Light-Weight Steel and Aluminium Structures

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LIGHT-WEIGHT STEEL AND ALUMINIUM STRUCTURES

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LIGHT-WEIGHT STEEL AND ALUMINIUM STRUCTURESFourth International Conference on Steel and Aluminium Structures

Edited by: P. Makelainen R Hassinen Department of Civil & Environmental Engineering Helsinki University of Technology Finland Espoo, Finland 20-23 June 1999

Organized by The Helsinki University of Technology

1999 Elsevier Amsterdann - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

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LOCAL ORGANIZING COMMITTEEP. Makelainen, Helsinki University of Technology, Chairman J. Fagerstrom, Helsinki University of Technology / Espoo City P. Hassinen, Helsinki University of Technology K. Hyry, TSG-Congress Ltd O. Kaitila, Helsinki University of Technology U. Kalamies, Finnish Constructional Steelwork Association J. Kesti, Helsinki University of Technology K. Kolari, Technical Research Centre of Finland M. Malaska, Helsinki University of Technology A. Talja, Technical Research Centre of Finland

LOCAL ADVISORY COMMITTEEP. Makelainen, Helsinki University of Technology, Chairman M. Mikkola, Helsinki University of Technology, Co-Chairman T. Cock, Skanaluminium L.-H. Heselius, Partek Paroc Oy Ab E. Hyttinen, University of Oulu J. Kemppainen, Outokumpu Steel Oy E.K.M. Leppavuori, Technical Research Centre of Finland R. Lindberg, Tampere University of Technology E. Niemi, Lappeenranta University of Technology K. Raty, Finnish Constructional Steelwork Association Ltd P. Sandberg, Rautaruukki Oyj

INTERNATIONAL SCIENTIFIC COMMITTEEP. Makelainen, Finland, Chairman M. Mikkola, Finland, Co-Chairman H.G. Allen, United Kingdom G.A. Akar, Turkey F.S.K. Bijlaard, The Netherlands Y. Chen, China A.M. Chistyakov, Russia K.P. Chong, USA J.M. Davies, United Kingdom D. Dubina, Romania B. Edlund, Sweden K.-F. Fick, Germany G.J. Hancock, Australia E. Hyttinen, Finland T. Hoglund, Sweden M. Ivanyi, Hungary G. Johannesson, Sweden B. Johansson, Sweden M. Langseth, Norway P.K. Larsen, Norway J. Lindner, Germany F.M. Mazzolani, Italy P. van der Merwe, South Africa T.M. Murray, USA J. Murzewski, Poland J.P. Muzeau, France R. Narayanan, USA E. Niemi, Finland T. Pekoz, USA J. Rhodes, United Kingdom J. Rondal, Belgium J. Saarimaa, Finland R. Schardt, Germany R. Schuster, Canada N.E. Shanmugam, Singapore M. Tuomala, Finland T. Usami, Japan W.-W. Yu, USA

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PREFACE ICSAS'99 - The Fourth International Conference on Steel and Aluminium Structures was a sequel to ICSAS'87 held in Cardiff, United Kingdom, to ICSAS'91 held in Singapore and to ICSAS'95 held in Istanbul, Turkey. The objective of the conference was to provide a forum for the discussion of recent research findings and developments in the design and construction of various types of steel and aluminium structures. The conference was concerned with the analysis, modelling and design of light-weight or slender structures in which the primary material is structural steel, stainless steel or aluminium. The structural analysis papers presented at the conference cover both static and dynamic behaviour, instability behaviour and long-term behaviour under hygrothermal effects. The results of the latest research and development of some new structural products were also presented at the conference. The three-day conference was divided into thirteen sessions with six of them as parallel sessions, and with five poster sessions. Five main sessions opened with a keynote lecture; four of these keynotes are published in these proceedings. A total of 76 papers and 30 posters were presented at the conference by participantsfi*om36 countries in all six continents. The Organizing Committee thanks the members of the Intemational Scientific Committee of the conference for their efforts in reviewing the abstracts of the papers contained in the Proceedings, and all the authors for their careful preparation of the manuscripts. The financial support given by the Finnish Constructional Steelwork Association Ltd, the Finnish companies Finnair Oyj, Outokumpu Steel Oy, Partek Paroc OyAb and Rautaruukki Oyj, the Nordic association Skanaluminium and the City Espoo are gratefully acknowledged. Special thanks are due to Local Organizing Committee Members Mr Jyrki Kesti, Mr Mikko Malaska and Mr Olli Kaitila for their most enthusiastic and effective work carried out for the success of the conference.

Pentti Makelainen Professor, D.Sc.(Tech.) Chairman of the ICSAS'99 Conference

Paavo Hassinen Laboratory Manager, M.Sc.(Tech.) Organizing Committee Member of the ICSAS'99 Conference

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CONTENTSSession Al: Structural Modelling and Analysis Keynote lecture: J.M. Davies (GBR), Modelling, Analysis and Design of Thin-Walled Steel Structures J.Y.R. Liew, H. Chen & N.E. Shanmugam (SIN), Stability Functions for Second-Order Inelastic Analysis of Space Frames B. Young & K.J.R. Rasmussen (AUS), Local, Distortional, Flexural and Flexural-Torsional Buckling of Thin-Walled Columns Poster Session PI: Structural Modelling and Analysis Y. Telue & M. Mahendran (AUS), Buckling Behaviour of Cold-Formed Steel Wall Frames Lined with Plasterboard B. Young & G.J. Hancock (AUS), Compression Tests of Thin-Walled Channels with Sloping Edge Stiffeners Y. Itoh, M. Mori & C. Liu (JPN), Numerical Analyses on High Capacity Steel Guard Fences Subjected to Vehicle Collision Impact A. Baptista, D. Camotim (POR), J.P. Muzeau (FRA) & N. Silvestre (POR), On the Use of the Buckling Length Concept in the Design or Safety Checking of Steel Plane Frames B.D. Dunne, M. Macdonald, G.T. Taylor & J. Rhodes (GBR), The Elasto/Plastic Behaviour and Load Capacity of a Riveted Aluminium/Steel Combined Member in Bending J. Tasarek (POL), Shear Buckling of Beam with Scaffold Web Session A2: Buckling Behaviour B.W. Schafer & T. Pekoz (USA), Local and Distortional BuckHng of Cold-Formed Steel Members with Edge Stiffened Flanges M. Kotelko (POL), Collapse Behaviour of Thin-Walled Orthotropic Beams M.C.M. Bakker, H.H. Snijder & J.G.M. Kerstens (NED), Elastic Web Crippling of Thin-Walled Cold Formed Steel Members J.P. PapangeHs & G.J. Hancock (AUS), Elastic Buckling of Thin-Walled Members with Corrugated Elements J. Kesti & P. Makelainen (FIN), Compression Behaviour of Perforated Steel Wall Studs 89 99 3

19

27

37 45

53

61

69 79

107

115 123

X N. Baldassino (ITA) & G.J. Hancock (AUS), Distortional Buckling of Cold-Formed Steel Storage Rack Sections including Perforations

Contents 131

Session A3: Beam-Columns Keynote lecture: R.M. Sully & G.J. Hancock (AUS), Stability of Cold-Formed Tubular Beam-Columns J. Lindner & A. Rusch (GER), Load Carrying Capacity of Thin-Walled Short Columns S. Kedziora, K. Kowal-Michalska & Z. Kolakowski (POL), Ultimate Load of Orthotropic Thin-Walled Beam-Columns J. Rhodes (GBR), Combined Axial Load and Varying Bending Moment in Beam-Columns A.M.S. Freitas & F.G.F. Bueno (BRA), Analysis of Thin-Walled Steel Beam-Columns

141 155

163

171 179

Poster Session P2: Sandwich Structures and Dynamic Behaviour P. Hassinen (FIN), Modelling of Continuous Sandwich Panels P. Rapp, J. Kurzyca & W. Szostak (POL), The Creep and Relaxation in Sandwich Panels with the Viscoelastic Cores J. Valtonen & K. Laakso (FIN), Impact Tests on Steel and Aluminium Road Side Columns J. Ravinger (SVK), Vibration of Imperfect Slender Web E.P. Deus, W.S. Venturini (BRA) & U. Peil (GER), A Cracked Model for Fatique Damage Detection and Evaluation in Steel Beam Bridges M. Al-Emrani, R. Crocetti, B. Akesson & B. Edlund (SWE), Fatigue Damage Retrofitting of Riveted Steel Bridges using Stop-Holes 205 211 189

197

215 223

Session A4: Analysis of Shells and Frames K.T. Hautala & H. Schmidt (GER), Buckling of Axially Compressed Cylindrical Shells Made of Austenitic Stainless Steel at Ambient and Elevated Temperatures W. Guggenberger (AUT), Nonlinear Analysis of General Steel Skeletal Structures Part I: Theoretical Aspects G. Salzgeber & W. Guggenberger (AUT), NonUnear Analysis of General Steel Skeletal Structures - Part II: Computer Program and Practical Applications

