Light-Weight Steel and Aluminium Structures

LIGHT-WEIGHT STEEL AND ALUMINIUM STRUCTURES

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LIGHT-WEIGHT STEEL AND ALUMINIUM STRUCTURES Fourth 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 COMMITTEE P. 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 COMMITTEE P. 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 COMMITTEE P. Makelainen, Finland, Chairman M. Mikkola, Finland, Co-Chairman H.G. Allen, United Kingdom G.A. A§kar, 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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CONTENTS Session Al: Structural Modelling and Analysis Keynote lecture: J.M. Davies (GBR), Modelling, Analysis and Design of Thin-Walled Steel Structures

3

J.Y.R. Liew, H. Chen & N.E. Shanmugam (SIN), Stability Functions for Second-Order Inelastic Analysis of Space Frames

19

B. Young & K.J.R. Rasmussen (AUS), Local, Distortional, Flexural and Flexural-Torsional Buckling of Thin-Walled Columns

27

Poster Session PI: Structural Modelling and Analysis Y. Telue & M. Mahendran (AUS), Buckling Behaviour of Cold-Formed Steel Wall Frames Lined with Plasterboard

37

B. Young & G.J. Hancock (AUS), Compression Tests of Thin-Walled Channels with Sloping Edge Stiffeners

45

Y. Itoh, M. Mori & C. Liu (JPN), Numerical Analyses on High Capacity Steel Guard Fences Subjected to Vehicle Collision Impact

53

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

61

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

69

J. Tasarek (POL), Shear Buckling of Beam with Scaffold Web

79

Session A2: Buckling Behaviour B.W. Schafer & T. Pekoz (USA), Local and Distortional BuckHng of Cold-Formed Steel Members with Edge Stiffened Flanges

89

M. Kotelko (POL), Collapse Behaviour of Thin-Walled Orthotropic Beams

99

M.C.M. Bakker, H.H. Snijder & J.G.M. Kerstens (NED), Elastic Web Crippling of Thin-Walled Cold Formed Steel Members

107

J.P. PapangeHs & G.J. Hancock (AUS), Elastic Buckling of Thin-Walled Members with Corrugated Elements

115

J. Kesti & P. Makelainen (FIN), Compression Behaviour of Perforated Steel Wall Studs

123

X

Contents

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

131

Session A3: Beam-Columns Keynote lecture: R.M. Sully & G.J. Hancock (AUS), Stability of Cold-Formed Tubular Beam-Columns

141

J. Lindner & A. Rusch (GER), Load Carrying Capacity of Thin-Walled Short Columns

155

S. Kedziora, K. Kowal-Michalska & Z. Kolakowski (POL), Ultimate Load of Orthotropic Thin-Walled Beam-Columns

163

J. Rhodes (GBR), Combined Axial Load and Varying Bending Moment in Beam-Columns

171

A.M.S. Freitas & F.G.F. Bueno (BRA), Analysis of Thin-Walled Steel Beam-Columns

179

Poster Session P2: Sandwich Structures and Dynamic Behaviour P. Hassinen (FIN), Modelling of Continuous Sandwich Panels

189

P. Rapp, J. Kurzyca & W. Szostak (POL), The Creep and Relaxation in Sandwich Panels with the Viscoelastic Cores

197

J. Valtonen & K. Laakso (FIN), Impact Tests on Steel and Aluminium Road Side Columns

205

J. Ravinger (SVK), Vibration of Imperfect Slender Web

211

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

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

233

W. Guggenberger (AUT), Nonlinear Analysis of General Steel Skeletal Structures Part I: Theoretical Aspects

241

G. Salzgeber & W. Guggenberger (AUT), NonUnear Analysis of General Steel Skeletal Structures - Part II: Computer Program and Practical Applications

249

Contents

xi

V.S. Hudramovych, A.A. Lebedev & V.I. Mossakovsky (UKR), Plastic Deformation and Limit States of Metal Shell Structures with Initial Shape Imperfections

257

M. Ohga, Y. Miyake & T. Shigematsu (JPN), Buckling Analysis of Shell Type Structures under Lateral Loads

265

N.E. Shanmugam (SIN) & R. Narayanan (USA), Strength of Thin Rectangular Box-Columns Subjected to Uniformly Varying Edge Displacements

273

I.H.P. Mamaghani (JPN), Elastoplastic Sectional Behavior of Steel Members under Cyclic Loading

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

293

D. McAndrew & M. Mahendran (AUS), Flexural Wrinkling Failure of Sandwich Panels with Foam Joints

301

R.F. Pedreschi (GBR), Design and Development of a Cold-Formed Lightweight Steel Beam

309

G.H. Couchman (GBR), A.W. Toma, J.W.P.M. Brekelmans & E.L.M.G. Van den Brande (NED), Steel-Board Composite Floors

317

Poster Session P3: New Structural Products K. Oiger (EST), Design of Glulam Arched Roof Structures with Steel Joints

327

A. Belica (LUX), Fixed Column Bases in Astron Structures

335

Z. Kurzawa, K. Rzeszut, A. Boruszak & W. Murkowski (POL), New Structural Solution of Light-Weight Steel Frame System, Based on the Sigma Profiles

343

H. Co§kun (TUR), Design Considerations for Light Gauge Steel Profiles in Building Construction

351

J. Murzewski (POL), Computer-Aided Design of Steel Structures in Matrix Formulation

359

J. Vojvodic Tuma (SLO), Construction of a 60.000 m^ Steel Storage Tank for Gasoline

367

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

377

xii

Contents

S.A. Alghamdi & M.H. El-Boghdadi (KSA), Design Optimization of Nonuniform Stiffened Steel Plate Girders - LRFD vs. ASD Procedures

385

H. Saal & U. Hornung (GER), Design Rules for Tank Structures - Different Approaches

399

W. Schneider, S. Bohm & R. Thiele (GER), Failure Modes of Slender Wind-Loaded Cylindrical Shells

407

A.M. Chistyakov, F.V. Rass, P.N. Konovalov & N.V. Chernoivan (RUS), Laminated Constructions on the Basis of Thin Metal Sheets in Building

415

B. Uy (AUS) & H.D. Wright (GBR), Local Buckling of Hot-Rolled and Fabricated Sections Filled with Concrete

423

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

433

K.J.R. Rasmussen (AUS) & J. Rondal (BEL), Column Curve Formulation for Aluminium Alloys

441

M. Matusiak & P.K. Larsen (NOR), An Experimental Study of Strength and Ductility of Welded Aluminium Beams

449

F.M. Mazzolani, A. Mandara (ITA), & M. Langseth (NOR), Plastic Design of Aluminium Members According to EC9

457

A. Starlinger & S. Leutenegger (SUI), On the Design of New Tram Vehicles Based on the Alusuisse Hybrid Structural System

465

Session A7: Aluminium and Stainless Steel Structures Keynote lecture: F.M. Mazzolani (ITA), The Structural Use of Aluminium: Design and AppHcation

475

F. Soetens & J. Mennink (NED), Aluminium Building and Civil Engineering Structures

487

K.F. Fick (GER), Design of Mechanical Fasteners for Thin Walled Aluminium-Structures

495

G. Sedlacek & H. Stangenberg (GER), Numerical Modelling of the Behaviour of Stainless Steel Members in Tests

503

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

515

Contents

xiii

T. Ala-Outinen (FIN), Stainless Steel Compression Members Exposed to Fire

523

A. Talja (FIN), Tests on Cold-Formed and Welded Stainless Steel Members

531

C. Faella, V. Piluso & G. Rizzano (ITA), Modelling of the Cyclic Behaviour of Bolted Tee-Stubs

539

J.S. Myllymaki & D. Baroudi (FIN), A New Method for the Characterisation of the Fire Protection Materials

547

P.P. Gedeonov & T.P. Gedeonova (RUS), Bloating Flame-Retardant Coatings on the Basis of Vermiculite for Steel Buildings Construction

555

Y. Orlowsky, K. Orlowska & T. Shnal (UKR), Fire Resistivity of Steel and Aluminium Constructions Protected by a Bloated Coating

561

Session A8: Connections R.A. LaBoube & W.W. Yu (USA), New Design Provisions for Cold-Formed Steel Bolted Connections

569

K. Kolari (FIN), Load-Sharing of Press-Joints in Thin-Walled Steel Structures

577

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

585 593

C.A. Rogers & G.J. Hancock (AUS), Bearing Design of Cold Formed Steel Bolted Connections

601

R.B. Tang & M. Mahendran (AUS), Pull-Over Strength of Trapezoidal Steel Claddings

609

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

617

Session B2: Aluminium and Stainless Steel Structures K.J.R. Rasmussen (AUS) & J. Rondal (BEL), Column Curves for Stainless Steel Alloys

627

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

637

R.M. Gon9alves, M. MaHte & J.J. Sales (BRA), Aluminium Tubes Flattened (Stamped) Ends Subjected to Compression - A Theoretical and Experimental Analysis

647

xiv

Contents

G. De Matteis, A. Mandara & F.M. Mazzolani (ITA), Interpretative Models for Aluminium Alloy Connections

655

F.M. Mazzolani, C. Faella, V. Piluso & G. Rizzano (ITA), Local Buckling of Aluminium Channels under Uniform Compression: Experimental Analysis

663

B. Boon & H. Weijs (NED), Local Impact on Aluminium Plating

671

J.S. Myllymaki & R. Kouhia (FIN), Creep Buckling of Metal Columns at Elevated Temperatures

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

689

J. Nieminen & M. Salonvaara (FIN), Long-Term Performance of Light-Gauge Steel-Framed Envelope Structures

703

M. Feldmann, C. Heinemeyer & G. Sedlacek (GER), Substitution of Timber by Steel for Roof Structures of Single-Family Homes

713

J. KuUaa & A. Talja (FIN), Vibration Performance Tests on Light-Weight Steel Joist Floors

719

A.Y. ElghazouH & B.A. Izzuddin (GBR), Significance of Local Buckling for Steel Frames under Fire Conditions

727

Poster Session P5: Composite Structures M. Shugyo & J.P. Li (JPN), Elastoplastic Large Deformation Analysis of Concrete-Filled Tubular Columns

737

J. Brauns (LAT), Resistance of Composite Section to Axial Loads and Bending: Design and Analysis

745

A.K. Kvedaras (LTU), Light-Weight Hollow Concrete-Filled Steel Tubular Members in Bending

755

C. Faella, V. Consalvo & E. Nigro (ITA), An "Exact" Finite Element Model for the Linear Analysis of Continuous Composite Beams with Flexible Shear Connections

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

xv

R.M. Schuster (CAN), Perforated Cold Formed Steel C-Sections Subjected to Shear (Experimental Results)

779

H. Pasternak & P. Branka (GER), Carrying Capacity of Girders with Corrugated Webs

789

J. Rhodes, D. Nash & M. Macdonald (GBR), An Examination of Web Crushing in Thin-Walled Beams

795

P. Konderla & J. Marcinowski (POL), Experimental Investigations and Modelling of Steel Grids

803

T. Yamao, T. Akase & H. Harada (JPN), Ultimate Strength and Behavior of Welded Curved Arch Bridges

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

821

E. Yamaguchi, Y. Goto, K. Abe, M. Hayashi & Y. Kubo (JPN), Stability Analysis of Bridge Piers Subjected to Cyclic Loading

829

P. Kujala & K. Kotisalo (FIN), Fatigue Strength of Longitudinal Joints for All Steel Sandwich Panels

837

T. Usami & H.B. Ge (JPN), Local and Overall Interaction Buckling of Steel Columns under Cyclic Loading

845

M. Yamada (JPN), Steel Shear Panels for Anti-Seismic Elements

853

A. Salwen & T. Thoyra (SWE), Results from Low Cycle Fatigue Testing

861

Keyword Index

869

Author Index

875

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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 STRUCTURES J 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 DEVELOPMENT The 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)

E^l

(1)

(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

* - [d)^ - Vr 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

p^

N

M

and

respectively. It is equally valid when used to allow second-order elastic GBT

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 "^ ./• "^ ./ M Figure 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;

p

if Xp > 0.673;

1.0

p

= 1.0 0.22 \ 1 P /

in which the plate slendemess Xp is given by:

P

(4)

_yb

N where

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:

N

M^ -^ M^

= (pS)fy, ^ 0.673:

p = 11.0 -

p = 1.0

(7)

^ 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,^ = 0

P _ - EI„—

ZQP^

(8)

?^ - E T — + GJ Ao

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: ZQP^

- EI,

PTP

(9) ZQP^

r^P^

ET— + GJ L2

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:

"

2

P^^-'

.ro^P^M,

(11)

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 FRAMES J 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:

M„A =

EI„

for n = z, y

S,„©„A+S2„0„A+C„„•^0 J

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

Ln

for n = z, y

1-' oy

V

M.=^^±^a b,„(0„A+0„By+b2„(0„A-0„B)'+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)

(6)

•8(S,„+S2„)

8q„

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 ^

andb^.-^-^-^-fe;^-^-^^:^^^^) 35(48 + q„y

^ '

^^"

_ 2(lIq^ + 33x48q^ +49x48^q„ + 3 5 x 4 8 ^ ^"

105(48+ q„y (7)

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

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

L,+ e

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 BEAMS An 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 both r

/ < 2 elements

L

1 element.

A

-% ^ L^^^>-^'

' ''

l-

— 1

1 0.2

1

1 1 0.4

1 0.6

M

1 element

^i

-l-\

Timoshenko and Gere (1961) Proposed element Cubic element 1

1 0.8

1

1 1 1.0

1 1.2

OuptM

M/Mcr 1.2 r

O.OOIPL

^ '-axis direction and buckling twist rotations (0^) about the z-axis. It follows from Eqn. 1 that flexural buckling in (w^, Ub) is uncoupled from flexural-torsional buckling in {vbMThe tangent rigidities ((EA)t, iESy)t, {EIy)u {EQt, (Eljt, (Elxjt) were obtained using an elastic nonlinear finite strip analysis, as described in Hancock (1985). In the analysis, a locally buckled cell of length equal to the local buckle half-wavelength (/) was subjected to increasing values of axial compression and at each load level, small increments of generalised strain were applied. This allowed the tangent rigidities to be obtained at each load level, as described in Rasmussen (1997).

LOCAL BUCKLING AND DISTORTIONAL BUCKLING ANALYSIS The elastic local buckling load (Ni) and the local buckle half-wavelength (/) were obtained using a finite strip buckling analysis, as described in Hancock (1978). The results of the two tests series L36 and L48 are listed in Table 1, where the tests are described in the following section of this paper. The local buckling andysis treated the section as a plate assembly, maintaining compatibility of

30

displacements and rotations at plate-junctions. The elastic distortional buckling load (Nd) was calculated according to Lau and Hancock (1987) and given in Table 1. This method of analysis assumed the web is partially destabilised by the uniform longitudinal compressive stress acting on it and providing elastic rotational and lateral restraints to the flange at the flange-web junction, as described by Bleich (1952). TABLE! MATERIAL PROPERTffiS AND BUCKLING DETAILS Test Series

L36 L48

Measured Material Properties E 00.2 (MPa) (MPa) 515 2.10x10^ 550 2.00x10^

Buckling Analysis Local Ni

/

(kN) 70.6 72.6

(mm) 75 75

Distortional Nu (kN) 125.3 101.3

FIXED-ENDED COLUMN TESTS The tests on fixed-ended lipped channel columns conducted by the authors at the University of Sydney are detailed in Young and Rasmussen (1998). The tests were performed on channels brake-pressed from high strength zinc-coated structural steel sheet with nominal yield stress of 450MPa. The test program comprised two series with different cross-sections, referred to as Series L36 and L48 according to their nominal flange width. The average values of measured cross-section dimensions of the test specimens are shown in Table 2 using the nomenclature defined in Fig. 2. Table 2 also includes the full cross-section area (A), major and minor axis second moment of area (4) and (ly) respectively, warping constant (/J, and torsional constant (7). The measured cross-section dimensions and the ultimate loads obtained from the tests of each specimen are detailed in Young and Rasmussen (1998). The material properties determined from tensile coupon tests are summarised in Table 1, where E is the Young's modulus and (7o.2 is the static 0.2% tensile proof stress. Overall and local geometric imperfections were measured on all specimens, except for the shortest specimen. The maximum local imperfections were found to be of the order of the plate thickness at the tip of the flanges. The maximum overall flexural imperfections about the minor axis at mid-length were 1/1100 and 1/1300 as measured in the longest specimen of length 3000mm for Series L36 and L48 respectively.

Figure 2: Definition of symbols

31 TABLE 2 CROSS-SECTION DIMENSIONS Test Series L36 L48

Bi

(mm) 12.5 12.2

Bf (nmi)

37.0 49.0

Bw

t*

n

(mm) 97.3 97.1

(mm) 1.48 1.47

(mm) 0.85 0.85

A (mm^) 280 314^

4

ly

L

(mm^)

(mm^)

(mm^)

J (mm^)

4.11x10^ 4.86x10^

5.38x10^ 1.04x10^

1.07x10^ 2.02x10^

2.06x10^ 2.26x10^

* Base metal thickness excluding zinc coating

COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS The fixed-ended lipped channel test strengths are compared with the bifurcation loads in Figs 3 and 4 for sections L36 and L48 respectively. The bifurcation loads are shown as Ncr on the vertical axis, non-dimensionalised with respect to the elastic local buckling loads (Ni) shown in Table 1. The figures include the overall flexural (F) and flexural-torsional (FT) bifurcation curves of both the locally buckled and undistorted cross-sections. The curves are shown in Figs 3a and 4a for a magnitude (Wo) of the local geometric imperfection (in the shape of the local buckling mode) of Wo = O.Olt and in Figs 3b and 4b for a magnitude of the local geometric imperfection of Wo = 0.2t. In addition, the elastic distortional buckling loads (Nd) non-dimensionalised with respect to the elastic local buckling loads are also shown in the figures. The distortional buckling loads are shown in Table 1. The failure modes observed near ultimate during testing are also shown in the figures. They include the local (L), distortional (D), minor axis flexural (F) and flexural-torsional (FT) modes. For Series L36, the flexural and flexural-torsional buckling loads of the locally buckled cross-section were nearly equal, as shown in Fig. 3. This result was supported by the overall buckling failure modes observed in the tests at column lengths L = 1500mm and 2000mm, which involved combined flexural and flexural-torsional buckling modes together with the local buckling mode. The Series L36 test strengths shown in Fig. 3 are lower than the bifurcation curves for both values of local imperfection. This is most likely a result of overall geometric imperfections. However, the tests performed on short columns are also likely to have been influenced by yielding before reaching the ultimate load. Distortional buckling was observed at specimens with short column lengths (L = 500mm and L = 1000mm). The test results for Series L48 are compared with the bifurcation curves and distortional buckling load in Fig. 4. Generally, the ultimate loads obtained from the tests are lower than or equal to the bifurcation loads for both values of local imperfection. The flexural-torsional bifurcation curves for both the distorted and undistorted cross-sections are clearly lower than the flexural bifurcation curves at all column lengths. This result is in agreement with the tests, where the flexural-torsional failure mode was observed for all lengths, except for short specimen lengths where yielding occurred before the ultimate load. Furthermore, the tests performed at column lengths less than or equal to 2000mm were influenced by distortional buckling. The load was well predicted by the distortional buckling analysis at column lengths L = 1000mm and 1500mm.

32

F, distorted FT, distorted F, undistorted FT, undistorted Tests

5000

2000 3000 Column length, L (mm) (a) w^f = 0.02 2.0

F, distorted FT, distorted F, undistorted FT, undistorted Tests

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

96

1000

2000 3000 Column length, L (mm)

4000

(b)w^r = 0.2

Figure 3: Non-dimensionalised load (NcM) vs column length (L) for fixed-ended L36 channel section

5000

33

^ F, distorted — FT, distorted — F, undistorted FT, undistorted Tests

1000

2000 3000 Column length, L (mm)

4000

5000

4000

5000

(a) Wo/t = 0.02

2000 3000 Colunm length, L (mm) (b)w^^ = 0.2

Figure 4: Non-dimensionalised load (NJNi) vs column length (L) for fixed-ended L48 channel section

34 CONCLUSIONS An overall flexural and flexural-torsional bifurcation analysis of locally buckled columns has been presented and applied to fixed-ended lipped channel sections. In addition, the elastic distortional buckling load and the local buckling load were obtained using a finite strip buckling analysis. The loads and failure modes predicted by these analyses are compared with tests conducted at the University of Sydney. Good agreement is found between the bifurcation analysis and the tests, except for short column lengths where yielding occurred before the ultimate load was reached. Furthermore, the tests performed at short column lengths were influenced by distortional buckling, and the load was well predicted by the distortional buckling analysis for test Series L48 (which had wider flanges than the test Series L36).

ACKNOWLEDGMENTS The comments of Prof. Gregory Hancock of the University of Sydney are appreciated.

