Structural Failure and Plasticity
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Structural Failure and Plasticity Proceedings of The Seventh Intemational Symposium on Structural Failure and Plasticity (IMPLAST 2000) 46 October 2000, Melboume, Australia
Edited by
X.L. Zhao and R.H. Grzebieta Department of Civil Engineering, Monash University, Claywn, VIC 3168, Australia
2000
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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 thinwalled 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 thinwalled 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 thinwalled 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 XiaoLing 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 CrashSafety 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 CoChairman Symposium Manager Founding Chairman
Raphael H. Grzebieta Department of Civil Engineering, Monash University XiaoLing 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. AIMahaidi 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. AfaghiKhatibi R. AIMahaidi 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 Hongik 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. TinLoi 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
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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 ThinPlate Members Some Points of Interest J. Rhodes
21
Failure Predictions of ThinWalled 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 ElasticPlastic BeamonFoundation 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
ElasticViscoplasticMicrodamage Modeling to Simulate Hypervelocity ProjectileTarget 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 RockShed
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 Ultrahigh 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 MultiMaterial 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 Airbacked 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 Presplit 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 LargeDeflection Design Technique for the Collapse and Rollover Analysis of ThinWalled 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
StressStrain Relationship for Confined Concrete in Various Shapes of ConcreteFilled Steel Columns K.A.S. Susantha, H.B. Ge and T. Usami
383
Experimental Behaviour of InternallyPressurized ConeCylinder Intersections Y. Zhao and J. G. Teng
389
FEM Analysis of Buckling of ThinWalled 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 ThinWalled 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 WeldInduced 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 StubColumn 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 QuasiStatic 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 ConcreteFilled 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 BeamtoColumn Connections with Weld Defects based on CTOD Design Curve Approach
495
K. Azuma, Y. Kurobane and Y. Makino
Simulation of Fracture Failure of Steel BeamtoColumn 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 SemiRigid 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 ThinWalled 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 CurvedStraight Pipe Configuration Conveying Fluid
539
A.M. Aldumaily
Axial Crushing of Frusta between Two Parallel Plates
545
A.A.A. Alghamdi, A.A.N. AljawL T.M.N. AbuMansour and R.A.A. Mazi
Strength Analysis of Buckled ThinWalled Composite Cylindrical Shell with Hydrostatic Loading
551
J. Brauns
Imperfection Sensitivity Function in Dynamic Response and Failure of 1D Plastic Structures
557
F.L. Chen and T.X. Yu
Straightening Effects of Steel Ibeams Failed by LateralTorsional 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 WeftKnitted Thermoplastic Textile Composites P. Xue, T.X. Yu and X.M. Tao
615
Fire Loading Nonlinear Analysis of ThreeDimensional 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 ConcreteFilled 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
Nonelastic Load Capacity of Compressed Steel Truss Member during Fire G. Ginda and W. Skowrohski Fire Resistance of Concrete Filled Steel Tubular BeamColumns 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 SideLap 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 ConcreteFilled 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 TJoint 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 ThinWalled TubetoPlate Joints using the Classification Method F.R. Mashiri, X.L. Zhao and P. Grundy
735
Cosserat and Nonlocal 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 TJoints 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
SnapThrough Analysis of Toggle Frame using the Software Package, NIDA, by 1 Element per Member S.L. Chan and J.X. Gu SecondOrder Inelastic Analysis of Steel Gable Frames Comprising Tapered Members G.Q. Li and J.J. Li
787
795
A Parallel ThreeDimensional ElastoPlastic Finite Element Analysis in a Workstation Cluster Environment Z. Ding, S. Kalyanasundaram, L. Grosz, S. Roberts and M. CardewHall
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 SemiInfinite 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 RigidPlastic 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 ISection 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 IMPLAST2000, 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 likeminded 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 cooperation 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 plastomechanics 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 cooperation 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 tenyear long memorable and fruitful association. On 1719 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 thermoplasticity"[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. 713, 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. 1114, 1993 with the title Plasticity and Impact Mechanics. This coincided with the 10th anniversary of the startup 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 1114 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
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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
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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 nonuniform 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 crosssection 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
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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 plastomechanics of structural elements like tubes of circular and noncircular crosssections, 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 [2325]. 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 plastomechanics of their postcollapse 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 nonexistent. 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 stressstrain 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 quasistatic 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 asreceived 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 asreceived aluminium tube deforms in diamond mode and when annealed, it deforms in a ring mode. On the other hand, asreceived strainhardened 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 stressstrain 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 asreceived 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 loaddeformation 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.
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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 semiapical 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 semiapical angles. As the frusta of low semiapical angles may fail in concertina mode that is perhaps why many of the studies available in literature seem to be for frusta of low semiapical angles. The frusta of semiapical 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 semiapical 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 loaddeformation curves of frusta collapsing due to the movement of rolling plastic hinge are shown in Fig. 11, wherein it is seen that the
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15
Fig. 12 Deformed shapes of frusta of semiapical 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 [3031 ], 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 hemispherical 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 colatitude 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 hemispherical 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. fptate
][
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 loaddeformation 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 loaddeformation 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 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. 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, nonmetals 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 twolayered 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 IMPLAST2000 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 postbuckling 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 pieinthesky. 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. POSTBUCKLING 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 thinwalled 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 postbuckling 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 postbuckling 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 thinwalled 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 postbuckling behaviour of thin plates is governed by two simultaneous nonlinear 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 inplane and outofplane 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 postbuckling behaviour of thinplates 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, v0.3 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 coldformed 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 postbuckling 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 simplysupported 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 postbuckling 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*Pt,
'~ ,,, 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 postbuckling 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 postbuckling 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 postbuckling behavieur required also a knowledge of the inplane 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 inplane 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 inplane 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 thinwalled 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 postbuckling 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 inplane 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 inplane. 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 postbuckling 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 coldformed 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 inplane 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 postbuckling range, and these equations give results in close agreement with existing theory in comparable cases. Loadout 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
~.
eI 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. Loadcompression behaviour in the far postbuckling 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 coldformed 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 coldformed 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 thinwalled member is much more complicated, however. Probably the first elastoplastic plate postbuckling 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 elastoplastic plate behaviour was carried out in the 1960s at Cambridge University, e.g. [38], [39]. Perhaps the major work here was that of GravesSmith 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 elastoplastie 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 computerorientated, using finite difference and finite element approaches. There were a number of attempts made to obtain simplified analysis of plate elastoplastic 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 nonlinear 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 elastoplastic 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 postyield 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)  2074
06
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 elastoplastic analysis with Frieze et. al. !
.,m, .':"
.
.0.8
...(_b/t)~ay/E= 1.59 061, P/PY ~
/
/ 
0'2l
i/
0
/
/
I
l,~rX;:,
I
x,~...
""
"~'~"~~'_~
T (blt)~/OY/E= 2"12
simol.VSUOl:)oneclplates.....
Stres.slree on unloaoea eoges vo.;,.*oO
_
Present m e t h ~
0:s
..... ;
,:s
~Ey
~
2.~
Figure 10. Comparison of approximate elastoplastic analysis with experiments
34
157
06 b 219 t
OS J~
Pv Present method Experiment (Rogers & Dwight) 0.2
(~s
1
1.5
~Y
Figure 11 Comparison of approximate elastoplastic 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(lv2) ~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 elastoplastic 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 postelastic capacity, while these plates with widththickness ratios of around 15 show experimentally fully plastic capacity. It is worthy of note that these were coldformed steel channels, and design codes for coldformed 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 Elastoplastic 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 postfailure behaviour of thinwalled 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 inplane 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 911 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 thinwalled 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 sideimpact 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 inplane 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 nondimensional 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 ThinWalled 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.McGrawHill, 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 19931_3:1996. Eurocode3:Design of Steel Structures  Part 1.3:General Rulessupplementary rules for coldformed 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 thinwalled strut, Civil Engineering, 48, 1953. 16 J. Rhodes. Secondary local buckling in thinwalled 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 postbuckling behaviour of rectangular plates with smt initial curvature loaded in edge compression, J. of App. Mech. 26, 1959. 26. N. Yamaki, The postbuckling 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. R40, 1959. 28 A.C. Walker. The posrbuckling 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 postbuckling 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 thinwalled structural members. ThinWalled Structures 9, 1990.  The N. W. Murray Symposium. 33 S. Wang and D. J. Dawe. Spline FSM postbuckling 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. Thinwalled structures, 32, 1998 35 K.S. Sivakumaran and N Abdel Rahman. A finite element analysis model for the behaviour of cold formed steel members. Thinwalled structures, 31, 1998
41 36 J. Rhodes. The nonlinear behaviour of thinwalled 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. Thinwalled 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/CStruct~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 elastoplastic 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. ThinWalled Structures 9, 1990. The N. W. Murray Symposium. 53. R. H. Grzebieta Research into failure Mechanisms of some thinwalled round tubes. Plasticity and Impact Mechanics. Ed N. K. Gupta. New Age International (P) Ltd., 1998 54 N.K. Gupta and R. Velmumgan. Axisymmetric 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 thinwalled structures. Clarendon Press~ Oxford, 1984. 59. K.W. Sin The collapse behaviour of thinwalled 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 thinwalled beams. PhD Thesis, University
62
of Strathclyde Glasgow, 1995 63. X. L. Zhao and G. J. Hancock. Plastic mechanism analysis of Tjoints 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. ThinWalled 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. ThinWalled 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 1115 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/Cat
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
1QR
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 thinwalled steel structures under cyclic loading Tsutomu Usami and Hanbin Ge Department of Civil Engineering, Nagoya University, Nagoya 4648603, Japan This paper deals with failure predictions of thinwalled steel structures subjected to cyclic loading. To trace inelastic behavior of steel structures, an elastoplastic large displacement analysis using a modified twosurface 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 thinwalled 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 thinwalled steel structures under cyclic loading. Elastoplastic large displacement cyclic analysis is carried out to study inelastic behavior and obtain the supply capacity. Cantilevertype 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 twosurface 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. STRESSSTRAIN 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 stressstrain 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 twosurface 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 twosurface 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 stressstrain 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 stressstrain 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 stressstrain curve; (2) Bauschinger effect, reduction of the elastic range of the unloading curve and cyclic hardening effect can not be taken into account.