233

241

249

Contents V.S. Hudramovych, A.A. Lebedev & V.I. Mossakovsky (UKR), Plastic Deformation and Limit States of Metal Shell Structures with Initial Shape Imperfections M. Ohga, Y. Miyake & T. Shigematsu (JPN), Buckling Analysis of Shell Type Structures under Lateral Loads N.E. Shanmugam (SIN) & R. Narayanan (USA), Strength of Thin Rectangular Box-Columns Subjected to Uniformly Varying Edge Displacements I.H.P. Mamaghani (JPN), Elastoplastic Sectional Behavior of Steel Members under Cyclic Loading

xi 257

265

273

283

Session A5: New Structural Products Y. Chen, Z.Y. Shen, Y. Tang & G.Y. Wang (CHN), Research on Cold Formed Columns and Joints Using in Middle-High Rise Buildings D. McAndrew & M. Mahendran (AUS), Flexural Wrinkling Failure of Sandwich Panels with Foam Joints R.F. Pedreschi (GBR), Design and Development of a Cold-Formed Lightweight Steel Beam G.H. Couchman (GBR), A.W. Toma, J.W.P.M. Brekelmans & E.L.M.G. Van den Brande (NED), Steel-Board Composite Floors 293

301

309

317

Poster Session P3: New Structural Products K. Oiger (EST), Design of Glulam Arched Roof Structures with Steel Joints A. Belica (LUX), Fixed Column Bases in Astron Structures Z. Kurzawa, K. Rzeszut, A. Boruszak & W. Murkowski (POL), New Structural Solution of Light-Weight Steel Frame System, Based on the Sigma Profiles H. Cokun (TUR), Design Considerations for Light Gauge Steel Profiles in Building Construction J. Murzewski (POL), Computer-Aided Design of Steel Structures in Matrix Formulation J. Vojvodic Tuma (SLO), Construction of a 60.000 m^ Steel Storage Tank for Gasoline 327 335

343 351 359 367

Session A6: Developments in Design Y. Itoh & H. Wazaki (JPN), Multimedia Database Using Java on Internet for Steel Structures

377

xii S.A. Alghamdi & M.H. El-Boghdadi (KSA), Design Optimization of Nonuniform Stiffened Steel Plate Girders - LRFD vs. ASD Procedures H. Saal & U. Hornung (GER), Design Rules for Tank Structures - Different Approaches W. Schneider, S. Bohm & R. Thiele (GER), Failure Modes of Slender Wind-Loaded Cylindrical Shells A.M. Chistyakov, F.V. Rass, P.N. Konovalov & N.V. Chernoivan (RUS), Laminated Constructions on the Basis of Thin Metal Sheets in Building B. Uy (AUS) & H.D. Wright (GBR), Local Buckling of Hot-Rolled and Fabricated Sections Filled with Concrete Session Bl: Aluminium Structures C.C. Baniotopoulos, E. Koltsakis, F. Preftitsi & P.D. Panagiotopoulos (GRE), Aluminium MuUion-Transom Curtain Wall Systems: 3-D F.E.M. Modelling of their Structural Behaviour K.J.R. Rasmussen (AUS) & J. Rondal (BEL), Column Curve Formulation for Aluminium Alloys

Contents

385

399

407

415

423

433

441

M. Matusiak & P.K. Larsen (NOR), An Experimental Study of Strength and Ductility of Welded Aluminium Beams F.M. Mazzolani, A. Mandara (ITA), & M. Langseth (NOR), Plastic Design of Aluminium Members According to EC9 A. Starlinger & S. Leutenegger (SUI), On the Design of New Tram Vehicles Based on the Alusuisse Hybrid Structural System Session A7: Aluminium and Stainless Steel Structures Keynote lecture: F.M. Mazzolani (ITA), The Structural Use of Aluminium: Design and AppHcation F. Soetens & J. Mennink (NED), Aluminium Building and Civil Engineering Structures K.F. Fick (GER), Design of Mechanical Fasteners for Thin Walled Aluminium-Structures G. Sedlacek & H. Stangenberg (GER), Numerical Modelling of the Behaviour of Stainless Steel Members in Tests Poster Session P4: Structures at Ambient and Elevated Temperatures R. Landolfo, V. Piluso (ITA), M. Langseth & O.S. Hopperstad (NOR), EC9 Provisions for Flat Internal Elements: Comparison with Experimental Results

449

457

465

475 487

495

503

515

Contents T. Ala-Outinen (FIN), Stainless Steel Compression Members Exposed to Fire A. Talja (FIN), Tests on Cold-Formed and Welded Stainless Steel Members C. Faella, V. Piluso & G. Rizzano (ITA), Modelling of the Cyclic Behaviour of Bolted Tee-Stubs J.S. Myllymaki & D. Baroudi (FIN), A New Method for the Characterisation of the Fire Protection Materials P.P. Gedeonov & T.P. Gedeonova (RUS), Bloating Flame-Retardant Coatings on the Basis of Vermiculite for Steel Buildings Construction Y. Orlowsky, K. Orlowska & T. Shnal (UKR), Fire Resistivity of Steel and Aluminium Constructions Protected by a Bloated Coating

xiii 523 531

539

547

555

561

Session A8: Connections R.A. LaBoube & W.W. Yu (USA), New Design Provisions for Cold-Formed Steel Bolted Connections K. Kolari (FIN), Load-Sharing of Press-Joints in Thin-Walled Steel Structures P. Makelainen, J. Kesti, W. Lu (FIN), H. Pasternak (GER) & S. Komann (GER), Static and Cyclic Shear Behaviour Analysis of the Rosette-Joint P. Makelainen & O. Kaitila (FIN), Study on the Behaviour of a New Light-Weight Steel Roof Truss C.A. Rogers & G.J. Hancock (AUS), Bearing Design of Cold Formed Steel Bolted Connections R.B. Tang & M. Mahendran (AUS), Pull-Over Strength of Trapezoidal Steel Claddings R.H. Fakury, F.A. de Paula, R.M. Gon9alves & R.M. da Silva (BRA), Investigation of the Causes of the Collapse of a Large Span Structure 569 577

585 593

601 609

617

Session B2: Aluminium and Stainless Steel Structures K.J.R. Rasmussen (AUS) & J. Rondal (BEL), Column Curves for Stainless Steel Alloys G. De Matteis (ITA), L.A. Moen, O.S. Hopperstad (NOR), R. Landolfo (ITA), M. Langseth (NOR) & F.M. Mazzolani (ITA), A Parametric Study on the Rotational Capacity of Aluminium Beams Using Non-Linear FEM R.M. Gon9alves, M. MaHte & J.J. Sales (BRA), Aluminium Tubes Flattened (Stamped) Ends Subjected to Compression - A Theoretical and Experimental Analysis 627

637 647

xiv G. De Matteis, A. Mandara & F.M. Mazzolani (ITA), Interpretative Models for Aluminium Alloy Connections F.M. Mazzolani, C. Faella, V. Piluso & G. Rizzano (ITA), Local Buckling of Aluminium Channels under Uniform Compression: Experimental Analysis B. Boon & H. Weijs (NED), Local Impact on Aluminium Plating J.S. Myllymaki & R. Kouhia (FIN), Creep Buckling of Metal Columns at Elevated Temperatures

Contents

655

663 671

679

Session A9: Design for Hygrothermal, Vibration and Fire Effects Keynote lecture: G. Johannesson (SWE), Design for Hygrothermal Performance and Durability of Insulated Sheet Metal Structures J. Nieminen & M. Salonvaara (FIN), Long-Term Performance of Light-Gauge Steel-Framed Envelope Structures M. Feldmann, C. Heinemeyer & G. Sedlacek (GER), Substitution of Timber by Steel for Roof Structures of Single-Family Homes J. KuUaa & A. Talja (FIN), Vibration Performance Tests on Light-Weight Steel Joist Floors A.Y. ElghazouH & B.A. Izzuddin (GBR), Significance of Local Buckling for Steel Frames under Fire Conditions

689

703

713

719

727

Poster Session P5: Composite Structures M. Shugyo & J.P. Li (JPN), Elastoplastic Large Deformation Analysis of Concrete-Filled Tubular Columns J. Brauns (LAT), Resistance of Composite Section to Axial Loads and Bending: Design and Analysis A.K. Kvedaras (LTU), Light-Weight Hollow Concrete-Filled Steel Tubular Members in Bending C. Faella, V. Consalvo & E. Nigro (ITA), An "Exact" Finite Element Model for the Linear Analysis of Continuous Composite Beams with Flexible Shear Connections

737

745

755

761

Session AlO: Special Features in Modelling and Design J. Outinen & P. Makelainen (FIN), Behaviour of a Structural Sheet Steel at Fire Temperatures

771

Contents R.M. Schuster (CAN), Perforated Cold Formed Steel C-Sections Subjected to Shear (Experimental Results) H. Pasternak & P. Branka (GER), Carrying Capacity of Girders with Corrugated Webs J. Rhodes, D. Nash & M. Macdonald (GBR), An Examination of Web Crushing in Thin-Walled Beams P. Konderla & J. Marcinowski (POL), Experimental Investigations and Modelling of Steel Grids T. Yamao, T. Akase & H. Harada (JPN), Ultimate Strength and Behavior of Welded Curved Arch Bridges

xv 779 789

795

803

811

Session B3: Response to Dynamic and Alternating Loads Y. Itoh, T. Ohno & C. Liu (JPN), Behavior of Steel Piers Subjected to Vehicle Collision Impact E. Yamaguchi, Y. Goto, K. Abe, M. Hayashi & Y. Kubo (JPN), Stability Analysis of Bridge Piers Subjected to Cyclic Loading P. Kujala & K. Kotisalo (FIN), Fatigue Strength of Longitudinal Joints for All Steel Sandwich Panels T. Usami & H.B. Ge (JPN), Local and Overall Interaction Buckling of Steel Columns under Cyclic Loading M. Yamada (JPN), Steel Shear Panels for Anti-Seismic Elements A. Salwen & T. Thoyra (SWE), Results from Low Cycle Fatigue Testing Keyword Index Author Index 845 853 861 869 875

821

829

837

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Session Al STRUCTURAL MODELLING AND ANALYSIS

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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) 1999 Elsevier Science Ltd. All rights reserved.