REFERENCES Bijlaard P.P. and Fisher G.P. (1953). Column Strength of H-Sections and Square Tubes in Postbuckling range of Component Plates. National Advisory Committee for Aeronautics, TN 2994. Bleich F. (1952). Buckling Strength of Metal Structures. McGraw-Hill Book Co., Inc., New York, N.Y. Hancock G.J. (1978). Local, Distortional and Lateral Buckling of I-Beams. Journal of the Structural Division, ASCE, 104:11, 1787-1798. Hancock G.J. (1981). Interaction Buckling in I-Section Columns. Journal of the Structural Division, ASCE, 107:1, 165-179. Hancock G.J. (1985). Non-linear Analysis of Thin-walled I-Sections in Bending. Aspects of Analysis of Plate Structures, eds D.J. Dawe, R.W. Horsington, A.G. Kamtekar & G.H. Little, 251-268. Lau S.C.W. and Hancock G.J. (1987). Distortional Buckling Formulas for Channel Columns. Journal of Structural Engineering, ASCE, 113:5, 1063-1078. Rasmussen K.J.R. (1997). Bifurcation of Locally Buckled Members. Thin-Walled Structures, 28:2, 117-154. Young B. and Rasmussen K.J.R. (1997). Bifurcation of Singly Symmetric Columns. Thin-Walled Structures, 28:2, 155-177. Young B. and Rasmussen K.J.R. (1998). Design of Lipped Channel Columns. Journal of Structural Engineering, ASCE, 124:2, 140-148.

Poster Session PI 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.

37

BUCKLING BEHAVIOUR OF COLD-FORMED STEEL WALL FRAMES LINED WITH PLASTERBOARD Yaip Telue and Mahen Mahendran Physical Infrastructure Centre, Queensland University of Technology, Brisbane QLD 4000, Australia

ABSTRACT Gypsum plasterboard is a common lining material for steel wall frame systems used in combination with cold-formed steel studs (C or lipped C-sections). However, the design of these wall frames does not utilise the full strengthening effects of plasterboard in carrying axial loads. Therefore an experimental study was conducted to investigate the local and overall buckling behaviour of the studs in these frames using a total of 40 full-scale wall frame tests and stub column tests. The tests included unlined, both sides lined and one-side lined studs. Test results were compared with predictions from the Australian standard AS 4600-1996 and the American specification AJSI-1996. This paper presents the details of the experimental study, the results, and comparisons with design code predictions.

KEYWORDS Buckling of studs. Plasterboard lining. Cold-formed steel wall frames. Full scale wall frame tests.

INTRODUCTION Gypsum plasterboard is a common lining material used in combination with cold-formed steel studs (C or lipped C-sections) for both the load bearing and non-load bearing walls. Li the design of load bearing walls, the support provided by the plasterboard in carrying the axial load is not considered. The current Australian Standard for Cold-formed Steel Structures AS4600 (SA, 1996) only considers the lining material to provide lateral and rotational supports to the stud in the plane of the wall. However, the American Specification (AISI, 1996) includes checking for column buckling between wallboard fasteners, overall buckling and for shear failure of the plasterboard. Design equations for lined studs were derived from the work of Simaan and Pekoz (1976) based on the shear diaphragm model. However, Miller and Pekoz (1994)'s tests on plasterboard lined stud walls showed that the results contradict the shear diaphragm model. The failure loads of lined walls were much higher than those predicted by the AISI (1986). All these imply that the behaviour of lined stud walls is not accurately modelled by the American Specification. Further, both design specifications ignore the possible improvement to the local buckling behaviour of slender plate elements of the stud. An experimental study was therefore carried out to address these problems using a total of 40 full scale wall frame tests

38 and stub column tests. This paper presents the details of this experimental study and the results. Experimental results were compared with AS4600 (SA, 1996) and AISI (1996) predictions, based on which appropriate conclusions and recommendations have been made.

EXPERIMENTAL INVESTIGATION Full Scale Wall Frame Tests The key parameters in these tests are plasterboard lining (thickness and type, no lining vs one side lining vs both sides lining), geometry of the stud section, stud thickness and grade, number of studs and their spacing in the frame. To investigate the effects of these parameters, a total of 20 full-scale wall frames consisting of three studs with studs spaced at 600 and 300 mm were chosen (Fig. la). This configuration was adopted as it represents a typical wall frame in a building. The height of the frames was set at 2.4 m to represent a typical wall in a building. Four frames were unlined. Eight frames were lined on one side while the remaining eight had lining on both sides. For the lined frames, the more commonly used 10 mm plasterboard was used as the lining material. The studs were made from two unlipped C-sections shown in Fig. lb and were fabricated from two grades of steel, a mild steel grade G2 (min. yield stress = 175 MPa) and a high tensile steel grade G500 (min. yield stress = 500 MPa). •^^^ Top Track 30 mm

I j1.15mm

n (a)

(b)

(a) Layout and Dimensions of Frames (b) Dimensions of C-section Studs Figure 1. Details of Full Scale Wall Frames Test frames were made by attaching the studs to the top and bottom tracks made of C-sections using a single 8-18 gauge 12 mm long wafer head screw at each joint. Plasterboard lining was fixed to the studs using Type S 8-18 x 30 mm screws at 220 mm centres (CSR, 1990). This is within the maximum spacing of 300 mm recommended by RBS (1993). The first screw was located 75mm from the edge of the tracks at both ends and is within a maximum distance of 100 mm recommended by RBS (1993). Table 1 presents the details of test frames. TABLE 1: DETAILS OF FULL SCALE TEST FRAMES

Frame Number 1 2 5 6 9 10 13 14

1 17 18

Stud (mm) 75 75 75 75 75 75 75 75 75 75

Steel Grade 02 0500 02 O500 G2 G500 G2 G500 G2 G500

Frame Number 3 4 7 8 11 12 15 16 19 20

Stud (mm) 200 200 200 200 200 200 200 200 200 200

Steel Grade 02 O500 02 O500 G2 G500 G2 G500 G2 G500

Stud Spacing (mm) 600

Lining Condition Unlined

600 Lined one side 300 600 Lined both sides 300

39 The test set-up for the full-scale tests is shown in Fig. 2. The test frame was placed in a vertical position within the support frame and adequately restrained. The bottom track of the frame was fixed to the steel beam support at both ends. At the top of the frames, timber blocks were used at each end of the frame to stop in-plane movement. Timber restraints were also used to prevent the frames and studs from moving out of plane, but allowed shortening of the studs to occur freely. Three hydraulic jacks were suspended off the top horizontal beam and were placed directly over each stud in order to apply a concentric load. A load cell attached to each jack enabled the load to be monitored during the tests. Loading plates were placed on the top of the track directly under the jacks to enable uniform load distribution to the entire stud cross-section. Any gaps between the stud and the tracks were packed with steel shims to ensure direct load transfer to studs from the loading plate. During each test, the axial compression load on each stud was increased until failure. When one stud failed, loading was continued for the remaining studs until they also failed one after the other. In this manner, three stud failure loads were obtained for each wall frame. This approach was used because the aim of this study was to investigate the behaviour of the studs and determine their failure loads as members of the wall assembly. It was not the intention to determine the failure load of the wall frame. Universal Beam

5 tonne Hydraulic Jacks

n

Timber restraints (typical) Reaction Floor

Support Frame

1

n

Figure 2. Full Scale Wall Frame Tests

Figure 3. Stub Column Tests

Stub Column Tests The main objective of these tests was to investigate the possible improvement to the local buckling behaviour of stud sections used in lined wall frames. It was considered that local buckling strength of C-section studs could improve depending on the slendemess of flange and web elements and the spacing of screws connecting the flanges to lining. Therefore a series of stub column tests was conducted on unlipped C-sections by varying these parameters (Table 2). Since the plasterboard lining restrains only the flanges, only the unlipped C-sections that are more susceptible to local buckling were considered in the study. The studs were made from seven unlipped C-sections as shown in Table 2 and were fabricated from 1.15 mm G2 grade steel. The b/t ratio of flanges thus varied from about 12 to 75. Since the maximum screw spacing recommended was 400 mm (RBS, 1993), it was varied from 65 to 260 mm in the tests. As for the frill scale wall frame tests, 10 mm plasterboard and Type S 8-18 x 30 mm long screws were used in the lined stud tests. The height of the studs was 600 mm in all the tests in order to minimise the end effects during loading and to eliminate overall column buckling effects. Only a single stud was used with 400 mm wide plasterboard lining on both sides as shown in Figure 3. For the lined studs, the height of lining was less than 600 mm (see Fig.3) so that the load could be appUed to the stud. Prior to the single stud tests, a few three-stud frames as in the frill scale frame tests, but with a 600 mm height, were tested to determine the adequacy of using single studs. A stud spacing of 300 mm was used in this series of three tests. For the 40 x 40 x 1.15 mm C-sections used in the threeframe tests, the failure loads were 18.5 kN for a screw spacing of 260 mm and 17.8 and 18.1 kN for a 220 mm screw spacing. These values compare well with the single stud failure load of 18.4 kN (see Table 2) and thus validated the use of single studs in the following tests shown in Table 2. The test specimens were kept between the fixed cross heads of a Tinius Olsen testing machine and loaded until failure. During each test, the local buckling and ultimate failure loads were observed.

40 TABLE 2: STUB COLUMN TEST DETAILS AND RESULTS

Stud size (mm)

Lining

4 0 x 1 5 x 1 . 1 5 Unlined Lined Lined Lined 40x60x1.15 T Jnlined Lined Lined Tvined 7 5 x 6 0 x 1 15 Unlined Lined

Screw Spacing (mm) _ 260

no 65

260

no 65 _

no

Failure Load (kN) 8.8 12.7 15.4 15.9 19.7 2L1 20.2 24.3 22.2 24.5

Stud size (mm) 40x40xL15

75x15x1.15 75x40x1.15

Lining

Screw Spacing (mm) _ T Jnlined 260 Lined Lined 130 65 Lined I Jnlined 1.30 Lined I Jnlined 130 Lined

75x90x1.15 Lined

130

Failure Load (kN) 17.5 18.4 19.2 21.3 13.9 17.8 20.8 24.2 25.1 24.5

RESULTS AND DISCUSSION Full Scale Wall Frame Tests Unlined Frames TABLE 3: ULTIMATE FAILURE LOADS OF UNLINED FRAMES

Kx = Kv=Kt=0.75 1 Expt. Section Kx = Kv =Kt=LO Expt. Failure Expt./ Pred. Expt./ Pred. Failure Capacity Pred. Failure Failure Pred. Mode Load (Ns) Load Mode Mode Load (kN) (kN) 0.80 5.6 FB 1.44 1 0.76 FB FB FB 1.36 2 20.3 1 5.3 3 FTB 4.3* 1.05 FB 1.90 7.8 1 0.97 1 1.76 FB 7.2 FB FB 2 2 44.6 0.89 1.61 6.6 3 FTB FTB 1 5.3* 0.86 FB FB FB 8.3 1.32 2 22.6 3 1.10 FTB 1.70 10.7 3 1.03 1.-59 10.8 FTB 1 FB FB 10.8 4 47.3 1.59 FTB 1.03 2 10.4 1.53 FTB _3 (L22 * Denotes values that were ignored in the subsequent computations and discussions. FB: - denotes flexural buckling; FTB: - denotes flexural-torsional buckling. Frame Number

Stud

Table 3 presents the failure loads from the tests and the predicted loads from the Australian standard AS 4600 (SA, 1996) for the four unlined frames. The predictions based on the American Specification (AISI, 1996) and AS4600 (SA, 1996) are identical for the unlined frames. The failure loads from tests are generally higher than those predicted by the codes. The predicted failure loads were first computed taking the effective length factors Kx, Ky and Kt as 1.0, (where Kx and Ky are the effective length factors for the buckling about the x and y-axes, respectively, and Kt is the effective length factor for torsion. But AS 4600 (SA, 1996) does not have any procedures to determine these factors. The AISI Specification (1996) states that these values can be determined using a rational method but shall not be less than the actual unbraced length. In these tests, timber restraints were used to prevent sway of the frames during the tests; therefore the effective length factor cannot be greater than unity. The predicted failure loads based on an effective length factor of 1.0 were found to be conservative as the top and

41 bottom tracks would provide some restraints to buckling about the x, y and z-axes (see Table 3). Hence various effective length factors were investigated. When a value of 0.75 was used for Kx, Ky and Kt the predicted loads agreed well with experimental results. This is similar to Miller and Pekoz's (1993) recommendation of 0.65 based on their tests on lipped C-sections. The observed and predicted failure modes are also given in Table 3. Li the computation of the failure loads the lowest load was selected based on the three possible failure modes. These were the elastic flexural buckling (FB), torsional and flexural-torsional buckling failures (FTB). hi general, the codes accurately predicted the failure mode. hi order to allow for any loading eccentricity that could have affected the test results, the AS4600 and AISI predicted loads were also calculated for a 2 mm eccentricity about both axes with Kx = Ky = Kt = 0.75. The range of experimental to predicted load ratios increased marginally to 0.84-1.20 compared with 0.76-1.10 reported in Table 3. This confirms the earher recommendation of 0.75 for Kx, Ky and Kt. Young and Rasmussen (1998a,b) have conducted extensive research into the behaviour and design of channel columns with pinned and fixed end conditions. Their research showed that fixed-ended columns can be designed by assuming the load to be at the effective centroid and by using an effective length equal to half the column length (Kx = Ky = 0.5). They also recommend that a column can be assumed fully fixed provided elastic rotational restraint exceeds three times the stiffness of the column (Ely/L). However, the end support conditions of wall frame studs in practice appear to be closer to a fixed end, but are not fully fixed. Failure loads from full scale wall frame testing and the need to use higher K factors (Table 3) confirm this. Therefore the effective length factors given in Table 3 are recommended, but further wall frame tests including rotational restraint measurements are required. Frames with Plasterboard Lining on Both Sides All the frames with plasterboard lining on both sides except 2 studs in Frame 18, failed by buckling between the fasteners at the top of the stud with the screws pulling through the plasterboard. As the load approached failure, the buckling between the two fasteners at the top of the stud (or the top screw and the track) increased causing the load to be eccentric. As the load increased further, it resulted in the screw pulling through the plasterboard. Once this occurred, the stud alone had reduced strength at this location and a sudden failure occurred. Figure 4 shows a typical failure of the studs while Figure 5 shows the buckling between fasteners. The failure of the plasterboard was observed to be localised and not distributed throughout. Miller and Pekoz (1994) also observed similar behaviour.

Figure 4: Typical Stud Failure between the top screw fasteners

Figure 5: Buckling between Screw Fasteners

Figure 6: Local Buckling in Stub Column tests

Experimental failure loads of the studs are summarised and compared with those predicted by AS4600 (SA, 1996) in Table 4. AS4600 (SA, 1996) requires that the ultimate strength of the studs under axial compression be computed by: (i) ignoring the lining or (ii) considering the in-plane lateral and rotational supports. However, it does not state what level of support can be used, hi the tests, the studs were connected to the tracks at both ends and therefore the rotation about the longitudinal stud axis and

42

the horizontal displacements in the x and y-axes at both ends were restrained. The studs, however, were free to rotate about x and y axes at both ends. Various combinations of the effective length factors were therefore investigated and the predicted failure loads are given in Table 4. TABLE 4: ULTIMATE FAILURE LOADS OF BOTH SIDES LINED FRAMES TO AS 4600 (SA, 1996)

Frame Number

Expt. Failure Load (kN)

Expt. Failure Mode

Section Capacity (Ns)(kN)

n

71 ^ ^5 8 79.0 41 7 1Q0 ^66 7?^

a a a a a a/Srr. a a

70'^ 446 77 6 47^ 703 44 6 77 6

14 15 16 17 18 1Q

Expt. Load/Ns 1 OS 080 007 088 0Q4 0 87 09Q

70 0.81 47.3 38.2 Case 1: Kx = Ky = Kt = 0.75; Case 2: K^ = 0.75, Ky = Kt = 0.1

Case 1 Expt./Pred. Load

Case 2 Expt./Pred. Load

^04 4 84 7 77 -^07 7 71 4Q4 7^0

1 16 1 07 0Q9 0Q7 1 03 1 09 1 00 0 85

3.64

Li computing the predicted loads, the effective length factors for the studs Ky (in-plane buckling), Kx (out of plane buckling) and Kt (torsional buckling) were initially taken as 0.75. This was based on the restraints used at the end of the studs as discussed earlier for the unlined frames. They were found to be inadequate as the failure loads were underestimated. In the latter computations a factor of 0.75 was maintained for Kx while a value of 0.1 was investigated for Ky and Kt. This is because the flexural buckling of the studs in the plane of the wall and twisting of the studs were expected to improve by lining the wall. In this case, a good correlation of experimental and predicted failure loads was obtained. An effective length factor of 0.1 corresponds to an effective length equal to the fastener spacing used in the lined frames. These results support the observation of buckling of the studs between the fasteners during tests (Fig. 5). The section capacities of the studs were also compared with the experimental failure loads and the two results were in good agreement (see Table 4). This result implies that the studs must fail by local buckling and/or yielding which was not the case during the tests, as all (except 2) studs failed by buckling between the top screws. Since only one screw spacing was adopted in the tests, it is difficult to conclude whether the failure loads of the studs with lining on both sides can be predicted by the section capacity. Finite element analyses will be used to confirm this result. The results based on assuming the appropriate effective length factors discussed above agreed reasonably well with the actual failure loads and the manner in which the studs failed. It is therefore reasonable to conclude that the failure load predictions of AS 4600 can be improved by using the effective length factors Kx= 0.75, Ky = Kt = 0.1 for the type of wall frames considered in this study. In the AISI Method (1996), the studs were checked for three failure modes and the lowest load was taken as the predicted failure load. These were the failure between the fasteners (mode (a)), failure by overall column buckling (mode (b)) and the shear failure of the lining material (mode (c)). Failure mode (a) requires the studs to be checked for buckling between the fasteners. An effective length factor Kf of 2 is used with the fastener spacing to allow for a defective adjacent fastener (AISI, 1996). In failure mode (b), the total length of the stud is considered. In this study using AISI rules, the same effective length factors used earUer (Kx = 0.75, Ky = Kt = 0.1) were adopted to check failure mode (b). For failure mode (a), Kf =2 was used whereas for failure mode (c) plasterboard was checked to ensure that the allowable shear strain of 0.008 (AISI, 1996) was not exceeded. The predicted failure loads and modes based on the AISI method are given in Table 5. Although reasonable estimates of the failure loads can be achieved when the effective length factors Ky and Kt were reduced to 0.1, the actual failure modes can only be predicted in 50% of the cases. There was no improvement in the resuhs when the effective length factor for failure mode (a), Kf was reduced to one. It only resulted in an

43

increase in the failure load for mode (a), which made mode (b) to govern. The AISI method therefore requires further improvement to ensure that both the failure loads and modes are accurately predicted. TABLE 5: ULTIMATE FAILURE LOADS OF BOTH SIDES LINED FRAMES TO AISI (1996)

Frame Expt. Failure Expt. Failure Number Load (kN) Mode

Section Expt. Capacity Load/Ns (Ns)(kN)

Case 1 Pred. Expt./ Failure Pred. Load Mode 2L3 20.3 1.05 b 1.20 a n 35.8 14 44.6 0.80 1.17 a c 22.0 15 22.6 0.97 b 1.06 a 41.7 16 47.3 1.51 a 0.88 c 19.0 17 20.3 b 1.04 a 0.94 36.6 18 44.6 1.20 a&c 0.82 c 22.3 19 b 1.08 22.6 0.99 a 20 47.3 0.81 C 1.38 38.2 a Case 1: Kx = Ky = Kt = 0.75, Kf = 2; Case 2: Kx = 0.75, Ky = Kt =0.1, Kf= 2

Pred. Failure Mode b b a a b b a

a

Case 2 Expt./ Pred. Load 1.16 1.06 1.02 1.03 1.03 1.09 1.04 0.94

Experimental results showed that there was little difference in the failure loads for the stud spacings (300 mm and 600 mm) and that the failure mode was independent of the stud spacing. Even though the stud spacing has been removed from the AISI specification (1996) the results imply that the shear diaphragm model assumed by AISI is not applicable to wall frames lined with plasterboard. Miller and Pekoz (1994) also made similar observations. Further research using tests and finite elements analyses are needed to study the effect of fastener spacing and the location of the last screw on the studs. Frames with Plasterboard Lining on One Side The failure of the studs was by flexural-torsional buckling (mode (b)) with the screws pulling through the lining at failure. Twisting of the web was observed and was more noticeable in the 200mm studs. As expected the unlined flanges of the studs were severely twisted. The lined flange was observed to deform/buckle between fasteners. At failure there was no crushing or tearing of the plasterboard. When the effect of the plasterboard was ignored as recommended by AS 4600, the predicted loads would be the same as those of unlined frames with Kx = Ky = Kt = 1.0 (Table 3) and thus conservative. When the lateral and rotational supports were considered as for frames with lining on both sides, a good correlation between predicted and failure loads was achieved for the 75mm studs (web b/t < 70), but not for the 200 mm studs. For the C sections, the AS 4600 predicted failure loads can be improved if the following effective length factors: Kx = 0.75, Ky = 0.1 and Kt = 0.2 are used. Since AISI (1996) does not include any provisions for one side lined walls, the failure loads and modes were predicted using AISI (1986). The same procedures in checking the studs for frames with both sides lining were adopted for the studs in this group. When the effective length factors for Ky and Kt were reduced, the failure loads were overestimated. The predicted failure modes changed from mode (b) to (a) for all the frames except one. Therefore the AISI specification cannot accurately predict the failure loads or the failure modes of studs lined on one side. This explains why the AISI (1996) does not include any design provisions for this case. Further details of comparisons of AS 4600 and AISI predicted loads and experimental failure loads could be found in Telue and Mahendran (1997). Stub Column Tests In all the tests, local buckling of flange elements was observed first. In the case of lined studs, it occurred between the screw fasteners (see Fig. 6). Following considerable post-buckling behaviour, the

44

collapse of the studs occurred through the development of local plastic mechanisms. Table 2 presents the ultimate loads achieved in each test. The use of plasterboard lining increased the failure loads in all the tests. However, the increase was not significant when the lining was fastened at the commonly used spacing of 260 mm. When the lining was fixed at closer centres, such as 130 and 65 mm, noticeable delay in local buckling of flange elements was observed, resulting in up to about 25% increase in failure loads. This means that the plasterboard lining has to be fastened to the studs at smaller spacing to be able to gain any additional strength. Therefore, it can be concluded that any improvement to local buckling behaviour can be ignored in the commonly used plasterboard lined wall frames unless they are fastened at considerably smaller spacing (-cr)min5 by considering a ficticious uniform column with the real column minimum inertia and maximum axial force. (ii) to treat the column as a set of uniform segments (which may or not require an approximation) and, similarly to what is done in frames, determine the BL of the "critical segment". It should also be noticed that, since the buckling mode configuration comprises now more than one analytical expression, it makes no sense to talk about "the distance between the column adjacent points of contraflexure" (however, this BL physical interpretation sfill applies, individually, to each uniform segment).