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Figure 2. Stress and strain relationships of material models under uniaxial cyclic loading 3. CYCLIC BEHAVIOR OF STIFFENED BOXSECTION 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 MIH, MKH, and 2SM models. Fig. 3 shows the lateral loadlateral displacement curves of a thinwalled column B 14 with Rf= 0.56 and ~ = 0.26 (Nishikawa et al. 1996). Here, R/is the widththickness ratio parameter, ~ is the column slenderness ratio parameter, and their definitions are given as follows. b I12(11p2) 4n2/g 2
~ /~
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, I ,I,
10
8
6
I i I , 4
2
, II 0
2
2SM 4
6
8
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816y Figure 4. Predicted hysteresis curves of a boxsection steel column: KD10
46
~ : 2hr7r1~~
H (2)
Failure Point

[email protected]@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 MIH and MKH models. However, the hysteresis loops of MIH become quite "fat" near the peak point due to the omission of the Bauschinger effect. In the case of the MKH model, the analytical result at the postbuckling stage overestimates the experimental result because the MKH 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 KD10 where Rf 0.35. This specimen is a thickwalled column. In the case of MIH [see Fig. 4(a)], the hysteresis loops of the analysis are much "fatter" than that of the test. The maximum strength and postbuckling capacity are largely overestimated. The reason for this is that it is very difficult for local buckling of such a thickwalled column to occur. On the other hand, the MKH 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 loadlateral 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 loadinward 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 !
!
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~ aim w
(a) at Hm,,x (b) at H95 Figure 6. Buckling modes of a stiffened boxsection column 2
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6
4. C Y C L I C BEHAVIOR OF PIPESECTION 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, BIH and BKH. The column has a slenderness ratio parameter of ~  0.26 and a radiusthickness 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 postbuckling stage. Figs. 8(b) and 8(c) show the corresponding lateral loadlateral displacement hysteresis curves obtained by using the BIH and BKH material models compared with the experimental result. The following phenomena can be observed: (1) The load carrying capacity at each halfcycle is overestimated by the BIH model; and (2) The computed hysteresis curve by the BKH 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 loadcarrying capacity. These differences are the result of drawback of the BIH and BKH material models. The Bauschinger effect is
48 neglected in the BIH model, and in the BKH 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 BKH and BIH 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 pipesection steel column.
4.2. Progression of local buckling
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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 BKH 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 pipesection column and eventually an elephantfoot buckling mode is formed. This phenomenon matches well with the actual mode of the steel bridge piers failed in the HyogokenNanbu earthquake. However, the extent of deformation of the BKH analysis is smaller than those of both the test and 2SM analysis. On the other hand, the buckling mode predicted by the BIH 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 BIH 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 BIH and BKH models predict unlikely buckling modes. _ Figure 11 illustrates buckled deformation of a thinwalled 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 1195 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 thickwalled 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 pipesection 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 boxsections or pipesections. 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 boxsection 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)
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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
,
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9
,,,t

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8~5y (b) 8~y Figure 10. Lateral loadlateral displacement curves of a thin'walled pipe'section column &~k" &
0.1
[   ' O    a t Hmax I    ~   a t H95
Oat 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 thinwalled pipesection column :
9
,
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9

,
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,
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l .
9
: Hnmx Failure P o i ~ ::::::::::::::::::::::::::::::::::::
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 1
1
(a)
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"::
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_
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Position: Point A, Oute'~r
40 20 0 (b) e]Ey Figure 12. Stress progression in a thinwalled pipe'section column
20
E~Ey
1 al~~ (7)
where Q is the local buckling strength of plate panels given by
51
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,t=16.8mm, hf4390mm
1
9
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.98
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0
9
;
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94
2
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30
t
.
.
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0
10
(a) ~Y (b) ~y Figure 13. Lateral loadlateral displacement curves of a thickwalled pipesection c o l u m n
'&
! 4Jat 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 thickwalled pipesection column
2.''1 ;:::::H:.:m:x: ~::.:::::.....::..........~jiure_ Point. .
.
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.
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.
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1 .
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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 thickwalled pipesection c o l u m n
Q=~y1 [P~/p24R, ]
(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 Tshape cross section which consists of one longitudinal stiffener and the adjacent subpanels. In the case of the pipesection 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 twosurface 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), 260267. 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 HyogokenNanbu Great Earthquake, JSCE, 599604. 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, 235244 (in Japanese). Shen C., Mamaghani IHP, Mizyno E., and Usami T. (1995). Cyclic behavior of structural steels. 11: theory. J. Engrg., Mech., ASCE, 121, 11651172. Usami T., and Ge H. B. (1998). Cyclic behavior of thinwalled steel structures  numerical analysis. Thinwalled structures, Vol. 32, 4180.
Impact Loading
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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 finiteelement 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. Rigidplastic 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 finiteelement 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 finiteelement 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 crosssection 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 finiteelement 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 onedimensional structure from a global, or design, perspective, although local threedimensional affects are important at the failure site. The impact failure of plates, which are nominally twodimemional 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, throughthickness failure in a plate underneath the impactor, throughthickness failure at the clamped edge of a plate, and throughthickness failure at the clamped edge of a plate and limited failure underneath the impactor.
Photographs of typical mode I to IV failures 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 laserDoppler 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 throughthickness 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. Forcetime 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 forcedisplacement histories for the plate in Figure 1.
3. NUMERICAL FINITEELEMENT STUDIES ON THE IMPACT RESPONSE OF BEAMS AND PLATES The finiteelement 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 finiteelement 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 finiteelement 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), 24852495, 1997. J. Yu and N. Jones, Numerical simulation of impact loaded steel beams and the failure criteria, Int. J. Solids and Structures, 34(30), 39774004, 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), 643667, 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), 11131137, 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), 141230, 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), 332342, 1997. J. Liu and N. Jones, Experimental investigation of clamped beams struck transversely by a mass, lnt. J. Impact Engineering, 6(4), 303335, 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 ElasticPlastic 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 massspring model is used to analyze elasticplastic BoFs under impact or pulse loading. A general analytical method for elasticplastic 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 beamonfoundation (BoF) systems has a large variety of engineering applications. Besides being directly applied to the case of actual foundationsupported structures or networks of beams, BoF is also used to form a simple but useful analogy to the structures consisting of thinwalled 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 rigidmass 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 rigidplastic 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 elasticplastic BoFs. By using a similar massspring model, this paper is mainly aimed to analyze the dynamic behavior of elasticplastic BoFs under impact or pulse loading.
2. Model and Formulation Consider an elasticplastic beam of finite length L and density per unit length o, resting on an elasticplastic foundation of Winklertype. A rigidmass 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
iI i
node 1 2
Ill i
i i+l (a)
nI
Q,]
Mi
A
1'
_ ~ )
y
.L4' T Mi.i
,;ii ..... Qi., * Qi Pi
(il)th bar nI
Fi
ith node
i ith bar
n (b)
Fig. 1 The massspring model of an elasticplastic 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 strainrate are neglected. It is also assumed that the foundation and the beam must always be in contact. A massspring 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 (nl) 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 (nl) massless rigid links of length L/(n1). The flexural deformation is represented by the relative rotation between adjacent rigid links and resisted by an elasticplastic rotational spring, which reflects the flexural rigidity of the beam. At each node, an elasticplastic spring linearly acting along the transverse direction of the beam is added to represent the effect of Winklertype 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 nondimensionalize 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 elasticplastic BoF can be formulated as
(3)
{W}= n(n  I)BI(AAT ~m} nBI {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(nl)
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)
Nondimensional 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 elasticplastic BoF, respectively. The nondimensional 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 = ~ (1or), 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~= (l13)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 nondimensional 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 RungeKutta 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 nondimensional 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 midspan 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 quasistatic 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 64, the elasticplastic 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 elasticplastic
65 Nondimensional 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 Nondimensional length x
Fig. 3 The evolution of EP 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 quasistatic 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 reexperiences 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 elasticplastic 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
•
.......
Nondimensional
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 Nondimensionallength x
Fig. 6 The evolution of EP 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 reexperiences an elastic vibration. In the quasistatic analysis [1], two collapse mechanisms exist, i.e. plastic yielding of foundation and "rigidbody" 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 massspring model is developed to analyze dynamic response of elasticplastic 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 rigidmass impact. References
[1 ] X.W. Chen and T.X. Yu, Elasticplastic beamonfoundation under quasistatic 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) 605612.
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, 2398686, Japan b Nippon Kokan Light Steel Co. Ltd., Tokyo, 1030012, 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 (W1.78kN) with spherical shape of diameter 22cm under the different dropping heights (velocities). The specimen is simply supported at
68
Figure 2. Weightdropping 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 (coldreduced carbon steel sheet, ISO35741986) 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/tr the prediction of ballistic limit velocily is:
I (t Y Vb  +#ptCi) 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.