MODELLING, ANALYSIS AND DESIGN OF THIN-WALLED STEEL STRUCTURESJ Michael Davies Manchester School of Engineering, University of Manchester Manchester M13 9PL, UK

ABSTRACT This paper provides an overview of the calculation models that are currently available, and used in codes of practice, for the design of thin-walled, cold-formed steel structures. Their limitations are discussed with particular reference to benchmark analyses provided by "Generalised Beam Theory".

KEYWORDS Beams, Buckling, Cold formed steel. Columns, Design, Generalised Beam Theory, Purlins.

INTRODUCTION In recent years, the practical usage of cold-formed steel sections has grown rapidly and there has been an evolution in the methods and procedures available for their design. This is evidenced in the various national and international standards, not least Eurocode 3 : Part 1.3 (CEN 1996). It is implicit that practical designers do not wish to use a sophisticated and detailed analysis, such as is available using the finite element method. They are looking for much simpler calculation procedures which can be carried out manually or, at worst, with a simple spreadsheet. However, some aspects of the behaviour of cold formed sections are extremely complex so that this is an aim which is difficult to meet. The primary alternative to deriving simple models with which to describe complex behaviour is to use testing. However, this has to be carried out full scale and is inevitably expensive. In the present state of the art, it is rarely justified and then only when a large number of identical elements are to be built so that the cost can be offset against the resulting economic gains. This paper attempts to give an overview of the current state of the art by reviewing the main phenomena which require to be modelled and the design models that are available. Sophisticated analysis or testing are yardsticks by which the success or otherwise of these models may be judged. It is shown that, although many of tiiese behavioral phenomena have their own specific design models, Generalised Beam Theory (GBT) can embrace most of the required characteristics within a single unified approach.

BEHAVIOUR OF COLD-FORMED STEEL SECTIONS Cold formed steel sections are characterised by the thinness of the material and this results in a number of failure modes or behaviour characteristics which, while they may be present in hot-rolled construction, are far less prominent. As an example, consider the cold-formed steel load-bearing cassette wall shown in Fig. 1, which may be considered to be the structural element of the external wall of a house or similar low rise construction. It is subject to axial compression (from the floors or roofs above), bending (in both directions from wind suction or wind pressure) and shear (from diaphragm action resisting the wind on walls at right angles).

Axial load from storey above

Wind load causing bending

Wind load causing shear

Figure 1

Load system in a cassette wall

The following phenomena must be modelled if this construction is to be safely designed. Most of these arise as a consequence of the buckling of thin-walled elements in compression: local buckling of plate elements of the section which are in compression buckling of intermediate stiffeners (which could be in any of the plate elements) buckling of lip stiffeners and interaction with plate buckling and cross-section distortion flange curling of the wide flange in either tension or compression. When this flange is in compression, there is interaction with local plate buckling and stiffener behaviour local buckling of the wide flange in shear. Another example is a singly symmetric upright in a pallet rack structure which carries bending about both axes as well as a substantial axial load. It has arrays of perforations which allow beams to be fixed using clips at levels which do not need to be predetermined. This results in: local buckling of plate elements aggravated by the presence of arrays of holes or slots lateral or lateral-torsional buckling with complex boundary conditions distortional buckling.

Finally, consider the case of a purlin, loaded through and to some extent supported by profiled steel cladding and continuous over one or more intermediate supports. Here the cladding partially restrains the beam against lateral torsional buckling and this must be taken into account if economic designs are to be obtained. However, both local and distortional buckling are also present and another significant problem concerns the behaviour at the intermediate support. First yield here does not constitute failure and considerable economic gains can be made if the design allows some elastic-plastic redistribution of bending moment. A similar problem arises in the design of profiled steel floor decking and roof and wall cladding. The above list is probably not exhaustive but it does paint a picture of the considerable number of phenomena that must be modelled if a modem design standard is to cover all aspects of the design of cold formed sections. Because of limitations of length, this paper will concentrate on the three generic buckling phenomena, and interactions between them, and leave other considerations such as behaviour in shear, flange curling and crushing at points of support for another occasion.

OVERVIEW OF BUCKLING PHENOMENA From a fundamental point of view, buckling phenomena can be divided into three categories, namely local, distortional and global. Local buckling is characterised by the relatively short wavelengtii buckling of individual plate elements while the fold lines remain straight. Although local buckling phenomena can be complex, they have been researched in detail ever since the early days of cold-formed sections and can be said to be well understood. These will be considered first. A similar situation pertains to global buckling which is characterised by "rigid body" movements of the whole member such that individual cross-sections rotate and translate but do not distort in shape. Euler buckling of columns and lateral torsional buckling of both beams and columns fall into this category. Here, many cases of practical significance can be analyzed by using explicit solutions of the governing differential equations. Distortional buckling is more problematic. It is characterised by distortion of the cross-section such that the fold lines move relative to each other. The practical significance of distortional buckling has only been recognised relatively recently, although considerable strides have already been made towards generating and validating suitable models for practical design (Davies and Jiang 1998b). It is generally obvious into which of the above categories the phenomena described above fall. However, some confusion arises with regard to the buckling of plates with a free edge, including the buckling of simple stiffening lips. Arguments may be made in favour of treating the buckling of these unstiffened plate elements as either local or distortional. From the practical point of view it makes little difference though design codes generally treat unstiffened plate elements as a special case of local buckling. However, the modem trend towards stiffening lips of more complex shape suggest that these may be better considered by theories applicable to distortional buckling. Each of these generic categories of buckling are capable of mutual interaction. Empirical models for the interaction of local and global buckling are included in most design codes but there is little fundamental knowledge of the interaction of distortional buckling with the other modes. It will be shown that Generalised Beam Theory (GBT) can consider each of these categories of buckling and has useful information to offer in each case. Buckling modes may be considered individually or in specified combinations. GBT shows to particular advantage in the case of distortional buckling and also in investigating specific interactions.

6 TOOLS FOR RESEARCH AND DEVELOPMENTThe currently available design models have been generated and validated by a combination of sophisticated analysis and testing. In recent years, testing is being used less and less as numerical methods of analysis become ever more sophisticated. Because of the complexity of the phenomena involved, classical methods based on explicit solutions of the governing differential equations exist for relatively few of the practical situations described above. It follows that the primary method available to researchers is the finite element method and, in principle, all of the phenomena described above can be modelled in this way. The primary building-block for the analysis of cold-formed sections is the second-order thin shell element which can accomodate the full range of section shapes and buckling phenomena. If a non-linear stress-strain relationship is incorporated into the analysis, such elements can model yielding and elastic-plastic buckling. Contact elements, connection elements and large deflection theory add to the huge range of facilities that are available to the analyst. Cold formed sections are, by nature, prismatic and this opens up the possibility of using analytical methods that are specifically designed for prismatic members. The finite strip method falls into this category and has the advantage over the finite element method of requiring less computer time and memory. It is, nevertheless, a numerical method which requires serious computing power and gives answers to specific problems. Another possibility, applicable to global buckling only, is to use the 7 degree of freedom prismatic member finite element first derived by Barsoum and Gallagher (1970). "Generalised Beam Theory" (GBT) is also applicable to prismatic members and has been compared to the finite strip method. However, it is much more than an alternative "method" and, in its nature as a new "theory", GBT can shed fundamental light on some of the phenomena being modelled. Furthermore, in certain cases, notably those associated with distortional buckling, second order GBT can offer explicit solutions to problems that could previously only be solved by numerical methods. By attempting to give a global view of the problems of modelling the behaviour of cold formed sections and then relating these, where possible, to GBT, this paper tries to define the current state of the art and to give GBT its rightful place within it.