Frame Members - Frame Stability Since only uniform frame members are considered and discussed here, the BL concept can be used and may be viewed as a "translation" of a "fi-ame stability statement" into a "member language". In fact, the definition presented earlier must now be changed to: "length of a ficticious simply supported uniform

63 column buckling simultaneously with the frame, for the loading under consideration" (Wood, 1974). Assuming the frame crhical load parameter XQT known, the BL of a compressed member i is given by (le)i=7rV(EI)i/Ni(Xer)

,

0)

which means that the ratio (le/L)i depends on the member geometry (L, EI) and also on the frame loading (Ni(A.cr), member axial force at the frame critical state). Physically, it can be shown (Gon9alves, 1999) that this BL still represents the distance between adjacent points of contraflexure of the member deformed shape, but now at the frame critical state. This definition implies that (i) it makes no sense to talk about BL of non-compressed members (IQ=CO if N=0) and (ii) the BL of members with high EI/N(A-cr) values may be unboundedly large (even in braced frames). Several widespread approximate methods to estimate frame critical loads (load parameters, to be precise) involve the previous determination of the individual members BL (Chen & Lui, 1991). They use expressions or graphs based on rigorous stability analyses of simple sub-frames containing the member under consideration. Although such methods are often usefril, they must be applied with caution, as there are limitations that may easily be overlooked by designers. In particular, it is worth mentioning that: (i) the methods only lead to the exact X,cr value if, among other conditions, the stiffness parameter (t)=(EI/NL2)0 5 is the same for all compressed members (the behaviour of the whole frame is then "reflected" in each sub-frame, i.e., all members would, individually, buckle simultaneously), (ii) in order to take into account the transversal members (usually named "beams") stiffness reduction, due to axial forces introduced by proportional loadings, an iterative procedure is required. When the members have different ^ values, the direct application of the methods based on the BL concept only leads to a lower bound of the critical load, (^cr)mm, and involves the performance of the following steps (Barreto & Camotim, 1998): (i) identification of the "critical member(s)", i.e., the member(s) with a lower (|) value (member(s) "triggering" the frame instability and "stabilised" by the remaining ones), (ii) use of adequate expressions and/or graphs to determine the "critical member" BL and calculate (A-cr)min- Naturally, the "quality" of the lower bound decreases for a wider (|) value range. Braced and unbraced frames are treated separately (different expressions/graphs) and one has 0l may occur even in braced frames. In frames displaying a wide range of (j) values, in order to obtain accurate estimates of A-cr on the basis of the previous determination of the individual members BL (each member assumed critical) it is necessary to resort to relatively complicated methods (e.g, Johnston, 1976, and Hellesland & Bjorhovde, 1996). In the authors opinion, a frame exact linear stability analysis is preferable in such situations (computer programs using either stability frmctions or the geometric stiffness matrix are easily available at present). Two simple illustrative examples are shown in figures 2(a) (braced frame) and 2(b) (unbraced frame). The frames are pinned-base and all members have the same length and flexural stiffiiess (L=5m and EI=21000 kNm2). In each case, one compares the stability resuhs (A-cr and columns BL), obtained by

64 means of (i) an exact linear stability analysis and (ii) an approximate method based on the determination of the columns BL using Annex E of EC3, for two situations: columns equally (Nci=Nc2^3A.) and unequally (Nci=5>- and Nc2=^) loaded. One observes that: 5X,

3^,

3?.^

^v

^1 C2

CI

/77^

/TTfW?

cr

cr

cr

(':)c,,.=(Oc,,r'»-35">

X ^

(b) /TmT?

/TM77

/Trrrn

/7nv77

X''=X''=507kN

A,^^= 0.92 A,^''= 2190 kN

X^''^ X^^= 3650 kN

5X, f

1

(a)

U

/7^ ' ^

3^,

-hX^

^1

cr

cr

(Ocrio5(Ocr«5n. (f) =i.05(i:v 9.75 m

cr

cr

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

/Trrm

A,''=A,''=507kN cr

(Ocr(Ocr9«5(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^—

Xcr= 5370 kN I le= 6.2 m t

X„= 2802 kN le= 3.85 m

j/5

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.

I 6

(ai)

///J///

^\

I.

I

\

0'"

1/15

I /

(32)

/7ny77

/77r777

\,= 606 kN le= 18.5 m

X„= 550 kN le=L

(a)

/rfrm

/rfPrn

/77?rn

/77r777

(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

, (4)


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Session A2 BUCKLING BEHAVIOUR

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

89

LOCAL AND DISTORTIONAL BUCKLING OF COLD-FORMED STEEL MEMBERS WITH EDGE STIFFENED FLANGES B.W. Schafer* & T. Pekoz^ ^ Simpson Gumpertz & Heger, Arlington, MA 02474, USA ' Professor, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA

ABSTRACT Cold-formed steel members with edge stiffened flanges have three important buckling phenomena: local, distortional, and global. Current North American specification (e.g., AISI) methods do not explicitly treat the distortional mode or account for interaction of elements in local buckling. Hand methods are presented for proper estimation of the critical buckling stress of compression and flexural members in both the local and distortional mode. Post-buckling behavior of edge stiffened flanges is examined. Phenomena unique to the distortional mode include: reduced post-buckling capacity and heightened imperfection sensitivity. A design method for strength prediction, based on the unified effective width approach, is proposed.

KEYWORDS local buckling, distortional buckling, cold-formed steel design, imperfection sensitivity

INTRODUCTION Buckling of cold-formed steel members with edge stiffened flanges may be generally characterized as occurring in one of three modes: local, distortional, or global (e.g., torsional, flexural etc.). The finite strip analysis shown in Figure 1 illustrates these three buckling modes for a member under pure compression.

I.3.V

1 0.33"

2.5/ =0.0284"

E

The local mode repeats at short wavelengths, generally involves only rotation at element half-wavclcngth (in) junctures (i.e., the elements appear "pinned" at each fold line) and hand methods in current Figure 1 Finite Strip Analysis of a use typically ignore any interaction amongst Member in Compression the elements. The distortional mode repeats at wavelengths from short to long depending on geometry and loading, generally involves rotation

90 and translation of multiple elements (but not the entire cross-section) and hand methods for prediction are relatively cumbersome. The global mode repeats at long wavelengths (often only one half-wavelength occurs in a given member) generally involves rotation and/or translation of the entire cross-section and hand methods for prediction are considered classical examples of buckling phenomena.

ELASTIC BUCKLING PREDICTION BY HAND CALCULATION Local Buckling In design local buckling is typically treated by ignoring any interaction that exists between elements (e.g., the flange and the web). Each element is treated independently and classic plate buckling solutions based on isolated simply supported plates are generally employed. The result of such an approach is that each element is predicted to buckle at a different stress. This approach shall be termed the "element model". Numerical methods, such as the finite strip method, or finite element method may be used to determine the local buckling stress of an entire member; however, for design purposes hand methods are desirable. In order to better predict the actual local buckling stress, but not result to numerical solutions a second class of local buckling solutions is introduced: the "semi-empirical interaction model". These local buckling solutions account for the interaction between two elements, but not the entire member. The solutions are approximations to finite strip analyses of two isolated members, e.g., the flange and the web. In general, it is found that the minimum local buckling stress of any two attached elements is a reasonable approximation of the entire member local buckling stress. Table 1 Proposed Models for Local Buckling Prediction ELEMENT MODEL

Flange: (/,,)/ Web: (/,.). Lip: (/,,)/

k=A k = {Q5^l,^Ae^^,^A)/iblh)' k = k,,Xb/df for 0 < 4^ < 1.1 for 1.1 < 4 , < 2

k,^ = \A^l - 0.254 + 0-425 k,^ = 134^ - 6 5 . 5 ^ +1314, - 8 0

SEMI-EMPIRICAL INTERACTION MODEL

Flange/Lip: (f,r)fi

k = (8.554,, " 110l){d/bf +(-1.594-, + 3.95)(^/^)+4 for 4,,\

\fh/b ^ ^- 1 D ^ • * . , ^, , . , .^ Figure 2 Importance of Rotational Restraint stress IS based on the rotational restraint at the . ^-j o. rr j T^I ,,^ . ^ J . , . m an Edge Stiffened Flange web/flange juncture. Consider a typical cross-section as shown in Figure 3 and the definition of the rotational stiffness. The rotational stiffness may be expanded as a summation of elastic and stress dependent geometric stiffness terms with contributions from both the flange and the web, Xj^^l i Buckling ensues when the elastic stiffness at the web/flange juncture is eroded by the geometric stiffness, i.e.. Writing the stress dependent portion of the geometric stiffness explicitly. Figure 3 k

Therefore, the buckling stress ( / ) is k

+k

k

+k

Complete expressions for the stiffness terms for members in flexure and compression are given in Table 2. The expressions for flexure are derived in Schafer and Pekoz (1999). The sfiffness terms for the flange are completed in the classic manner - assuming the flange acts as a column undergoing flexural-torsional buckling with springs along one edge. The expressions for the web stiffness are derived by truncating the solution for the buckling of a single finite strip.

92 Table 2 Proposed Model for Distortional Buckling Prediction P R O P O S E D M O D E L FOR DISTORTIONAL B U C K L I N G PREDICTION

Jed -"-

7r

Flange Rotational Stiffness:

+1- U^, (^.-/^.y

(*~.l=f7 ^' I L

-^yo{xo-h. K^>f

'>/;

+'^' + )''

Flexural Member: Critical Length and Web Rotational Stiffness

47r\{\-v^) ( /^ \fA^o-h,f+C.f—7^(^o-h,f\

^cr'

^ K n

/*

720

Et 12(1-v^) ^h [LJ

^tiw,,

~

htn^ 13440

60

240

{LJ

(45360(1-^,,,)+62160f^l ^U%n'^{^

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

; r ^ + 2 8 ; r 2 | - I +420

Compression Member: Critical Length and Web Rotational Stiffness

%

(67t'h[\-V^) -J

V

Et 6h{\-v^)

l^ihAlO

~

L

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

J

15

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

Average St. Dev.

0.74 0.12 'predicted

Average St. Dev.

0.75 0.13

Mpredicted/Mlocal

'predicted

0.97 0.06

•^predicted

'^disl.

0.95 0.08

0.90 0.05 ''local

Distortional Buckling Proposed Method

''local

'predicted

''dist.

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 Flanges

Q'

'

o o

Pcr.lncul

25

50

75

e

dit

bit

o PcrJist

4.00-19.0

90

1.82-0.25

6.25-

12.5

45

1.94-0.96

5.00-25.0

90

1.58-0.27

6.25

- 25.0

45

1.76-0.51

6.25

- 37.5

90

1.34-0.18

6.25

- 37.5

45

1.73-0.35

6.25

- 50.0

90

1.40-0.14

6.25

- 50.0

45

1.75-0.23

o

Disionional Mechanism IJislortional Mechanism + LiKal Yielding Mixed - Mechanism [)epcnds on Imp. I^K-al Mechanism + Distortional Yielding 1 AK'al Mechanism

{

O

O

o

^X.

e

ifX.

© e

o

X

X X

0 o X m

X X

X X

100

O O K X X

• o O ®

X X

XX X

X X

^

X

Figure 5 Failure Mechanism Finite element analysis also reveals that the post-buckling capacity in the distortional mode is less than that in the local mode. Consider Figure 6, for the same slendemess values the distortional failures exhibit a lower ultimate strength. Similar loss in strength is experimentally observed and summarized in Hancock et al. (1994). Note for Figure 6, ifct)mechanism is the buckling stress, either local or distortional, that corresponds to the final failure mechanism. As shown in Figure 5 (fcr)mechanism

i s n O t C q u a l tO t h C m i n i m u m o f (fcr)local

and

(fcr)dist.'

Geometric imperfections are modeled as a superposition of the local and distortional mode. The magnitude of the imperfection is selected based on the statistical summary provided in Schafer and Pekoz (1998). The error bars in Figure 6 demonstrate the range of strengths predicted for imperfections varying over the central 50% portion of expected imperfection magnitudes. The greater the error bars, the greater the imperfection sensitivity. The percent difference in the strength over the central 50% portion of expected imperfection magnitudes is used as a measure of imperfection sensitivity:

(/) - ( / ) '25%imp.

\[{a.

25%imp.

15% imp.

v-^«/75%/mp./

xlOO%.

95 A contour plot of this imperfection sensitivity statistic is shown in Figure 7. Stocky members prone to failure in the distortional mode have the greatest sensitivity. In general, distortional failures are more sensitive to initial imperfections than local failures. Areas of imperfection sensitivity risk are

-

1.8 1.6

"^"^^

L *

'^^^^^V'

1.4

r

1.2

fet

A 0.6

^^\

MEDIUM'^

)

1

fy

A •: \v)-'

0.8

Winter's Curve • Local Buckling Failures O Distortional Buckling Failures T f J-

0.6 0.4

error ban indicate (he range of suengtlis obaerved between imperfection magnitudes of 2S and 75% probability or exceedance

•

.^

\

12/

.

MEDIUM 5%^

0.2

LOW

,

^^o^~\

-10%-;^

1

>/A7(a Figure 7 Imperfection Sensitivity Figure 6 Failure Strength assigned. INTEGRATING DISTORTIONAL BUCKLING INTO DESIGN The current North American specification approach for the capacity of a member involves determining an effective area or section modulus to account for local buckling. The reduction is based on an empirical correction to the work of von Karman et al. (1932) completed by Winter (1947). The extension of this approach to all members of the cross-section is based on the unified approach of Pekoz (1987). The resulting effective section is used to (1) calculate the capacity due to local buckling alone and (2) determine the reduced section properties for use in global buckling modes in order to account for interaction between local and global modes. When considering distortional buckling in design one must consider whether distortional buckling should be treated in a manner similar to local buckling, global buckling, or in an entirely new way. If distortional buckling is a separate failure mode it may be treated as such (e.g., the method of Hancock et al. 1996). If distortional buckling can interact with global modes then an effective width type of approach that accounts for local and distortional buckling would be appropriate - this is the method currently suggested. Further, the results of the previous section show a direct competition between local and distortional buckling that must be accounted for. If distortional buckling is considered then the critical buckling stress of an element (flange, web or lip) is no longer solely dependent on local buckling, as is currently assumed in most specifications. In order to properly integrate distortional buckling, reduced post-buckling capacity in the distortional mode and the ability of the distortional mode to control the failure mechanism even when at a higher buckling stress than the local mode must be incorporated. Consider defining the critical buckling stress of the element as:

(/J=min[(/„L,,/?,(/„X,,] The slenderness (for an applied stress equal to the yield stress) is: For strength, if the reduced distortional mode governs, then effective width would be: b^ff =pb where p = 7 ^ / l ( l - 0 . 2 2 ^ ^ / 1 ) For Rd < 1 this method provides an additional reduction on the post-buckling capacity. Further, the method also allows the distortional mode to control in situations when the distortional buckling

96 stress is greater than the local buckling stress. Thus, Rd provides a framework for solving the problem of predicting the failure mode and reducing the post-buckling capacity in the distortional mode. The selected form for Rd based on Figure 5 and 6 and the experimental results of Hancock et al. (1994) is: ^ 1.17 ^ Rj = mini 1,1 7 + 0.3 where/l, = ^ / , / ( ^ ) ^ ^ . ^ . If numerical methods (finite strip analysis) are not used to determine the critical buckling stress in the local and distortional modes, then the models proposed herein are suggested for use. The above approach was examined for the strength capacity of laterally braced flexural members. Experimental data of Cohen (1987), Desmond (1981), Ellifritt et al. (1997), LaBoube and Yu (1978), Moreyra (1993), Rogers (1995), Schardt and Schrade (1982), Schuster (1992), Shan et al. (1994), and Willis and Wallace (1990) on laterally braced lipped channel and Z sections was gathered and examined - see Schafer and Pekoz (1999). Using the proposed hand methods for calculation of the local and distortional buckling stress (for local buckling the interaction model is used) a test to predicted ratio of 1.07 with a standard deviation of 0.08 was determined for the 190 experiments. In addition to properly accounting for distortional buckling individual cases are observed where including the local buckling interaction yields markedly better results. For example, local buckling initiated by long lips (long edge stiffeners) and local buckling with highly slender webs and compact flanges are examples where including the interaction is observed to improve the strength prediction markedly. Currently work is underway to investigate similar approaches for compression members and also to investigate the possibility of directly using finite strip analysis results on the entire member instead of the current element by element approach. CONCLUSIONS Cold-formed steel members with edge stiffened flanges have three important buckling phenomena: local, distortional, and global. Current North American specification methods do not explicitly treat the distortional mode or account for interaction in local buckling. Distortional buckling deserves special attention because it has the ability to control the final failure mechanism and is also observed to have lower post-buckling capacity and higher imperfection sensitivity than local buckling. New hand methods are developed to properly estimate the critical buckling stress in both the local and distortional mode. A design method for strength prediction, based on the unified effective width approach, is discussed. The design method uses the new expressions for prediction of the local and distortional buckling stress. Proper incorporation of the distortional buckling phenomena is imperative for accurate strength prediction of cold-formed steel members. ACKNOWLEDGEMENT The sponsorship of the American Iron and Steel Institute in conducting this research is gratefully acknowledged. APPENDIX I. REFERENCES American Iron and Steel Institute (1996). AISI Specification for the Design of Cold-Formed Steel Structural Members. American Iron and Steel Institute. Washington, D.C. Cohen, J. M. (1987). Local Buckling Behavior of Plate Elements, Department of Structural Engineering Report, Cornell University, Ithaca, New York. Desmond T.P., Pekoz, T. and Winter, G. (1981). "Edge Stiffeners for Thin-Walled Members." Journal of the Structural Division, ASCE, February 1981. Elhouar, S., Murray, T.M. (1985) "Adequacy of Proposed AISI Effective Width Specification Provisions for Z- and CPurlin Design." Fears Structural Engineering Laboratory, FSEL/MBMA 85-04, University of Oklahoma, Norman, Oklahoma. Ellifritt, D., Glover, B., Hren, J. (1997) "Distortional Buckling of Channels and Zees Not Attached to Sheathing." Report for the American Iron and Steel Institute.