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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, 199. 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, 141230. 4. Young, C.W., Depth Prediction for Earth Penetrating Projectiles, Proc. ASCE, 1969, 95, SM3,803817. 5. Johnson, W., Some Conspicuous Aspects of the Century of Rapid Changes in Battleship Armours ca 18451945, lnt. J. lmpact Engineering, 1988, 7, 261284. 6. Goldsmith, W., and Finnegan, S.A., Penetration and Perforation Process in Metal Targets at and Above Ballistic Limits, Int. J. Mech. Sci., 1971, 13,843866. 7. Gupta, N.K., and Madhu, V., An Experimental Study of Normal and Oblique Impact of HardCore Projectile on Single and Layered Plates, Int. d. Impact Engineering, 1997, 19, pp. 395414. 8. Lambert, J.P., and Jonas, G.H., Ballistic Research Laboratory, BRLR1852(ADA021389), 1976. 7. Gupta, N.K., and Madhu, V., An Experimental Study of Normal and Oblique Impact of HardCore Projectile on Single and Layered Plates, Int. J. Impact Engineering, 1997, 19, pp. 395414. 8. Lambert, J.P., and Jonas, G.H., Ballistic Research Laboratory, BRLR1852(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, Fumcho, Chilcasaku, Nagoya 4648603, Japan bD~almaent of Civil Engineering, Nagoya University, Furocho, Chikusaku, Nagoya 4648603, 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 largescale 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 reexamined 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 fullscale 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 fullscale 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 largescale 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 reexamined 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 fullscale 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 fullscale 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, threedimensional finitemlement code LSDYNA3D is capable for simulating the vehicle impact onto the guard fences['0. The analysis results are further compared with the fullscale 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 fullscale 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 Htype 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 subbeam 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 subbeam 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 elastoplastic material following the von Mises yielding condition. The swain hardening and strain velocity are taken into consideration the stressstrain relationship. The concrete constructed in the curb is assumed as a general elastoplastic material. This means that the concrete is in the general elastoplastic condition while the concrete in the compressive side reaches the yield point and only the cutoff 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 laddertype 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 elastoplastic stressstrain 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 elastoplastic material following the von Mises yielding condition, and the siresstrain relationship is perfectly elastoplastic. The aluminum used for the cargo body is assumed in a multipiece linear stressswain 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"
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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, 128 model, 4416 model and 8832 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 4416 model and 8832 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 4416 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 subbeams 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 subbeam $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
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(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, 598617. 4. Reid, J., Sicking, D., Paulsen, G., Design and Analysis of Approach Terminal Sections Using Simulation, Journal of Transportation Engineering, ASCE, 122(5), 1996, 399405. 5. Miller, P. and Camey, J., Computer Simulations of Roadside Crash Cushion Impacts, Journal of Transportation Engineering, ASCE, 123:5, 1997, 370376. 6. Reid, J., Sicking, D., and Bligh, R., Critical Impact Point for Longitudinal Barriers, Journal of Transportation Engineering, ASCE, 124:1, 1998, 6572. 7. Hallquist J., LSDYNA3D 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, 5360, 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, 821828, 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 Kenichi G. MATSUOKA a aDepartment of Civil Engineering, Muroran Institute of Technology, 271 Mizumoto, Muroran, 0508585 Japan b Technical Research Institute, Mitsui Construction, Co. Ltd., 5181 Komaki, Nagareyama, 2700132 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 Shearbending 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 13, 3 2.4 163.1 0.63 A36I,S 13, 3 0.018 3.6 68.8 108.7 0.84 A48I,S 13, 3 (A) 4.8 81.5 0.67 B24I, S I  2 , 2 2.4 78.4 1.00 B36I,S I  3 , 3 0.008 3.6 52.3 52.2 1.33 B48I, S 1  3, 3 ( B ) 4.8 39. 2
Figure 1. Dimensions of RC beams
Photo 1. Experimental setup
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 shearbending 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 midspan of beam by means of a freely falling method using a 300 kg steel weight. Here, two types of loading method are applied:
89
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(d) B24I
(b) As6X
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/x
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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 onefiftieth 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 midspan deflection (hereafter, deflection) are continuously recorded by using wideband 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 halfsine 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 halfsine 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 hMfsine
90
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(~) 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 halfsine 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~
Ra 6
"12
Deflection,
150~,, 100[
= O.o" 5 0 ~ "
. . . .
!
, 'v=;./, 1
V= 2m/$ _
o" 50~
"
l
Pa
eL
,
i
' Vf''lm/sI
'
,
,
18
8' (mm) ,
t
24
18
24
0
6
12
18
i

V= 3m/s
Deflection, 8 (mm)
24
(a) A36I
i '

V= 1m/$
 :Pa , :R8
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) B48I Figure 4. Hysteresis loops P6, R6 for A36/B48 I beams tion are almost the same irrespective of loading method, respectively. 3.2. Hysteresis loops of P6 and R6 Hysteresis loops of impact force  deflection P~ and reaction force  deflection R6 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 P6 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 P6 and R6 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 R6 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 halfsine wave; 2)The midspan 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 AsiaPacific 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 AsiaPacific 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, 1049001 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 [ 15]. 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 spacetime finite elements and the solution of the dynamic equations is obtained either by global integration or stepbystep. 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 stepbystep 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 spacetime 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 pathdependent. 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 stepbystep integration, we have selected the direct differential approach due to its easier implementation. For the atonce 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 spacetime 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 timeelement equation as D e z e = R e,
z e = ( t , t+Atz)'
tz = (tu, tO, tQ )
(9)
Eq.(9) may be solved stepbystep, or assembled as Dz = R to be solved globally, i.e, atonce. 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 righthand side of the equation Dz = R after multiplying the vector U c of those conditions, resulting KU=RDcU 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 controlCase 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 controlCase 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 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 strainrates, 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 viscositytemperaturevolume strain relation has been obtained for Copper from limited high stainrate impact data [3, 69], 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~ ~ 102 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 Microdegradation 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 preexisting ones, grow and join together forming microcracks that subsequently connect with other voids progressively forming macrocracks and fragments. For high strainrate microdamage, the stresstemperature driven nucleation rate process is considered most important [5, 1012], whereby
m, [ (m,l_ 1
= eM
Failure,
e M 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 postfailure analysis. The reduced compliance matrix has, along its diagonal, Ex, Ez, Gxz, and Gxy as the only nonzero 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 4layered 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 = 2.54x103 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 102 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 forcetime 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 laminastacking. 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 E07 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 9.5E07 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 E07 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 impactdamage 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, 612 (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, SpringerVerlag, 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 (kNm)
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 kNm, it could withstand higher thrust of 128.0 kNm. 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 nonbum 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 cooperation 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/9086 (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 UltraHigh Velocity Impact S.Pazhanivel*, V.K. Sharma Terminal Ballistics Researh Laboratory, Sector30, Chandigarh 160 020, India. The impact of an ultrahigh 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 UltraHigh 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(xl)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 (EEo), are specified as a function of the volume compression Xl = (VoV)/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 RankineHugoniot relations that the pressure 'P' generated in a shock of compression 'x~' is c 2 x1
P=
(4) V o (1SXl) 2
as described by G.B.Benedek [2]. Where, Vo Specific volume of unshoeked metal & VSpecific volume of shocked metal The increase in internal energy (EEo) produced by a compression 'xl' is given by 89c2xl 2 EEo =
(5) (1sxl)2
4. SEMIEMPERICAL 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 semiempirical 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 (TvTo)
(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 closepacked 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 (TTO = 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  PPc Therefore, when metal subjected to pressure, the heat required ( ~ ) to vaporise the metal is calculated from Pn = 71( c / V ) (TTo+Eo/c)
(s)
En c (TTo)
(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 ~a1) 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 & UShock 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~losionvisavis 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 (aluminiumRoUed Homogeneous Armour; aluminiumaluminium) 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
EEo (MJ/gg) 18.02
4'1
T(~
P (Opa) 219
10450
~ [;~O]ii~lii
Table 3 9 Comparison Of Energy Required (EEo) for Impact Explosion of metals Calculated By The Author With Shockey's Results [5] Metal
.....Author calculation Energy required for explosion metal upon impact (EEo) 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 (EEo) 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 UltraHigh 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. 573579,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 :
[email protected]/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 explosiontarget 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 projectiletarget 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 highspeed 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 xrays 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 highspeed oscilloscope is used as recording equipment. These electrical contactors are in open circuit connected to an RC network. On arrival of the high an~litude pressure wave, the contactors get closed and RC 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
1.611.62
6.9
2.89
19.7
G Pa
TNT
100
CompositionB
RDX60, TNT40
1.68
7.8
2.95
25.88
Torpex
RDX41, TNT41,
1.81
7.6
3.07
25.69
1.85
7.45
3.1
25.04
Al18 Pentolite
PETN50, TNT50
159 2.3 Other applications Shock attenuation studies (i.e. the decay of shock pressure and shock velocity with the increasing tlAckness 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 RC 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
Fig1 (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 multigap 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 crystalbaseA blast pressure gauges have been developed at TBRL. Blast gauge having a pile of twelve Xcut 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 shadowgraph 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 Xray 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. Xrays are generated based upon the principle of field emission. The wavelength of the Xrays emitted at 300 kV is of the order of 0.04/~. High intensity Xray 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 highspeed photography, are studied by this technique.
