, the general solution of the homogeneous equation is assumed to be separable into functions of space and time,

Wren (x,y,t) = Wmn (x, Y)Tmn(t) = E - e -~c~ Q~ n~__l[sm . mnx sin n~3'] m=l a b

[ a0mnei 1-~-~~

+ b0mne-i 1-~~~

]

(2)

in which Win, (x,y) is the sp__atial function, Tmn(t) is the temporal function, ~m. is the natural frequency of the system, 7 is the damping ratio, aomn and b0m, are constants, m and n are numbers of modes in x and y direction respectively. Since Eq. (1) is a linear partial differential equation and the boundary conditions are linear, the superposition of any number of solutions in the form of Eq. (2) is also a solution of Eq. (1). Therefore, since the set of fundamental solutions is complete, an appropriate application of the eigen function expansion theorem gives the general solution in the form of the superposition of the fundamental solutions for each value of m and n. Thus, the deflection for free vibration is given by oO

oo

w mn(x, y, t) = E Y',Wren(x, Y)Tmn(t)

m=l n=l

= ~

m=l 1

sin

sin~

a

e -~~

(3)

r 1-X~2t~nt + boron e-i 1-~720)mnt]

2.2 Particular Solution By using the orthogonality conditions of the eigen functions, integrating over the rectangular region 0<xS.
w(x, y,t) = E E sin

m=ln=l

a

I

sin nTtye_YtOmnt amn e b

+ bmne_i

~

mnt

Y Y P(X'Y'~)isindm~x xlsin~ dy sin~/1- ~2mmn ( t - x) m=ln=l 0L phQmn 0 a 0 --b ~/i- 7/2tOmn where Qmn in Eq. (4) is a normalization factor

+ (4) x

867 The general solution presented above may be integrated to determine the response of the plate to an arbitrary applied surface load p(x,y,t).

(5)

p(x, y,t) = P0 c~176 8{x-Ix0 + X l H ( t - t 0 ) ] }~5(Y - v t )

in which H is the Heaviside generalized unit step function and x,=x2-x0. Substituting the load function given in Eq. (5) into the general deflection solution, Eq. (4) becomes

w(x, y,t) = Y Y~ sin m=ln=l

sin a

r r 0coso !!I

+ 2 m=ln=l [_0[. PhQmn

e-~(Omntamne,

mnt + bmne ~/1-~(o

b

sin - s i n ~ 6 [ x - [ x 0 + xlH(t-t0)}5[y-vx a b

xdy

1]

(6)

[ e-YC~ (t-x) __ COma 1 ~]1-7 2 ~mn sin~/1 Y2 ( t - x ) dx

Thus, this problem may be treated in two parts. The first part involves a harmonically oscillating concentrated transverse load moving in the y direction at a constant x position Xo. The second part involves a harmonically oscillating concentrated transverse load moving in the y direction at constant x2. The two parts of the problem are related through the initial conditions. The motion of the plate at t--to due to the load at X=Xobecomes the initial condition of the plate for the subsequent instantaneous change at t=to due to the load at x=x2. Using the above analysis, the motion during an interval of time including the sudden change in x direction can be computed. Assuming the motion has achieved the steady state at xo, prior to the sudden change in direction, the motion at t=to may be easily computed. This motion at t-~ determines the initial conditions for the second part of the problem when the load is at the x2 position. The steady state deflection response is readily obtained by neglecting all the transient terms. This includes not only terms from the homogeneous solution, but also any transient terms from the particular solution obtained from the Duhamel convolution integral method used to solve the non-homogeneous problem. These transient terms from the particular solution are related to the initial effect of the applied loading function of the system. 3. NUMERICAL EXAMPLE Using the procedure of the last section, some results have been obtained for a floor slab with a dynamic human load. Three types of phenomena are presented affecting the total dynamic deflection: the effect of changing the load's frequency co, the effect of changing the damping ratio (y), and the effect of x,, the magnitudeof the sudden change in the x position. To determine the role of the damping ratio (V), four cases have been calculated for which y equalled 0, 0.02, 0.03 and 0.05, representing the damping factors for building structures. The position of the human load is varied from x0=2.5m to x0 = 7.5 m. The velocity of the human load is taken as v - 1.94 m/see (a typical dancing speed), the weight of the m

868 person is P0=1000N and t0=5 sec, being the instant when the dynamic deflection of the slab is near its maximum. The following numerical results have been calculated for the case of a thin concrete floor slab simply supported at all edges with dimensions and characteristics as folows" x=15m, y=20 m, p=2400 kg/m3, h=120 mm, E =30 GN/m 2 and o=0.01. Fig. 1 shows the maximum dynamic deflection at mid-span of the slab (7.5, 10) due to the sudden change in x position of a concentrated load of varying amplitude (c0=l 5 rad/sec) as a function of the magnitude of the sudden change in x position, xt and of the damping ratio 7From the figure, it is obvious that the maximum dynamic deflection is a function of the magnitude of the sudden change in the x position. As expected, the maximum dynamic deflection increases as the change in the x position increases for all values of damping ratio. Fig. 2 shows the maximum dynamic deflection response spectra based on the first nine modes at mid span of the slab. Note in this figure that if the value of the natural frequency of the system is approaching the natural frequency of the load, the maximum dynamic deflection increases for all values of damping ratio. Damping plays its usual role of reducing the maximum steady state deflection. Analysis of this nature is particularly useful for applications, where constraints are imposed on the allowable maximum dynamic deflection. 0.035

5.00E-04 ~"

4.50E-04

" .o ~

4.00E-04

~1 0

3.00E-~

._o E m c

2.50E-04 2.00E-04

"o

1.50E-04

m E

1 .I~E-I~

E c

3.50E-I~

0.03

i

_1

0.025

"10

1

,

.~_ E e-

0.015

X

0.0~

E

o.o2

0.005

0

"9'o

O.OOE+O0 , 0

o

2

4

xl (m) 6

=

damp ratio=O

J.

damp ratio=O.03 - - w - - d a m p ratio=O.05

- . m - - d a m p ratio=O.02

Figure 1. Maximum dynamic deflection at mid-span as a function of X~.

-0.005

. . . . . . . . . .

!

L

i

,

~

- -

.-

20

~

~damp

4p

60

BO

=

ratio=O

~

~damp

1:

100

= (racV~ec) ratio=O.02

~dk--darnp ratio.--O.03 ~ N - - d a m p ratio=O.05

Figure 2. Maximum dynamic deflection response spectra at mid-span computed based on the first 9 modes.