97 Hancock, G.J. (1995). "Design for Distortional Buckling of Flexural Members." Proceedings of the Third International Conference on Steel and Aluminum Structures, Istanbul, Turkey. Hancock, G.J. (1997). "Design for Distortional Buckling of Flexural Members." Thin-Walled Structures, 27(1). Hancock, G.J., Kwon, Y.B., Bernard, E.S. (1994) "Strength Design Curves for Thin-Walled Sections Undergoing Distortional Buckling." Journal of Constructional Steel Research, 31(2-3), 169-186. Hancock, G.J., Rogers, C.A., Schuster, R.M. (1996). "Comparison of the Distortional Buckling Method for Flexural Members with Tests." Proceedings of the Thirteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO. HKS. (1995) ABAQUS Version 5.5. Hibbitt, Karlsson & Sorensen, Inc. Pawtucket, RI. LaBoube, R.A., Yu, W. (1978). "Structural Behavior of Beam Webs Subjected to Bending Stress." Civil Engineering Study Structural Series, 78-1, Department of Civil Engineering, University of Missouri-Rolla, Rolla, Missouri. Lau, S.C.W., Hancock, G.J. (1987). "Distortional Buckling Formulas for Channel Columns", ASCE Journal of Structural Engineering, 113(5). Moreyra, M.E. (1993). The Behavior of Cold-Formed Lipped Channels under Bending. M.S. Thesis, Cornell University, Ithaca, New York. Pekoz, T. (1987). Development of a Unified Approach to the Design of Cold-Formed Steel Members. American Iron and Steel Institute Research Report CF 87-1. Rogers, C.A., Schuster, R.M. (1995) "Interaction Buckling of Flange, Edge Stiffener and Web of C-Sections in Bending." Research Into Cold Formed Steel, Final Report of CSSBI/IRAP Project, Department of Civil Engineering, University of Waterloo, Waterloo, Ontario. Schafer, B.W., Pekoz, T.P. (1998). "Computational Modeling of Cold-Formed Steel: Characterizing Geometric Imperfections and Residual Stresses." Journal of Constructional Steel Research, 47(3), 193-210. Schafer, B.W., Pekoz, T.P. (1999). "Laterally Braced Cold-Formed Steel Members with Edge Stiffened Flanges." ASCE Journal of Structural Engineering, 125(2), 118-127. Schardt, R. Schrade, W. (1982). "Kaltprofil-Pfetten." Institut Fur Statik, Technische Hochschule Darmstadt, Bericht Nr. 1, Darmstadt. Schuster, R.M. (1992). "Testing of Perforated C-Stud Sections in Bending." Report for the Canadian Sheet Steel Building Institute, University of Waterloo, Waterloo Ontario. Shan, M., LaBoube, R.A., Yu, W. (1994). "Behavior of Web Elements with Openings Subjected to Bending, Shear and the Combination of Bending and Shear." Civil Engineering Study Structural Series, 94-2, Department of Civil Engineering, University of Missouri-Rolla, Rolla, Missouri, von Karman, T., Sechler, E.E., Donnell, L.H. (1932). 'The Strength of Thin Plates In Compression." Transactions of the ASME, 54, 53-51. Willis, C.T., Wallace, B. (1990). "Behavior of Cold-Formed Steel Purlins under Gravity Loading." Journal of Structural Engineering, ASCE. 116(8). Winter, G., (1947) "Strength of Thin Steel Compression Flanges." Transactions of ASCE, Paper No. 2305, Trans., 112, 1.

This Page Intentionally Left Blank

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

99

COLLAPSE BEHAVIOUR OF THIN-WALLED ORTHOTROPIC BEAMS M. Kotdko^ ^ Department of Strength of Materials and Structures, Technical University of Lodz, 90-924 Lodz, Stefanowskiego 1/15, Poland

ABSTRACT Influence of orthotropic properties on ultimate load and post-failure behaviour of thin- walled beam is investigated in the paper. An orthotropic, homogeneous material is considered. The analysis accounts for a rigid-elastoplastic material, which displays an orthotropic strain hardening. The plastic mechanism analysis is performed using the energy method based on the rigid-plastic theory with additional assumptions concerning orthotropic properties of the beam material. A fiilly plastic moment at a yield-line is evaluated using Hill yield criterion. Solution of the problem is based on the principle of virtual works. Two postulates which define an approximate relation between the plastic strain at the yield-line (treated as a yield strip of finite width) and the angle of relative rotation of two adjacent walls of the global plastic hinge ( plastic mechanism) are applied. The ultimate bending moment ( upper bound estimation) is determined as an ordinate at an intersection point of two curves: the post-buckling path obtained fi^om an approximate postbuckling analysis and the failure path gainedfi-omthe plastic mechanism analysis. The dependence of the collapse behaviour and ultimate bending moment upon the orthotropy ratio is investigated.

KEYWORDS thin-walled structures, orthotropic material, collapse, load-carrying capacity

1. INTRODUCTION Collapse behaviour of a thin-walled structure is a factor of a great importance. A designer should know a character of a potential catastrophe as far as the safety of the structure is concerned. From the other point of view the knowledge about the energy of plastic deformation is necessary in the design process of structural members, particularly designed as absorbing elements, e.g. energy absorbers against automobile collisions. Finally, the combination of the non-linear, postbuckling analysis with the analysis of the plastic mechanism allows one to establish a failure parameter approximately, e.i. to estimate the upper hound load-carrying capacity of the structure. Considering each of three issues mentioned above a researcher faces with the substantial problem of the evaluation of the failure structural path. A solution of the collapse

100 behaviour problem is based upon the rigid - plastic theory [1] and may be accomplished using the energy method. The method consists in the evaluation of the energy of plastic deformation, after establishing the geometry of the plastic mechanism of failure ( a global plastic hinge). As a result one obtains the relation between the generalised force and the generalised displacement at the global plastic hinge. The rigid-plastic theory assumes mainly a rigid-perfectly plastic [1] behaviour of the structural isotropic material. Recently, the rigid-elastoplastic material behaviour ( with linear strain hardening) have been taken into account in the solution of the plastic mechanism problem in thin-walled beam subject to bending [2]. However, nearly all steel and aluminium alloys display orthotropic properties, particularly after cold-forming or rolling. Some of sheet metals made of both steel and aluminium alloys are of strong orthotropic properties in the plastic range so that it induces the necessity to incorporate the orthotropy factor into the analysis of plastic mechanisms of failure. The problem of anisotropic plastic properties of sheet metals caused by complex manufacturing processes like multistage rolling and stretching is comprehensively discussed by Szczepihski [7]. Influence of orthotropic properties on ultimate load and post-failure behaviour of thin- walled beam is investigated in the paper. An orthotropic, homogeneous material is taken into consideration. The analysis accounts for a rigid-elastoplastic material, which displays an orthotropic strain-hardening. The plastic mechanism analysis is performed using the energy method based on assumptions of the rigid-plastic theory, extended by additional assumptions concerning orthotropic properties of the beam material.

2. SUBJECT AND BASIC ASSUMPTIONS OF THE ANALYSIS The subject of investigation was a thin-walled, rectangular and trapezoidal box-section beam under pure bending ( Fig. 1). The beam cross-section was a rectangle or an isosceles trapezoid. The bending moment was acting in the plane created by the axis of the cross-section symmetry and the longitudinal axis of the beam. The analysis was carried out on the basic assumptions, which were as follows: - the failure of the beam was initiated by buckling of the flange subject to compression and also the first yield was assumed to occur in the compressed flange or, in a particular case, in the flange intension so th?it the flange mechanism was expected, - kinematically permissible (true mechanisms ) were taken into account only, i.e. plastic mechanisms were assumed to be well developed and membrane strains in walls of the global plastic hinge were neglected [1], - plastic zones were concentrated and could be regarded either as stationary or travelling yieldlines of the global plastic hinge [2], - the rigid-perfectly plastic (Fig.2a) or rigid- elastoplastic behaviour displaying a linear strain hardening (Fig.2b) was assumed for an orthotropic material. X I ^

^^T

I

f—r

D V Figure 1: Thin-walled beam under pure bending.

101 - one of the principal directions of orthotropy was assumed to coincide with the longitudinal beam axis. b)

a)

CTol

CJo2

Figure 2: Material characteristics, a) - rigid-perfectly plastic, b) - rigid-elastoplastic with linear strain hardening Considered material characteristics are simplifications of real material behaviour as shown in Fig. 2. Apart fi-om uniaxial tensile diagrams for two principal directions of orthotropy, two other characteristics are neccessary when considering an orthotropic material. There are pure shear test diagram and an uniaxial tensile test diagram for a selected direction inclined with respect to principal directions of orthotropy.

3. YIELD CRITERION AND BASIC RELATIONS IN THE ELASTO-PLASTIC RANGE The most effective yield citerion for the considered case is a Hill function [3]. The effective stress in the elasto-plastic range takes form a' = a,al + a^al -a.^a^a^ +3a^ r%

(1)

where a; - 03 are parameters of anisotropy which should be determined in four independent tests. Initial parameters of anisotropy are as follows

a.n =•

(2)

3rL

for 0 = 45°: ano = aio +a2o + 3a3o - 4a33o, (2a) where aio ,020, cieo are initial yield stresses determined in tensile tests for x, y and 9 directions, respectively, while Xno is an initial shear yield stress - determined in pure shear test. When x is chosen as reference direction, then we obtain aio=l

"2o

2

3r^

*iio

2

(2b)

The initial yield stress corresponding to the direction y (where y is an angle between the considered direction and the principal direction x) is evaluated as

102 —2

^lo

px

^° aio cos"* y + a2^ sin"* y - aijo sin^ y cos^ y + O.TSIjo sin^ (2y) It has been also proved that the effective stress in y direction may be expressed in the similar way, when X is a reference direction and aio = ai=l (4)

^ cos"^ y+a2 sin"* y - a,2 sin^ y cos^ y + 0.15a^ sin^ (2y)

For a strain hardening material (Fig.2) a change in an actual yield stress (eflfective stress) depends upon a value of plastic work performed in a given direction. For the linear strain hardening behaviour this work amounts

where a = (TQ + E^s^ and E^* is a tangent modulus in the considered direction. In order to obtain an equivalent change in effective stress the plastic work performed in an arbitrary direction has to be of the same value WP = WxP = W / = WPe=45o=WPxy

(6)

Taking into account (6) in (5) actual parameters of anisotropy are obtained as follows

a'

5,=

(El IE'

a'-I

a' «2 =

«33

w'--crl) + o-?o

(^;'IE '\a' - 0.673

X,+\ where

K=yl7/fZ COMPARISON OF TEST RESULTS AND DESIGN RECOMMENDATIONS In the case of test specimens CS1.3-F, CC1.2-F and CC1.5-F, whose web parts were removed, the flange parts behaved independently and their buckling mode was torsional-flexural. According to Eurocode 3, the strength should be determined in this case using column curve c (a = 0.49). The comparisons of test results and predicted values are shown in Figure 9. In order to compare different design methods. Figure 9 also presents the compression resistance values calculated by using all the other column curves and by using design methods for distortional buckling as well. As Figure 9 shows, the Eurocode 3 column curve c gives over 40% conservative values for all specimens whose web part was removed.

129 1.60

1,40 1,20

I 1.00 j ^

0,80

-*- EC3-a=0.49 •^ EC3-a=0.34 -*- EC3-a=0.21 -*-EC3-a=0.13 -^AUS-dist •*• Schafer-dist - ^ EC3-dist.

2 0,60 -[ 0,40 j 0,20-} 0,00 CS-1.3F

CC1.2F

CC1.5F

Figure 9: Comparison of test results and predicted values for "flange" specimens A comparison of the test values with predicted values for the whole section are shown in Figure 10 and 11. Figure 10 shows the comparison when the lips of web-stiffened C-sections failed inwards and Figure 11 when the lips failed outwards. Figures 10 and 11 indicate that the design methods for distortional buckling offer quite good predictions for web-stiffened C-sections, but they are about 20% out for sigma-sections. The Eurocode 3 method gives a slightly higher compression strength than the method presented in the Australian Standard or in the Schafer method. It seems that determining elastic buckling stress by using the thickness (or stiffness) reduction for the perforated part of the web of the sigma-section leads to too high buckling stress values. One obvious reason for this is the much lower axial stiffness of the perforated web than that used in the model. It seems that this is an important factor for sigma-sections, whose distortional buckling behaviour differs from the behaviour of Csections. The distortional buckling stress is also sensitive to the local buckling stress level of the web (Lau and Hancock, 1987) and thus a "thickness reduced" model can lead to an inaccurate result. M

1,601,40 1,20-

. ..

^^^-C^ ^ ^^ , v ^^ ^ ^ .^ ^, ^ ^ ^^4

|i,oo^

0,80

z

0,60 -

. . .

,

"V -r'.-^g

y^ >ZL

^ ^=**=^ ^O'i:^^:

-

"

r==^

0,400,200,00-

1

CS13

r

CC12

1,60-

A

1,40-

0=-^

',—-% T^

1,20-

A

- ^ EC3-a=0.34 -o- EC3-a=0.21 -»-EC3-a=0.13 ^AUS -*- Schafer - ^ EC3-dist CC15

Figure 10: Comparison of test results (Failure type inwards) and predicted values for whole specimens

1,000,80-

. . .

.J

%—

0,600,400,20 0.00CC12

£, '-^

_

i

A

•- - - - -1 —X

1

- ^ EC3-a=0.34 -^ EC3-a=0.21 -^EC3-a=0.13 ^AUS - ^ Schafer ->^EC3-dist CC15

Figure 11: Comparison of test results (Failure type outwards) and predicted values for whole specimens

130 CONCLUSION A test programme on web-perforated steel wall studs has been described. The tests were performed on web-perforated sigma-sections and web-stiffened C-sections between fixed ends. Two test series were performed for each section type. In the first series, the section was tested as a whole and in the other test series, the perforated area was removed. The tests showed that even a small restraint given by the perforated web has an influence on compression capacity. The test results of web-stiffened C-sections also showed that ultimate compression capacity is sensitive to the direction of initial imperfection. The failure load of the specimens, whose lips failed inwards, was about 14 - 16% higher than the failure load of identical specimens whose lips failed outwards. An analytical prediction for compression capacity was performed. The elastic distortional buckling strength was determined by using the generalized beam theory by replacing the perforated web part with plain plate of the same bending stiffness. The ultimate compression capacity was determined by using the design curves given in design recommendations, such as Eurocode 3 and the Australian and New Zealand Standard. A comparison between the test results and predicted values showed that the method used gives reasonable results for web-stiffened C-sections but it overestimates the compression capacity of sigma-sections. The compression capacity of the specimens, whose perforated web was removed and whose flange parts behaved independently, was over 40% higher than the predicted value according to Eurocode 3 using column curve c for torsional buckling. ACKNOWLEDGMENTS This paper was prepared while the first author was on a one-year study leave at Manchester University. This leave was supported by The Academy of Finland. The authors would like to thank Professor J.M. Davies for his contribution. The facilities made available by the Division of Civil Engineering are gratefully acknowledged. Thanks are also due to the Finnish companies Aulis Lundell Oy and Rautaruukki Oyj for supplying test specimens. REFERENCES Standards Australia / Standards New Zealand (1996). Cold-formed Steel Structures. AS/NZS 4600:1996. Davies, J. and Jiang, C. (1995). GBT - Computer program, public domain, University of Manchester. Eurocode 3 (1996), CEN ENV 1993-1-3 Supplementary Rules for Cold Formed Thin Gauge Members and Sheeting. Lau, S. and Hancock, G., Distortional Buckling Formulas for Channel Columns, Journal of Structural Engineering, 113:5, 1063-1078. NISA, Version 6.0 (1996). Users Manual, Engineering Mechanics Research Corporation (EMRC), Michigan. Schafer, B. and Pekoz, T. (1998). Direct Strength Prediction of Cold-Formed Dteel Members using Numerical Elastic Buckling Solutions. 14^^ International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri U.S.A., pp. 69-76.

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

131

Distortional Buckling of Cold-Formed Steel Storage Rack Sections including Perforations N. Baldassino^ and G. Hancock^ ^ Department of Mechanical and Structural Engineering, University of Trento, Italy • Department of Civil Engineering, University of Sydney, Australia

ABSTRACT The types of cold-formed sections commonly used in steel storage rack uprights are generally susceptible to distortional buckling. Design provisions have recently been included in the Australian Standard AS4084 (1993) for Steel Storage Racks, and in the Australian/New Zealand Standard AS/NZS 4600 (1996) for Cold-Formed Steel Structures to account for distortional buckling of lipped channel sections with additional rear flanges and lips. However, it is not clear how the effect of perforations (holes) influences distortional buckling and whether local and distortional buckling interact. The current European FEM document (1997) for the design of steel storage pallet racking systems permits rational analyses only for sections without perforations. For sections with perforations, an approach based on a suitable testing technique should be used. The paper describes a series of tests on commonly used steel storage rack sections which has been carried out with the aim of investigating the previously mentioned interactions. Comparison of the test results with gross section models, net section models and effective section models are presented and discussed.

KEYWORDS Cold-formed members, distortional buckling, flexural-torsional buckling, local buckling, gross section, perforated section, effective section, testing technique.

INTRODUCTION The upright columns of steel storage racks are generally manufactured from channel members. In accordance with the USA practice, bracing members are welded to the uprights so that simple lipped channels are used. Otherwise, in Europe and Australia, additional flanges (called rear flanges) are attached to the lips to allow bolted braces to be connected to the uprights. In some cases, additional lips are located at the ends of the rear flanges and normally point outwards. Under compression, the

132 uprights may buckle in a local (L), flexural (F), flexural-torsional (FT) or distortional (D) buckling mode as shown in Figures 1(a), 1(b), 1(c) and 1(d), respectively. Combinations of these modes are also possible, and these combinations are called interaction buckling modes. The individual modes of buckling have been extensively investigated and summaries of these studies are reported in Timoshenko and Gere (1961) for local, flexural and flexural-torsional buckling, in Trahair (1993) for flexural-torsional buckling and in Hancock (1998) for distortional buckling. The interaction of the buckling modes is less well understood. For the interaction of local and flexural or flexural/torsional buckling, a method called the "unified method" has been recently developed and verified by Pekoz (1987). However, the interaction between the distortional mode and the local or the flexural-torsional modes is the subject of a research project currently underway in co-operation between the Universities of Sydney (AUS) and Trento (I).

a)

b)

c)

d)

Figure 1: Typical buckling modes for thin walled sections The upright sections usually contain holes and/or perforations at regular intervals to allow beams and bracing to be easily attached without bolts or welds. The influence of the holes has been in some cases investigated focusing attention on local buckling. The usual approach to design perforated members requires short length column tests (stub column tests) to define a suitable stress reduction factor (form factor Q) which accounts for reduction due to both local buckling and perforations. In accordance with the Rack Manufacturers Institute Specification (RMI, 1997) and the Australian Standard AS4084 (1993), the form factor Q is defined as: Q=

-^netjmin^y

(1)

where P^ is the ultimate load of the stub column specimen, fy is the experimental yield stress of the material (obtained from tensile coupon tests) and Anet,min represents the minimum net cross-sectional area. The European rack design specification (FEM, 1997) uses a similar formula except that the symbol Q is replaced by % and A^gt^jn is replaced by the gross (full) area Ag. The purpose of this paper is to investigate the effect of the perforations on the different buckling modes. Herein the results of a series of tests on one type of rack upright section are presented. The column length ranged from short length (stub columns), which underwent mainly local instability to intermediate length sections, which underwent distortional instability and longer length sections, which underwent flexural-torsional instability. Moreover, the influence of load eccentricity was also investigated since the particular section tested changed its buckling mode depending on the load eccentricity.

THE EXPERIMENTAL PROGRAMME A commercial type of rack column has been selected, owing to the results of a preliminary finite strip elastic analysis [Papangelis & Hancock, 1995], which pointed out that the elastic distortional buckling stress evaluated on the gross section is very close to the one associated with local buckling mode.

133 Moreover, both values of the stress are practically coincident with the one associated with the yielding of the material. The geometry of the considered rack section is shown in Figure 2. ITi

L

J 5^

ON

[mm]

_

80

Figure 2: Geometry of the considered rack section The experimental program comprised compression tests (in total 49 specimens) on both perforated and non-perforated members. Specimens were prepared in accordance with the criteria followed for stub column tests, as indicated in the FEM recommendations (1997). The area of the perforated cross-section (Anet,min) was 86.23% of the gross area (Ag) of member. The mean measured yield stress of the perforated specimens was 358 MPa and for the specimens without holes was 376 MPa. The resulting squash loads based on the measured yield stress multiplied by the gross area are 136.5 kN and 143.4 kN, respectively. To investigate the different buckling modes, several specimen lengths have been considered, ranging between nominal values of 285 mm and 1185 mm. In perforated members, the number of pitches of the perforations varied between 4 and 16. Moreover, the influence of the eccentricity of the load has been considered too. The relative position between the centroid of the plates welded to the end of the column has been varied with respect to that of the gross section. Nominal eccentricities ( e j of 0, +2.5 and +5 mm have been considered. Zero eccentricity means that the centroid of the plates is coincident with the one of the gross section of the column. The "convention" for definition of positive and negative eccentricities is presented in Figure 3.

G

• G=C

C

-*Io

^ Positive eccentricity

Negative eccentricity

C : Load application point G : Centroid of the gross section Figure 3: Convention for definition of the eccentricity Specimens with a length of 510 mm are commonly used for commercial stub column tests. Imperfections due to the welding operation during the workshop preparation of the specimens could induce large load eccentricity and could influence the test results. Hence specimens with 510 mm length and load eccentricity varying between +10 and -5 mm have been considered.

134 The compression tests have been carried out up to the collapse of the specimen: during the test, the applied load and the displacement of the end plates were monitored [Baldassino & al., 1998].

THE TEST RESULTS The test results are shown in Figure 4, in terms of ultimate load (P^) versus nominal length of the member (1), with reference to the different nominal load eccentricity. The specimens are labelled as WH or H for members without and with holes, respectively. A first appraisal of test results indicates a non-negligible influence of the perforations on the value of collapse load. In particular, it can be noted the very similar load reduction trend independently of the specimen length or eccentricity. This trend is confirmed also by Figure 5, which reports the ultimate load versus the actual eccentricity (e) for the 510 mm length specimens. The maximum load reduction observed is approximately 32.4% and the effect of the area reduction taken together with the yield stress reduction is 18.8%, hence the effect of the perforation and local buckling produces an additional reduction of 13.6%. Furthermore, the specimens with holes appear slightly more sensitive to the load eccentricity compared with the ones without holes.