7.1 Applications Some of the studies of interest carded out by flash xray 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 Xray radiographs showing expansion of 32 mm fragmenting shell fragments. Fig. 5 shows xray 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 ExplosiveDriven Metal Plate, Propellants, Explosives, Pyrotechniques 11, 1622 (1986). 4. R.H.Cole, Underwater Explosions, Princeton University Press, Princeton, NJ, 270285 (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 metalforge 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 Metalforge was analysed using a piezoelectric accelerometer and strain gauges. The total stress of the upper platen consists of two parts. One is prestress, 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 prestress caused by the assembly is a major factor in the initiation of cracks. A threedimensional 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 MetalForge which had failed. It was important to investigate this problem in order to reduce the possibility of failure in the future. Online testing and stress analysis were performed to identity the mechanisms responsible for the failure of the upper platen. The maximum shock power of the MetalForge was one hundred tonmetre. 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 metalforge 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 YD12 Piezoelectric Accelerometer
Upper Platen HP9000320C Computer Workstation
L
F
HP 35665A Dynamic Signal I Analyzer
Type YE5852 Conditioning Amplifier ~r TEAC XR50C 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 9000320C Computer Workstation Figure 3.
'~
Circuit
~
DPM600 Dynamic Strain Amplifier TEAC XR50C 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 AA 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 BB (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 BB and CC (shown in Figure 6) were analysed. The results for Point 1 at cross section BB are presented in Table 2.
t (sec)
Figure 6.
Strain distribution on the surface of the critical section BB.
169 Table 2 Measured and calculated results for point one at cross section BB Acceleration Shock Prestress 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 prestress and a shock stress. One can consider that the prestress remains unchanged in the platen after assembly and does not vary with time. According to the specification, the calculated prestress 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 prestress. The analysis reveals that the prestress plays a significant role in the creation and growth of cracks and is the key factor to the upper platen damage. A threedimensional 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 twentynode 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 prestress 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
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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 multimaterial 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 multimaterials, 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 multimaterial 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 MultiMaterials, 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 multimaterial 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 Multimaterial 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 RayTracer 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 MULTIMATERIAL EULERIAN LAGRANGIAN COUPLED TECHNIQUE The multimaterial 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 MultiMaterial 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 MultiMaterial model and takes about 1/20 of the execution time of the MultiMaterial model to run. The execution time of the MultiMaterial 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 MultiMaterial approach is used. The number of elements required in the finite element model for the airexplosive parts in the multimaterial 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  MultiMaterial Method
178 Table 2 Finite Element Para.metersOn.e Quarter Model Technique Component Nodes Elements Lagrangian structure 5876 5636 Multimaterial 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) ChapmanJouget 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 MultiMaterial 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] LSDYNA USER'S MANUAL, Version 950, May 1999, Livermore Software Technology Corporation, USA. [2] Marco J., et al, Second International LSDYNA3D 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, Sector30, Chandigarh160020, 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 (13). 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 8120 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
xobserved 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
xobserved points
xobserved 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 r0.95 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
, ~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 midplate displacement, dmax,peak midplate 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. Leastsquares 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 farfield 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, LSDYNA User's Manual (Nonlinear 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 (USAVersion 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. 89107, 1989. [4] L.C. Hammond, R. Grzebieta, Structural Response of Submerged AirBacked 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. 345359, 1997. [8] G.I. Taylor, The Pressure and Impulse of Submarine Explosive Waves on Plates, Underwater Explosion Research, Vol. 1  The Shock Wave, pp. 11551173, 1950.
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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 nonlinear 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 singlesided 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 overpressures 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 overpressure 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 fillet713LL 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 crossbrace 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 nonlinearities 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 VonMises 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 stressstrain 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 CowperSymonds overstress model was used in the numerical analyses for one of the tests.
213
3.3 Pressuretime curves Figure 3 illustrates the pressuretime 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 pressuretime 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 pressuretime 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 loaddeflection diagram behaviour on predicted loaddeflection 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 BlastLoaded Square Plates. Int J. Of Impact Engng, Vo122, No. 4, pp449467, 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, pp273291, 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 Jintegral analysis. Initially the operational requirements are established, followed by the explanation of the design criteria and their implementation to X80 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 postexplosion 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 worstcase naval transverse bulkhead for the operational requirements and subsequently used in the design criteria. 2.1.
Preairblast 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 drydocking. 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.
Airblast load
The airblast 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 postairblast load has to be supported by the transverse bulkhead.
2.3.
Postairblast load
The postairblast 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.
Preairblast 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 Jintegral procedure will be used to confirm that no rupture occurs.
3.2.
Airblast load
The airblast load is equivalent to, by the Hopkinson scale method, 7 kg of CompB 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 X80 and is discussed in Raymond et. a/.(1999).
Figure 1. Blast pressure history of 7 kg of CompB 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 airblast situation, material data is required. This material data can be obtained by undertaking Hopkinson bar tests, with the use of constitutive equations such as JohnsonCook or CowperSymonds (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 Jintegral procedure.
3.3.
Postairblast load
The postairblast 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 airblast load, is required to maintain structural integrity due to the flooding load. The Jintegral 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.
Jintegral procedure
The Jintegral 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 elasticplastic stress situations such as high toughness steel in naval transverse bulkheads. Therefore, a path independent Jintegral application to determine if the naval transverse bulkhead has ruptured or not is an extension Of current practices. The Jintegral cannot be applied to the whole transverse bulkhead arrangement at once. The solution is to apply the path independent Jintegral 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. Doublesided 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 Jintegral 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 subprogram will calculate the Jintegral 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 Jintegral value will be verified against empirical results in Hrovat and Hoffman (1999).
221
The Jintegral will be calculated using the method for a singleedge 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 singleedge crack plate under remote uniform tension. The Jintegral can be evaluated by the following equation, from EPRI (1981)
J = fl (ae)~7 + ~
(a~b,n
'
(1)
where ct, n are the RamsbergOsgood 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. Singleedge 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 Jintegral 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 Jintegral is calculated by the above method. This calculated Jintegral value is related to a JR 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 Jintegral 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 JR 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 X80 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 X80 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 airblast 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 X80 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 1041 "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 elasticplastic 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, SSC382, 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. 277286 8. OPNAV Instruction 9070.1, Chief of Naval Operations, Washington, 1988 9. Raymond, I., Chowdhury, M., Kelly, D., Design criteria for X80 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. 1934. 11. Rice, J. R. A., Journal of Applied Mechanics, 35, June, pp. 379386, 1968 12. Shih, C. F., and Hutchison, J. W., Journal of Engineering Materials and Technology, 98, October, pp. 289295, 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 AsiaPacific Conference on Shock and Impact Loads on Structures, 1997, pp. 569576. 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 highmodulus 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 [14]. 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 twodimensional 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 twodimensional 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 fighthand 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 quasilinear 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
dS 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:
(hlw,
(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 CourantFriedrichsLewy 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/TR86/044L (Limited Release) (1986). 3. J. R. Vinson and J. A. Zukas, On the Ballistic Impact of Textile Body Armor, ASME Paper No. 75APM12 (1975). 4. B. PargaLanda and F. HernhndezOlivares, 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. PargaLanda, Inst. Phys. Conf. Ser. No. 102: Session 11 (1989) 565. 9. G.B. Whitham, Linear and NonLinear Waves, WileyInterscience, 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
LargeScale 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 602083111 USA 2U.S. Army Engineer Research and Development Center, Waterways Experiment Station 3909 Halls Ferry Road, Attn: CEERDSDR Vicksburg, MS 391806199 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 largescale 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 T3E1200. The concrete, reinforcing steel, and C4 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 preanalysis phase, a parallel analysis phase that includes explicit message passing among each partition on separate processors, and a postanalysis 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 workconjugate 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 elastoplastic 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 C4 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 explosivestructural analysis uses the microplane constitutive model for the
232 concrete, an elastoplastic model for the reinforcing steel, and a JWL equationofstate model for the C4 explosive. Ignition of the explosive is treated by a programmedbum 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 T3E1200 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 T3Ean analysis that would take over 41 days on a single processor. Because of a communicationcomputation
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 dependenteffective 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 largescale 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 DAAH04952003/contract number DAA95C0008, 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" StressStrain Boundaries and l:inite Strain; II" Data Delocation and Verification",,/. Engng Mech. 122(3) 245262 (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) 179186 (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 5386894, 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 95035, 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 presplit 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 presplit 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 Presplit 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 presplit 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 presplitting 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 nonblasting 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 insitu stress is neglected In the case of a single hole, the pressurising will create cracks initiating from preexisting 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 2D 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 nondegenerate type, hence its solution cannot be obtained analytically. The GaussChebyshev 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 insitu 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 preexisting surface flaw of length a0 be po.ack.Obviously, if pcrack
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 SideImpact 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[nwvI
.........
> 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
to
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 newmethod 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
tt"
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 VlxV2x and VlyV2y. 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){ (VaxVbx)2+(VayVby) 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 ax2.7 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 sideimpact 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 energyabsorbing 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 energyabsorbing capacity and various failure modes of the polymer foam laminated structure are investigated. Since the shape of the forcedeformation 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 SETUP
2.1 Impact Tests of Laminated Block The samples were laminated panels of dimensions 100x100x26 mm with PVC polymer skin and expendedpolyurethane (EPU) foam core of density 96 kg/m 3. The tests
340 were conducted in Dynatup 8250 dropweight 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 foamcore laminated panels. An initial linearelastic 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 linearelastic 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 overloading 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 foamcore 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 impactinduced dynamic deformation and damages. 3.3 Structural Impact Response of Bicycle Helmets Twentyseven new helmets were tested. The loaddeflection 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 nonnegligible 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 degreeoffreedom massspring 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 elasticplastic 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 crosssectional 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)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 or 0
\~0 where D=40.4/s and q=5 [5] and the tested static stressstrain curve is given in Fig. 1. Only axisymmetric deformation is assumed. A commercial code ABAQUS  Explicit was used and a twodimensional model was constructed using 2691 fournode 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 forceat 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 CowperSymonds 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 quasistatic 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 strainrate 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) N7491 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, noncombustibility, 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 [14], 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 quasistatic 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 [ 13], 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 nonfilled 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 nonfilled 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 nonfilled extrusions. For nonfilled 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 crosssectioned 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 035
forDIt Pn where R 2 = (1.2  0.005 L / D) //2 0.2
_~ 1.0
for D / t 50 and ~a is the average ductility per strut cycle. In contrast to the Astrut algorithm, the critical flexural energy in the Bstrut algorithm varies with strut cycles since it is a function of the average ductility per strut cycle.