Table 1. Computed natural frequencies of the floor slab system. m

1 2 3 4 5 6

m, n=l O)mn (rad/sec) 8.394657 24.5124 51.3753 88.98337 137.3366 196.435

m,n=3 COmn ~mn (rad/sec) (rad/sec) 17.46089 32.57127 33.57863 i 48.68901 60.44153 75.55192 98.0496 113.16 146.4028 161.5132 205.5012 220.6116 m, n=2

m, n=4

O~mn (rad/sec) 53.72581 69.84355 96.70645 134.3145 182.6677 241.7661

m, n=5

COmn (rad/sec) 80.9245 97.04224 123.9051 161.5132 209.8664 268.9648 ,

m, n=6 COmn (rad/sec) 114.1673 130.2851 157.148 194.7561 243.1093 302.2077 ......

869

0.0003 .

.

.

.

.

.

.

.

.

.

.

-

0.0002

-

-

-

0.0001

--

0

. . . . . .

I.I

"~ -0.0001 -0.0002 .

-0.0003

-o.0oo4

time (sec) to= 5 s e c . . . . . . . . . . . . . . . . . . . .

-0.0005 0

5

10

15

20

25

30

Figure 4. Total dynamic deflection at position (7.5,10) for co= 15 rad/sec of the concentrated moving load, with v=1.94 m/sec, x0=7m, Po=1000N, ~,=0.05,to=5 sec and x2=7.5m.

870 Fig. 3 shows the dynamic deflection mode shapes for the first nine modes subjected to a concentrated transverse dyn_amic load with loading parameters as follows" co=15 rad/sec, x0=7m, x2=7.5m, P0= 1000N, y= 0.05, computed at t=6 see. Fig. 4 shows the motion at the position (7.5, 10) for the case of o)=15 rad/sec. The value of to is chosen such that it occurs when the dynamic deflection of the plate is near a maximum. Note in this figure that at t=t0, a large transient motion begins that eventually decays to the steady state motion due to the load at x--x:. 4. CONCLUSION Based on the above study, the following conclusions can be drawn. The conditions for system resonance occur when a certain relationship between the system load parameters and the system natural frequencies is met. For any particular special case of interest, the resonance conditions can be obtained by considering the expression for the total dynamic deflection. When the deflection expression is maximised, resonance may be said to occur. The resonance conditions are most clearly exhibited when there is no damping. The maximum deflection response spectra for various values of the load's frequency may be used in the design of a floor slab system to determine the maximum response deflection, and to avoid the resonance condition. The maximum dynamic deflection is a function of the magnitude of the sudden change in the x position. The contribution of individual modes to the total dynamic deflection decreases as the mode number increases. For the system considered, nine modes in the total dynamic deflection is sufficient to determine the total dynamic deflection. The time at which the load suddenly moves to its new position should be chosen such that it occurs when the dynamic deflection of the plate is near its maximum. If this to coincides with the time at which the dynamic deflection occurs, the total dynamic deflection after the sudden change in the x position of the load will be a maximum. RELATED REFERENCES

1. Alisjahbana, S.W. "Dynamic Response of Large Space Building Floors to Dynamic Moving Loads", Proceedings 3rd Asia Pacific Conference on Shock and Impact Loads on Structures, Singapore, November (1999). 2. Alisjahbana, S.W. "Rotating Annular Disk Response to Circumferentially Moving Loads with Sudden Position Changes", Proceedings 2"d Asia Pacific Conference on Shock and Impact Loads on Structures, Melbourne, Australia, November (1997). 3. Wangsadinata, S.M. "Rotating Annular Plate Response to Arbitrary Moving Load", Ph.D. thesis, University of Wisconsin, Madison, USA (1992). 4. Wangsadinata, W., Alisjahbana, S.W. "Vibration of Large Space Building Floors", Proceedings 1st International Conference on Structural Dynamics, Bandung, Indonesia, November (1996). 5. Weisensel, G.N. "Annular Plate Response to Circumferentially Moving Loads with Sudden Radial Position Changes", The International Journal of Analytical and Experimental Modal Analysis 5 (4), Oct. (1990).

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

871

Effect o f cables on the behavior of I-section arches Yanlin GUO and Jinsan JU Civil Engineering Department, TsLnghua University, 100084, Beijing, P.R. China The paper presents the effects of cables on the behavior of I-section arches. An elastic large deflection (displacement) finite element formulation with beam and cable elements is employed, in conjunction with arc-length method and Newton-Raphson iteration in order to trace a full load-deflection path. Three kinds of loading types, i.e. the concentrated load at the top of arch, the full-span uniformly distributed load and the half-span uniformly distributed load are considered, and two kinds of cable disposition forms as shown in fig. 1 and fig.2 are involved in the theoretical investigation. It is found that the cables affect significantly the arch's behavior, including its load-carrying capacity and its load-displacement paths. Some conclusions are drawn from the numerical results obtained. 1.

INTRODUCTION

A major trend in structural design is to improve the efficiency of structure and construction by reducing its material consumption. This trend leads structural engineers to adopt and invent new structural forms or improve and optimize the existing ones to reach purpose. With rapid development of new structural theory and constructive skill, more and more new structural forms have been found which were widely used in the large span space structure. An arch is a good structural form that can not only supply enough space but also span large distance. However, the half-span distributed load is considered to be the worst load case for the arch because it results in a lower load-carrying capacity. To improve arch behavior under the half span distributed load, this paper presents a new structural form, namely the arch hybridized by cables. The cables are diagonally arranged in the form shown in fig. 1 and fig.2. They are crossly cormeeted to four points where two points are the abutments of arch and other are linked to inside surface of arch, the approximate locations of 1/4 and 3/4 span of arch where the greater deflection amplitudes take place for the arch subjected only to the halfspan distributed load [1,2]. This paper presents the theoretical investigation to its nonlinear instability behaviors of the arch hybridized by cables, and a full load-deflection path is obtained by employing an elastic large deflection finite element method. The results obtained show that the cables not only could control effectively the arch deflection amplitude but also could increase greatly its limit load-carrying capacity especially in the case of the half-span distributed load. Therefore, it is recommend strongly to be use when the half span distributed load become predominant in structural design, and it may span greater space than the arch without cables.

872 2.

ASSUMPTIONS The effects of cables on the stability behavior of I-section arch and its nonlinear buckling behavior is investigated in the study. The cable disposition forms are shown in Fig. 1 and Fig. 2. The following assumptions are made in the analysis: (1) analysis is totally in elastic range, and (2) consider the behavior of the arch only in the plane of loading.

3.