GENERALISED BEAM THEORY (GBT) Generalised Beam Theory is a unification and generalisation of the familiar 1st and 2nd order theories for the behaviour of prismatic beams and columns and makes a fundamental contribution to structural mechanics. Space precludes a detailed description of GBT which has been adequately described elsewhere (eg Davies and Leach 1994a and 1994b, Davies and Jiang 1998a). However, as an aid to the discussion of design models which follows, some of its main characteristics are emphasised. GBT operates in terms of displacement modes which are chosen to be "orthogonal" which means that they are uncoupled in first-order analysis. This ensures that the ftindamental modes of axial displacement (mode 1), bending about the principal axes (modes 2 and 3) and torsion (mode 4) are isolated from each other. Fundamental local and distortional modes (modes 5 and above) are similarly identified. In second-order analysis, these displacement modes become buckling modes which may or may not become coupled depending on the nature of the problem and the wishes of the analyst. GBT has two parts. The first is essentially an analysis for section properties which includes the familiar properties such as cross-sectional area, second moment of area about the principal axes, torsional and warping constants, etc which are associated with the global (rigid body) modes 1-4. It also includes other section properties associated with local and distortional modes, second order effects

etc which may be less familiar or have no obvious meaning in conventional structural mechanics. The calculations for this first part of GBT can be rather complex and generally require the use of a computer program. As this is fundamental to the practical use of GBT, the author and his colleague Dr Jiang have placed the software for this calculation for open cross sections in the public domain. It is available from them at the address given at the head of this paper or via e-mail at jmdavies@fs 1. eng. man. ac.uk. The second part of GBT utilises these section properties, together with the fundamental differential equations, in order to obtain solutions to specific problems. In the general case, numerical methods have to be used and the finite difference method has generally been used for second-order problems. This gives accurate solutions in a small fraction of the time required by the finite element or finite strip methods. Furthermore, some very simple explicit solutions can be obtained using half sine wave displacement functions. These have particular application in generating usable design models. Thus the critical stress resultant and the corresponding half wavelength for single mode buckling in mode 'k' due to a stress resultant W applied in mode 'i' are (Davies and Leach 1994b):^'^^ = - ^ ( 2 V / E H: ^ + G *D) (1)

E^l

(2)

In these equations, E and G and the elastic and shear moduli respectively and the remaining terms all section properties. Equations (1) and (2) allow a particularly simple calculation to be made any individual buckling mode, including the distortional modes. No other method known to authors allows the distortional modes to be isolated in this way. When two or more modes included in the analysis, the solution of an elementary eigenvalue problem is required.

are for the are

It should also be noted that, when generating the section properties in the first part of GBT, free movement of the section may optionally be restrained. This allows, for example, lateral movement of the top flange of a purlin to be restrained or the stiffening effect of the sheeting to be simulated by an elastic torsional restraint of specified stiffness. Restraints of this nature alter the fundamental deformation modes in interesting ways but do not otherwise change the second part of GBT.

THE AYRTON-PERRY EQUATION Second-order GBT gives rise to elastic buckling loads whereas practical cold-formed sections generally fail in a combination of buckling and yielding. Combined buckling and yielding can, of course, be considered using non-linear finite element or finite strip analysis but this is very cumbersome. However, it is now apparent that solutions that are sufficiently accurate for all practical purposes can generally be obtained by combining the theoretical load (or stress) for elastic (bifurcation) buckling with the corresponding yield load (or stress) using the Ayrton Perry equation:X = ; -TTT ^ut X ^ 1

* - [ ) - Vr d^with 4) = 0.5[l + a(X - 0.2) + V]

(3)

where x oc X

= the reduction factor for buckling with respect to the unbuckled capacity = an imperfection factor = the relative slendemess in the relevant buckling mode

In Eurocode 3: Part 1.3 (CEN 1996), this equation is applied to the flexural buckling of columns (clause 6.2.1) and to the lateral torsional buckling of beams (clause 6.3) with X equal to and M respectively. It is equally valid when used to allow second-order elastic GBT

N

p^

N

solutions to be used to give reliable estimates of the failure loads for both beams and columns in a wider range of practical situations. For both beams and columns, OL can take one of a range of values (0.13, 0.21, 0.34, 0.39) depending on the cross-section under consideration and its susceptibility to residual stresses, imperfections etc.

MODELS FOR COLD-FORMED SECTION DESIGN Effective width and effective cross-section The primary "building block" for cold formed section design is the concept of "effective width" which is illustrated in Fig. 2. Slender plate elements in uniform compression are designed to operate in the post-buckled condition. The complex stress distribution may then be simplified to the two stress blocks shown with the same maximum stress and stress resultant but with reduced width bgff. The reduced properties of effective pla^e elements in compression may then be combined with the full width of plate elements in tension to give an "effective section" for use in stress calculations.Actual stress distribution 7 Simplified equivalent stresses^/^eff/2

/

M I "^ ./ "^ ./ MFigure 2

1^

Effective width of a plate element in uniform compression

The usual effective width formula is the semi-empirical formula due to Winter (CEN 1996): pb where if Xp ^ 0.673; if Xp > 0.673; in which the plate slendemess Xp is given by: p p 1.0 = 1.0 0.22 \ 1P / P

(4)

_yb

^

Nwhere f^h E k^ = = = =

1.052^ t ^ Ek

(5)

compressive stress in the plate element critical stress for elastic buckling of the plate element Young's modulus buckling factor = 4.0 for a simply supported plate in uniform compression = 0.43 for an outstand plate element with one edge free

In the above equations, the theoretical value of a^r for the elastic buckling of a long uniformly compressed plate with simply-supported longitudinal edges leads directly to k^ = 4.0. Other stress and boundary conditions can also be substituted and the approach remains valid with different values of k. However, in a complete cross-section, the buckling stress may be enhanced by the elastic support that the buckling element receives from other elements of the cross section that are not at their buckling stress. This may lead to some rather complex considerations. Eurocode 3: Part 1.3 (CEN 1996) gives a comprehensive table of values of k^ for different stress distributions across a plate element with either both edges simply supported or one edge simply supported and one edge free. However, it ignores the interaction with other elements of the crosssection. Conversely, BS 5950: Part 5 (BSI1987) has a less detailed treatment of the alternative stress conditions but does allow account to be taken of the enhancement of k^ for a limited range of crosssections. The American code (AISI1996) has an even more restricted treatment of these phenomena. It follows that none of the available code of practice models is fully comprehensive or totally accurate over the whole range of cross-sections and stress distributions which may be encountered in practice. Second-order GET, however, is capable of providing accurate values of acr for any cross-section under any stress distribution. It is merely necessary to know the relevant section properties and to solve the governing equations for the relevant load case. From the theoretical point of view, it is possible to take this process a stage further. What is really required is not a^r but the stress distribution in the post-buckled condition. GET has a third-order (large deflection) capability which can model this directly. However, this has only been attempted at the research level and there is little information in the public domain. Interaction of local buckling with global buckling of columns Local buckling is common to all types of cold formed section members and therefore potential interaction with other buckling modes is common. Most codes adopt a simple model for dealing with the interaction between local and global buckling of colunms in which the capacity of the section in the absence of global buckling in the Ayrton-Perry equation is based on the effective rather than the gross cross-section. This is found to give results that are adequate for all practical purposes. From the fundamental point of view, investigating the interaction between local plate buckling and global column buckling is difficult because it is necessary to consider the post-buckling behaviour of the plate elements. Third-order GET offers possibilities here that have not been fully explored.

10 Edge and intermediate stiffeners There is a group of buckling problems that may advantageously be modelled by treating an appropriate part of the cross-section as a compression member with a continuous elastic restraint representing the influence of the remainder of the section. The buckling of lip and intermediate stiffeners falls into this category. Some types of distortional buckling provide other examples. Eurocode 3: Part 1.3 (CEN 1996) uses this procedure for both lip and intermediate stiffeners, as shown in Fig. 3. The stiffness 'K' of the continuous elastic restraint is given by u/6 as illustrated for C and Z sections in Fig. 3(c). This value of K is then used in the classical equation for the buckling of an infinitely long axially loaded beam on an elastic foundation (Timoshenko and Gere, 1961) in order to calculate the theoretical buckling stress and hence the relative slendemess X: ^/KE^

and

Vb

(6)

N

where A^ and I^ are the cross sectional area and second moment of area respectively of the stiffener. This relative slendemess can then be used in the Ayrton-Perry equation with a = 0.13, as described above, in order to predict the reduction factor x for buckling. This reduction factor then gives a reduced thickness of t^ed = xt for the stiffener.

a) Actual system

b) Equivalent system

Compression

Bending

Compression

Bending

c) Calculation of 5 for C and Z sections

Figure 3

Buckling models for stiffeners based on beam on elastic foundation theory

Experience suggests that models of this type can be successful provided that the calculation of the restraint is realistic. Here, the calculation is often complicated by local buckling in the plate elements adjacent to the stiffener. A recent calibration study by Kesti (1998) on C-sections with lip stiffeners has compared the results given by Eurocode 3: Part 1.3 with comparable results obtained using the Australian code (AS 1996), which is in effect the model developed by Lau and Hancock (1987) which is considered in the next

11 section, and GBT. Kesti found that Eurocode 3 gave rather variable results for the critical buckling stress, the ratio of EC3/GBT varying within the range 0.62 - 1.85. However, this scatter reduced significantly when the Ayrton-Perry equation was used to compare the corresponding ultimate loads. Much better correlation was obtained between the method given in the Australian code and GBT. Distortional buckling under axial compressive load A more general model for distortional column buckling, which was originally developed by Lau and Hancock (1987), is now well established and is shown in Fig. 4. In contrast to GBT, in which the whole cross-section is considered, the analytical expressions are based on aflangebuckling model in which the flange is treated as a compression member restrained by a rotational and a translational spring. The rotational spring stiffness k^ represents the torsional restraint from the web and the translational spring stiffness k^ represents the restraint to translational movement of the cross section.Flange Shear Centre