3. TEST CASE AND EVALUATION Psuedostatic pushover analysis of the Xframe [5] was considered to evaluate the performance of the Marshall Bstrut in predicting the reduction in the strut capacity with strut cycles. The Xbraces of the lower and the upper bays of the frame were modeled with the Marshall Bstrut elements. All the other members were modeled with elastic beam elements. For the full evaluation of the structural collapse mechanisms, inelastic portal beamcolumn modeling of the legs would be required. A cyclic displacement profile [5] was imposed at the top right corner of the frame. Geometrically nonlinear analysis was performed. Figure 2 shows the maximum frame load versus pushpull cycles for the Xframe. 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 Astrut 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
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.
.
40 20
~
.
.
.
.
.
.
.
.
.
.
.
f
f
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~ 40
  ,
~...~.,
 6 0
i
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_
. . . . . _.
!
!
i'
!
!
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3
4
5
6
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!
,~ '
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i
7
!
8
i

!
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!
!
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 Xframe 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 postbuckling 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
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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 Bstrut for strut behavior is conservative with respect to the experimental results. 5. ACKNOWLEDGEMENTS The support of the president Maini, R., and the executive vicepresident 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 3D BeamColumn Elements, Report No. UCB/EERC82/20, 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/EERC80/27, UC, Berkeley, California, August 1980. 6. Sherman, D.R., Ultimate Capacity of Tubular Members, Report CE15, University of Wisconsin, Report to Shell Oil Company, August 1975. 7. Sherman, D. R., Post Local Buckling behavior of Tubular Strut Type BeamColumns: An Experimental Study, Univ. Wisc., Milwaukee, Report to Shell Oil Company, June 1980. 8. Sherman, D.R., Interpretive Discussion of Tubular BeamColumn 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 forcedisplacement 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 forcedisplacement 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 elastoplastic 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 forcedisplacement hysteretic loop model require theoretical calculation of forcedisplacement 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 FORCEDISPLACEMENT 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 isoparametric 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 forcedisplacement 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 forcedisplacement 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)
Z120 Z130
108x4.5 x 1250 108x4.5 x 1250
312.4 312.4
77.1 77.1
1.07 1.07
200 300
Z220
114x6.0 x 1250
319.3
77.1
1.04
200
Z230
114x6.0 x 1250
319.3
77.1
1.04
300
Z320 Z330
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 120
(ram)
434
~~
~
x~'tO
~"30
N
tO
'~
0
o lO
at)
,,
///~,
//~;
4O
~
io ;o 
~(mm )
(ram)
(b)Specimen Z130
c,~. 40
,
4D
40
,
~
,,
O
20
r

el) W ~( mill )
~
...
~  ~ " : & "  o ~
~
to
so
(c)Specimen Z220
0
I)
~ " 
6[mm
~ 
:& . . . .
~ .... i .... ~ ....
6 (men)
)
(d)Specimen Z230 r~
2o
~20
lo
Z 2O .~0 40 6O /',( m : )
(e)Specimen Z320
8 (m~
435
//
/,,'./"
,
.~o ~
.,or
~'~
.ao
~,~
....................................
80 ~
40
~O
[]
~n
lO
80
80
~,~,
r6.'////
,~7
1130
8 (i..i) (f)Specimen Z330 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:iIO
GO~020
0
~0
I I ! I
~0 a O 8 0 t 0 [ ~.; ( m m *]
(ram)
Specimen HSC32 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 FORCEDISPLACEMENT HYSTERETIC LOOP CHARACTERISTIC We can find some characteristics of forcedisplacement 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 forcedisplacement 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 beamcolumns. 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
Stubcolumn 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 stubcolumn 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 postbuckling 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 weldfailure 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 builtup section members are widely used in the steel structures. Up till now, the connection welds of the welded builtup 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 postbuckling of plate assembly was observed. During the 1985 Mexico City earthquake, one of three identical 22storey 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 openWeb 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 widththickness ratios of the stiffened plates are all about 20. A WBS column of a boilersupporting 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 stubcolumn failure tests and the nonlinear analysis results. The influence of the widththickness ratio, size of fillet weld and the form of the box cross section (square or rectangular) to the weldfailure 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 stubcolumn 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 smallsized 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 S1 200 S2 200 R1 200 R2 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
)/
1specimen; 2dial gauges; 3base; 4testing machine cross head; 5electrical gauges. Figure 2. Test setup
l
11 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. Stressstrain curve for S1
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. Stressstrain curve for S2
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. Stressstrain curve for RI
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. Stressstrain curve for R2
10
441
0.54 036I ~
8t
o
'
~
~ 0.1
0.36~L..
=1700KN = 1850KN v1900KN .~ 1950KN . ,', , 20p0K~l 600 700
5oo
Column 200x200x10x600 hf=4.5mm
Figure 7. The distribution of direct stress at welds for S1
propagation in the comer welds was observed. The results of the stressstrain 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. WELDFAILURE DUCTILITY RATIO OF WBS The modified nonlinear (including material and geometrical nonlinearity) 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 widththickness ratio, size of fillet weld and the form of the box cross section to the weldfailure ductility ratio of WBS under axial loading was conducted by this method. From the statistical analysis, a formula for the weldfailure 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 widththickness 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 weldfailure ductility ratio of WBS can be derived from equation (2):
442 hI > 1.618,~x/e.y + 0.22t +O.09b/t14.56
(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 weldfailure 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.852870 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.181182 3. S.Kumar and T.Usami, An EvolutionaryDegrading Hysteretic Model for ThinWalled Steel Structures, Engrg. Struct. Vol.18 No.7 (1996), pp.504514 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 2931, Vol.1 (1997), pp.205212 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 2931, Vol.1 (1997), pp.213220 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, 2527, July, Vol.3 (1995), pp.20992104
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 XiaoLing 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 coldformed C450 (450 MPa nominal yield stress) square hollow sections (SHS). Four different section sizes were chosen for the outer tubes with a widthtothickness ratio ranging from 16.7 to 25.0. One section size was chosen for the inner tube which had a widthtothickness 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 seabed 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 highrise 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 voidfilled 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 coldformed 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 inline 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 crosssection 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 loaddeflection 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 momentversuscurvature graphs are shown in Figure 2 (b) where Mpouter is the plastic moment capacity of the outer tube and kpoutcr is the curvature corresponding to Mpoater. 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 "(hi2" rexti)" t i
A ...... ,e=
(b2.to)24.(r,2o~.r,~,o)
b~4.(ri2
_~.r~,i)
where beo = b o  2. rex,~
if ~ _ 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 Isections. 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 crosssection 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.
491 120
I I I
P .
O
O
oo
.
.
.
_
I
i
6)
L A~
I I
i
L Beam crosssection
!
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 ,.,i21 9
I
i
$11 9
"
I
t~
o
I
!
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'
O
i "
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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 S7$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
SI$355 S2Weldox S3$355 S4Weldox S5$355 S6Weldox S7$355 S8Weldox 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
,
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o
o
,,
,
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i
, , ,
I
=
~ , , 
10 15 Displacement [ mm ]
500
t" /t']'"
~
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300

!
6oo~ L !///.   : . ~~" ' .... :~\
'~176
_
o i + Z,S1$355
1 ___!
z
,,.,
i
I ,,

,,/s3s35~
s wo, ox
.
2oo ~

.,~
. . . . . . . . . .
_ ", 
,

S7$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
TI$355 T2Weldox T3$355 T4Weldox T5$355 T6Weldox T7$355 T8Weldox T9$355 T10Weldox
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 forcedisplacement 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 $355 (Table 2). The tearout of the web block resulted in a splitting force in the web that caused bending in the
493
I000 9 0 0 I . . . . i .... . . '\~ /  T. .5 .$.3 5 5~'"' ~l' '' 51 /s S "
'~ ,
800 700
ooo
!r
~176 i ,
100 0
. . . . . . ./T3$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
~T4~Neldox J
,oo
25
0
3oo 200 100 0
i/_
_
\\
,oo'~176
1
I,~,~lt,,
10 15 20 Displacement [ mm ]
""
T8Weldox

..