FINITE ELEMENT FORMULATION

Instability behavior of the arch hybridized by cables is a main point of this study, which includes linear buckling problem and nonlinear buckling problem. For the former, the eigenvalue can be obtained by following foiTnula: ([k] + A[s]){~,} = 0

(1)

where [k] is the initial stiffness matrix, 2 is the eigen value, [s] is the stress matrix, and {~u}is the eigen vector. For the nonlinear buckling problem, it is emphasized on the geometric nonlinear buckling behavior of the arch hybridized by cables on the basis of the above assumptions. The NewtonRaphson approach and the arch-length method [3] ate combined and employed to obtain a full load-displacement path. Of course, a cable element and a beam element are applied in finite element formulation. For the beam element or the cable element, the application of the principle of virtual displacements leads to the following nonlinear equations: {r

= {F}- I [B] {r}dV - 0

(2)

in which {F} is a vector of equivalent nodal forces resulting from body forces and surface forces, [B] is the incremental strain-displacement matrix, {T} is the vector of stress resultants and {~(~)} represents a vector of nodal residual forces. These relations constitute the basic equations in the solution of nonlinear equilibrium equations. For each iteration, the displacement increments for the structure {AS} are obtained by solving the linearized system of equations: [4]

{r = [K~ ]{zx6} (3) where [K r ] is termed the global tangent stiffness matrix which is formed by assembling elastic element stiffness matrix and geometrical matrix. 4.

NUMERICAL EXAMPLES

4.1 General

The nonlinear buckling analysis of the arch hybridized by cables is carried out under three loading conditions mentioned above, namely, the concentrated load at the top of arch, the half-span uniformly distributed load and the full-span uniformly distributed load. The behavior of the arch both with and without cables is theoretically investigated in the large deflection range[5,6]. The I-section arch is considered in the analysis, composed of web of-358x 13mm and flanges of-400x21mm, where steel elastic modulus E=2.06E5 MPa. The cable is a

873 combination of 2'D15(7,5) steel wire with elastic modulus E=2.0E5 MPa. The span of the arch is chosen as 40m, with two ends hinged. Numerical results are presented below to demonstrate the effects of cables on the deformation process, stability behavior and limit load of the arch. q

q

i

~p § § * +

I FI

Fig. 1 Cable disposition form-1

n

Fig.2 Cable disposition form-2

4.2 Behavior of arch hybridized by cables

Table 1 gives the limit loads of the arches with and without cables in the cases of two different cable disposition forms. It is found from the table that, in the case of the top concentrated load, the limit load of the arch for the cable disposition form-1 is slight higher than that of the arch without cables, while the limit load of the arch for the cable disposition form-2 is less than that of the arch withottt cables. Fig.3 and Fig.4 are the load-deflection curves at the apex of arch with and without cables, respectively, for the cable disposition form-1. It is seen that the limit load-carrying capacities and deformation shapes between with and without cables are almost same. This implies that the cable disposition form-1 do not make any good effect for the predominant top concentrated load. Load-cart. 'ing capacity of the arch with and without cables disposition Cable disposition Cable Arch without cables Load case form-2 form-1 3300kN 3360kN 3358kN Concentrated load Half-span uniformly 149.6kN/m 149.7kN/m 109.5kN/m distributed load Full-span uniformly 209.8kN/m 191.8kN/m 195.2kN/m distributed load Table 1

Unlike the top concentrated load, it is a much more different in the case of half span distributed load. The cables are very effective and they do play a significantly good part to increase half span limit load. The limit loads of the arches with cables are much higher than those of the arches without cables. The limit loads of the arches with cables are 149.7kN/m for cable disposition form-l, and 149.6kN/m for cable disposition form-2, while limit load of the arch without cables is only 109.6kN/m. This is because the cables constrain effectively the lateral deformation that results directly from the action of half-span limit load, and then enhance its half-span limit load significantly. It is seen from fig.5 and fig.6 that the behavior of arches with and without cables is quite different in the load displacement curves. The

874 deflection amplitude of former is 0.338m much less than the latter (3.1m). There also appeared in the figures some sharp points on the load-deflection path in Fig.6, which is only caused by the tension-only character of cables. 1.5

o

1.5 1.0 0.5 0.0

-0. 5

N-o.s

1 0.5

"g"

* -1.0

-1

-1.s

"~ -1.5 -2

|

25

B(m)

-2. 5

-2. 0 -2. 5

B(m)

Fig.3 Load-deflection curve of apex of arch

Fig.4 Load-deflection curve of apex of arch

It is noted that two limit points A and C will be found in Fig.5 and Fig.6 while tracing the load-displacement curve. The first limit point, namely point A, corresponding to the first loading step as shown in Fig.7, is generally defined as a limit load that has a practical importance. Second limit point, namely point C in Fig.5 and Fig.6, corresponding to the fifth loading step as shown in Fig.7, is much higher than the point A. The second limit point exists theoretically, but there is less practical significance. 9

t

.

,~

O.

0.8 O. 0.3

.~ 0. ",~-0. 2 --0.7

Fig.5

~. ~ . x . .

"~-0. 6 B(m)

Load-deflection curve of apex of arch

-0,

Fig.6

5

,..

10

,

I 5 ~

. 25

8 B(m)

Load-deflection curve of apex of arch

Fig. 7 shows the deformation comparison between with cables and without cables in the case of the half-span uniformly distributed load. It is found that all the deformation shapes are almost same for the arch with and without cables, but the limit loads are much more different one another, as mentioned in the table above. The deflection amplitudes of the arch with cables are greatly reduced, although the arches behave as the same deformation shapes. The main reason is that the deformation of the arch is constrained strongly by cables at the point of 1/4 span length from the left end of the arch just subjected to the distributed load in the right half span of arch. In the case of full-span uniformly distributed load, it is found from the table that the arch for the cable disposition form-1 will carry a slight lower load than that without cables only by 1.7%, and for the cable disposition form-2 it will can3' a slight higher load than without

875 cables only by 7.5%. Obviously the main reason of increase of the load-carrying capacity is that the additional horizontal cable forms a small arch on the top part of the original arch, and further prohibits the connecting points of the horizontal cable to move free out-forward. This has been verified by the limit load of the cable disposition form-1 in the table that has not the additional horizontal cable and therefore there is a slight lower limit load than the cable disposition form-2.