200

400

600

800

1000

1200 1 [mm]

Figure 4: Experimental results related to specimens with and without perforations

-

6

-

3

0

3

6

Figure 5: Experimental results related to specimen with length of 510 mm

135 With reference to the H specimens, it can be observed that an increase of the ultimate load is associated with positive eccentricities, independently on the specimen length (Figure 4). The value of the load eccentricity also affects the failure mode: the local buckling failure mode was prevalent at negative and zero eccentricities, while the distortional buckling failure mode was mainly associated with large positive eccentricities. Furthermore, combined local and flexural-torsional buckling was observed as a typical failure mode for specimens with longer lengths.

THE AUSTRALIAN/NEW ZEALAND DESIGN APPROACH The prediction of structural performance of cold-formed steel members subject to different loading conditions is a problem which has been solved for some commonly used sections. The approach proposed in some international standards for cold formed steel member design is mainly related to non-perforated members. As previously mentioned, the influence of perforations on the failure mode and on the load carrying capacity can be significant and, as a consequence, suitable rules should be defined and used in design. To investigate the applicability of the approach proposed by the international standards for perforated members, in the following it is proposed to compare the experimental results and the ultimate loads evaluated on the basis of the Australian/ New Zealand Standard AS/NZS 4600 (1996). Attention has been focused on perforated members and on the distortional and flexural-torsional buckling modes. The relationships proposed by the AS/NZ 4600 (1996) - Section 3.4 are briefly presented as follows: - Singly-symmetric sections subject to torsional or torsional-flexural buckling (Clause 3.4.1) (2)

Nc=Aef„ where:

f

2l

V

J

c fn = 0.658'^

f„ =

fy

1.5

(4)

Nc is the ultimate load of the specimen, A^ is the effective area at the critical stress fn , which account for both local buckling and material yielding, fy is the yield stress of the material, X^ is the non-dimensional slendemess and fo^ represents the minimum value between the elastic flexural, torsional and torsional-flexural buckling stresses. - Singly-symmetric sections subject to distortional buckling (Clause 3.4.6) The nominal member capacity (Ng) is the lower of the values evaluated in accordance with Eqn. 2 and

136

Nc = Af„ = Afy 1 - 4fod

Nc = Af„ = Afy 0.055

(5)

focl>-

fy

+ 0.237

fy (6)

where A is the full (gross) area of the cross-section, f^ is the critical stress for distortional buckling, fy is the yield stress of the material and f^^ represents the elastic distortional buckling stress of the cross-section. In accordance with the aim of this research, three different values of the area (gross, net and effective area) have been taken into account to evaluate the ultimate load (N^) based on the equations previously presented (Eqns. 3-6). Furthermore, both zero eccentricity and effective eccentricity have been considered. In Eqns. 3-6, fy has been considered as the mean yield stress obtained from the tensile coupon tests while foe and fod have been evaluated in accordance with the AS/NZ 4600 (1996) and by means of an elastic buckling analysis, respectively. The effective area (A^) in Eqn. 2 has been estimated on the basis of RMI Specification (1997) as:

Ae = l - ( l - Q ) v^yy

•^ net, mm

(7)

where the Q-factor has been deduced on the basis of the experimental results (Eqn. 1). In particular, the two different Q-values for specimens with and without perforations chosen as the maximum experimental values calculated for each group have been adopted (Q=1.00 for W H specimens and Q=0.828 for H specimens). The comparison between the experimental results and ultimate load estimated on the basis of the AS/NZ 4600 (1996) is summarised in Tables 1 and 2, where the specimens are identified by H (perforated section), by the value of the actual load eccentricity and by the length. For each test the observed failure mode (F.M.) is presented. It should be noted that, for those specimens that failed in the distortional mode, the mean value of the ratios Pu/(Anet,minfn) (Equs. 5-6) is 0.928 with a coefficient of variation of 0.0312. For those specimens which failed in flexural-torsional mode, term Pu/(Aefn) computed by Eqns. 2, 3, 4 and 7 is 0.991 with a coefficient of variation of 0.0332. It is clear that the interaction between flexural-torsional buckling and local buckling failure mode including the effect of the perforations via the Q-factor is an accurate method. The same is not true for the distortional buckling mode where the effective area in Eqns. 5 and 6 has been taken as Anet,min- It appears that there is some interaction between local and distortional buckling. An alternative proposal is to replace A in Eqns. 5 and 6 by A^ given by Eqn. 7 with f^ the critical stress for distortional buckling. In this case, for the specimens that failed in the distortional mode, the mean value of ratios of the test maximum load to those based on Eqns. 5 and 6 is 1.087 with a coefficient of variation of 0.0235. This value is slightly conservative and assumes interaction between local and distortional buckling.

137 TABLE 1 COMPARISON BETWEEN TEST RESULTS AND A S / N Z ESTIMATED FAILURE LOADS

Distortional failure mode e = effective e=0 1 Specimen

F.M.

H/+5.2/510 H/+4.45/510 H/+4.15/510 H/+3.75/510 H/+5.4/737 H/+4.2/963

fkN] 93 D 93 D 92 D D 95 L,D 87 83 D D,FT 81

|H/+4.75/1187

Mean value [standard deviation

| |

Pu

Pu

Pu

Pu

Pu

Agf„

A f ^net,min^n

Agfn

A f '^net.min^n

Aefn

0.780 0.780 0.773 0.795 0.814 0.831 0.849 0.800 0.0269

0.905 0.904 0.897 0.921 0.944 0.964 0.985 0.928 0.0312

1.069 1.068 1.060 1.089 1.099 1.111

0.769 0.891 0.770 0.893 0.765 0.887 0.787 0.912 0.778 0.903 0.788 0.913 0.791 0.918 0.777 0.901 0.0098 0.0114

1.129 1 1.087 1 0.02351

TABLE 2 COMPARISON BETWEEN TEST RESULTS AND A S / N Z ESTIMATED FAILURE LOAD

Flexural-torsional failure model 1 Specimen H/+l,95/510 H/+2,65/737 H/+2,15/960 |H/+2,25/1185

Mean value [standard deviation

F.M.

Pu

[kNl L,F 95 L,FT,D 92 L,FT 82 L,FT 76

Pu

Pu

A net,min fn

Aefn

0.851 0.865 0.817 0.820 0.838 0.0204

1.019 1.028 0.962

1

0.954 1 0.991

0.0332 J

CONCLUSIONS A test program on cold-formed rack section uprights with lip-stiffened rear flanges has been planned and executed to assess the effect of load eccentricity, length and perforations on the different modes of buckling. Three buckling modes were observed in the tests: local, distortional and flexuraltorsional. Interaction between these modes also occurred. The ultimate load for perforated sections undergoing flexural-torsional buckling interacting with local buckling is well predicted by the method described in the RMI Specification (1997), AS4084 (1993) and AS/NZS 4600 (1996). However, the design approach for distortional buckling in AS/NZS 4600 (1996) is conservative only for perforated sections, if the distortional buckling critical stress is multiplied by the effective area, which accounts for local buckling of a perforated section. The use of the minimum net area rather than the effective area with the critical stress for distortional buckling will produce unconservative results. This indicates that there was interaction between local and distortional buckling for the perforated sections tested.

138 ACKNOWLEDGEMENTS The authors greatly appreciate the skilful work of the technical staff of the Laboratory of the Department of Mechanical and Structural Engineering of the University of Trento and express their particular thanks to Mr. S. Girardi for his assistance in the experimental work. The authors wish to thank Prof R. Zandonini (University of Trento) and Prof C. Bemuzzi (Technical University of Milan) for their useful suggestions and advice. The contribution of the Italian Rack Company Transima Italiana S.p.A. in providing the cold formed steel members is appreciated. The financial support of the University of Sydney for the second author to work in Italy is also appreciated.

REFERENCES AS4084. (1993). Steel Storage Racking, Standards Australia. AS/NZS 4600. (1996). Cold-formed Steel Structures, Standards Australia. FEM. (1997). Recommendation for the Design of Steel Pallet Racking and Shelving, Section X of the Federation Europeenne de la Manutention. Baldassino N., Bemuzzi C. and Zandonini R. (1998). Experimental and Numerical Studies on Pallet Racks. Proceeding of the Conference "Professor Otto Halase-Memorial Session", Technical University of Budapest, TU Budapest publ. (to be published). Hancock, G.J. (1998). Design of Cold-Formed Steel Structures, 3rd Edition, Australian Institute of Steel Construction, Sydney, Australia. Papangelis J.P. and Hancock G.J. (1995). Computer Analysis of Thin-Walled Structural Members. Computers and Structures Vol. 56, No 1,157-176. Pekoz T. (1987). Development of a Unified Approach to the Design of Cold-Formed Steel Members, American Iron and Steel Institute, Research Report CF87-1. RMI. (1997). Specification for the Design, Testing and Utilization of Industrial Steel Storage Racks, Rack Manufactures Institute. Timoshenko SP. and Gere J. (1961). Theory of Elastic Stability, McGraw-Hill Book Co. Inc., New York, USA. Trahair N.S. (1993). Flexural-Torsional Buckling, E & F N Spon, London, UK.

Session A3 BEAM-COLUMNS

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

141

STABILITY OF COLD-FORMED TUBULAR BEAM-COLUMNS R. M. Sully' and G. J. Hancock' ' Hyder Consulting Australia, Sydney, NSW, Australia ' University of Sydney, NSW, Australia

ABSTRACT Cold-formed tubular beam-columns of slender cross-section may undergo buckling in both the local buckling mode and overall (flexural) buckling mode. The paper describes a major research program investigating the beam-column strength of cold-formed square hollow sections. A large experimental program was undertaken and has been described in detail in earlier papers. Simulation of the local and overall stability was performed using the finite element program ABAQUS. The paper describes how the slender beam-columns were modeled using ABAQUS including the selection and calibration of geometric imperfections and residual stresses. The influence of welding induced imperfections on the local stability behavior is also investigated since it was found in the experimental program that local instability adjacent to welded connections severely weakened the sections. Accurate simulation of the sudden drop in strength following local instability is described in the paper. Methods for detecting this point in a conventional nonlinear frame analysis without the full finite element mesh to discover local instability are presented. The paper provides a broad overview of the use of ABAQUS for investigating the interaction of local and overall instability.

KEYWORDS cold-formed, square hollow section, beam-column, residual stresses, local buckling, interaction buckling, finite element methods.

INTRODUCTION Sully and Hancock (1998) have described an experimental program to investigate the behavior of slender cold-formed square hollow section (SHS) beam-columns. The test program included section axial (STC) and bending (PB4PT, PBINT) capacity tests, column tests (LC), and both short (SIT) and long (B1,B2) beam-column tests. The long beam-column tests were conducted at two different ratios of end moment (P) and followed an earlier program conducted on compact SHS beam-columns (Sully

142 and Hancock (1996)). A greater variety of tests would have been desirable but was not possible given the time and financial constraints of the program, and the physical restraints of the test rig. It was thought that a finite element analysis would be able to expand upon the experimental testing program and aid in the full description of cold-formed slender SHS beam-column behavior. This paper describes the use of a finite element analysis program to accurately model the full range of experimental tests conducted and fully described in Sully and Hancock (1996) and Sully and Hancock (1998). In this paper, the symbol 's' after a test number refers to the slender sections (Sully and Hancock (1998)), and a symbol 'c' after the test number refers to the compact sections (Sully and Hancock (1996)). Finally there is a discussion on the prediction of the strain at which local buckling will occur in the inelastic range and a simple model for predicting these values is discussed.

FINITE ELEMENT PROGRAM Two finite element packages were used in this study. They were ABAQUS and NIFA. The NIFA program was developed at the University of Sydney (Clarke (1993)), and is able to model material and geometric non-linearity for fi-ames. NIFA only has beam elements which means it is unable to predict local buckling. Hence the need for a criterion to predict the point at which inelastic local buckling

Most of the finite element modeling described in this paper was completed with the commercially available "ABAQUS" version 5.4, fi-om Hibbert, Karlsson, and Sorenson Inc. ABAQUS is a widely used finite element package with a large variety of apphcations. A brief description of the major applications used in this analysis follows. The analysis required both material and geometric nonlinearity. Geometric nonlinearity was obtained using the standard Newton technique for solving nonlinear equilibrium equations. Material nonlinearity is handled by employing a standard von Mises or Hill yield surface model. The material properties were input using multi point curves obtained fi-om the tensile coupon results. Two methods of control were used. They were displacement control and the modified Riks method. Displacement control was used for the analysis of the stub column section, while the modified Riks method was used for all the section bending and beam-column problems.

General Model Attributes Four finite element models were developed to compare with the experimental data and validate the method for further analytical use. They were; (i) Stub column or axial section capacity model, (ii) Section bending capacity model, (iii) Section interaction model, and (iv) Member interaction model. The cross section dimensions used in the model were the centre line dimensions of the as-measured section described in Sully and Hancock (1998) (i.e. 199.36x199.36x5.029). The length used varied with the test model. For the section capacity tests, a length of 1000 mm was used as this was the astested length for the stub columns. For the long column and long interaction tests, a length of 5500 mm was used, again to reflect the actual test specimen pin-ended length. All four models have a number of common attributes including choice of element, mesh size, material properties, use of symmetry along the longitudinal axis, and size and shape of local imperfections. In order to derive an optimum model each of the common model attributes was studied parametrically. A short description of each variable chosen for the final finite element models now follows. Element Choice

143

A shell element was chosen above any of the solid elements for use in this finite element analysis. The shell elements were able to give accurate results for much less computational effort given the slender dimensions of the SHS sections studied. The S4R5 element was chosen for the analysis. The S4R5 element, as described in the ABAQUS user manual (ABAQUS (1993,1994)), is a four node shell element with reduced integration using five degrees offi*eedomper node. A pre-processor was written that was able to create ABAQUS input files with overall and local initial imperfections. Use of Symmetry Use of symmetry is a common practice to allow for reduced computational effort. A common geometry was wanted for all models to allow the input to be generated by a single pre-processor as far as was possible. Early runs with stub columns of half a cross section and half length, in displacement control were not able to model the local buckling at midspan that was observed experimentally. This problem was rectified by using the whole specimen length with the half cross section. The half cross section, fiill length models proved to be very good in allowing the models to fail in the modes observed experimentally. This use of symmetry along the length only also allowed for accurate modeling of beam-columns with different end loading conditions. Mesh Size Mesh size was varied to obtain a model that accurately predicted test results while using the minimum computational effort. One of the first studies conducted was into the mesh size required for this analysis. The mesh was defined by four quantities; (i) the number of elements across the flanges, (ii) the number of elements across the web, (iii) the number of elements in each comer, and (iv) the number of elementsfi-omtop to bottom of the model. The number of elements was varied both around the cross section and along the length while trying to maintain an aspect ratio on any plate element of 2:1. It was found that by having two elements in each comer, four elements in each flange, eight elements in the web and seventy per metre length, accurate modeling of the test results could be achieved. Material Properties Actual material stress strain curves were used in developing the model. These were obtained by curve fitting to actual tensile coupon results fi-om both the comer and flat material around the SHS test specimens (Sully and Hancock (1998)). Table 1 shows the curves used for the comer and flat material in the finite element models. For comparison some finite element models were also tested with elastic perfectly plastic material curves. Residual Stresses The residual stresses in the test specimens were measured and are reported in Sully and Hancock (1996). It was found that the axial residual stresses were very low and naturally summed to be zero around the cross section. However the bending residual stresses were quite high (Sully and Hancock (1998)). The axial residual stresses were ignored. Initially the bending residual stresses were incorporated by using an elastic perfectly plastic stress strain curve (yield stress equal to the measured 0.02% proof stress) and residual stresses were introduced into the plate element gauss points. This proved to be very difficult to achieve. The altemative was to use the multi-point curves from derived form the tensile coupon results and use no residual stresses at the plate element gauss points. It was

144 found that using this method the averaged effect of the bending residual stresses was taken into account, the method was much simpler and accurate results were achieved. TABLE 1 MATERIAL STRESS-STRAIN CURVES USED IN VERIFICATION RUNS

Flats 1

1

Stress (MPa) a 0 100 200 250 300 323 333 343 350 361

Comers Strain 8

0.000000 0.000497 0.001115 0.001535 0.002050 0.002500 0.003000 0.004000 0.006000 0.010000

Stress (MPa) a 0 250 300 350 400 448 477 493 502 520

1 Strain 8

0.000000 0.001280 0.001600 0.002000 0.002475 0.003125 0.004000 0.005000 0.006000 0.010000

'

Size and Shape of Imperfections Three types of imperfections were used in the validation finite element models. Two are local imperfections, the third is an overall imperfection; 1. Sympathetic imperfection. At a cross section, this imperfection consists of a sinusoidal wave form bowing out on two opposing sides, and bowing in on the other two opposing sides of the cross section. The magnitude of the bowing is equal on all four sides. This is called 'sympathetic' because this imperfection is in the shape of the local instabiUty seen at failure in the experimental work, and is thought to precipitate failure. 2. Unsympathetic imperfection. In cross section this imperfection consists of a sinusoidal wave form bowing out on all four sides. The magnitude of the bowing on one set of opposing sides is equal to 0.95 of the magnitude of the bowing on the other set of opposing sides. This was done to ensure that some inherent instability exists in the system, as would occur in reality. 3. Overall imperfection. This imperfection is not a local imperfection, as are the two above, but consists of a half sinusoidal wave bowing out along the longitudinal axis. All three imperfection types are illustrated in Figure 1. The half wavelength of the imperfections, along the members was also varied. The actual experimental specimens were subject to some form of local imperfection that precipitated a local instability in the centre of the specimens. Describing the imperfections as either sympathetic or unsympathetic with a certain magnitude and a certain half wavelength is oversimplifying the real situation. However for the sake of modeling slender cross section response, it is necessary to use an equivalent imperfection. That is an imperfection that is able to give a response in the model that is equivalent to the actual response.

145

Sympathetic

Unsympathetic

Overall

Figure 1. Imperfections used in finite element models. The equivalent imperfection proved to be different for the different tests. For the stub column tests, a sympathetic imperfection with a maximum magnitude of 0.0002t (t is the thickness of the plate element), or an unsympathetic imperfection with a maximum magnitude of 0.02t to 0.4t gave the best correlation. For the bending tests, which were not as sensitive to the size of the local imperfections, a maximum magnitude of 0.002t gave good correlation. For the short interaction tests, a combination was used with a sympathetic mode with a maximum magnitude of 0.0002t superimposed onto an unsympathetic mode with a maximum magnitude of 0.02t. The long column and long interaction test simulations were given a local imperfection in a sympathetic mode with the maximum magnitude of 0.002t, and an overall imperfection of 0.00IL, where L was the length of the test. All local imperfections mentioned above had a half wavelength equal to the depth of the section. Loading and Boundary Conditions For the stub column simulations, the nodes at the bottom of the model were fixed while those at the top of the model were put under displacement control. This gave a very stable loading and unloading regime. For the bending test, short interaction test and long interaction test models, fixed against translation while the central top node wasfi-eeto translate axially. The nodes at the top and the loading conditions had to be changed. In these cases, one point in the centre at the bottom was bottom of the structure were then loaded to give an overall moment and/or axial load. In all cases, symmetry was imposed along the sections.

Results Comparison of the validation runs with actual test results are shown in Figures 2 to 6. Figure 2 shows the results of finite element (FE) tests conducted for the stub column tests (STCls & 2s) with various local imperfections. As can be seen fi-om the figure, the point at which local buckling occurs is very sensitive to the size of the initial local imperfection chosen. It is also clear, by including a FE model with elastic perfectly plastic material properties, that it is very important to use the actual material properties in order to get good agreement with test results. The FE model results are slightly conservative in predicting the ultimate capacity, but predict the shape of both the loading and unloading portion of the curves very well. Figure 3 shows a similar graph for the FE section bending tests. Here two quite separate tests are described; PBINTs and PB4PTs. A more complete description of the differences in these tests is found

146 1400 1

f^^'^\

1

1200

-

/ jf

\

/J

1000

/

:=«800

\

N

\

\

V ^ *»^

If

* "*^ .

"~"

•*-

iM do/t = 0.()04 do/t = 0.()02 do/t = 0()002

400

straight 31s & 2s — - - — EPP

200

2

3

4

5

6

Axial Shortening (mm)

Figure 2. Comparison of stub column FE models with test results.

100

-.^ * V V^

y

^-^^^^^PB4PTs

' v^

iPBINTs do/t = 0.4

40

&

do/t = 3.2

J

do/t = 3.02 do/t = 3.002 t

20

esults

L» 0.02

0.04

0.06

0.08

Curvature (1/m)

Figure 3. Comparison of section bending FE models with test results.