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 34 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 67 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 610% 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/CSAS16.1M89 "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, DoubleAngle Beam Connections. Engineering Journal, AISC, 15(4) 1978. 7. Ricles J. M. and Yura J. A., Strength of DoubleRow BoltedWeb 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 highstrength steel angles. Engineering Journal, AISC, 32(3) 1995. 9. Orbison J. G. et al., Tension plane behavior in singlerow 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 ultimatetoyield 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 4221, Kumamoto 860 0082, Japan bDepartment of Architecture and Civil Engineering, Kumamoto University, Kurokami 2391, Kumamoto 860 8555, Japan
This paper concerns the assessment of significance of weld defects in beamtocolumn connections. Four fullsized beamtocolumn 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 crosssectional 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 nonlinear 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 beamtocolumn 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 BEAMTOCOLUMN JOINTS WITH WELD DEFECTS
2.1 Specimens and Loading Procedures Four fullsize beamtocolumn connections, two with wide flange section eolurrms in grade SS400, designated as BH1 and BH2, and two with box section columns in grade STKR400, designated as BS3 and BS4, were tested. All the specimens were made of onesided 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 stressstrain, 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
~ ~
XX' Section   ~ "
E
Weld Defect
[ o,! ~ !
I0 Wel, Deft
X
L~mm to, BH! S ~ i m ~ .
\11 I~ i
L=8mm for BH2 Specimen ~.x~ ~
/'
M L
I
J rqr
, w
f
L
[ r~
..0 .
,,
•
3s%
,.0
~.._I . . . . . . . . . . . . . . . . . . . . .
L
J
,.o

.... 9
: i ElS00xS00x22 ;; 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,"
BHI BH2 BS3
33.2 27.5 8.3
16.8 13.9 8.0
53.4 42.4 23.0
22.6 10.5 10.5
BS4
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
'
   BH1 BH2    BS3 BS4,, , 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 postprocessing were carded out using the ABAQUS generalpurpose finite element package. The models were constructed from 8noded 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 loaddeformation 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 BH1 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
....
_ _
BH1
BH2
BS3
BS4
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 beamtocolumn joints in fullscale 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 semiellipse for a surface crack, or as the half crack length for a through crack. ~ was calculated from Jintegral 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 overestimated. 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 _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 (3I) and (32) 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 Table2. 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 BS3 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 BS4, ~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 beamtocolumn 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., 1526, 1998 3. JWES, Method of Assessment for Flaws in Fusion Welded Joints with Respect to Brittle Fracture and Fatigue Crack Growth, JWES 28051997, 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, 647657, 1997 5. JWES, Evaluation Criterion of Rolled Steels for Low Ternperaalrc Application, JWES 30031995, 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 beamtocolumn 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 beamtocolumn 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 beamtocolumn 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 HyogokanNanbu 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 beamtocolumn cormections[4] and on the 'real behavior' of beamcolumns 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 socalled multispring model[7l which is effective to simulate the inelastic behavior of steel members subjected cyclic loads including varying axial force and bidirectional moments. The model takes short segment of the beamcolumn end adjacent to joints and divides it into several elements. The elements are modeled as elastoplastic 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 multilines shown in Figure 2, and the hysteresis loop obeys RambergOsgood 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 beamtocolumn 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 Noncrack skeleton ~ .N.....
P~I
,
/ ~'~"" elast~ plas[ic spring " *~'
......
1
functiofi
od
Compression
Figure 1. Composed spring model
Figure 2. Noncrack skeleton curves and Hysteresis loop
,^
~~~ii~ column
.beam "/" " i backing metal ~
1
L Figure 3. Beamtocolumn 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 stressstrain 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 noncrack range or crack range, RambergOsgood 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 beamtocolumn 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_ A Figure 8. Equivalent elastoplastic spring (after SimSes da Silva [7]).
Figure 9. Typical forcedisplacement 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 semirigid 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 [12] evaluated the structural behavior of bolted semirigid connections in the column minor axis. A nonlinear 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 semirigid connections. A semirigid 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 SemiRigid 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 VigaColuna em Estruturas de A~o Submetidas a Flexao no Eixo de Menor In&cia", MSc Dissertation, PUCRio, in portuguese (1999). 3. ANSYS, Version 5.4, Basic Analysis Procedures Guide, Second Edition. 4. Eurocode 3, ENV  199311, "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 BeamtoColumn MinorAxis Joints", Numerical Simulation of SemiRigid 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
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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 thinwalled 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 nonlinearities. 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 "stressfree" 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 thinwalled 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 FiniteElement model of one of these specimens could be built where the recorded data was used for calibration. This FEmodel 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 centrelines. Therefore only one quarter of the specimens had to be represented in the FEmodel 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 FEModel
2.2. Modelling the weld In a preliminary study, the cooling process of the weld was modelled taking into account the highly nonlinear 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 elasticplastic model; and secondly a bilinear model which took strain hardening into account. :9r
~
...~o
1
i
i
Measured stresses [ 1 ]
,sol
9
='
 Stresses
FErnodel ..
so .loo o
~_ 2o

~
'
~
4o
Distance a~r0ss Width
[.m m . ]. .
Figure 2. Weldinduced 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 outofplane 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 weldinduced 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 wallthicknesses and a buckling and postbuckling 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 prebuckling 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 wallthicknesses (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 wallthickness. 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 stressfree 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 prebuckling phase.
4.4. Strain Hardening Elasticplastic material behaviour was compared to a bilinear 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 weldinduced residual stress fields and displacements. Differences occurred in the postbuckling behaviour of the box where plastic hinges are developed and strainhardeningreserves result in a significant gain in postbuckling strength (Figure 7). Again the amplitude of the initial geometric imperfection had hardly any effect on the loaddeflection 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 weldinduced 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 bilinear material model. Prebuckling 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 postbuckling behaviour in the FEmodel 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
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 LeWu 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 ElastoPlastic Columns with Initial Crookedness in Parametric Resonance, Proceeding of JSCE, No.356/I3, 207210 (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), 24342448 (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 J2incremental and J2deformation 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 J2incremental and J2deformation theories of plasticity. Numerical values of buckling loads are obtained for sandwich plates of 24ST3 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 J2deformation theory of plasticity. However, p u t t i n g e  0 yields the J2incremental 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 rz and Oz planes respectively, the (outofplane) 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 oP + 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{rOS + ( 0r
r
G Ow }, Qo = f daozdz= ~~{ G  r q ~ = f d a r z d z = ~~{r +~rr
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 "h7/(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 nonuniform 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 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 (24) 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 crosssection is smaller than that in the loading period. Therefore, the imperfection sensitive regions for rectangular pulse are the same as the steploading 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 lowvelocity impact tests on thinwalled metallic notched circular tings with arcshaped 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 freefree 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.
+  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 (IV) figures (IV), 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 1wave 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 (ac) 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 (IV). 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 (IIV). 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 (IV) contain curve 1, which is the input stressstrain curve at /" = 4000 s ~ for material I. Curves 24 are the output of the 1, 2, and 3wave analyses. For the configuration (III) without an input bar the 2, and 3wave 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 2wave analysis can be understood from the reflection of a pulse from the sample in the rodsamplerod 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 2wave 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 nonstationary 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 2wave 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 2wave 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 shockwave character of loading. In contrast, the quasiisentropic 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. Strainrate curves
602 7 and 7a in Fig. 5(111) correspond to the two and oneside 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 2wave analysis gives very close results for the materials. Results of the 1wave analysis are shown in Fig. 5(VIa). Fig. 5(VIb) 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 3wave 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 PochammerChree oscillations but by the launching conditions as well. (ii) The flow stress obtained with SHPBs is determined by local stress strainrate properties of the material. However, a careful interpretation of data on the initial part of the stressstrain 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. 938. 2. G.T. Gray III, HighStrainRate Testing of Materials: The SplitHopkinson Pressure Bar, LAUR974419, 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 StressStrain 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, Sector30, Chandigarh160020, India The metallic jets produced from explosivedriven 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 quasiperiodic 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 jetelongation and particulation were recorded by multichannel Flash Radiography. The jet length and diameter, jet breakup time, number of fragments and their size were calculated from the experimental data. The value of IIirseh 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 breakup 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 BREAKUP 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 breakup mechanisms have been suggested and calculations of break up time have been made by following analytical as well as empirical approaches. IIirsch4'5 has suggested a very simple breakup 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 breakup 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 breakup time is measured from the start of the elongation process of the jet. This mechanism of breakup has been supported by the porosity found in the particles recovered from the elongated jet ai~r its breakup. Very recently Curtis et.al. 6 have proposed an empirical jet breakup model and calculated breakup time for the shaped charge jets. The breakup 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 IIirsch 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 breakup times have also been calculated from experimental data by using Hirsch breakup time formula. 3. EXPERIMENTAL SETUP Flash Radiography is the most widely used experimental technique to record the shaped
6O5 charge jets. In the present study a three channel flash xray 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 xray tubes placed inside the small holes in thick concrete walls. The radiograph of the jet was recorded at three different times on the separate xray films. The xray 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 breakup 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 breakup 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 breakup 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 breakup 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 breakup 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 breakup 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 breakup times for the entire jet lengths which are the values corresponding to the slowest velocity point plotted here. Very large breakup 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 breakup times were also observed by Waiters and Summersv for copper jet. The Hirsch cumulative breakup 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 breakup time calculated from Hirsch formula as given in equation (1) and the breakup 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 breakup time, initial jet diameter and jet breakup 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 Breakup Breakup (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 longnecking 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, 13 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, 2328 Sept 1996 P369 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, 2629 Sept 1993 P49 9. J.E. Backofen Jr. and E.Hirsch, Proc. Of the 13~ Int. Symp. On Ballistics, Stockholm, Sweden, 13 June, 1992 P359.