3~

\ ~ ~ ~ ~ ' ~ 2

l

Fig.7

Deformation Process of the arch with and without cables with loading

5. CONCLUSIONS The following conclusions are drawn from the numerical results obtained: (1) The cables do play a great and effective role in controlling arch deformation development, improving arch failure mode and increasing arch stability. They could affect not only limit load-carrying capacity but also post-buckling equilibrium path significantly. (2) For the cable disposition form-l, the cables will markedly enhance the half-span limit load, greatly reduce deformation amplitudes and strengthen structural global stiffness, while they hardly make great effect for the full span distributed load and the concentrated load cases. For the cable disposition form-2, its application is recommend in the case of predominant full span distributed load. REFERENCES

1. Guo Yanlin, Zheng Haoran, Failure Mechanism of Arch Roof Structure with Color Corrugated Steel Sheet Section From Full Size Test, Engineering Mechanics, (additional), 1997. 2. Guo Yanlin, Zheng Haoran, Crinkle Collapse Mode of Arch Roof Structure with Color

876 Corrugated Steel Sheet Section, Industrial Construction, Vol.27, No.11, 1997. 3. Victor Gioneu, Buckling of Reticulated Shells: State of the Art. International Journal of Space Structures, Vol. 10 No. 1 1995. 4. J.G Teng, Y.F.Lou, Analysis of Bifurcation Buckling in Shell of Revolution After Axisymmetric Snap-through, Proceeding of Asia-pacific Conference on Shell and Spatial Structures,Beij ing,China, 1996. 5. Karim Abedi, Gerard A. R. Parke, Dynamic Propagation of Local Instability in Singlelayer Braced Dome, Proceeding of Asia-pacific Conference on Shell and Spatial Structures, Beijing, China, 1996. 6. Du Shougun, Sun Jianheng and Xia Hengxi, Stability Behavior Investigation of Braced Barrel Vaults, Space Structure 4, Thomas Telford, London, 1993.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

877

S h a k e d o w n of three layered p a v e m e n t s S.H. Shiau and H.S. Yu Ph.D. Student and Associate Professor Department of Civil, Surveying and Environmental Engineering The University of Newcastle, NSW 2308, Australia A lower bound shakedown formulation is presented by combining displacement and stress finite elements for layered pavements. Both elastic and residual stress fields required in the shakedown analysis are assumed to be linearly distributed. The proposed formulation is first verified with a homogeneous isotropic half space. The variation of shakedown limits with different material properties for three layered pavements are investigated in detail. Some design charts for layered pavements are also presented. 1. INTRODUCTION The purpose of shakedown analysis is to derive the shakedown limit load for any given layered pavements subjected to repeated loading. Pavements operating at loads above this limit will fail due to the accumulation of plastic strains in each load cycle. However, those operating at loads below this shakedown limit would behave elastically after some initial permanent displacements. As direct calculation of the exact shakedown load is difficult, it is necessary to estimate the best lower or upper bounds to the shakedown limits. The lower bound shakedown limit can be obtained by using the Melan's static shakedown theorem together with an optimisation procedure. As the lower bound shakedown loads are conservative estimates of the true shakedown load, they are therefore more useful in the design of pavements. The objectives of the present paper are: (1) to implement the lower bound shakedown theorem using finite elements and linear programming; (2) to perform a parametric study for shakedown of three layered pavements; and (3) to develop charts that can be used for the design of three layered pavements. 2. FINITE ELEMENT FORMULATION 2.1 Plane strain pavement model

Following Sharp and Booker (1984), a trapezoidal traffic load distribution is assumed in a vertical plane along the travel direction. It is further assumed that the resulting deformation is plane strain by replacing the wheel load as an infinite wide roller (Figure 1). The normal stresses (Pv) and longitudinal shear stresses (PH) are related by a surface friction parameter ~=PrdP~ For simplicity, this coefficient of surface friction ~ is assumed to be constant so that the longitudinal shear stresses are also trapezoidal (Figure 2).

Pv

B/a-2 Travel direction

_

y

Figure 1. Idealized pavement model

x

Figure 2. Trapezoidal load distribution for plane strain model

878 2.2 Melan's static shakedown theorem Melan's static (i.e. lower bound) shakedown theorem states that "If the combination of a time independent, self-equilibrated residual stress field oij and the elastic stresses ;to 0. can be found which does not violate the yield condition anywhere in the region, then the material will shakedown". ~, Shakedown load factor

Element Equilibrium

-x

r'

Stress boundary

I

+

Residual stressfields

\ Displacement boundary

Discontinuity Equilibrium

Figure 3. Application of Melan's shakedown theory Supposing that the elastic stresses are proportional to a load factor ;t as shown in Figure 3, the total stresses are therefore ~j = 200. + o~/j, where 2 is the shakedown load factor, oij are the elastic stresses and ff//j are the residual stresses. By insisting that both the total stresses and the residual stresses do not violate a linearised Mohr-Coulomb yield condition in the mesh, Melan's static shakedown theorem can be implemented using finite elements and linear programming techniques. 2.3 Finite elements The displacement finite element method is used to determine the elastic stress field. In this paper, the 6-noded triangular displacement elements are utilised so that the elastic stress field is linearly distributed across each element as shown in Figure 4(a). Number of . . . . . . . . Number ot

]Number

Figure 4(a) Displacement finite element mesh

of elements --- 576

--r

-= lrr - " 0

Figure 4(b) Stress finite element mesh

In the stress-based finite element mesh, the 3-noded triangular elements are used so that the residual stress field is also linear with coordinates and varies through an element according to 3

3

a r = E N , a r ~ ; ~ = EN,o~yi; ~ i=1

i=1

3

= ENK~i

(I)

i=1

where t ~ , ~yi and r~i are the nodal residual stresses and N i is the linear shape function. Statically admissible stress discontinuities are permitted at shared edges between adjacent stress

879 triangles. Unlike the usual form of the displacement finite element, each node is unique to a particular element and more than one node shares the same coordinates as shown in Figure 4(b). If E denotes the number of triangles in the mesh, then there are 3E nodes and 9E unknown residual stresses. 2.4 Mohr-Coulomb yield criterion The Mohr-Coulomb yield criterion for plane strain condition can be approximated as a linear function of the unknown stresses and details of this linearization can be found in Sloan (1988) and Yu and Hossain (1998). The linearized yield surface must be internal to the original MohrCoulomb yield circle to ensure that the solutions obtained are rigorous lower bounds. In each element, both the elastic and residual stress variations are linearly distributed across the mesh. The yield conditions in terms of both residual and total stresses will be satisfied everywhere within an element as long as the yield criterion is enforced at corner nodes. As a result, the condition of not violating the yield criterion in the mesh can be replaced by enforcing the following inequality constraints: (a) yield criterion at comer nodes by the residual stresses (b) yield criterion at corner nodes by the total stresses. This differs from the formulation by Yu and Hossain (1998) where the total stresses need to be enforced at several sampling points within each element. 2.5 Linear programming The best lower bound shakedown limit load is obtained as a solution to a large linear programming problem: the maximisation of the shakedown load factor ;t subject to the constraints on the stresses due to: (1) element equilibrium; (2) discontinuity equilibrium; (3) stress boundary condition; and (4) yield criterion. The problem can then be stated as Minimise - k ; Subject to A 1 x = B 1 and A 2 x _< B 2. where A1 is the matrix of equality constraints, A2 is the matrix of yield constraints, B1 and B2 are the respective vectors containing strength properties. The modified active set strategy developed by Sloan (1998) is employed to solve the above linear programming problem. A detailed discussion of the theory and implementation of the active set algorithm can be found in Best and Ritter (1985). 3. APPLICATION 3.1 Verification of the numerical formulation For the purpose of verification, only the vertical load is considered in this section. The results of dimensionless shakedown limits with the variation of soil internal friction angle from 00 to 30* are presented in Figure 5. It is shownthat the values of SPv/c derived in this paper are very close to those obtained by Sharp and Booker (1984) and this is particularly true for the case of a purely cohesive clay which gives a value of 3.696. 0 h

I

- - - - _w- -t m- m- m ~ ~.~' First yield load

15 \

____L___L

Collapse load (Strip !ooting,Pmadtl, 192 I) /

-~ I0..