Figure 4 Analytical model for distortional column buckling Lau and Hancock showed that the translational spring stiffness k^ does not have much significance and ^ the value of k^ was assumed to be zero. The key to evaluating this model is to consider the rotational spring stiffness k^ and the half buckling wavelength X, while taking account of symmetry. Lau and Hancock gave a detailed analysis in which the effect of the local buckling stress in the web and of shear and flange distortion were taken into account in determining expressions for k^ and X. This gives rise to a rather long and detailed series of explicit equations for the distortional buckling stress. Notwithstanding their cumbersome nature, these are now included in the Australian code (AS 1996). Davies and Jiang (1996a) carried out a systematic comparison of the results given by this model and those given by GBT. As with all such models, the outcome is rather sensitive to the value of k^. A modest refinement of the expression for this value improved the comparison, after which the model shown in Fig. 4 was found to give excellent accuracy. It should be noted that distortional buckling proved to be rather sensitive to the boundary conditions. The models discussed above are based of a half sine wave displacement function and this gives a lower bound value of the buckling stress. Unless great care is taken with the end conditions, stub column tests are likely to give higher values of the failure stress and are, therefore, potentially unsafe. In practice, it is not possible to make a fixed-ended column test sufficiently long to determine the lower bound distortional buckling stress. Distortional buckling in bending The buckling behaviour of beams bent about the major axis differs from that of columns in a number of respects. Figure 5(a) shows a typical cold formed section beam. Ignoring considerations of local buckling, which do not add anything to the argument here, the section has 6 natural nodes and therefore there are 6 orthogonal modes of buckling. These are shown in Figure 5(b) and are 4 rigidbody modes and 2 distortional modes.

12

"n \-r

" " ^ ^ ^

'=.

J 1.1

(a) cross-section Figure 5

(b) 6 orthogonal modes Buckling modes for a lipped channel section beam

When the beam is bent about the major axis, it is well known that individual lateral and torsional modes have no significance and the only rigid-body buckling mode is a combination of modes 3 and 4, namely lateral torsional buckling. In the same way, the distortional modes 5 (symmetrical) and 6 (antisymmetrical) have no individual significance and the only distortional mode is a combination of the two such that most of the distortion takes place in the compression flange with the flange in tension playing a minor role. Assuming again that the bucking mode is a half sine wave, GBT again allows a simple calculation for the case of pure bending. Analytical expressions for the distortional buckling of thin-walled beams of general section geometry under a constant bending moment about the major axis have been developed by Hancock (1995). These analytical expressions were based on the simple flange buckling model shown in Fig. 6 (together with an improvement proposed by Davies and Jiang 1996b) in which the flange was again treated as a compression member with both rotational and translational spring restraints in the longitudinal direction. The rotational spring stiffness k^ and the translational spring stiffness k^ represent the torsional restraint and translational restraint from the web respectively. In his analysis, Hancock again chose the translational spring stiffness k, to be zero.Shear centre

kJ(a) Hancock's model Figure 6

j,- 0.673: ,

p = 11.0 -

^ U

The elastic buckling moment M r for local or distortional buckling may be readily obtained from either ^ the finite strip method or GBT using half sine wave displacement functions. The proposals of Schafer and Pekoz (1998) offer two possible additional refinements to the basic procedure described above. Noting that there may be decreased post-buckling capacity in the case of distortional buckling, a reduction factor is suggested for this case. Alternatively, a modified equation may be used for p in order to obtain better agreement with the experimental results. The above proposal has been calibrated against the AISI(1996) specification for a total of 574 test results for unrestrained beams obtained by 17 researchers and covering a wide range of section shapes. It is shown that the initial form of the method is conservative and at least as accurate as the AISI specification. The reduction factor for distortional buckling does not improve matters but the second proposed improvement results in a distinct improvement on the AISI design rules. Global buckling of columns Cold-formed section columns generally have a single axis of symmetry and fail in either flexural or torsional-flexural buckling, possibly with interaction with either local or distortional buckling. Discounting, for the present, these possible interactions, the design model used by all codes of practice is to use the classical equations of structural mechanics to determine the theoretical elastic buckling stress of a pin-ended member buckling in a half sine wave. The influence of yielding of the steel is then taken into account by using the Ayrton Perry equation as discussed above. Other boundary conditions are taken into account on the basis of "effective length". The most general case embraced by the conventional theory for column buckling is that of a section with no axis of synmietry loaded through its centroid. Using a familiar notation, the critical load PTF

15 of a section of length 'L' buckling in a combination of torsion and flexure is given by:.2

PIP

- EI,^ P _ - EI ?^ - E T + GJ Ao = 0 (8)

ZQP^

Yo^T

where yo and ZQ are the coordinates of the shear centre and IQ is the polar second moment of area about the shear centre. Completely analogous equations can be set up using GET by considering the three rigid body modes 2, 3 and 4 (bending about the two principal axes and torsion) with an applied load ^W (axial load) which is constant over the length of the member and assuming that all modes buckle in a half sine wave with the same wavelength. GET, of course, not only offers this elegant account of the "rigid body" buckling theory but also allows these rigid body modes to be combined with the local and distortional modes. We may note here that if, the section has one axis of symmetry, yo = 0 and minor axis buckling becomes uncoupled. The equation for the buckling load then simplifies to:PTP

- EI,ZQP^

ZQP^

(9) r^P^ ET + GJL2

which is the equation usually given in codes of practice. Lateral-torsional buckling of beams bent about the major axis The torsional-flexural buckling of unrestrained beams is complicated because sections may have two axes of symmetry (I-sections), a single axis of symmetry (C-sections) or may be approximately pointsymmetric (Z-sections). Furthermore, the stress resultant causing buckling (bending moment) is not generally constant along the length of the member. Eurocode 3 (CEN 1996) avoids these complicated considerations by giving the design equations in terms of M^, the elastic critical moment of the gross cross-section for lateral torsional buckling about the relevant axis. The designer is, therefore, left to wrestle with the mysteries of lateral torsional buckling without any help from the code. BS 5950 (ESI 1987) gives the following equation for equal flange I-sections and symmetrical channel sections of depth D bent in the plane of the web and loaded through the shear centre: M^ (10) 20 rD

where the expression within the brackets [ ] may conservatively be taken as 1.0. Similar expressions are given for Z-sections bent in the plane of the web and T-sections. Cb is a semi-empirical coefficient which takes account of the variation of bending moment along the member which may be

16 conservatively assumed to be unity. As in all similar cases, the interaction between buckling and yielding is taken into account using the Ayrton-Perry equation. The above equation arises directly from a solution of the governing differential equations for a member subject to a uniform bending moment and buckling in a half sine wave. Evidently, there will always be severe limitations on the number of situations which can be modelled by explicit solutions of rather complex differential equations and, in any case, solving such equations is not to the taste of many practising engineers. Yet again, GBT can come to the rescue by offering relatively simple yet precise solutions to all such problems. When the interaction of lateral-torsional buckling and local buckling is significant, the analysis becomes highly problematic. For example, local buckling of the compression flange of a C-section purlin immediately renders this flange "less effective" than the tension flange so that the section, which originally had a horizontal axis of symmetry, becomes completely unsymmetrical. The author knows of no simple model for this complex situation. However, test results have been reproduced very successfully by GBT and the Ayrton-Perry equation (Davies and Leach 1996). In practice, completely unrestrained beams are rare because beams generally receive restraint from the members that they support. In many cases, this restraint is sufficient to prevent lateral-torsional buckling so that design may be based on the moment of resistance of the cross-section without any need to consider global buckling. Much more interesting are situations where this restraint is partial, as typified by a purlin supporting profiled metal sheeting. With the proliferation of cladding types, there has recently been considerable interest in developing design models for partially-restrained beams and Eurocode 3: Part 1.3 (CEN 1996) includes one of these which is related to the model described by Pekoz and Soroushian (1982) discussed above. With GBT, the effect of continuous restraint from the sheeting is included in the section properties and this clearly provides a yardstick model whereby other models may be assessed - either for distortional buckling, as discussed earlier, or for lateral-torsional bucklmg, as considered in this section. Lateral-torsional buckling of beams bent about the minor axis British Standard 5950 Part 5 "Code of practice for design of cold formed sections" (BSI 1987) contains the following statement: "Lateral buckling, also known as lateral torsional buckling, will not occur if a beam is loaded in such a way that bending takes place solely about the minor axis..." This statement is incorrect. However, a recent paper of some distinction (Buhagiar et al 1994) which attempts to study this subject and point out the error is also incorrect. Such is the potential for misunderstanding in what at first sight appears to be a relatively simple subject. Davies and Jiang (1998a) show that "Generalised Beam Theory" (GBT) offers a simple and relatively foolproof account of the problem. By comparing the classical solutions with the solutions given by GBT, the true nature of the lateral-torsional buckling modes of thin-walled beams bent about the minor axis is revealed. We consider the coupled instability of GBT modes 2 and 4 (bending about the z-axis and torsion) subject to ^W = MLT (bending about the y-axis). The classical solution (Buhagiar et al 1994), is: (11)

"

2

P^^-'

.ro^P^M,

17 where jSy is a somewhat complex section property which is given explicitly by GBT. The problems in the earlier paper arose primarily because of errors in calculating 13y. With the aid of GBT, Davies and Jiang (1998a) show that the global buckling mode of a beam bent about its minor axis is almost a case of pure torsion so that it is generally sufficient to use the simpler equation:_2 T -1

EC + GJ

(pure torsional buckling)

(12)

thus avoiding the complications of calculating jSy.