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 RuhrUniversity Bochum D44780 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')
(a174174 = c174174 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
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 hypoelasticity 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 integrableexacfly 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 elastoplasticity kinematics. The determinants of both elastic and plastic parts of F are positive. Furthermore we assume a natural, stressfree 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)'Fe1,
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 selfconsistent, 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), 10771086. GREEN, A. E. and NAGHDI, P. M. A general theory of an elasticplastic continuum. Arch. Rat. Mech. Anal. 18 (1965). LEE, E. H. Elasticplastic deformation at finite strains. ASME J. Appl. Mech. 36 (1969), 16. XIAO, H., BRUHNS, O. T. and MEYERS, A. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124 (1977), 89105. 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), 40014014. ZAREMBA, S. Sur une forme perfection6e de la th6orie de la relaxation. Bull. Intern. Acad. Sci. Cracovie (1903), 594614. JAUMANN, G. Geschlossenes system physikalischer und chemischerDifferentialgesetze. Akad. Wiss. Wien Sitzber. IIa (1911), 385530. HILL, R. On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16 (1968), 229242; 315322. HILL, R. Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. Soc. London A 326 (1970), 131147. HILL, R. Aspects of invariance in solid mechanics. Advances in Appl. Mech. 18 (1978), 175. HENCKY, H. Uber die Form des Elastizita'tsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys. 9 (1928), 215220. MEYERS, A. On the consistency of some eulerian strain rates. ZAMM 79 (1999), 171177. XIAO, H. Unified explicit basisfree expressions for time rate and conjugate stress of an arbitrary Hill's strain. Int. J. Solids Structures 32 (1995), 33273340. XIAO, H., BRUHNS, O. T. and MEYERS, A. Hypoelasticity model based upon the logarithmic stress rate. J. Elasticity 47 (1977), 5168. BERNSTEIN, B. Hypoelasticity and elasticity. Arch. Rat. Mech. Anal. 6 (1960), 90104. SIMO, C. and PISTER, K. S. Remarks on rate constitutive equations for finite deformation problem: computational implications. Comp. Meth. Appl. Mech. Engng. 46 (1984), 201215. BRUHNS, O. T., XIAO, H. and MEYERS, A. Selfconsistent eulerian rate type elastoplasticity models based upon the logarithmic stress rate. Int. J. Plasticity 15 (1999), 479520.
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 weftknitted 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 coknitted samples along the wale, course and 45 ~ directions are all significantly nonlinear. The tensile behavior is superior in the wale direction to those in the course and 45 ~ directions. The deformation mechanisms in mesoscale were identified experimentally by insitu 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 netshape/nearnetshape 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 inplane stiffness and strength of knitted fabrics were found to be inferior to woven, braided, and unidirectional materials with an equivalent proportion of inplane fibers, but to be superior to continuous or short fiber random mats composites [ 1]. However, inplane 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 coknitted 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 energyabsorbing 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) coknitted 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 setup was cooled by water to room temperature. Fig.1 Schematic diagram of the Tensile specimens were cut into narrow strips of weltknitted 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 setup 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 insitu 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 coknitted 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 coknitted textile composites along the wale, course and 45~
It is evident that the PP/PET coknitted samples exhibit strong nonlinear behaviors and the tensile properties and material constants are all orientationdependent. When extended along the loading direction, PET/PP coknitted 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 MESOSCALE
MECHANISMS
PET/PP
THERMOPLASTIC
Fig. 4 The deformation process of the PET/PP coknitted 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 45 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 stressstrain displayed a linear relationship. With the
Fig. 5 The deformation process of the PET/PP coknitted 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 coknitted 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 stressstrain 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  Hole1
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 coknitted samples. From Figures 45, 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 mesoscale mechanisms are identified. The results show that the tensile curves of PET/PP coknitted samples along the wale, course and 45 ~ directions are all significantly nonlinear. The tensile behavior in the wale direction is superior to those in the course and 45~ directions. By insitu 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, ICCM11, 1997. I:108133. 2 K.H. Leong, P.J. Falzon, M.K. Bannister and I. Herszberg, Composite Science and Technology, 58(1998), 239251. 3 S. Ramarkrishna, H. Hamada, N.K. Cuong, Z. Maekawa. ICCM10, 1995, IV:245252. 4 T.X. Yu, X.M. Tao and P. Xue, Composite Science and Technology, 60(5), 785800. 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, 177180 (2000), 339344. 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, 177180 (2000), 745750. 7 S. Ramarkrishna, N.K. Cuong and H.R. Hamada, Journal of Reinforced Plastics and Composites, 1997, 16(10), 946966. 8 X.P Ruan and TW Chou, Journal Composite Materials, 1998, 32(3), 198222. 9 C.R. Rios, S.L. Ogin, C. Lekakou and K.H. Leong, ICCM12, 1999, 10351041.
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, 45061 Opole, Poland
The paper presents the concept of analysis of 3D 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 hypoelastic RambergOsgood formula and Dorn creep theory) and Plem proposition [4] (for string). Both models are approximated in calculations by hyperbolic NortonBailey 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 twostage 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 RambergOsgood and time
624 dependent strains (thermal creep strains) according Dom theory [3, 7]: T + OWl(T) + a[o lit(T) 1 xg2 (T) +
ala[ m1 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), strainhardening coefficient ~t(T) and material constants W, , W2 , ~g3 are temperaturedependent. 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)  ZenerHollomon's parameter, eo  stressdependent strain, 0  Dom's parameter, 0o  limit value of Dorn's parameter. A simplified, but sufficiently accurate for fire engineering purposes, BaileyNorton formula can replace the equations (1) and (2)
=A(T,T)o n(T)
+aT,
(3)
where A and n denote temperaturedependent material functions. Two stages method of identification [3] is applied for determination of the pair of parameters A and n from BaileyNorton 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 twosided finding the logarithm of BaileyNorton 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 MarqurdtLevenberg [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 onedimensional 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(T20~
). 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 BaileyNorton model by twostages 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)
= 104),
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 ) ,~~',/~N ~
_,7/,I_H,'
'
~::.,_
l.C
~ii~~! '~gll"l.,~ ' ~  i ~ ~ ~ ~ ~ 0.0 _//,,./.~'~~, :. 0.5 '~t'/~,.~.i!.i!!i.' 1.C 0.5
(c) Type C20 (d) Type C40 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 C20 30 IFTypeC40
,, ; : .....
/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 flangeweb 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 flangeweb 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 halfwaves in the panel of the column base (see Figure 9(b)).
672 (3) Type C20 In the case of specimen Type C20, 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 flangeweb junctions occurred in this part at 5+11~ (see Figure 9(c)). (4) Type C40 Specimen Type C40 showed the same behavior of as specimen Type C20 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 C40 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 boxsection 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 C20 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 boxsection 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 thickwalled and lessstiffened 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 sidelap 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 Centreville, Montreal, Quebec, H3C 3A7, Canada.
Singlestorey 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 nonductile 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, buttonpunched and welded deck sidelap connections. The results of monotonic, cyclic and quasistatic 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 sidelap connections that are typically found in steel roof decks of singlestorey 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 StLawrence 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 CSAS16.1 [2] and CSAS136 [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 lowrise 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 sidelap 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 sidelap specimens, with screwed (1014x7/8"). buttonpunched (10 mm diameter) and welded (25 mrn length using a 41010 MPa welding rod for 23 sec at 200 V) connections, were tested in the structures laboratory at l~cole Polytechnique using the test setup 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 sidelap 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 quasistatic 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 quasistatic 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 buttonpunched or welded sidelap connections, as well as a modified version of this deck section, which allows for the use of screwed sidelap 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
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.
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Load#Y Rigid Support ~ Connection Plate Cell ~ B Guide Plate
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Figure 2. DecktoDeck SideLap Connection Test SetUp 20 i
20 
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500 1000 1500 2000 2500 3000 Time (see)
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Figure 3. Test Specimen QuasiStatic and 0.5 Hz Displacement Protocol
3. TEST RESULTS
Information regarding the displacement versus load behaviour for typical 0.76 mm welded, screwed and buttonpunched sidelap connections, cyclically tested at 0.5 Hz, is provided in Figures 46, with specific data for all load types listed in Table 1. (Note: each specimen consisted of 2 connectors). The complete loaddisplacement 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 buttonpunched connections did not vary significantly with load type (see Table 1), although the Pu results for the welded sidelaps ranged in value from 4.838.05 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 buttonpunched
676 WELD
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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
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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
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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
Buttonpunch Buttonpunch Buttonpunch Buttonpunch Buttonpunch Buttonpunch Buttonpunch 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 Quasistatic Mono Mono 0.5 Hz 0.5 Hz 3 Hz 3 Hz Quasistatic Mono Mono Quasistatic 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 pullout 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 buttonpunched 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 (131152 kN mm) in comparison with the buttonpunched connectiom (104137 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 (101608 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 buttonpunched sidelap 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 buttonpunch connections yielded the lowest values. However, when comparing the energy absorption per unit of force the buttonpunched connections outperformed their counterparts. Further fullscale seismic cantilever tests of steel roof deck assemblies would be beneficial to better understand the relative performance of these sidelap 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 ColdFormed 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, ZincCoated (Galvanized) or ZincIron AlloyCoated (Galvannealed) by the HotDip 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, 7221, Roppongi, Minatoku, Tokyo, Japan b Railway Technical Research Institute, 2838, Hikarimachi, Kokubunjishi, 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 hightension 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 (CFT1, CFT2). 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 CFT1 specimen, the displacement level was increased, while the displacement amplitudes were kept constant in 7 6 y for CFT2 specimen. 4
EXPERIMENTAL RESULTS
4.1. LoadDisplacement Relationship Loaddisplacement 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 CFT1 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
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682
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Fig.7 Strain Distribution of ST Specimen 0.20 ~y .1 + + 3 x +4 0.15 I 4~+5 + + 6
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0.20
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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 CFT2 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 CFT1 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~ !