1(

~'

,

~ ~

.~

~,~

0 10 20 30 ~ Figure 5. Effect of internal friction angle upon dimensionless shakedown limits

O. 0.0 0.2 0.4 0.6 0.8 1.0 Figure 6. Effect of coefficient of surface friction upon dimensionless shakedown limits

880

As shown in Figure 6, the dimensionless shakedown limit decreases dramatically with the coefficient of surface friction/~. This is likely due to the existence of high elastic shear stresses in the top layer which will possibly result in a type of shear failure of that layer when the value of /~ is high. In this Figure, it is also found that the results from this formulation yield smaller values than that in Sharp and Booker (1984) at higher frictional angle of soil. The present formulation uses many elements to model both the elastic and residual stress fields and therefore should give more accurate shakedown loads. Figure 7 shows the distribution of horizontal residual stresses with depth for the case of -- 0,/a -- 0. The value of ~/c reaches a maximum at D/B=0.21 with ~ - 0.9c. This value converged to zero at a depth of D/B=0.8. This result is in good agreement with the experimental data presented in Radovsky and Murashina (1996) on the residual stress field. Figure 8 presents the principal residual stress vectors beneath the loaded area and shows that only ~ exists in the medium under the traffic load. For the plane strain condition assumed in this paper, residual stresses vary only with the depth and are uniform over any horizontal plane (i.e. travel direction). .m

B

IL~

0.00.~~~ -0.25-

ZoneI exsiting residuIstres~s

-0.50t = o- I -1.00:

~

t

.

0.00 0.25 0.50 0.75 1.q Figure 7. Distribution of horizontal residual stresses with D/B

Figure 8. Principal residual stress vectors beneath the loaded area

3.2 Parametric study on shakedown of three layered pavements Figure 9 shows the problem notation of a three layered pavement. A purely cohesive subgrade (r - 0) is considered in this section and the internal friction angles for the first and second layers are assumed to be 50 and 20 degrees respectively. The influence of relative stiffness ratio (El/E2) on the dimensionless shakedown limit for different values of relative strength ratio (C1/C2) for both soft and stiff subgrades is presented in Figure 10. It is clear from the figure that, at a given value of relative strength ratio, there exists an optimum relative stiffness ratio at which the resistante to incremental collapse is maximised. Further increase in relative stiffness ratio does not contribute to an increase in the shakedown limit. Results for the case of a stiff subgrade are higher than those from a soft subgrade.

Figure 9. Problem notation of three layer pavements

881 30 25

30

] StiffSubgrade

7

5

.

hi/B=0.25, h2/B=0.75, Ph/Pv=0

"

20 LP..._~ cz 15

"

~c, ~~ ./c2=so " ~

zP,

cz

lO

.

.

.

Soft Subgrade . . . .

20

hi/B=0.25, h2/B=0.75, Ph/Pv=O---

15 .... c~/c2=5o

lO

:11C2=10

.

v o

5

5

,

a * quid~ ~ , m D ~ , , El

Ez

Q

..

Cl/C2 z l 0

Ct~C2=~)Q ') ~) ~) ,, . . . . . .

e, m

El

Ez 1.00 10.00 100.00 1000.00 1.00 10.00 100.00 10t ~O.OO (a) Stiff clay for Subgrade E2/E3=0.O01, C2/C3=1.0 Co)Soft clay for Subgrade E2/E3=1, C2/C3=2.5 Figure 10. Influence of El/E2 and C1/C2 on dimensionless shakedown limits 0

.'

....

9

As shown in Figure 11, it is important to note that for a lower value of El/E2 the dimensionless shakedown limit ceases to increase at a particular value of C1/C2 for both soft and stiff subgrades. This may indicate a transfer of failure mode from the top layer to the bottom layer when the value of El/E2 is low. Thus, further increase in basecourse strength will not improve the shakedown capacity. In Figure 12, the dimensionless shakedown limit increases with the increase in surface layer thickness (hi/B) for both soft and stiff subgrades. 15

~

[

[

I

20

[---7

S b r.o

//

Stiff clay for Subgrade E2/E3=0.001, C2]C3=1.0 . . ,, ,D Soft clay for Subgrade E2/E3=l, C2/C3=2.5

15

xp.

Lo,/..//

r"/(~)"l/'ta)

I

r

I r

" oO Ii

I_..

~

c2

II

(a) "hl/B=O.5, h2/B--0.75 (b) 9hi/B-0.25, h2/B-0.75

10

0

2

4

6

8

C1

10 N

0

1.00

Figurel 1. Influence of C1/C2 and internal friction angle on dimensionless shakedown limits (El/E2= 10)

, 10.00

100.00

E2

1000.00

Figure 12. Influence of El/E2 and hl/B

on dimensionless shakedown limits (C1/C2=2)

3.3 Design chartsfor three layeredpavements Current pavement design methods can be checked using the shakedown-based design method through charts like Figure 13. For a given three layered pavement with known material parameters, the shakedown limit can be determined from this chart and compared against the design load. The design load has to be less than the shakedown load to ensure that accumulation of permanent strains will not occur. In the case that the design load is much smaller than the shakedown load, the design of pavement is conservative and the adjustment of basecourse thickness is possible through this chart.