CONCLUSIONS In the design of cold-formed sections for axial load and bending, there are three generic types of buckling which have to be considered, namely local, distortional and global. Each of these has its own characteristic design model. Thus, local buckling is best modelled by an effective width approach. Distortional buckling is best approached by models based on beam-on-elastic-foundation theory. Global buckling can be tackled by explicit solutions of the governing differential equations. For cold-formed steel colunms and beams with the proportions typically used in practice, distortional buckling may often be critical. In practical design, it is also the most difficult to deal with. Generalised Beam Theory (GBT) provides a particularly appropriate tool with which to analyze distortional buckling in isolation and in combination with other buckling modes. It also provides a yardstick with which other simplified methods may be assessed. In general, there is little interaction between the distortional and global modes and it is sufficient to consider the critical distortional mode in isolation. GBT then provides an explicit expression for the critical buckling stress and half wavelength whereas the alternative approaches attempt to calculate these quantities on the basis of simplified models based on a rotation of the compression flange about its junction with the web. These models lead to quite complex calculations but are potentially quite accurate. They are, however, rather sensitive to the rotational stiffness assumed to represent the interaction of the flange with the remainder of the section. Initial assumptions have been shown to require refinements which are discussed in the paper. Although the design approaches to local and global buckling are more mature, it should not be assumed that adequate design models are available for all situations likely to arise in practice. The paper discusses the limitations of the available models and shows that GBT has much to offer here also.

REFERENCES AISI. (1996). Specification for the design of cold-formed steel structural members. American Iron and Steel Institute. AS. (1996). Cold-formed steel structures. (Revision AS 4600-1988). Australian / New Zealand Standard. Committee BD/82. BSI. (1987). BS 5950: Part 5, British Standard: Structural use of steelwork in building: code of

18 practice for design of cold formed sections. British Standards Institution. CEN. (1996). Eurocode 3: Part 1.3, Design of Steel Structures: General rules: supplementary rules for cold formed thin gauge members and sheeting, ENV 1993-1-3. Barsoum R. S. and Gallagher R.H. (1970). Finite element analysis of torsional and torsional-flexural stability problems. Int. J. for Numerical Methods in Engineering. Vol. 2. 335-352. Buhagiar. D., Chapman J.C. and Dowling P.J. (1994). Lateral torsional buckling of thin-walled beams subject to bending about the minor axis. The Structural Engineer. 72, No. 6. 93-99. Davies J. M., Jiang C. and Leach P. (1994). The analysis of restrained purlins using Generalised Beam Theory, 12th Int. Speciality Conf on Cold-Formed Steel Structures, St. Louis, Missouri. 109120. Davies J. M. and Jiang C. (1996a). Design of thin-walled columns for distortional buckling, 2nd Int. Speciality conf. on Coupled Instabilities in Metal Structures, CIMS 96. Liege. 165-172. Davies J. M. and Jiang C. (1996b). Design of thin-walled beams for distortional buckling, 13th Int. Speciality Conf. on Cold-Formed Steel Structures, St. Louis, Missouri. 141-153. Davies J. M. and Jiang C. (1996c). Design of thin-walled purlins for distortional buckling, TWS Bicentenary Conf. on Thin-Walled Structures, Strathclyde, Glasgow. Davies J. M. and Jiang C. (1998a). Generalised Beam Theory (GBT) for coupled instability problems. Part IV of "Coupled Instabilities in Metal Structures", Ed. J Rondal, International Centre for Mechanical Sciences, Courses and Lectures No. 379, Springer Wein New York. 151-223. Davies J. M. and Jiang C. (1998b). Design for distortional buckling. / Construct. Steel Res. 46, Nos. 1-3. 174-174. Davies J. M. and Leach P. (1992). Some Applications of Generalised Beam Theory, 11th Int. Speciality Conf. on Cold-Formed Steel Structures, St. Louis, Missouri. 479-501. Davies J. M. and Leach P. (1994a). First-Order Generalised Beam Theory. J Construct. Steel Research. 31. 187-220. Davies J. M., Leach P. and Heinz D. (1994b) Second-Order Generalised Beam Theory. / Construct. Steel Research. 31. 221-241. Davies J. M. and Leach P. (1996). An experimental verification of the Generalised Beam Theory applied to interactive buckling problems, Thin-Walled Structures. 25, No. 1. 61-79. Hancock G. J. (1995), Design for distortional buckling of flexural members, Proc. Third International Conference on Steel and Aluminium Structures, Istanbul, (also in Thin-Walled Structures, 27, 3-12. 1997). Kesti J. (1998). Local and distortional buckling of thin-walled colunms. To be published. Lau S. C. W. and Hancock G. J. (1987). Distortional Buckling Formulas for Channel Columns, Journal of Structural Division. ASCE. 113(5). 1063-1078. Pekoz T. and Soroushian P. (1982). Behaviour of C- and Z-purlins under wind uplift, Report No. 812, Dept.of Civil Engineering, Cornell University, Ithaca, NY. Schafer B.W. and Pekoz T. (1998). Direct strength prediction of cold-formed steel members using numerical elastic buckling solutions. Thin-Walled Structures, Research and Development. Proc. 2nd International Conf. on Thin Walled Structures. Singapore, Dec. 1998, Elsevier. 137-144. Timoshenko P. and Gere J. (1961). Theory of Elastic Stability, McGraw Hill book company, New York.

Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) 1999 Elsevier Science Ltd. All rights reserved.

19

STABILITY FUNCTIONS FOR SECOND-ORDER INELASTIC ANALYSIS OF SPACE FRAMESJ Y Richard Liew, H Chen, and N E Shanmugam Department of Civil Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260

ABSTRACT This paper outlines the key concepts and approaches from the recent work on second-order plastic hinge analysis of three-dimensional (3-D) frame structures. An inelastic beam-column element has been developed for analysing steel frame structures composed of slender members subjected to high axial load. The element stiffness formulation is based on the use of the stability interpolation fiinctions for the transverse displacements. The elastic coupling effects between axial, flexural and torsional displacements also are considered. A computer program has been developed and it can be used to predict accurately the elastic flexural buckling load of columns and frames by modelling each physical member as one element. It can also be used to predict the elastic buckling loads associated with axialtorsional and lateral-torsional instabilities, which are essential for predicting the nonlinear behaviour of space frame structures. The member bowing effect and initial out-of-straightness are considered so that the nonlinear spatial behaviour of structures can be captured with fewer elements per member. Material nonlinearity is modelled by using the concentrated plastic hinge approach. Formation of plastic hinge between the member ends is allowed in the element formulation. Numerical examples including both geometric and material nonlinearities are used to demonstrate the robustness, accuracy and efficiency of the proposed analytical method and the program.

KEYWORDS Advanced analysis, buckling, nonlinear, plastic hinges, frames, space frames, spatial structures, and stability.

INTRODUCTION A method for an accurate analysis of rigid and semi-rigid plane frames composed of members with compact section, fiiUy braced out-of-plane, have been developed and verified by tests (Chen and Toma, 1994; Chen et al., 1996; Liew et al., 1997c). This method fiilfils the requirements for prediction of member strength and stability, with some constraints, satisfying the conventional column and beamcolumn design limit-state checks. Although there have been much work proposed on second-order plastic hinge analysis of 3-D structures, the issues related to different formulations and their accuracy

20

and efficiency in solving large frameworks are not addressed well. The research work presented in this paper is the authors' continual effort to extend their work from advanced analysis of 2-D frames (Chen et al., 1996) to 3-D frames (Liew et al., 1997a and b). In the proposed approach, the 3-D frame element is developed using virtual work equations following an updated Lagrangian formulation. Stability interpolation frmctions, which are derived from the equilibrium equation of beam-column, are used for the transverse displacements. The force recovery method is based on the natural deformation approach (Gattass and Abel, 1987), which is consistent with the nonlinear plastic hinge analysis. Material nonlinearity is modelled by using the concentrated plastic hinge formulation (Orbison, 1982; Chen et al, 1996), which is based on the plastic interaction between the axial force and biaxial moments. In the solution procedure. Generalised Displacement Control method (Liew et al., 1997a) is implemented to perform the geometrical nonlinear analysis as the method is effective in overcoming the numerical problems associated with softening, snap-through and snap-back limit points. The theory and the computer program developed are verified for robustness, accuracy and efficiency through several examples which include both geometric and material nonlinearity (Liew et al., 1997a).