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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
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,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[lcaxial (x)direetion[ [iiii!iill 1 [ ,~,,~ ]oaxial(x)directiorl[ o ~ 0.251"]+axial (x)direetionlI!i:ili:ii] .... / [ohoop (y) direetior~ Ji!i~:~!!i!.~.~.[ "'~ 0.20t I..o_hoop (V)direetion~. (a) Upper fogtYf the buckling
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Fig.10 Strain History of CFT1 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~ I4
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
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i It
'
, , I Ilia i i! I]H
, CFT1 ,~~..] 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, IA, pp.222223.(1998) 2. Murai, S., Okuda, T. and Nakamura, H., Report of the Institute of Industrial Science, The University of Tokyo, 29(6),pp. 115. 3. Nishimura,T. and Miki, C., Proc. of JSCE, No.279, pp.2944 (1978). 4. Tateishi K. and Murata K., Proc of the EASEC 7, pp.949954 (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 concretefilled steel tubes under large deformation cyclic loading Chen Lee, Raphael H. Grzebieta, and XiaoLing Zhao Department of Civil Engineering, Monash University, Clayton, Victoria 3800, Australia This paper reviews recent studies concerned with evaluating the capacity of concretefilled 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 coneretefiUed 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 concretefilled 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 concretefilled 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 concretefilled 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 concretefilled tubes with a larger widthtothickness 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 concretefilled steel box columns with large widthtothickness 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 concretefilled coldformed RHS columns under axial cyclic compression and tension loading. The steel tubular braces were subjected to cyclicdirect or cyclicincremental 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 LoadDisplacement 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. CyclicIncremental 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 loaddisplacement 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 concretefilled columns significantly improve earthquakeresistance capability. In a further study, Ge and Usami [7] examined similar partially concretefilled 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 concretefilled 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 concretefilled steel tubes. However, the strength of onlycyclicloaded 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 ConcreteFilled Steel Box Columns (Usami and Ge, 1994)
Fig 7. Cylic Loading Path for Concretefilled 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 concretefilled steel box column. In a loaddisplacement envelope curve, the collapse point was defined as the postpeak 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 postpeak 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 halfcycle 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 concretefilled circular, square and rectangular section tubes for different cyclic loading paths; 2. To reevaluate 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 Concretefilled Tubular Braces." J. Struct. Engrg., ASCE, 114(7), 14881506. 2. Ge, H.B. and Usami, T. (1992). "Strength of Concrete Filled ThinWalled Steel Box Columns: Experiment" J. Struct. Engrg., ASCE, 118(11), 30363054. 3. Zhao, X.L., Grzebieta, R.H., Wong, P. and Lee, C. (1999). "Concrete Filled ColdFormed C450 RHS Columns Subjected to Cyclic Axial Loading." Proc., 2"d Int. Conf. Advances in Steel Structures, Hong Kong, China. 429436. 4. Fukumoto, Y. and Kusama, H. (1985). "Local Instability Tests of Plate Elements under Cyclic Uniaxial Loading." J. Struct. Engrg., ASCE, 111(5), 10511067. 5. Maison, B.F. and Popov, E.P. (1980). "Cyclic Response Prediction for Braced Steel Frames." ,Z. Struct. Engrg., ASCE, 106(ST7), 14011416. 6. Usami, T. and Ge, H.B. (1994). "Ductility of ConcreteFilled Steel Box Columns under Cyclic Loading." J. Struct. Engrg., ASCE, 120(7), 20212040. 7. Ge, H.B. and Usami, T. (1996). "Cyclic Tests of Concrete Filled Steel Box Columns." J. Struct. Engrg., ASCE, 122(10), 11691177. 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, 51,297319. 9. Popov, E.P. and Black, R.G. (1981). "Steel Struts under Severe Cyclic Loadings." J. Struct. Engrg. Division, ASCE, 107(ST9), 18571881. 10.Dafalias, Y.F. and Popov, E.P. (1975). "A Model of Nonlinear Hardening Materials for Complex Loading." Acta Mechanica, 21, 173192 11. Haijar, J.F. and Gourley, B.C. (1997). "A Cyclic Nonlinear Model for ConcreteFilled Tubes I: Formulation." J. Struct. Engrg., ASCE, 123(6), 736744. 12. ECCSCECMEKS. (1986). Study on Design of Steel Building in Earthquake Zones, ECCS, Brussels, Belgium. 13. American Society of Civil Engineers, 1995, ANSI/SCE 795 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 tenspan 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 2.05.0384*). 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 (CHHI111781"). 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 MSK64 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 35 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 3132 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 2324 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 2.05.0384*. (Building norms and regulations). Bridges and culverts. Minstroy (Ministry of Construction) of Russia, M, 1996 (in Russian). 2. CHnl111781". (Building norms and regulations). Construction in seismic regions. Minstroy (Ministry of Construction) of Russia, M, 1996. (in Russian):
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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 crosssectional 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 planestrain 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 planestrain critical stress intensity factor, which is known as Kic. In thin sections, where planestress 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 crosshead 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 SetUp
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 _ ML where M is the postfailure "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 rockburstprone 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 forcedisplacement curves indicating the postfailure 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 postfailure energy consumption for dynamic failure compared to static failure (Atype) (ii) rocks that exhibit a decrease in energy consumption for dynamic failure compared to static failure (Btype) The sandstone, shown in Figure 3, for example belongs to Atype behaviour; the rock salt belongs to Btype 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 postfailure energy consumption but this dependence attenuates with increase of WL. The explanation of this WpdWL behaviour, typical for different rocks, is given in [4, 5]. A sevenfold 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. Postfailure portions of the dynamic (1) and static (2) forcedeformation 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. Postfailure 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 Btype 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 thinwaUed 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 postfailure energy consumption for dynamic failure compared to static failure (Atype). 2) rocks that revealed a decrease in energy consumption for dynamic failure compared to static failure (Btype). 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, 389403 (1965). 2. Cook, N.G.W., eL al. Rock mechanics applied to the study of rockbursts. J. So. Afr. Inst. Min. Metall. V. 66, 436528 (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, 1827 (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 Tjoints 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 threedimensional 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 semielliptical crack. Stress intensity factors (SIFs) for tubular Tjoints 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 macrocracks and the behavior of crack growth in tubular joints subjected to the load have been investigated[I3]. 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 timeconsuming work for the FE analysis of joints. In the case of the threedimensional 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 threedimensional crack. SIFs for tubular Tjoints 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 Tjoint 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 xy and yz 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 ydirection. 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 beachmarking 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. Semielliptical crack
Figure 2d. Mesh with a surface crack atweld toes
The tubular Tjoint 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 Tbutt 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 3D Tbutt mesh with a refined weld toe and crack mesh. A series of mapping on the Tbutt 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 Tjoint 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 20node quadratic brick elements (C3D20R) are used elsewhere. 15node quadratic triangular prism elements (C3D15) are used in the crack front. There are 3316 elements and 16703 nodes in this model. Semielliptical 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)[22v+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 quarterpoint 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 endnodes D, E, and at the 88 B, C in the quarterpoint 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 ~: = (34v) for plane strain and ~: = (3v)/(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 Tjoints with a semielliptical 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 semielliptical crack surface. FE1 and FE2 are used to model other parts of tubular Tjoints including chord and brace. The subdomain BE1 is then connected to FE1 and likewise BE2 to FE2. In this model, 820 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
o4o
9
BEFE 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 Tjoints with a surface crack. Therefore, it means that in the stresssWain 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 threedimensional 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 semielliptical 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 nonJcontrolled crack growth, Engineering Fracture Mechanics, Vol. 57, No. 6, (1997) 689699,. 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) 129147. 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, Threedimensional surface crack generator, v3.1, FRCR003, (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. IIintona, 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 agerelated 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 dogbone coupon assembled using a pair of cadmiumplated 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 7075T6 bare aluminium alloy; specimens were machined in the longitudinal (rolling) direction. All specimens were protected with a chromate conversion coating, applied to Specification MILC5541 "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 humidity2/
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 timehistory 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 'snapthrough' type dynamic instability happens in the upward and downward directions respectively. When instability happens, the structural tangent matrix becomes nonpositive. 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. Timehistory 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 timehistory 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 timehistory 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 nonpositive, 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 postbuckling 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 postbuckling state and the postbuckling 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 SingleLayer 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. Haddenday. 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, pp249264. 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),pp409412. 8. Y.B. Yang and M.S. Shieh, Solution Method for Nonlinear Problems with Multiple Critical points. AIAAJournal, Vol.28,No.12" (1990) pp21102116. 9. G.Z. Voyiadjis, G.Y. Shi. Nonlinear Postbuckling Analysis of Plates and Shells by FourNoded Strain Element, AIAA Journal,Vol.30,No.4(1992),ppl1101116.
Structural Failure and Plasticity (IMPLAST 2000) Editors: X.L. Zhao and R.H. Grzebieta 9 2000 Elsevier Science Ltd. All rights reserved.
787
Snapthrough Analysis of Toggle Frame using the software package, NIDA, by I element per member SiuLai CHAN and JianXin 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 beamcolumn 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 overlooked 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 highprecision element. 1
INTRODUCTION Secondorder analysis for checking of the large deflection or the PA 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 P5 effect, referring to the secondorder 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 secondorder 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 secondorder 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 midspan 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 midspan, 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., xMr 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 x0 and xL, v~=0, we have,
v,
_ M r sin(#kx) (~ sin~
Lx M2 sinkx x + vmoSmL "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+11qv. ~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