882

I .oo

hl/B O.75

0.5O

0.25 0.25

0.$0

0.75

1.00

1.25

1 .50

,,

IE1/E2=IO'E2/E3=5'C1/C2=5'C2/C3=5'Ph/Pv=O I !12]][I Figure 13. Contours of dimensionless shakedown limits as an example chart for the thickness design of three layer pavements using shakedown analysis (Soft subgrade) For a given design traffic load and various material properties for the surface course, base course and subgrade, charts such as Figure 13 can also be used to determine the minimum thickness for basecourse so that the pavement designed would 'shake down' under the given design traffic load. 4. CONCLUSIONS A formulation using finite elements for both elastic and residual stress fields has been developed in this paper to perform lower bound shakedown analysis of layered pavements. Unlike most existing shakedown formulations, in this paper the elastic and residual stress fields are modelled respectively by displacement and stress-based finite elements. Selected results presented in the paper suggest that the new formulation gives accurate shakedown solutions and can be used easily to develop design charts for multilayered pavements. REFERENCES

Best, M.J. & Ritter. K. (1985)Linearprogramming: active set analysis and computer programs, Prentice-Hall, New Jersey. Prandtl, L. (1920) "Uber die harte plastischer korper", Gottinger Nachricten, Math. Phys. K1, pp74-85. Radovsky, B.S. and Murashina, N.A. (1996) Shakedown of Subgrade Soil Under Repeated loading, Transportation Research Record, n 1547, pp 82-88. Sharp, R.W. & Booker, J.R. (1984) Shakedown of Pavements Under Moving Surface Loads, Journal of Transportation Engineering, Vol 110, pp 1-14. Sloan, S.W. (1988) Lower Bound Limit Analysis Using Finite Elements and Linear Programming, Int. J. of Numerical and Analytical Methods in Geomechanics, Vol 12, pp 61-77. Yu, H.S. and Hossain, M.Z. (1998) Lower Bound Shakedown Analysis of Layered Pavements Using Discontinuous Stress Fields, Computer Methods in Applied Mechanics and Engineering, Vol 167, pp 209-222.

Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.

883

Laser application to surface deformation and material failure Sandra H. Slivinskya , Peter Kuglerb , Harry Drude b and Rainer Schwarzeb a Air

Force Research Laboratory, Albuquerque NM, USA

b Fiedler Optoelektronik GmbH, Lutzen, Germany

A two-dimensional laser scanning device has been applied to rubbery materials to measure local surface deformations during tensile testing. The device scans the sample with two laser elements while the tensile tester gathers force/displacement data. Post-test data analysis tracks local vertical and horizontal surface displacements at small incremental steps along the bulk stress-strain curve. This technique has been applied successfully to elastic and highly filled, inert solid propellant-like materials. The eventual objective is to predict tensile failure in large-scale structures based on local deformations.

1. INTRODUCTION Previous publications (1,2) have discussed results of a one-dimensional scanning system to measure local longitudinal elongation. Current work has developed a two dimensional system. Both of these devices will be discussed in this paper and examples of the type of data available with these devices will be given. The system utilizes optical and electronic equipment, coupled with software to acquire and present the tensile test data for analysis. In application, this system will provide important data for failure analysis of any material bonds. Current techniques result in an average test result, whereas the laser technique results in a detailed knowledge of local material behavior over the entire sample, as it is placed in tension.

2. ONE-DIMENSIONAL SCANNING SYSTEM The one-dimensional laser scanning system formed the basis for the development and use of the two-dimensional system. A schematic is shown in Figure 1. Referring to Figure 1, a collimated beam of a semiconductor laser in continuous wave (cw) - mode is directed centrally onto a rotating mirror in the scanner. The deflected beam is passed through a cylindrical lens

884

and scans the surface of the test specimen along a fixed scanning length or angle. The power of the laser is in the range of several milliwatts and the wavelength is in the visible (630690nm) range. This makes it possible for the scanning range to be adjusted visually by an operator. At pre-selected areas of interest on a sample, highly reflective contrasting stripes are applied. The technique for applying these stripes consists of using a faint black primer, followed by a white or silver metallic coating through an adhesive masking foil. The use of the mask results in well-defined sharp stripes. Diffuse scattering of the scanning laser light occurs at these metallic coating marks or stripes. The scattered light is collected by a lens onto a photodiode and converted into an electrical signal. Cylindrical

I TestSample

L Lens

Contrasting ~ Stripes [--~

~ting

-~

Lens [ x ~

I

Scanner I 1 Laser I

,

i--~_ Photodiode with Amplifier

Figure 1. One Dimensional Scanning System Schematic The signal is subsequently amplified and formed into a train of digital pulses. The angular velocity of the scanning system is recorded as well as the relative locations of all the reflective points on the sample by a multi-stop counter device. The reference length between the single stripes for the unstressed sample is recorded at the beginning of the test. As the sample is placed in tension, the distance between the reflective points or stripes increases as the test proceeds. These changes are analyzed on-line and subsequently transformed into strain values between the stripes on the sample. The precision of the measurement is a function of the scanning length, degree of contrast, scattering of the constant fraction trigger, scanning speed, short term scanning stability, noise in the laser output, and resolution of the multi-stop counter. The optimization of the counter occurs with an internal clock of up to 74MHz. For this system and for the particular samples used, the scanning length is 160mm with a measurement accuracy or resolution of 2 microns, an active scanning time of about 1.5 msec, and a minimum distance from stripe to stripe of

885 lmm. The average deviation for unstressed samples is 0.005% for a reflective length of 80mm, for a stripe thickness of lmm. For a sample placed under tension, detailed information is obtained along the length of sample, between pairs of stripes. So, instead of obtaining a single average value for a test, a pair-wise mapping is obtained of the entire material surface. The importance of this is shown below. Additional details of the scanning system, detection of the optical signal, and signal processing have been previously discussed (3).

3. TWO-DIMENSIONAL LASER SCANNER With the one-dimensional scanner as the basis, the hardware and on-line measurement software for a two-dimensional multi-line scanner was developed. The basic principles of laser-scanning along a marked sample and the use of the multi-stop-counter were maintained. The scanning unit was expanded from one beam to six beams, each scanning the sample on an individual trajectory. The basic principle is shown in Figure 2.

• ~~"

~J Micrometerscrews

~

Ru~R-~.samplewith

~ l' Tr-an.mi~r ~

2-dimensional scanner l [ ~,.::~

[

~

/

Figure 2. Perspective Rear View of the Test Set up The six individually adjustable laser diodes are arranged on a half-circle and directed to the center axis of the rotating mirror within the scanner. By proper angular adjustment, one beam is just leaving the area of interest on the bottom as the next beam is beginning its scan on the

886

top of the next area. micrometer screw.

The position of individual lasers can be adjusted by means of a

The resolution of the multi-stop counter can be selected up to 40MHz. To ~ m p l i s h the desirable two-dimensionalresult, two scanners are necessary. The software, which accepts the measurement, is hardwareinterrupt controlled and runs :as a real-time job under DOS, on a 1O0 MHz Pentium processor. As in the one-dimensional scanner, the most relevant parameters of the scanning during the test are presented on-line on the screen. The output from the two independent scanners is combined, resulting in a single solution.

4.

EXAMPLES OF DATA WITH THIS DEVICE

In Figure 3 we show stress-strain data which is nommlly available from a mechanical properties test of a sample in tension. A single result for this physical property across the 0.3-

~

4!

0.2 i- ~X,

2.5

.......:...................- ......................................................... ~ |.