0.1

147 in Sully and Hancock (1998). To summarize; PBINTs had very large local imperfections at the loading points due to the heavy welding of the section. The size of these imperfections was of the order of 2 mm. Test PB4PTs was conducted in a four point bending test configuration and was free of the large imperfections observed in PBINTs. As can be seen from the figure, the FE model results are able to accurately predict the loading and unloading behavior and the ultimate load reached. It is also observed that by introducing a large imperfection as was observed in the experimental program (djt = 0.4) the FE model was able to predict this behavior as well, up to the point of collapse. Figure 4 shows the results of the long column (LCls) results. Here the FE model results are able to predict the test results very closely. They appear to also be able to pick up the portions of the experimental curve that is missing from the test results due to the lack of stiffriess and control in the test rig. The ultimate load is predicted very closely. We also see from the results that inclusion of initial imperfections is very important to the accurate prediction of local buckling and overall failure. The NIFA model had no local imperfection and shows the unloading path associated with overall buckling well. A similar ABAQUS model gave identical results up to the point of local buckling occurring (at a larger displacement than for Lcm2 and Lcm3). The models Lcm2 and Lcm3 (ABAQUS models) have initial imperfections added to them, causing collapse at displacements approximate to those observed experimentally. For Lcm2 bjt = 0.002 and for Lcm3 bjt = 0.004. Figures 5 and 6 show the FE model results compared to the two long interaction test series. Figure 5 has the Bl series (P = -1) (BlRls, BlR2s, BlRBs) and Figure 6 the B2 series results (P = -1/2) (B2R1, B2R2, B2R3) . In both figures it is observed that the ultimate load predictions are slightly conservative. The load path predictions are good although there is some difficulty in accurately predicting when local buckling will occur. The model results for test BlRls show that there is some sensitivity, in high load ratio conditions, to the inclusion and size of local imperfections. It is observed in the B2 series results that test B2R3s failed at a moment similar to PBINTs rather than PB4PTs. This test, like PBINTs, had the large local imperfections at the end introduced during the welding of the end plates.

r

LCs

400

200 (

NIFA model

• ^ . ^

Lcm2

-1

""^-^^^

I

V

*>v^

(\

{ {

ol )

50

100

150

200

Axial Shortening (mm)

Figure 4. Comparison of column FE model with test results.

250

148

1

No Imperfections

1000

Local Imperfection NlFA runs St Results (B1)

B1R1S

800 z.

TzT;;^-^

•^"^^^^ Jj^ ^^^^^' 200

-"•"*' ^^^^

*B1R2s

,^ %

^^^^^^

1^'BTRas

PBINTs • 20

40

PB4PTS

•

60

Central Moment (kNm)

Figure 5. Comparison of beam-column FE models (p=-l) with test results.

1000

ABAQUS runs 800 3t Results (B2) B2R1S 600

1 r^^^r-^.^^

B2R2S ^ ^ ^ , , ,

B2R3S 200

PBINTs • 20

40

PB4PTS •

60

Maximum Moment (kNm)

Figure 6. Comparison of beam-column FE models ((3=-l/2) with test results.

100

149 The validation runs for the stub column, short interaction and long interaction models were slightly conservative. This may be due to the material properties used. The model had comer and flat material sections with properties derived directly from tensile coupon tests. The flat material was taken as being equal to the coupons cut from the centre of the face. This assumption neglects small portions of material close to the comers that would have a higher yield stress, as the transition from the flat to the comer material was made. From the validation mns, it is also clear that the effects of section slendemess are more pronounced for shorter length specimens. The short tests were all govemed by local instability. The longer tests, however failed locally after the maximum load had been reached. Local imperfections are therefore much more important in short length members or more generally, members where strength approaches section capacity. Furthermore local imperfections appear not to significantly influence the loading path taken by a member but only determine when local instability occurs i.e. when the transition from overall instability to a spatial plastic mechanism occurs. The importance of local imperfections in determining the section strength of thin walled members is very evident in Figure 3 which shows the section bending results. Here the test PB4PTs shows a higher capacity than test PBINTs. As discussed in Sully and Hancock (1998) this is thought to be due to the local imperfections that were introduced into the member during the fabrication process. As can be seen from Figure 3, if a large local imperfection is introduced into the FE model then we can follow the results of test PBINTs very closely. This shows that the section bending capacity of SHS members is sensitive to the large imperfections that might be introduced by the welding of connections.

INELASTIC LOCAL BUCKLING The advanced analysis NIFA (Clarke (1993)) has been shown to predict the overall behavior of compact sections, and frames with good accuracy, provided the proper material and geometrical properties are used. It was also shown in Figs 5 and 6 that it was capable of predicting slender behavior as well, up to the point of local buckling. The option of using an advanced analysis in design is now only available for compact sections. There are however other uses for such an analysis. If a rational approach was developed for predicting actual local buckling strain, for a given type of cross section, then an advanced analysis could be used for the design of stmctures with other than compact sections. This would avoid the problems of designing the overall system using a second order elastic analysis, and then using equivalent second order member mles for checking the individual members. Much more accurate predictions of the behavior of a stmcture at limit state, and under serviceability requirements could be gained than are now possible using a second order analysis. It can be seen that there is great potential for improved design in many fields by obtaining accurate information about the inelastic local buckling behavior of plates, loaded into the strain hardening range. A study was conducted to look at local buckling of SHS members into the strain hardening range. The stub column and bending section capacity finite element models used previously were used to study local buckling for a number of plate thicknesses. Both models were used because the test resuhs described in Sully and Hancock (1996) showed a different extreme fibre strain at local buckling for the stub column tests (STClc and STC2c) than occurred in the bending test (PB4PTc) . The two stub column tests, STClc and STC2c, locally buckled at strains of 0.0129 and 0.0157 respectively. The plastic bending test locally buckled at an extreme fibre strain of 0.0210. The difference in local

150 buckling strains is thought to be due to the restraint offered by the webs to the flanges, and the nonuniform strain distribution through the section, for the bending case. For the stub column, all of the plates are under the same strain and are loaded uniformly. In the bending case, the web is acting to restrain the flanges, and the flange itself has less strain on the inside surface than on the outside surface. Eight finite element analyses were conducted for both the stub column and the section bending capacity models. All sixteen models were a 200x200 SHS with varying thicknesses, and comer radii. For the models, the outer comer radius (RJ for each mn was made equal to 2.5t. Values of b/t from 55 to 20 (using the AS4100 definition) were chosen. Each of the mns was given a sympathetic imperfection with a magnitude djt = 0.0002, and half wave length equal to the depth of the section. The multi-point stress-strain curves, described earlier, for the 200x200x5 SHS were used in these models, with the yield stress equal to 350 MPa and the modulus equal to 200000 MPa. Results for the analyses are shown in Tables 2 and 3. TABLE 2 RESULTS FROM STUB COLUMN MODEL ANALYSES

Run IbstOl lbst02 lbst03 lbst04 lbst05 lbst06 lbst07 IbstOS

t (mm) 9.091 7.407 6.250 5.405 4.762 4.255 3.846 3.509

b/t

6o

P.b

Sib

Sib

20 25 30 35 40 45 50 55

(mm) 0.0018182 0.0014814 0.0012500 0.0010810 0.0009524 0.0008510 0.0007692 0.0007018

(kN) 2609.2 2120.3 1766.7 1483.7 1300.6 1109.9 1010.0 822.0

(mm) 15.71 11.22 7.44 4.48 3.94 3.00 3.05 2.25

0.01570 0.01120 0.00744 0.00448 0.00394 0.00300 0.00305 0.00225

TABLE 3 RESULTS FROM SECTION BENDING CAPACITY ANALYSES

Model IbbeOl lbbe02 lbbe03 lbbe04 lbbe05 lbbe06 lbbe07 lbbe08

t (mm) 9.091 7.407 6.250 5.405 4.762 4.255 3.846 3.509

b/t

So

20 25 30 35 40 45 50 55

(mm) 0.0018182 0.0014814 0.0012500 0.0010810 0.0009524 0.0008510 0.0007692 0.0007018

M,, (kNm) 187.8 153.0 127.8 107.7 91.16 76.82 63.58 54.63

Pib

Slb

(1/m) 0.21099 0.14439 0.08959 0.05865 0.04083 0.03010 0.02649 0.02665

0.021099 0.014439 0.008959 0.005865 0.004083 0.003010 0.002649 0.002665

It was assumed that local buckling precipitated the drop in load in all cases. The values of P,i,, and Mj,, are therefore the maximum load and moment respectively for the various tests. For the stub column models the axial deflection at maximum load (5,i,), and the resulting local buckling strain s,b are recorded in Table 2. Likewise the values of curvature at local buckling p,b, and the corresponding local buckling surface strain s,i, are recorded in Table 3.

151 Results of finite element analyses compared with design rules and experimental values of critical buckling strain, for the pure axial load and pure bending cases Results for all of the analyses are shown in Figures 7 and 8. Here the values of b/t have been non dimensionalised to represent plate slendemess; using the AS4100 relationship X^ = b/tV(Fy/250). Values of local buckling strain for the compact (Sully and Hancock (1996)), and slender series (Sully and Hancock (1998)) stub column and bending capacity tests have been included for comparison where 'c' denotes compact and ' s ' denotes slender. The results for the compact series tests show good agreement with the ABAQUS results; tests PB4PTc and STC2c lie on or close to the predicted curves, test STClc is slightly below, so the average of the stub column tests would be slightly less than the predicted curve. For the slender series tests, the results for tests STCls, STC2s, and PB4PTs lie on the same value, below the predicted value, with test PBINTs lying below the other three values, as expected. The tests verify the trend shown by the predictions; that for local buckling of compact and non-compact sections, there is a difference in the local buckling strain for bending and pure axial load cases. This difference increases as the plate slendemess decreases. The finite element results show some variability in the high plate slendemess regions. This is probably due to sensitivities within the model. To avoid possible problems with the different comer radii used in the analyses, as a result of the different thicknesses, the mesh density used for all of the models described in this chapter was twice that for the model described earlier. Even with the increased number of elements, different buckled shapes were observed between the analyses. The fluctuations observed for the model results in Figure 7 may not have occurred if the models had all failed in similar displaced shapes. Drawn on Figure 7 also is a proposed bilinear design mle; a single linear function for plate slendemess values greater than 40, and two different linear functions for slendemess values less than 40, with the 25

\ PB4PTC 20

s rctQ

— - - — Ibst results

\\

— - — - Ibbe results

- ^ 15 s rc2c

o

•

I's curve

^

design rule

\

•

125 series

O

200 series

10

'^ » ^PB4PTs. i>TC1s, STC2S

FB I N T s " "

10

20

30

40

:a».^jr=^

60

Plate Slendemess (^e)

Figure 7. Results of FE models compared with design rules and experimental values of critical local buckling strain, for pure axial load and pure bending cases.

152 higher Une for bending cases and the lower line for axial cases. The equations of these lines are; 8,b = 0.032 - OmOlX,

Axial case, for X, < 40

8ib = 0.046 - O.OOIOSX,^

Bending case, for X, < 40

and, •• 0.00667 - 0.000667X,

for X, > 40

From Figure 7 the equations form a lower bound for values of plate slendemess greater than 40, the yield limit (AS4100), and that they give a good approximation for both the bending behavior and axial behavior. It can be seen that for plate slendemess values less than 20 the straight line equations start to become more conservative. In AS4100 a value of X^ of 30 corresponds to the plastic limit. So the curves derived give a good approximation to local buckling strain for SHS sections that are not compact. Also shown in Figure 7 is the relationship for inelastic local buckling proposed by Bleich (1952);

•Ai

l2{l-u)'''\hJ ' ^^^ where r| equals E/E, Et being the tangent modulus. Bleich's formula was used in conjunction with the modified Plank material curve (Sully (1996));

where \i is equal to a/a^ , c = 0.8,d=1.05,E was taken as 200000 MPa, and a^ equal to 350 MPa, and a =1.165a.. The derivative of Equation 2 is then; do-

E(l-d/i)' (3) ds 1 - 2c// + cd/i from which values of r| were obtained, and hence values of G„, for comparison in Figures 7 (and 8 described following). As can be seen Bleich's formula gives good predictions of critical buckling strain for high values of plate slendemess, but is overly conservative for values less than the yield limit. Results of finite element analyses compared with design rules and experimental values of critical buckling strain for the beam-column cases The same comparisons are made with results from the interaction tests in Figure 8. Here the results show a much wider scatter than was observed for the stub column and section bending capacity tests. The values of local buckling strain shown for the interaction tests represents the strain at the extreme fibre, and were calculated by comparing NIFA finite element mns with the test results. For test B2Rlc the value of strain was obtained from the strain gauge readings taken during testing. For the interaction tests, the local buckling strain appears to increase as the ratio of moment to load increases, with the exception of test B2R3s which was noted to have collapsed early. Figure 8 shows that the mles derived for stub column and section bending capacity cases form a lower bound to the interaction test results. It

153 25

20

\

— - - — Ibst results — - — - Ibbe results ch's curve des gn rule

2

125 B2R1°

\

V) O)

a

125 B2 series

•

200 B1 series

O

200 82 series

B1R3

•

g 10 OQ

B2R2 •

„,„^ B1R2 B1R1

* vO B2R1 *^—«^„__^ >

B2Rr—^—Xj^;^^^*^:^'-"-

30 40 Plate Slenderness (^e)

50

-"--^•---.J

60

70

Figure 8. Results of FE models compared with design rules and experimental values of critical local buckling strain, for beam-column cases. also shows that the interaction cases need specific rules to estimate local buckling, depending on the ratio of the applied moment to the applied load. This requires further investigation. At the present time advanced analyses are only allov^ed to be used in designing frames made from compact members. The difficulties of applying elastic methods of design to describe the behavior of cold-formed members has been highlighted previously in this paper. Obviously using an advanced analysis to design structures made fi-om semi-compact or even non-compact members as well as compact sections would produce much more reliable results, and a better understanding of actual behavior. The problem with such an approach is that the point at which local buckling will occur in any of the members must be known. Finite element programs that use plate elements are capable of doing this for individual members but not for entire systems as the process of modeling complete systems becomes far to complex. Finite element programs employing beam elements are very accurate in predicting overall behavior but lack the ability to model local instabilities in component members. By employing a simple design rule as described in this paper, this problem can be overcome allowing advanced analyses to be used for members of all types of cross section slenderness. The pilot study described in this paper has shown that for the stub column and pure bending cases, design rules capable of predicting local buckling strains for a wide range of section slenderness are possible. The study also highlights the problem of combined loads, showing that under such circumstances, the problem of plate local buckling becomes more complex, requiring more advanced rules than were used for the pure axial and pure bending cases. Further studies are required in this area. At this stage, the pure compression curve should be used as a conservative estimate.

154 CONCLUSION This paper has described the use of finite element analysis programs to accurately predict the behavior of cold formed slender SHS beam-columns. Using the ABAQUS software the local and overall instability of SHS members was able to be predicted very closely. The accuracy of the analysis was dependent on the model attributes. This paper has described the major parameters and model attributes studied. The paper has also described the problems associated with premature collapse of slender section SHS members due to large imperfections at the member connections. Such imperfections may resultfi-omthe welding process used to attach the connections. Advanced analyses used for full three-dimensional analyses are not able to be used for non compact or semi compact cross section because there is no ability to model local buckling of the members. A pilot study into the prediction of inelastic local buckling was presented in the paper. Accurately predicting the strain at which a semi or non-compact member will locally buckle would allow advanced analyses packages which use beam elements to be used for all types of cross section. The difference between the local buckling strain of axially loaded and bending members was described but needs to be investigated more fully.

REFERENCES ABAQUS (1994), ABAQUS Theory Manual, ABAQUS/Standard User's Manual, Volumes 1 and 2 (version 5.4). Hibbert, Karlsson, and Sorenson, Inc, Pawtucket, RI, United States. Clarke, M.J. (1993), "Plastic-Zone Analysis of Frames", Chapter 6 in Advanced Analysis of Steel Frames; Theory, Software and Applications, eds. W.F.Chen and S.Toma, CRC Press, Inc., Boca Raton, Florida, pp. 259-319,1993. Standards Association Australia (1998), Steel Structures, Standards Association of Australia, AS41001998. Sully R.M. and Hancock G.J. (1996), Behavior of Cold-Formed SHS Beam-Columns, Journal of Structural Engineering, Vol. 122, No. 3, March 1996. Sully R.M. and Hancock G.J. (1998), The behaviour of cold-formed slender aquare hollow section beam-columns, Proc. of Eighth International Symposium onTubular Structures, Singapore, August, 1998, Balkema. Sully R.M. (1996), The Behaviour of Cold-Formed Rectangular and Square Hollow Section BeamColumns, PhD thesis. University of Sydney, School of Civil and Mining Engineering.

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

155

LOAD CARRYING CAPACITY OF THIN-WALLED SHORT COLUMNS J. Lindner * and A. Rusch ^ Technische Universitat Berlin, Fachgebiet Stahlbau, Sekr. Bl, Hardenbergstr. 40 A, D-10623 Berlin

ABSTRACT Thin-walled columns are most frequently used in frame systems. The load carrying capacity is usually governed by the stability check. In many cases additionally an interaction between overall flexural buckling and local plate buckling occurs, where local buckling is taken into account by effective widths. If the column is loaded centrally, there are differences in opinion on the kind of reduced cross section and the question whether an unintentional eccentricity must be taken into account or not. The effect of such eccentricity is especially interesting for short columns. A current research project at the Technical University of Berlin deals with these questions in two different ways. Firstly tests on short columns with welded thin-walled cross sections are carried out. Experimental measurements result in global and local load deflection curves and the ultimate load. Addionally the number of buckling waves can be determined from these tests. Additionally finite element calculations are carried out to confirm the test results. Here the postcritical plate buckling behaviour and the influence of imperfections are taken into account. The comparison of the test results with different design formulae shows that in codes the cross-section check is not sufficient. KEYWORDS Columns, Thin-Walled Structures, Coupled Instabilities, Effective Widths, Postcritical Behaviour, Imperfections, Load Carrying Capacity, Finite Element Method. TEST PROGRAMME Tests on thin-walled I-sections were performed, where three-sided simply supported plates had a strong influence on the load-carrying. The loads were applied by end plates welded to the columns. The bearings were designed to realize pin-ended support condition to allow rotations about major and minor axes, but to avoid lateral displacements and twist rotations. The load was applied centrally. So the support conditions corresponded to a so called "static loading" (Fischer et al. (1996)). The columns are relatively short but they do not correspond with stub columns because of the boundary conditions and the length.

156 The material properties, real dimensions, local and overall geometric imperfections and - in some cases - the residual stresses were measured prior to testing. All tests were performed under displacement control in an universal testing machine. A selection of the test results is summarized in Table 1. Both the overall and the local deformations of the specimens were gauged during the test. The stress distribution in the post-critical range was measured with strain gauges. TABLE 1 EXPERIMENTAL DATA

Specimen

b

f^

t

hw

[N/mm^l fmml [mml [mml 185 1.9 40 1072 150 186 1.9 80 150 1060 185 1.9 120 150 1065 154 1.5 40 150 1070 175 1.5 40 150 1025 154 1.5 120 150 1063 175 1.5 80 150 1013 194 1.5 80 150 1030 185 1.9 80 200 1057 222 1.5 80 150 1059 154 1.5 80 200 1055 175 1.5 80 200 1028 1' cold rolled material: yield stress fy defined with 0.2 % offset method

L

Nu,test

[mml 1200 1200 1200 1200 1000 1200 1200 1000 1200 1200 1200 1000

rkNl 82.1 100.9 101.1 49.5 51.6 60.2 55.7 59.7 110.3 71.4 64.7

67.2

1 |

Already at low forces the flanges of the I-sections start to buckle locally in multiple half-waves. If the load is applied centrally, all flanges are buckled locally. There is only little influence of the overall imperfections on the difference of the local buckling amplitudes between the "compressive" and "tension" flange. Figure 1 shows the stress distribution to be approximately symmetric. Tensile stresses Gx are present at the free flange ends. Between the beginning of local buckling and the final failure of the column, the loading can be increased considerably. The failure is always characterised by a sharp amplitude increase of one buckling half-wave. Additionally local plastic deformations are observed.

b^

\i - -200

|

B— —o— —B

o

B

B

1 T T

—U T

1

^N = 20.1kN

1

T T

T

f

f

\ N = 39.7 kN jN^ = 59.7kN

[N/mm^

1

1

-200 -100

r—1^— H 0

100 200

1

A ^

200 100

^

0

1

1

-100 -200

i-

Figure 1: Experimental stress a^ distribution at different load increments, specimen 1030

157 FEM-CALCULATION For comparison, finite element analyses are carried out with the FEM-program ADINA 7.x. 4-node isoparametric shell-elements are used for the geometric and physical non-linear calculation. The ultimate load is determined with the Riks Arc Length method. Figure 2 shows a modelled column. Test and numerical calculation are in good agreement.