FINITE ELEMENT FORMULATION Many of the nonlinear formulations presented in the literature are based on stiffness or displacement method, for its relative ease in implementation. Virtual work formulation is often used to define the nonlinear coupling effects between the axial, fiexural and torsional displacements, which are essential for an accurate estimate of the second-order effect in space frames. However, many researchers adopt the cubic interpolation ftmctions to approximate the transverse displacements along the element length. Such displacement fields do not satisfy equilibrium conditions within the member. Therefore they cannot be used to predict accurately the fiexural buckling load of columns with various end conditions by modelling the member as only one element. The frame members have to be sub-divided into several elements in order to achieve the desired level of accuracy. This will inevitably increase the cost and time of computation. Stability Function Approach In the proposed formulation, the element force-displacement relationships can be expressed in terms of stability ftmctions, derived from equilibrium considerations. The stability ftmctions account for the effect of axial force on the bending stif&iess, and hence can be used to predict accurately the P-5 effect and the elastic fiexural buckling load of columns and frames by modelling each physical member as only one element. For low axial load case, i.e., |P/Pj 0.4, the corresponding members should be divided into two or more elements to limit the error in stiffness terms to be less than one percent. The proposed frame element, based on stability functions approach, can be used to predict accurately the P-6 effect and the elastic flexural buckling load of columns and frames by modelling each physical member as one element. Member Bowing Effect and Initial Out-of-straightness Considering a member with initial out-of-straightness, the member basic force and deformation relationship can be written as: EI for n = z, y^0 J

MA =

S,A+S20A+C-

MB = T ^ I I S,0. +S,0 - c j ^ ' 2n nA In nB ^On

Ln

for n = z, y

V

1-' oy

M.=^^^ab,(0A+0By+b2(0A-0B)'+b^-^(^-0^)+b,^0 n=z.y

As shown in Fig. 2, MnA and MnB are the end moments, and 0nA and 0nB are the total end rotations; Mx is the torsional moment, and 0x is the total twist; P is the axial force, and e is the relative axial displacement. Lo is the initial length of member, r^ = ^ ( l y + l J / A is the polar radius of gyration; Sin and S2n are stability functions. The bowing functions bin and ban relate the change of member chord length due to the curvature shortening, and they may be written as (Oran, 1973):

b

_ (S,+S,Xs,-2)8q8(S,+S2)

(6)

the coefficients CQ^ , bvsn and bwn account for the effect of member initial out-of-straightness and may be written as (Chan and Zhou, 1994): _2q(llq;+42x48q+35x480 105(48+ q j ^ ' ^^" ^ ^" _ 2(lIq^ + 33x48q^ +49x48^q + 3 5 x 4 8 ^ 105(48+ qy (7)

andb^.-^-^-^-fe;^-^-^^:^^^^)35(48 + qy

and 6n is the amplitude of initial out-of-straightness at the middle span, 6y = V^QZ ^ ^ ^y = '^moz The actual shape of initial out-of-straightness may be arbitrary. However, it is assumed to follow a parabolic shape in the above beam-column formulation. The bowing functions can be used to capture the nonlinear behaviour of structures with slender members. It also gives good prediction on the

22 behaviour of structural members that are loaded far into the post-collapse region with fewer elements per member, including the effect of initial out-of-straightness on the stiffiiess of frame member. Tangent Stiffness Formulation In the proposed 3-D formulation, the incremental equilibrium equation of an inelastic beam-column element can be summarised as follow:

([k]-^[k3]+lkj4-[kj){du}=[kj{du}={df}

(8)

in which {du} is the incremental displacement vector; {df} is the incremental force vector; and [k], [ks], [kp], [ki] and [kj are the element stiffness matrix, bowing matrix, plastic reduction matrix, induced moment matrix and tangent stiffness matrix, respectively. In Eq. (11), the plastic reduction matrix, which represents the material nonlinear effect, is derived through the concentrated plastic hinge formulation (Orbison, 1982; Chen and Toma, 1994; Chen et al., 1996). The torsional effect on cross-sectional plastic strength are not considered, and hence the proposed plastic hinge model may not be accurate for analysing inelastic lateral-torsional behaviour, although the elastic behaviour can be captured accurately. When a plastic hinge is formed, the force point on a cross-section will move on the plastic strength surface. From a numerical point of view, it is necessary to calculate the element incremental forces from the previously knovm equilibrium configuration. This is particularly important for a plastic cross-section to keep the state of force point on the plastic strength surface. Therefore an Updated Lagrangian formulation is suitable for such operation. The natural deformation approach proposed by Gattass and Abel (1987) is adopted for the element force recovery. In this approach, the element incremental displacements can be conceptually decomposed into two parts: the rigid body displacements and the natural deformations. The rigid body displacements serve to rotate the initial forces acting on the element from the previous configuration to the current configuration. Whereas the natural deformations constitute the only source for generating the incremental forces. The element forces at the current configuration can be calculated as the summation of the incremental forces and the forces at the previous configuration. The induced moment matrix is generated by finite rotations of semi-tangential torsional moment and quasitangential bending moment to yield the true equilibrium condition that satisfies the rigid body tests.

M2.6)

0.0

0.2

0.4

0.6

0.8

Compression P/Pe

L,+ e

Fig. 1 Accuracy of stiffness matrix terms based on cubic interpolation function.

Fig. 2 Member basic forces and deformations

23 Formation of a Plastic Hinge within the Element Length In some occasions, a plastic hinge may form within the member ends. A tedious and approximate procedure is to model each frame member with several beam-column elements. However,tiWsmethod will increase the overall degrees of freedom of the structure, and it becomes computationally expensive. Moreover, only a few members in a structure will have plastic hinges forming between the member ends. The proposed analysis can model the formation of plastic hinge between the element ends with minimum computational effort. Based on the member initial out-of-straightness, the deformed element shape and the forces at element ends and the force state within the element length can be established by taking the equilibrium of axial force and moment at the internal cross-section. The element length is divided into six segments with equal length. The cross-sectional forces are then checked at five points between the element ends. A plastic hinge is said to have formed when the plastic strength is reached at any of these points. The analysis will automatically subdivide the original element into two sub-elements at the plastic hinge location. The internal hinge is then modelled by an end hinge at one of the sub-element. The stifhiess matrices for the two sub-elements are determined. The inelastic stiffiiess properties for the origmal element are obtained by static condensation of the "extra" node at the location of the internal plastic hinge. Since the static condensation process is only performed at the element level, it does not involve much computational cost.

ANALYSIS OF COLUMNS AND BEAMSAn axially-compressed cantilever as shown in Fig. 3 is used to illustrate the capability and limitation of the proposed method in solving large rotation and large displacement problems. The cantilever is assumed to be inextensible and elastica with E = 1,1 = 1, and colunm length L = 1. To approximate the inextensibility of the cantilever, the cross-sectional area is assigned a large value of A = 1000. A perturbation load of the moment type is introduced at the free end in order to initiate lateral buckling. The cantilever column is modelled as one and two elements. As shown in Fig. 3, the loaddisplacement curves obtained by using one element do not compare well with the theoretical solutions by Timoshenko and Gere (1961). When two elements are used in the analysis, the chord rotations at the element ends are reduced and the load-displacement curves compare well with the "exact" theoretical solutions. It is also observed that the use of one or two cubic elements is not accurate enough to capture the nonlinear load-displacement behaviour unless more cubic elements are used. Figure 4 shows a beam with rectangular cross-section under the action of equal end moments with bothr

/ -^'^ A

2 elonents/

\ / /*

^

(Ocrio5(Ocr5n. (f) =i.05(i:v 9.75 m

(Oc,,.= (Oc>,ri'-^"

(Ocr(Ocr95(Ocr(C)cr 20.2 m

Figure 2: Illustrative examples - (a) braced frame and (b) unbraced frame (i) for equally loaded columns, the approximate method yields exact results (all members are critical), (ii) for unequally loaded columns, the approximate method yields a conservative braced frame ^cr estimate (s=8.2%). In the unbraced frame, an axial force redistribution leads to the exact result. Concerning the column BL, they are either overestimated (braced case) or exact (unbraced case), (iii) (le)c2=9.3m>L=5m for the braced frame (exact value). Finally, one should also mention that, in unbraced frames with a "weak" member (much lower (|) value), the critical mode may be triggered by the sole instability of such member. This "local mode" is similar to a braced mode and the corresponding BL are not adequately predicted by the approximate methods (an exact stability analysis is required). Figure 3 shows an illustrative example of this situation (L=5m, for all members, and EI= 105000 kNm^). As the left column stiffness is reduced five times, the critical mode changes from "global sway" to "local almost non-sway" and its (le/L) value, yielded by an exact analysis, changes from 1.24 to 0.77. The "leaning columns", studied next, are a limit situation of this behaviour.