0

I

.~

_'

-

"

9

,

~

0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 t.e.sited~ Sum (%), Figure3. Strmsvs. S u a i n F o r ~ v e Sere

'

2

1.5 I

0..

0

-

3

.

6

.

.

. -

9

.

.

.

12

.

.

15

"

18

21

24

Tune (m:) F~ure4. Sumsv~ SaainForEedl SlipPair

sample is the result of the test as shown. The impor+amce of the devices described in this paper is that detailed local deformation information becomes available. This is shown in Figure 4, the s t r e s s - ~ curve. The multitude of curves results from information ga~ered from stripe to stripe, so that a kind of local mapping of the stress-strain curve becomes available for any

887

Figure 5. Schematic of Typical Sample

24

"l ............

22

1

20

"I ..........................................

"I

~ ..................

: ..... i ..... " ..... " ...... : ..... ! .................

i ................................

. . . . . . . . . .

1877

f596 16:.09""

: ...............

.................

: ..... : .............

~..................

........................

16 "1 o~

~............................

......................

: ....

: ..... i ..... : ..... " ........................

......................

" ...... ........... 9

..... :,,-1-9.84

.......................

- ...... .........................................

"i'"~

...........

""

: .............

'

i ............

m~,2~

;i5:'$3"'1"6

".'36 ...... i ..........

'

'16""

'

: ..........................

" .................

: i : ~ i I : : : :

'~,0" .,n ....~2,~? ............~................~.,2,5.3 .....~.,..~..9..........:...........;...... ..~i~:6" 14

8

I

"1

4 ,_~

:"

':

~i .................................... .46

:

.....:..............

....

1.6.....'

.~....i.il. ...i;. i iiiii.ii2111111.il.21; .i. .~..~:,, ................ ::..... .......... . .... ::_i_.:....: .... .... ................ : ..... :...?i2121.1'..ii?2)2112211112;:=.iiii:.221111.i:/il.i..i.~iillli:

0

4.

i

"

3

.

5

.

7

.

.

.

.

9

.

i

!!

13

,

'

15

:

17

"

:

19

,,

'

21

23

:

25

'

0,31'

27

.II

29

Zone Figure 6. Logitudinal Strain Mean Value 2x3 Beam Trajectories to the Vertical Center Axis given test. This type of detail, in its application, makes it possible to locate specific areas of weakness or poorly bonded areas. Since these samples are normally constructed with the same materials and at the same time as full size articles, they may serve as a quality assurance test. To further illustrate the types of results obtainable with this device, a typical sample is shown in Figure 5 and typical strain results are shown in Figure 6. The sample consists of rubbery material, of dimensions shown, with bonds (glued areas) indicated, and aluminum end tabs. The thinner rubbery areas are stiffer than the thicker areas. The aluminum end tabs are glued to the rubbery material with the same glue as used to make the bonds. These samples are

888 typical of those used in tension testing in the propellant industry, but the technique is applicable to any bondline. This application uses a tension test. For the typical strain curve, the results are qualitatively interpreted as follows.

5. RESULTS Focusing on the center region of the strain curve in Figure 6, in the 15, 16 zone region, the maximum at the center (A in Figure 5) is the effect of the center bondline, the two adjacent maxima are the effects of the bonds between the thick material and the stiffer thin material (B,C). The last two slight increases are in the transition between the rubbery material and the aluminum end tabs, (D,E). This curve is produced on-line by the software and is the result of taking the mean value of the six measured beams across the sample. As described above, this detailed information of the strain, replaces a single curve in more typical tests of this type. From this information, it becomes possible to study detailed effects at bondline interfaces and determine the role of local deformations in the bond failure process.

6. CONCLUSION The two-dimensional laser scanner has been used on rubbery samples with well defined bondlines. The choice of sample was made for a particular bond problem in the solid propellant industry, but the device is applicable to any bonded area. The result of performing a tension test is used as an illustration of the local surface deformation observed with the device, but the device can be applied to other types of mechanical property tests, such as shear, torsion, compression, etc. Other types of tests have been performed and work in hysteresis effects is planned. This local test can be applied to areas which are suspect for possible failure. The process could be automated for large articles. The ultimate objective is to correlate the local surface deformation to bulk stress-strain properties in large structures and to predict bond failures.

REFERENCES

1 2

3

Sandra H. Slivinsky, H. Peter Kugler, and Harry Drude, (May 1997), "Service Life of Solid Propellant Systems," AGARD-CP-58. Sandra H. Slivinsky, H. Peter Kugler, and Harry Drude, "Laser Application to Surface Deformation and Analysis of Rubber Materials," Werkstoffprufung 1997, Deutscher Verband Fur Materialforschung Und-Prufung E.V., p. 139. N. Eisenreich, C. Faby, R. Fisher, A. Geissler, H.P. Kugler and F. Sims (1987) Propellent Explosive Pyrotech, 12, 101.

889

AUTHOR INDEX

Aalberg, A. 487 Abu-Mansour, T.M.-N. 545 Adley, M.D. 229 /~gfirdh, L. 115, 133 Akers, S.A. 229 Alghamdi, A.A.A. 545 Alisjahbana, S.W. 865 Aljawi, A.A.N. 545 A1-Jumaily, A.M. 539 Ando, T. 87 Andrade, S.A.L. de 513 Arizumi, Y. 527 Azuma, K. 495,711 Bailey, C.G. 635 Bakoss, S. 165 Baldassino, N. 859 Barros, F.B. 807 B6da, Gy. 585 B6da, P.B. 585 Beilin, D. 591 Bemuzzi, C. 859 Beutel, J. 479 Bignell, P. 367 BiUon, H.H. 223 Bola, M.S. 179,247, 267, 603, 747 Brauns, J. 551 Bridge, R.Q. 419, 521 Bruhns, O. 609 Bryant, R.H. 407 Bullen, F. 367 Buri, P. 255 Cao, J.J. 473 Cardew-Hall, M. 801 Cardoso, J.B. 93 Carrasco, C. 109 Castro, J.A. 93 Chart, S.L. 787 Chellapandi, P. 275 Chen, B.K. 851 Chen, F.L. 557 Chen, X.W. 61 Chen, Y. 501 Chen, Y.Z. 61 Chirwa, E.C. 311

Chowdhury, M. 217 Chudnovsky, A. 729 Cimpoeru, S.J. 325 Clark, G. 765 Cottam, R. 571 Crespo, G. 99 da Silva, J.G.S. 103 Danielson, K.T. 229 Deam, R. 571 Deng, C.-G. 781 Deshpande, P.U. 73 Dhamari, R. 765 Ding, Z. 801 Drude, H. 883 Du, S. 473 Dyskin, A.V. 235 Et~is, J. 109 Elchalakani, M.