Figure 2: Deformation pattern of a modelled column at ultimate load VERIFICATION CONCEPTS Having a centrally loaded thin-walled I-section, coupled instabilities are caused by the column length and the local slendemess of the single plates. The effects of the local plate buckling are accounted for by the established method of effective widths. For the sake of comparability the "Winter Formula" for the four-sided simply supported plate Eqn. 1 is always used.

b'-

0,22

Jcr/cj

JoTa

(1)

Further calculations v^th other formulations for the effective widths of the three-sided simply supported plate have been reported earlier in Lindner and Rusch (1998). Kalyanaraman et al. (1977), (1978), Fischer and Konowalczyk (1988) argue that the three-sided simply supported plate has a relatively a higher post-critical load-bearing capacity than the four-sided simply supported plate. On the other hand the German standard DIN 18 800-2 (1990) requires a smaller effective width. For overall stability, the influence of local buckling on the stiffness must be determined. There are difference in opinion to be found in the literature and the international codes as to whether and how a reduced stiffness must be taken into account or not. The European standard EC 3 Part 1.3 (1996) goes back to the Q-factor approach which was introduced in the American standard AISI (1968). The Q-factor method first determines the effective cross-section A' using the real yield stress fy. The second step is an overall stability check with a reduced yield stress fy' = Q fy where Q = A7A. Since 1986 the American specification AISI (1986) has applied the Unified Approach. First, the overall stability is checked without accounting for the influence of local buckling and then the overall buckling stress is used for the calculation of the effective cross-section. A completely different approach is chosen in the DIN 18 800-2 (1990). The overall stability failure is determined with the effective stiffness using second order theory. The effective section properties are to be determined by using the stresses calculated before. Due to this remarkable contrast to the other codes, the DIN 18 800-2 (1990) requires an iterative process. The second order calculation always

158 yield in increased complexity, because the considered imperfections eo lead to additional bending moments. This has the effect of an unintentional eccentricity on the reduced cross-section which can be accounted for by an additional imperfection in the design formula (Figure 3). The test column lengths are selected carefully in order to make sure that the influence of overall buckling on the cross-section capacity is minimized. In addition, the measured overall geometric imperfections are smaller then the commonly used value of L/1000 in ultimate load calculations. Thus, it can be assumed that the experimental ultimate loads Nu,test are close to the cross-section capacity. Furthermore, the additional imperfection of DIN 18 800-2 caused by the unintentional eccentricity is independent of the column length. Consequently, it may be expected that this effect plays a more important role for short columns.

DIN 18800 T 2

EC3/AISI

N

'OH

A' y-y/z-z

buckling about

y-y

Z-2

Figure 3: Effective cross-sections for centrally loaded columns Table 2 shows the results calculated for the sections of Table 1 including the cross-section capacity which is determined in two different ways, (i) The sum of the individual load-carrying capacity of flanges and web is used, (ii) Eqn. (1) is evaluated with the critical buckling stress GCT of the whole member. For the design concepts the member GCT is always used. TABLE 2 ULTIMATE LOAD OF DIFFERENT DESIGN CONCEPTS

1

Specimen

Nu,test

1072 1060 1065 1070 1025 1063 1013 1009 1057 1059 1055

fkNl 82.1 100.9 101.1 49.5 51.6 60.2 55.7 61.6 110.3 71.4 64.7 67.2

1028

EC 3 Part 1.3 (1996) [kNl 67.1 79.0 83.9 40.4 44.5 49.7 50.0 51.0 80.6 56.9 46.8 50.3

AISI (1986) fkNl 70.8 79.0 84.2 41.7 45.4 49.2 49.5 50.5 79.8 56.5 46.3 49.8

DIN 18 800-2 (1990) fkNl 50.6 55.7 58.2 29.5 32.7 34.0 34.4 35.7 57.7 38.8 34.0 37.1

cross-section capacity (i) (ii) [kNl [kNl 63.4 74.7 78.0 80.2 82.5 85.4 38.8 43.6 42.5 47.0 48.2 49.8 50.2 49.5 51.0 50.4 80.8 79.4 57.4 57.1 46.8 46.6 50.3 50.3

|

1 1

159 All design concepts clearly underestimate the test results. It can be stated that the low values of DIN 18 800-2 are influenced by the additional imperfection due to the unintentional eccentricity. Furthermore, the cross-section capacity is lower than the experimental ultimate load. The use of the favourable buckling curves (Kalyanaraman et al. (1977), (1978), Fischer and Konowalczyk (1988)) cannot remove this fact. A changed distribution of the effective v^dth (Fischer et al. (1996)) would reduce the unintentional eccentricity about the weak axis, but it has no influence on the cross-section capacity of a centrally loaded double symmetric I-section.

CROSS SECTION CAPACITY An extensive FEM study has been performed in order to determine the cross-section capacity for several sets of parameters. A stub column with the length of one plate buckling half-wave is modelled. The half-wave length Li is defined as the length, where the critical buckling stress acr of the whole member reaches the first minimum (Figure 4). The tests have shown that the columns buckle in n halfwaves if the columns are approximately n half-waves long. 10(:r

ir^ \

/\

^^

/

^crt

—1

/

/

/

/

Ji^/2 + ai | l - - ^ y / 2 + ai„C'/3 + bHnCV4-ai-^CC*

(5)

where: X - load parameter, X„ - critical value of /I, W^^= a^X,^ /2 energy of prebuckling state and coefficients ao, ai, am, bm are given by formulas of Byskov method. By substituting the expansion (4) into equations of equilibrium (3), junction conditions and boundary conditions, the boundary value problems of zero, first and second order can be obtained. The zero approximation describes the pre-buckling state while the first approximation, that is the linear problem of stability, enables us to determine the critical loads of global and local value and their buckling modes. This question can be reduced to a homogeneous system of differential equilibrium equations. The solution of the first order approximation enables to determine the critical values performing a minimisation with respect to the number of axial half-waves. The second order boundary problem can be reduced to a linear system of non-homogeneous equations whose right-hand sides depend on thefirstorder displacement and force fields. In the presented method the plates with linearly varying prebuckling stresses along their widths are divided into several strips under uniformly distributed compressive (tensile) stresses. Instead of the finite strip method, the exact transition matrix method is used in this case. The pre-buckling solution of the i-th orthotropic plate consisting of homogeneous fields is assumed as:

u[^>-f^-XiV,

y['^=v,y,A,,

(6)

where: Ai is the actual loading. This loading is specified as the product of a unit loading system and a scalar load factor Aj. Numerical aspects of the problem being solved for the first and the second order fields, resulted in the introduction of the following new orthogonalfianctionsin the sense of boundary conditions for two longitudinal edges: a,«=v[,^,>+ViU«,

b,« = u[^,>+v«

c.«=u«,

d[''>=vp>,

1

'^1

>

ei«=wf'.

h|^'=E;(wl^,>. where : k = 1,2

(7)

f ; « = w|,'^>

^i =

bi'

i.^4x

167 The system of the ordinary differential equilibrium equations (4) for the first and the second order approximation is solved by the modified transition matrices method in which the state vector of the final edge is derived from the state vector of the initial edge by numerical integration of the differential equations in the transverse direction using the Runge-Kutta formula by means of the Godunov orthogonalization method (Bidermann, (1977)). Consideration of displacements and loads components in the middle surface of walls within the first order approximation as well as precise geometrical relationships enabled the analysis of all possible buckling modes including "mixed" buckling mode (Camotim and Prola (1996), Dubina (1996), Kolakowski et al. (1997)).

3. BASIC RELATIONS IN THE ELASTO-PLASTIC STATE In the studies concerning the stability of structures in the elasto-plastic range it is essential to describe in the analytical way the uniaxial stress-strain curves of a material. For orthotropic materials there are four independent elastic constants (E^, Ey, v, Gxy) to be found for each component plate. In the theory of plasticity many approximated relations are proposed: perfectly plastic relationship, linear hardening, non-linear hardening that may be represented by analytical formula. As pointed out by Hill there are four independent characteristics to be known for orthotropic material (in plane stress case). Three of them correspond to the uniaxial stress-strain curves for principal and 9=45 degrees directions of the strength plane of the material. The fourth characteristic corresponds to the pure shear test. In this work the material stress-strain curves corresponding to the linear elastic - perfectly plastic and linear hardening relations have been taken into account during analysis. It is assumed that orthotropic material obeys Hill's Yield Criterion that for a plane stress state can be written as (Hill (1948)): (^cff

= ^i^l + ^2^1 - ai2^x^y + 3a3T^.

(8)

In this expression the parameters aj -^ ei^ are anisotropic parameters which depend on the material stress-strain curves and initial yield limits in particular directions. For stress-strain hardening material the uniaxial yield stresses vary with increasing plastic deformations and therefore the anisotropic parameters should also vary since they are functions of current yield stresses. So the parameters aj ^ a3 must be determined in each step of calculations for considered characteristics of the material. The plastic stress-strain relations are described by Prandtl-Reuss equations, where the plastic strain increment is defined as: d8P=A^,

(9)

where Q is the plastic potential which is assumed to be the yield function for an associated flow rule and A is a scalar positively defined. For finite increments as considered in the analysis the relations between stress and strain increments in the elasto-plastic state are:

168 ^^ix =

^ [ ^ i x

+ ViTliABj - A ( S i ^ + VjTliSi

)],

^^iy = V. 27[^iy + ^i^ix - ^(Siyy + V^Sj^)], (l-'HiVi )

(10)

where: Sixx = -(2aiai^ - aijGiy),

Sjyy = -(2a2aiy - ^u^^iJ,

S-^^ = 2&^Xi^.

Further it is assumed that all assumptions of large deflection plate theory still hold. The forms of displacement ftmctions in the elasto-plastic range are assumed to be the same as in the elastic case but their amplitudes may take any values, constrained by geometric boundary conditions. The solution of non-linear elasto-plastic problem is reached by an incremental and iterative procedure. The problem is solved in an analytical-numerical way where the Rayleigh-Ritz variational principle is applied (Gradzki, Kowal-Michalska (1988)). During the investigation the plate response on the small increment of loading is examined. The potential energy V in each point of a component plate is a sum of elastic and plastic energy. For the purposes of minimisation (Graves-Smith (1968), Little (1977)) only the changes of the energy are considered: AG:

f

Kr^

\

AV = AV, + AVp = JJJI Gij + ^ k j d n +jjll CTij + ^ U d P \

V

2

(11)

,

where: Q, P - elastic and plastic volumes of a plate (i,j=x,y) The increment of a potential energy can be expressed in terms of strains determined in the elastic range. The value of AV is calculated in a numerical way. The volume of each plate is divided on pxqxs cubicoids. The values of energy increments calculated in each of cubicoids are summarised for a whole plate and next for a column. According to the Rayleigh-Ritz method the independent parameters of strain functions are found by minimisation of expression (11). In numerical procedure the path of loading (elastic or plastic deformation, unloading) has to be assumed at a start in each step of calculations. Thus the results are charged with some errors which have to be corrected in next step of calculation. To avoid the cumulating of errors during succeeding steps the response of a column to the fairly small increment of loading is examined.

4. DISCUSSION OF PRELIMINARY RESULTS On the basis of the analytical considerations the computer programme has been elaborated. The programme allows to calculate the stress and strain fields in the elastic range for orthotropic beamcolumns under eccentric compression. The numerical calculations in the elasto-plastic range are conducted taking into account the forms of elastic strain fields with free parameters c,.

169 0.9 0.8 0.7 r- A-ndb—'P' = ^

(2)

In the equation above, P depend of the geometry of the cross section, Pkx and Pky are the critical loads of bending and Pkt is the torsional critical load.

From ey=0, the equilibrium equation become an equation of the second degree and one of the roots of this equation is the flexural-torsional critical load, given by the Eqn. 3. Usual code prescriptions indicate this equation to axial compressed load and bending m x and/or y combined.

(Pkx -Pkt)-A/(Pkx - P k t ) ' -4aPkxPkt %T

2a

a = l-(xo/io)^

(3)

Previous studies were doing by Sarmanho (1995) and Batista & Sarmanho (1996), using experimental results beam-columns stiffened channel section, Batista (1987) and Loh & Pekoz (1985), submitted to compress and double eccentricity load. The results to the flexural-torsional critical load, without taking account the y eccentricity were good compared with the experimental ones. Nevertheless to unstiffened channel profiles the test results were placed very conservative. The set of experimental results used in to present investigation it is composed by beam-columns with simple and double eccentricity load. The cross section profiles are unstiffened channel.

182 TABLE 1 CRITICAL LOADS OF BUCKLING OF A COLUMN WITH MONOSSIMETRIC SECTION. Batista (1987).

'^

Oil

Section

Load Position

Critical load buckling. g(P) equation Ll-oo g(P)

X ^

Xo^O Yo^O

Q

e Xo^O Yoi^O

:^ 0

y

^KFT

=z 0

X

^y^

0 KR-'^KY length:Ll

L1>L2

L3>L2>L1

Xo^O YoffeO L3>L2>L1

L1>L2

Ag(P)

A,g(P)

e

X

= 0

A^N

ey=0

Xo^O Yoi^O

KFT "

''KY

EXPERIMENTAL RESULTS AND ANALYSIS The set of experimental results is composed by 5 test specimens of unstiflfened cold-formed chamiel beam-columns. The relation between eccentricity and the radius of gyration of the cross-section Cx/ix and Cy/iy variesfrom0.18 to 1.23.

183 The specimens were test by biaxial compressive loading. Figure 1 shows the test scheme.

REACTION BEAM

/ L PIN-ENDED

SPECIMEN

HYDRAULIC ACTUATOR

T

TEST SLAB

1

Figure 1: Test scheme.

Table 2 shows the summary of the main characteristics of the tested specimens and also show R l , R2 and R3 coefficients, related to the interaction equation as presented m the Eqn. 4, where the safety strength resistance factors are not included. These coefficients show the relative importance of the different collapse modes of the beam-column.

Rl - column buckling; R2 - beam ultimate limit state by lateral buckling or section plastification, principal axis x; R3 - beam ultimate limit state by lateral buckling or section plastification, principal axis y.

184 CmyMy Rl+R2+R3=—+ ^"^^^ Pn Mnx.(l-P/Pkx)

(4)

-=1 Mny.(l-P/Pky)

TABLE 2 EXPERIMENTAL RESULTS

1

1

U4 1

U5 1

Ul

U2

U3

(74x40)

(74x40)

(80x60)

(80x60)

(80x60)

#1.50

#1.50

#2.0

#2.0

#2.0

ex (mm)

-10.30

0.00

3.50

10.00

-10.00

ey (mm)

30.00

37.00

17.00

17.00

6.00

ex / ry

0.81

0.00

0.18

0.51

0.51

ey / rx

0.99

1.23

0.51

0.51

0.18

L (mm)

850

850

1300

1300

1300

fy(MPa)

300

300

300

300

300

PEy(KN)

102.82

102.82

177.70

177.70

177.70

Peq(2)(KN)

83.89

74.63

142.87

131.17

174.52 1

R1_AISI

0.36

0.27

0.23

0.18

0.80

R2_AISI

0.38

0.35

0.13

0.10

0.17

R3 AISI

0.25

0.39

0.64

0.73

0.03

Pu AISI (KM)

14.44

10.60

15.11

11.64

53.06 1

Pu eq(2) (KN)

14.15

10.61

15.15

11.68

52.81

Pu exp (KN)

16.80

13.44

47.53

35.05

64.82 1

Puexp

1.19

1.27

3.14

3.00

1.23

1

1.16

1.27

3.15

3.01

1.22

1

1

1

Pu eq(2) Puexp PuAISI

185 L - beam-column length; Pu exp - Experimental ultimate load; Pu AISI - AISI ultimate load; Pu eq (2) - Ultimate load; using Eqn. 2; P Ey - Euler critical load; P eq (2) - Flexural-torsional critical load; fy - stress yielding

CONCLUSION The main goal of these tests was to analyze the influence of the eccentricity in the load capacity of coldformed steel profiles with monosymmetric and unstifFened section. For all specimens, the theoretical flexural-torsional buckling load, calculated according to Eqn. 2, is lower than the Euler critical load due to the influence of ey eccentricity. But the influence in the ultimate load is not important. The coefficient of Rl has the same weight that the beams coefficients (R2 and R3), nevertheless the ultimate load is similar for all the specimens. On the other hand, applying the AISI escification, without taking account the ey eccentricity, is appropriate, resulting facilities and good results.

ACKNOWLEDGMENTS

The authors thank FAPEMIG and USIMINAS for the financial support.

REFERENCES

- AISI (1196). LRFD Cold Formed Steel Design Mcamal. American Iron and Steel Institute. Batista E. M. (1987). Essais de Profits C etU en acier plies a Froid. MSM rapport n** 157 Universite de Liege. - Batista E. M. and Sarmanho A. M. (1996). Ultimate Limit State of Thin-Walled Steel Beam-Columns. In: Rondal J., Ehibina D and Giouncu, V. (eds.) Proceedings 2nd International Conference in Coupled Instabilities in Metal Structures (CIMS'96). London: Imperial College Press, 237-245. - Loh T. S. and Pekoz T. (1985). Combined Axial Load and Bending in Cold-Formed Steel Members. Internal Report Structural Engineering Department, Cornell University.

186 - Eurocode 3 (1992). Design of Steel Structures Part. 1.3, Cold Formed Thin Gauge Members and Sheeting. - Sarmanho A. M. (1995). Resistencia Nominal de Vigas-Colunas Compostas de Perfis Metalicos Esbeltos. Internal Report (in Portuguese), COPPE/UFRJ. - Sarmanho A M. C. and Bueno, F. G. F. (1997). Estudo da Resistencia de Vigas- Colunas Compostas de Perfis Metalicos Esbeltos - REM Revista Escola de Minas, Ouro Preto - MG, Vol. 50, 61-65.

Poster Session P2 SANDWICH STRUCTURES AND DYNAMIC BEHAVIOUR

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

189

MODELLING OF CONTINUOUS SANDWICH PANELS p. Hassinen Helsinki University of Technology, Laboratory of Structural Mechanics P.O.Box 2100, FIN - 02015 HUT, Finland

ABSTRACT Load-bearing capacity of continuous multi-span sandwich panels is limited by the criteria given for the serviceability and ultimate limit states. The distribution of the stresses in the faces and the interaction between the bending moment and the support reaction have influence on the serviceability limit state load. The inelastic bending resistance after the first failure at intermediate supports increases the ultimate limit state load. Design models and proposals in the paper base on the analytical and experimental work with sandwich panels the core of which is made of structural plastic foams or of rock-wools. Properties of the core materials are discussed and results from compression-shear tests for core materials are presented. KEYWORDS sandwich panel, continuous, intermediate support, resistance, interaction, modelling, material, serviceability, ultimate, lunit state design

INTRODUCTION Lightweight three-layer sandwich panels are conmion building components used to cover walls and roofs of buildings and to separate and isolate spaces inside the buildings. The faces of typical panels are made of flat or profiled thin metal sheets, the thickness of which vary from 0.5 to 0.8 mm. The structural and insulating core layer is composed of structural foams or of mineral wools. The layers are bonded together continuously to provide a load-bearing composite structure. Typically, the sandwich panels are assembled on purlins, which provide a continuous transverse support at distances determined on the basis of the resistance and loads of the panel. Because of this, the sandwich panels can in the design calculations in most cases be assumed to be beam-type simply-supported or continuous structures. The response and resistance in the width or in the axial direction of the sandwich panel is normally not utilised in the design and use of multi-span panels.. The load-deflection path of continuous multi-span sandwich beams contains a linear part, a non-linear part caused by the local failures at intermediate supports and finally, a second more or less linear part ending to a failure load caused by a compression failure in the face or a shear failure in the core. Figure 2 shows the load-deflection behaviour of two steel sheet faced sandwich panels with a

190 polyurethane foam core. The cross-section of the specimens and the static system in the test are shown in Figure 1. A narrow support width, Ls = 70 mm in the test, causes a compression failure in the core at the mid-support, which results in a small unloading part in the load-deformation curve. If the support width is large, Ls = 200 mm in the test, the first failure mode at the intermediate support is a buckling and bending failure in the internal compressed face, at which a buckling wave extends over the whole width of the panel at an edge of the supporting beam. The failure mode produce a sudden change in the load-deformation path. If the span length is small and the core material brittle under the tensile and shear stresses, a shear failure mode m the core may dominate the load-deflection behaviour of the multi-span sandwich beam. In this case the first shear failure determines also the ultimate loadbearing capacity of the panel. For comparison, the load-deflection curve for the corresponding onespan sandwich panel is also presented in the Figure 2.

]L/8J

L/4 j L = 3600

tfL. -4—

L-3600

Figure 1: Cross-section and static system and the loading of the two-span poljoirethane cored sandwich panels. Thicknesses of the face sheets are ti=0.45 mm (external) and t2=0.46 mm (mtemal). 50 L,=200mmN

L, = 70 mm

10 20 30 40 50 60 Deflection in mid-spans and compression at mid-support w [mm]

70

Figure 2: Deflections in the mid-spans and compressive displacements on the mid-supports of twospan and one-span sandwich panels with steel sheet faces and a polyurethane foam core shown in Figure 1.