H

I ^-nf^X= 2802 kN le= 3.85 m j/5

Xcr= 5370 kN I le= 6.2 m t

Figure 3: Illustrative example - frame with a "weak" column

Leaning columns Leaning columns are compressed members pinned at both ends and located in unbraced frames (figure 4). Concerning their influence on the frame stability, which has led to a fair amount of research and some controversy (e.g., Picard et al., 1992, and Cheong-Siat-Moy, 1986 and 1996), one should mention that: (i) the leaning columns possess no lateral stiffness, regardless of their flexural stiffness EI. (ii) the presence of a leaning column always reduces the frame overall stability, as it introduces a destabilizing effect. This can be clearly seen by looking at its "negative stiffness matrix", given by [K]=-(N/L) 1 -1 -1 1

(3)

65 where the horizontal end displacements are the degrees of freedom, (iii) it makes no physical sense to talk about the BL of a leaning column associated to the frame overall stability. There are no points of contraflexure (the column remains straight) and an isolated leaning column is unstable, (iv) a "pure local mode" may be triggered by the instability of a leaning column (figure 4(a2)). The leaning column BL is then equal to its length and the other columns remain undeformed. The previous remarks show^ that, whenever an unbraced frame contains leaning columns (e.g., the frames depicted in figure 4), its A-cr value should be obtained from an exact stability analysis (using the matrix presented in (3)). The BL of the laterally stiff members may then be calculated (notice, however, that it is possible to conceive a frame without laterally stiff compressed members, in which no member has a meaningful BL - figure 4(b)). Using the BL concept in leaning columns, although possible (Cheong-SiatMoy, 1996), seems to the authors somewhat artificial and does not appear to bring any distinct advantage. Figure 4(a) presents a simple illustrative example of a frame with a leaning column (L=5m, for all members, and EI=21000 kNm^). For uniform stiffness the critical mode is global (figure 4(ai)) and the leaning column destabilising effect may be estimated by noticing that removing its axial force increases ^cr from 606 kN to 1195 kN (the stiff column BL decreases from 18.5m to 13m). If the leaning column stiffness is sufficiently reduced, the critical mode becomes local (figure 4(a2)) and, obviously, le=L.

I6

I.0'"(ai)

^\I //7ny77

I(32)

\

1/15/77r777

///J///

/rfrm

/rfPrn

/77?rn

/77r777

\,= 606 kN le= 18.5 m

(a)

X= 550 kN le=L

(b)

Figure 4: Illustrative examples - frames with leaning columns

COMPRESSED MEMBERS RESISTANCE Rigorously, the in-plane resistance of a plane frame (out-of-plane deformations prevented) should be verified by performing an accurate second-order analysis, which must include all the relevant imperfections, and checking whether its members cross-section capacity (elastic or plastic) is exceeded or not. However, all the existing codes of practice allow an indirect and approximate verification procedure, which consists of isolating the frame members and checking their individual resistances. Each member is acted by internal forces and moments determined by combining the end values, obtained from a global analysis of the frame, with the directly applied forces. The presence of compression in the frame members, together with the displacements produced by the initial geometrical imperfections and primary moments, induces additional internal forces and moments (second-order effects), both in braced (P-5 effects) and unbraced (P-A and P-5 effects) frames (Chen & Lui, 1991). It is a common procedure to calculate the member design end internal forces and moments by means of a first-order linear elastic analysis. In unbraced frames, these internal forces and moments normally incorporate the P-A effects, obtained by an appropriate amplification of the sway moments using the factor (1-^sd/^cr)"^ (EC3, 1992). This means that the P-5 effects must be taken into account during the verification of the members resistance. Although the BL concept plays a crucial role in this procedure, its use is not completely clear in some situations.

66 For the sake of simplicity, the use of the BL concept to verify a member resistance is interpreted and discussed in the context of "elastic analyses of class 3 members" (ultimate limit state defined by the onset of yielding and "exact" results provided by a second-order elastic analysis). For members with laterally restrained ends and subjected to compression Nsd and uniaxial bending Msd, a rather physically meaningful and accurate interaction formula, recently proposed by Villette (1997), is given by N Sd A (Cm-Msd+Nsd-ep) W., l-(Nsd/Ner) 1/;

+'^' + )''

Flexural Member: Critical Length and Web Rotational Stiffness^cr'

47r\{\-v^) ( /^ \fA^o-h,f+C.f7^(^o-h,f\ Et12(1-v^) ^h [LJ 60{LJ

^ K n720

/*

240 (53 + 3(l-^,,,)>r^

^tiw,,

~

htn^ 13440

(45360(1-^,,,)+62160f^l ^U%n'^{^ ; r ^ + 2 8 ; r 2 | - I +420

Compression Member: Critical Length and Web Rotational Stiffness (67t'h[\-V^)V

%-JJ

Et6h{\-v^)

l^ihAlO

~

L

15

E = Modulus of Elasticity G = Shear Modulus v= Poisson's Ratio t = plate thickness h = web depth ^ = (fi-fiVfi stress gradient in the web Lm = Distance between restraints which limit rotation of the flange about the flange/web junction

Af, Ixf, lyf, Cwf, Jf = Section properties of the compression flange (flange and edge stiffener) about jc, y axes respectively, where the JC, >' axes are located at the centroid of flange with jc-axis parallel with flat portion of the flange Xo = X distance from the flange/web junction to the centroid of the flange. hx = x distance from the centroid of the flange to the shear center of the flange

93 The distortional buckling methods proposed for compression members by Lau and Hancock (1987) and flexural members Hancock (1995) and Hancock (1997) perform in a manner similar to the proposed method except in the cases where the geometric stiffness of the web is "driving" the distortional buckling solution (e.g., distortional buckling in which essentially the flange is restraining the web from buckling). The explicit treatment of the role of the elastic and geometric rotational stiffness at the web/flange juncture and the expressions for the web's contribution to the rotationanl stiffness are unique to the method presented here. Verification In order to verify the proposed buckling models a parametric study of members in either flexure or compression is performed. The geometry of the studied members is summarized in Table 3 and the results are given in Table 4. The results are determined by comparison to finite strip analysis. For calculation of the local buckling moment or load (M or P) the minimum buckling stress of the elements is used to compare to the finite strip solution. For local buckling prediction use of the minimum element buckling stress for the entire member (element model) is quite conservative. Use of the semi-empirical interaction model that accounts for any two attached elements is generally a reasonable local buckling predictor. For distortional buckling prediction the proposed method is a reasonable predictor, but not without error. For cases with slender webs the proposed distortional buckling solution correctly converges to the web local buckling stress, Hancock's method conservatively converges to zero buckling stress.

Table 3. Geometry of Members used for Verifcation*d/t h/b h/t b/t max min max min max min max min count 30 15.0 2.5 32 Schafer (1997) Members 90 30 90 3.0 1.0 Commercial Drywall Studs 4.6 1.2 318 48 70 39 16.9 9.5 15 AISI Manual C's 7.8 0.9 232 20 66 15 13.8 3.2 73 18 20.3 5.1 50 AISI Manual Z's 4.2 1.7 199 32 55 15 20.3 2.5 170 7.8 0.9 318 20 90 * for members in flexure only Schafer (1997) members are studied

Table 4. Performance of Elastic Buckling Methods*Local Buckling Element Model Interaction Model'^predicted ''^ local Mpredicted/Mlocal

Distortional Buckling Proposed Method^predicted '^disl.

Average St. Dev. Average St. Dev.

0.74 0.12'predicted ''local

0.90 0.05'predicted ''local

0.95 0.08'predicted ''dist.

0.75 0.13

0.97 0.06

1.07 0.05

' finite strip analysis does not always have a minimum for both local and distortional buckling, comparisons are only made for those cases in which finite strip analysis revealed a minimum in the appropriate mode.

94 POST-BUCKLING BEHAVIOR To investigate the post-buckling behavior in the local and distortional modes, nonlinear FEM analysis of isolated flanges is completed using ABAQUS (HKS 1995). The boundary conditions and the elements used to model the flange are shown in Figure 4. The material model is elastic-plastic with strain Roller" Support hardening. Initial imperfections in the local and distortional IX)I' 23 restrained mode are superposed to form the initial imperfect geometry. A longitudinal through thickness flexural residual stress of 30% fy is also modeled. The geometry of the members investigated is summarized in Table 5. The thickness is 1mm and/y = 345MPa. It is "Pin" Support 1X)1- 1-3 restrained observed that the final failure mechanism is consistent with the distortional mode even in cases when the distortional Figure 4 Isolated Flange buckling stress is higher than the local buckling stress. (fixed at flange/web juncture) Consider Figure 5, which shows the final failure mechanism for all the members studied. Based solely on elastic buckling one would expect the local mode to control in all cases in which {fcr)iocail(fcr)dist. < 1 - as the figure shows, this is not the Table 5. Edge Stiffened FlangesPcr.lncul bit 25 dit 4.00-19.0 6.2550 12.5 Q' o o o PcrJist 1.82-0.25 1.94-0.96 1.58-0.27 1.76-0.51 1.34-0.18 1.73-0.35 1.40-0.14 1.75-0.23 o o e e o 0 o X m X X XX X ^ X X X o O ' O O K X X Disionional Mechanism IJislortional Mechanism + LiKal Yielding Mixed - Mechanism [)epcnds on Imp. I^K-al Mechanism + Distortional Yielding 1 AK'al Mechanism O O

e90 45 90 45 90 45 90 45

{

^X.ifX.X X X X

5.00-25.0 6.25 - 25.0 - 37.5 - 37.5 - 50.0 - 50.0

X X X

75

6.25 6.25

100

6.25 6.25

Figure 5 Failur