463

Fedczuk, P. 623 Figovsky, O. 591 Flockhart, C.J. 201 Franco, J.R.Q. 807 Friis, J. 209 Fujiwara, K. 333 Fukuehi, O. 67 Galybin, A.N. 235 Ge, H.B. 43,383 Gentile, C. 527 Gel31er, U.J. 319 Ghojel, J. 629 Ghojel, J.I. 641 Ghojel, J.I. 647 Ginda, G. 653 Goel, R.A. 127 Grobbelaar, W.P. 185 Grosz, L. 801 Grundy, P. 735 Grzebieta, R. 443 Grzebieta, R.G. 325 Grzebieta, R.H. 463,685 Gu, J.X. 787 Gu, X.L. 839

890 Guo, Y. 871 Gupta, N.K. 3, 73,413

Hammond, L.C. 201 Hart, B.K. 375 Han, L.H. 659 Hancock, G.J. 699 Hanssen, A.G. 401 Hansson, H. 115, 133 Hetherington, J.G. 261 Hinton, B.R.W. 765 Hirai, I. 845 Hoe, A. 311 Hopperstad, O.S. 401 Hu, W. 723 Huang, X. 457 Hui, S.K. 339 Ikeda, K. 121 Ishihara, H. 845 Ishikawa, N. 67 Islam, Md.S. 729 Issa, Mahmoud A. 407, 729 Issa, Mohsen A. 407, 729 Itoh, Y. 79 Iwashita, T. 711 Iwatsubo, K. 667 Jaggi, I.J.L. 241 Jeong, C.H. 375 Jiang, Z.D. 501 Johnson, A. 345 Jones, C. 55 Jones, N. 55 Ju, J.S. 871 Kajita, Y. 67 Kalyanasundaram, S. 801 Kammel, C. 319 Kaushik, D.R. 281 Kelly, D. 217 Khan, M.M. 353 Khosla, P.K. 145 Kim, C.W. 375 Kindervater, C.M. 345 Kishi, N. 87, 121 Kohlgrtiber, D. 345 Konishi, T. 579 Konno, H. 121

Koudela, K.L. 195 Krauthammer, T. 195 Krige, G.J. 353 Krishnamurthy, K.S. 139 Kr6ger, M. 353 Kubo, M. 563 Kugler, P. 883 Kumar, P. 127 Kumari, R. 179,241 Kurobane, Y. 495, 711 Kwan, Y.F. 813 Lal, H. 179,241,247, 255, 267 Langseth, M. 401 Lapovok, R. 571 Larsen, P.K. 487 Lee, C. 685 Li, D.S. 437 Li, G.Q. 795 Li, J. 165 Li, J.J. 795 Li, Z.-X. 781 Lie, S.T. 759 Likverman, A. 691 Lima, L.R.O. de 513 Liu, C. 79 Louca, L.A. 209 Lu, G. 395,451,457 Lu, L.-W. 527 Ltitzenburger, M. 345 Magnusson, J. 133 Mahajan, P. 139 Mahmoud, E.H. 261 Makino, Y. 495, 711 Malik, K.K. 145 Marco, J. 173,295 Marshall, P.W. 425 Mashiri, F.R. 735 Matsuoka, K.G. 87, 121 Mayrhofer, C. 301 Mazi, R.A.A. 545 Meyers, A. 609 Mikami, H. 87 Mills, T.B. 765 Mittal, R.K. 139 Moita, P.P. 93 Moil, T. 705 Mtihlhaus, H.-B. 741 Murakami, S. 579 Muramta, K. 679

891 Murray, N.W.

325

Nara, S. 579 Neuenhaus, D. 319 Nguyen, N.T. 819 Nuriek, G.N. 185 O'Daniel, J.L. 195 Ogushi, M. 667 Okazawa, S. 775 Oseguedo, R. 109 O'Shea, M.D. 521 Packer, J.A. 473 Palaniehamy, M.S. 413 Pande, P.H. 145 Papados, P.P. 229 Pasternak, E. 741 Patterson, N.L. 629 Pazhanivel, S. 151 Perera, N. 479 Pireher, M. 419, 521 Potapov, J. 591 Prakash, S. 747 Raymond, I. 217 Resnyansky, A.D. 597 Rhodes, J. 21 Roberts, S. 801 Rogers, C.A. 673,699 Rose, L.R.F. 717 Routman, Y. 827 Russo, S. 765 Saehdeva, S.S. 157,267 Saitoh, T. 679 Samali, B. 165 Saroha, D.R. 603 Schaumann, P. 507 Sehwarze, R. 883 Searancke, E.J. 311 Sedlaeek, G. 319 Seidel, M. 507 Seliverstov, V. 691 Sethi, V.S. 127, 157, 179, 241,247, 267, 275 Shankar, K. 765 Sharma, A.K. 275 Sharma, V.K. 151,281 Sharp, P.K. 765 Shen, C.W. 833 Shen, Z. 839

Shen, Z.-Y. 781 Shestoperov, G. 691 Shiau, S.H. 877 Shimazu, Y. 579 Shrivastava, S.C. 533 Singh, G. 603 Singh, M. 747 Skowrofiski, W. 623,653 Slivinslej, S.H. 883 Sdrengan, K. 425 Srivastava, V. 281 Strait, L.H. 195 Sugiyama, N. 563 Sun, F.F. 839 Sun, H.H. 833, 851 Suneja, H.R. 747 Susantha, K.A.S. 383 Suzuki, S. 79 Szuladzinski, G. 287 Takemoto, K. 67 Takla, M. 813 Tang, X.B. 833 Tao, X.M. 615 Tarasov, B.G. 753 Tateishi, K. 679 Teng, J.G. 389 Thambiratnam, D. 367,479 Tran, H.H. 295 Tremblay, R. 673 Usami, T.

43, 383, 775

van Schalkwyk, W. 353 Vellasco, P.C.G. da S. 513 Velmurugan, R. 413 Verma, M.M. 255 Verma, R.K. 145,247 Wang, B. 395,759 Wang, C.H. 717, 723 Wang, Z. 431 Wong, M.B. 629 Wong, S.M.P. 311 Wu, Y. 165 Xiang, Z.H. 759 Xiao, H. 609 Xue, P. 615 Yabuki, T. 527 Yamamuro, T. 667

892 Yamao, T. 667, 845 Yu, H.S. 877 Yu, T.X. 61,339, 457, 557, 615 Zandonini, R. 859 Zhang, J. 451 Zhang, J.J. 437 Zhang, S.X. 851 Zhang, W.Y. 437 Zhang, Y.C. 437 Zhang, Y.J. 501 Zhao, X.L. 443,463, 659, 685, 735 Zhao, Y. 389 Zhen, Y.H. 431