191 In the design, two different loading cases for continuous multi-span sandwich panels can be separated. The positive support reaction is exposed by the snow and wind pressure loads and it causes compressive contact stresses between the sandwich panel and the supporting structure. The loaddeflection behaviour in the positive support reaction loading case is illustrated in Figure 2. The negative support reaction, exposed by the wmd suction load and some temperature differences between the faces, introduces tensile forces in the fastenings between the sandwich panel and the supporting structure. The first failure in this case is typically a compression failure in the external face, which is strongly influenced by the transverse load in the fasteners at the support. The load-deflection curve has similarities with the loading case called the positive support reaction described above. The failure modes caused by the negative support reaction are not studied in more details in this paper, even though the case is very important in the design work in practice. In the design of continuous multi-span sandwich beams, the loads causing the first failure mode, which determines the serviceability limit state, and the second failure, determining the ultimate limit state, have to be known. For the serviceability limit state load - the real distribution of stresses in the face and - the interaction of the bending moment and support reaction at the intermediate supports are of the most importance. On the ultimate limit state load - the remaining inelastic bending resistance at the intermediate support after the first failure may have a large influence.

BENDING MOMENT DISTRIBUTION AT INTERMEDIATE SUPPORT On an intermediate support the lower face of the sandwich panel is loaded by an axial compressive load Ns = Ms/e, where Ms is the bending moment and e is tiie distance between the centroids of the faces. In the thin faced sandwich panels Ms corresponds the total bending moment carried out by the sandwich panel; M = Ms. In the sandwich panels with profiled faces the moment Ms constitutes a part of the total bending moment, the another part being introduced by the face profiles; M = Ms + Mpi + MF2, where Mpi and MF2 are the bending moments carried through by the external and internal face profiles. In addition to the axial compressive force Ns, the face layer and the core at an intermediate support are loaded by the transverse support reaction F. The shear force V caused by the internal and external loads introduces shear stresses mainly in the core layer. The distribution of the bending moment Ms at the intermediate support depends on the support pressure distribution. Figure 3 shows measured tensile stress distributions in the upper face of the test panels introduced in Figure 1. The results show the tensile stress distribution and so, also the bending moment distribution Ms = on Api e, to have a curved shape on the support, which indicates that the support pressure is in some extent distributed over the support width Lj. Thus, the support reaction can not be described by two line loads located at the edge of the supporting beam, which would result in a constant bending moment value on the support. On the other hand, the describing of the support reaction with a line load located on the centre line of the support, yields to a triangular distribution of the bending moment Ms and the tensile stress an , and thus, gives conservative results. If the support pressure is assumed to be uniformly distributed over the support width, the reduction of the bending moment Ms at the intermediate support is AA/^ = F LJS , where F is the support reaction and Ls the support width. This reduction has some practical importance for sandwich panels with relatively large support widths compared to the span lengths. For simplicity, this reduction has not been taken into

192 account in the design in practice and not either in the further comparisons in this paper, which yields to the results being on the safe side in practice.

-100 -35 0 35 100 Distance from the mid-line of the mid-support [mm] ^^^^^^^m^— Support plate

-200 -100 0 100 200 Distance from the mid-line of the mid-support [mm] — ^ M M M ^ g M — Support plate

Figure 3: Experimental tensile stress distributions opi = Ns/Api in the upper face on the intermediate support for a support width a) Ls = 200 mm and b) Ls = 70 mm.

INTERACTION AT THE INTERMEDIATE SUPPORT The first failure mode at an mtermediate support may be a consequence of a shear fracture or a crushing failure in the core or a buckling failure in the compressed face. The shear failure mode is typical to short span panels, the core crushing failure to medium span panels and the face buckling mode to large span panels. Numerical simulations have shown that the failure at intermediate support is a combination of the three failure modes, from which the dominating mode can be visually observed in the experiments. Numerous attempts have been made to write design equations for the three failure modes and for the combinations of them. The theoretical models for the compressed face layer are based on beam-column models, in which the face layer is supported by the elastic core layer and loaded by an axial force Ns and a transverse support reaction F simultaneously. However, the comparisons have not shown acceptable agreement between the theoretical models and the test results (Bemer 1995, Martikamen&Hassmen 1996). An equation for the core crushing failure can be derived from the condition kwWmax ^ fcc, in which kw is the Winkler's foundation coefficient and Wmax the maximum local deflection of the face at the support caused by Ns and F. The local deflection Wmax can be calculated on the basis of the above mentioned beam-column model, in which the non-linear mteraction between the forces Ns and F shall be taken into account. The sunplified design model given in Eqn. 1 is formally based on the equation for the core crushing failure mode (Martikainen&Hassinen 1996). , Figure 2. The corresponding moduh of elasticity of faces are El, E2. The cross - section area of the core equals A = ab. The shear force Q at the cross - section of the panel equals Q= AT. Multiplying the equation (1) by ^ and substituting Q = AT, we obtain the constitutive equation for shear of the core: 0+

' — = — ^ — A y - \ - ' ^ A-^. Go+G dt G^+G G^+G dt

(2) ^^

The constitutive equation for one - way bending of the panel is as follows, Allen (1969): dy d w M = B^-B-—,

(3)

where: A/ - the bending moment at the panel cross - section, 5 - the flexural rigidity of the panel, w - the deflection of the panel. The flexural rigidity equals: B = ^1^2^1^2^ / (^l^l + ^ih)

3. THE CREEP ANALYSIS Let's take into consideration a simply supported panel, subjected to an unifoimly distributed load q assumed to be constant in time. If the load q is apphed to the panel and then kept constant, instant elastic deformations take place. The instantaneous shear strain y(x,0) and transversal deflection w(x,0) are defined by:

^

^ 24B\

I

IG^A

^

The shear force Q and the bending moment M equal:

2(^,0 = ^[^-^J

and M(x,0 = ^ | ( / - 4

Substituting the functions Q and M defined above into equations (2) and (3), and taking into account that 5 2 / ^^ = 0, for / > 0, we get the following solutions:

y(x,0 = Y(x,0) + y ( x , 0 ) ^ f l - e x p | ^ - - / | | ,

(6)

199 w(x,0 = w ( x , 0 ) + ^ ^ ^ [ ^ l - e x p | ^ - ^ ? j j .

(7)

When time t tends to inJSnity then the deformations j{x,t) and M^{x,t) tend to their asymptotic functions y(x, oo), w{x, oo), which can be calculated from (6) and (7): \ , / y^ (x,0) m .^ ^ , .w(x, / . ^ oo) ^ _ ,=. w(x,0) / . m . + -q^{l-^) y (x, cx)) =/ l 1++^ —^ a«(i 2GA Only the shear deformation of the core increases with time and the asymptotic functions y(x, oo) and w{x, oo) are finite should be noted.

4. THE EXPERIMENTAL DETERMINATION OF THE RHEOLOGICAL CONSTANTS The instantaneous shear modulus G^, the retarded shear modulus G and the viscosity constant rj have been established from an experimental test. A simply supported panel ISOTHERM SC 80 subjected to a constant in time and uniformly distributed load over the panel has been apphed. The specimen has been randomly sampled from a population of panels manufactured in a lot production by Metalplast - Obomiki. The span of the panel has been chosen as / = 4000 mm with the width of b= 1100 mm. The cross - section geometry has been hmited the face thicknesses of ^ = 0,57 mm and the core thickness of« = 80 mm. The core is made from urethane foam of nominal density 42,7 kg/m^. The faces are made from steel. The following loads are apphed: the weight load g= 13,97 daN/m, and the external load p = 77,99 daN/m. The midspan deflections for two sides of the specimen have been measured daily up to 46 days using deflection dial gauges. By w{t) the average midspan deflection of the panel at time t has been denoted. The following results have been obtained: - the instantaneous deflection caused by the load/?, (Figure 3): w(0) = 11,94 mm, (8) - the creep deflections w{t) - w(Q) caused by the total load g + p shown with a broken hne in Figure 4, where fluctuations following changes in the temperature have been noticed. The instantaneous shear modulus G^ has been calculated from the formula (5) for x = 0,5 /, / = 0, and q=p^ 77,99 daN/m. Go=

^ pf-

J-

(9)

485

Calculating: A = 8,8 lO^ mm^, and B = 3,77 lOl 1 N mm^, from (8) and (9) it has been obtained: G^ = 3,52 N/mm^ The creep deflections w(t)-w(0) in the form of

shown in Figure 4 have been approximated by the function y{t)

y{t) = 2,S0^l-e-^'^^^^\

(10)

200

The ftmctionj^(0 by means of the least squares method has been reached. On the other hand, from (7) for x = 0,5l and for ^ = g + /? we have:

The comparison of the formulae (10) and (11) resulted in: il±£)I^ SAG

= 2,S0

and

^ = 0,217. Ti

Hence, we calculate G = 7,46 N/mm^

and

n = 2,97 lO^ ^ ^ mm

= 34,4

Kl^, mm

Thus, the fitting of the rheological model to the empirical data has been accomphshed. Figure 5.

5. THE RELAXATION ANALYSIS Let's consider a two-span continuous sandwich panel with no external load and with the forced and fixed displacement S of its central support. In such a panel the bearing reactions, the inner forces as well as the deflections between supports will change with time. Finding how the above mentioned values will change in a long period of time has been our goal. Consider an auxihary simple supported panel with point load P at the midspan. Figure 6. Let P be constant in time. Using the constitutive equations (2) and (3), the shear strains y(x,t) and the deflections w(x,t) can be obtained as follows: P y(x,t) = ^ ^ 2AG(t)

for -^

Wy,0 = — ( 3 / ^ - ^ ^ ) + ^ ^ ^ 125V / 2AG{t)

0<x(x,a)] ^^^^'^^^k

for for

0<x5(a)

(^)

where 21,060 160,320

Crack initiation | Neck-rivet in the constant moment region 1 Neck-rivet outside the constant moment region 1 Neck-rivet in the constant moment region

368,500 1,725,120 3,920,700

1,134,280 2,002,000 5,317,350

>161,800 69,900 >707,850

Neck-rivet in the constant moment region 1 Flange-rivet outside the const, moment region Neck-rivet in the constant moment region |

Crack length, mm 250

n=368,500

200

:n=872,600 150

^

^

I

Total failure

n

J3_ n=761,600 50

500,000

600.000

J

700,000

800,000

900,000

1000,000

1,100,000

1,200,000

Number of cycles, N

Figure 5: A typical fatigue crack propagation scenario (stringer 12A) Considering the data presented in Table 2, it can be concluded that: 1) Despite the fact that these stringers first had been subjected in the bridge to almost 100 years of loading and environmental effects, all test results lie well above the fatigue design curve for riveted connections (C=71 in Eurocode 3 / AASHTO category D), see Figure 6. (It is worth mentioning that the design curve was based on tests with small virgin specimens.) In fact, only for two stringers the number of cycles to failure was somewhat below the mean fracture value (1,79 million) given by the design curve. Those two where previously tested by Akesson with 20 and 10 million cycles at a stress range of 40 and 60 MPa respectively, see Table 1. 2) An important observation regarding the fracture behaviour of riveted stringers is that a substantial number of cycles were required for fatigue cracks to appear in the second L-bar after the first one had been completely fractured (denoted as Nredund in Table 2). This inherent, redundant structural behaviour of built-up riveted stringers makes their fatigue life considerably longer and more or less rules out the occurrence of brittle fracture in this type of girders. The results from the second set of the stringers are presented in Table 3 together with the data obtained by Out et al. (1984). The values of the two terms of Eqn. 4 are also included for comparison. In this table, Narrest IS the number of cycles from that the stop-hole was drilled until observed reinitiation of the crack from the edge of the hole.

229 stress range, a^ [AfPa] 200

x^71

150 100

^

crao

ox

o

0.737C = 52 MPa -

bU

• Akesson o Kadir X Al-Emrani

10

10^^

'

105

•

l'0«^

107 '

Figure 6: Test results from this investigation and those presented by Akesson. TABLE 3 TEST RESULTS WHERE FATIGUE CRACKS WERE ARRESTED BY DRILLING STOP-HOLES, IN COMPARISON WITH AN INVESTIGATION PERFORMED BY OUT ET AL (1984). Stringer

12A 3A lA Out et al.

Or \MPa\

Cross-section reduction

100 100 60 57.2

17 33 27 20

J^J arrest ^ noin

>67,000 34,280 215,350 2,200,000

4.2 5.61 4.95 4.9

2.23 2.23 3.26 4.1

In order to have a better understanding of the value Narrest given in Table 3, it was transferred into "an additional monitored service life" in years based upon the statistics obtained from the loading history for the bridge before it was taken out of service. Taking stringer 1A as an example, the corresponding time was calculated to 2.1 years. This value (although it is seemingly "large") is well underestimated because the stress ranges in most riveted railway bridge stringers in service today are well below those applied in the tests. In fact, the maximum stress range due to traffic loading obtained from strain measurements on 15 riveted railway bridges in Sweden, as presented by Akesson, was 42 MPa. Finally, considering Figure 7 and the results presented in Table 3, a suggested conclusion (due to the limited amount of data to fit the curve in Figure 7) is, that a more or less "permanent" crack arrest can be achieved if the maximum stress at the hole edge does not exceed the yield strength of the material.

73

_J

O UJ to

2.-Plola-Kirchhoff

Biot strain m

materially exact

• Ji ••llll illll

=^(0*i)/a Green-Lagrange

II

stresses and stress transformations

lllllll• I •••111 •iii

•ill Blot stress

CO

TBIOT = [S^-EBIOT

>

Biot stress (materially inexact)

^ "

E

73

I i

TBIOT=^-EBIOT

pseudo-Biot stress

O

12

^ ^ 3

.

2

=[ei;e2;e3]

2

_^23

2

2

Box 1: Consistent quadratic approximation of the kinematic rotation tensor of the cross-section 2

2

r,

^1 = r + 2(2'3-23') = ^1

^2 = v'-(i,3(i + f/') + (|,jW + ^ Tj = w + (i)2(i + (/') + -(t)iy + ^

Eu

= r;

= r;

^ 2 = 2' + 2^4>3'l-3l') ^ 3 = ^3' + 2^l'2-l
Pth

256 256

9.1 8.2 7.6 22.0 23.1 24.2 25.8 29.9 26.9 9.4 8.8

10.2 9.3 8.6 25.2 25.7 25.9 30.3 32.6 32.6 10.1 9.3

256

8.3

8.6

321 321 321 321 321

321

^cxp/Pth

Tons Tons 0.89 0.88 0.88 0.87 0.90 0.93 0.85 0.92 0.83 0.93 0.95 0.97

RESULTS AND DISCUSSION Results presented in Table 1 show generally good correlation between the analytical and experimental values of ultimate load. The analytical method is found to overestimate the failure load for all specimens. It can be seen from the comparison between the two values that the experimental and analytical values are close for most of the test specimens the maximum deviation being 19% in the case of B9. The mean value of the ratio between the experimental and theoretical values (Pexpt/ Pth ) is 0.9. A nominal 10% reduction in ultimate capacity to allow for the effect of residual stresses will result in more realistic predictions, consistent with experimental results. Such a correction is adequate for all practical purposes. It can, therefore, be concluded that the proposed method is sufficiently accurate to predict the ultimate load capacity of short rectangular box-columns with or without perforations. The effect of inserting small diameter openings (not exceeding one-third of the plate width) could be assessed by comparing pairs of test specimens (Bl v^th BIO; likev^se, B2 with Bl 1 and B3 with B12). All these specimens have a/t values of 140-150 and (hole diameter/plate width = 0.33). In all these cases, the introduction of small diameter holes has not significantly reduced the ultimate compressive resistance of the box columns made of plates having large values of plate slendemess (a/t = 140 -150). Perforated plates, however, will exhibit a reduction in stiffness. These results are consistent with other test data published elsewhere, for wide plates with perforations. The results are significant for box columns made up of plates having high slendemess values (e.g. box colunms made of cold rolled steel), wherein it is possible to introduce small service openings (not exceeding d/a of 0,33) without fear of loss of resistance. Hot rolled steel box columns, however, are made up of more stocky plates, having significantly lower (a/t) values and are illustrated by the results for other specimens.

280 Comparison can be made by studying the other series of pairs of tests, B6 and B8; as well as B5 and 39 and B4 and B7. In all these cases, the plate slendemess (a/t) values are significantly smaller, being set at 80. The average value of drop in observed resistance due to the introduction of a hole for these three pairs is 16%, with the maximum drop being 19%. These tests demonstrate that box columns made up of hot-rolled plates with medium values of plate slendemess (a/t) will - indeed - exhibit an observable drop in resistance, when holes are introduced. These observations are significant for practical box columns, which are likely - in general - to be made up of plates having a/t values of less than 80. The values given in the table show also the effect of local buckling of the component plates on ultimate load. Comparing the failure loads for specimens wdthin each group subjected to different degree of eccentricity it is seen that those specimens under uniform edge displacements carry the largest load. Typical load - axial displacement plots for selected specimens are shown in Fig. 4 in which curves for box-columns B4, B7, Bg, and B9 are plotted so that the effect of plate openings and degree of eccentricity on the ultimate load behaviour can be studied. General observation of the curves show that those specimens containing perforations are less stiff compared to those with solid plates. It is observed from curves corresponding to B^, Bg and B9 that specimens subjected to eccentrically applied load have experienced larger deformation compared to the one under uniform edge displacements. CONCLUSIONS An approximate but simple method based on effective width concept has been proposed to predict the ultimate load capacity of short rectangular box-columns. Energy method is used to derive equations for effective widths of component plates in a box-column assumed to be simply supported and free to pull-in along the unloaded edges whilst the loaded edges remain clamped. The loaded edges are subjected to xmiformly varying edge displacement. The effective width equations have been used to compute the effective cross-sectional area of box-columns and hence the ultimate load capacity. Box-column specimens tested earlier have been analyzed by using the proposed method and the predicted failure loads have been compared with the corresponding experimental failure loads. It is found from the comparison that the proposed method is capable of predicting the ultimate loads with sufficient accuracy. It is also found that the presence of openings results significant reduction in the load carrying capacity and the cross-sections under varying load are less stiff and carry less load compared to those under uniform loading. r—m^

1.0

2.0

3.0 4.0 5.0 DEFLECTION (mm)

6.0 7.0

Figure 4 Load - Axial Displacement Curves for Selected Test Specimens

281 The theoretical method proposed in the paper is extremely useful for rapid assessment of the ultimate resistance of box columns but gives somewhat unconservative predictions when compared with observed test results on model box columns. This could well be due to the fact that the analytical model has not taken account of residual stresses due to welding, which could be significant in small scale models.

REFERENCES Kalyanaraman V and Ramakrishna P. (1984). Non-Uniform Compressed Stiffened Elements, University ofMissouri-Rolla, 75-92. Narayanan, R. and Shanmugam, N.E. (1979). Effective Widths of Axially Loaded Plates, Journal of Civil Engineering Design, 1, No. 3: 253 - 272. Narayanan, R. and Chow, F.Y. (1984). Ultimate Capacity of Uniaxially Compressed Perforated Plates, Thin-Walled Structures, Vol. 2, 241-264. Narayanan R. and Chan S.L. (1985). Ultimate Capacity of Plates Containing Holes under Linearly Varying Edge Displacements, Computers and Structures, 21:4, 841-849. Rhodes J. and Harvey J.M. (1971). Effect of Eccentricity of Load or Compression on the Buckling and Post-Buckling Behaviour of Flat Plates, InternationalJournal of Mechanical Sciences, Vol. 13, 867-879. Rhodes J., Harvey J.M. and Fok C. (1975). The Load-Carrying Capacity of Initially Imperfect Eccentrically Loaded Plates, InternationalJournal of Mechanical Sciences, Vol. 17, 161-175. Ritchie, D. and Rhodes, J. (1975). Buckling and Post-buckling Behaviour of Plates with holes. Aeronautical Quarterly, November, 281-296. Shanmugam, N.E. and Narayanan, R. (1998). Thin Plates Subjected to Uniformly Varying EdgeDisplacements, Proceedings, Second International Conference on Thin-Walled Structures, Singapore, 2-5 December, 1998,449-457. Timoshenko S. (1961). Therory of Elastic Stability, McGraw-Hill Book Company, Inc, NewYork. Walker A.C. (1967). Flat Rectangular Plates Subjected to a Linearly Varying Edge Compressive Loading, Thin Walled Structures, Chatto and Windus, London

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

283

ELASTOPLASTIC SECTIONAL BEHAVIOR OF STEEL MEMBERS UNDER CYCLIC LOADING Iraj H. P. Mamaghani Department of Civil Engineering, Kanazawa University, 2-40-20 Kodatsuno,Kanazawa 920-8667, Japan.

ABSTRACT The present paper is concerned with the cyclic elastoplastic sectional behavior of steel members subjected to the combined axial force and bending moment by employing the two-surface plasticity model in force space (2SM-FS), which takes accurately into account cyclic elastoplastic behavior of structural steels such as the yield plateau, Bauschinger eflFect and strain hardening. First the basic concepts of the 2SM-FS are addressed. Then the procedure for determining of the model parameters is presented and the values of the model parameters for circular, I and box sections are given. Finally the accuracy of the 2SM-FS is verified by comparing the cyclic elastoplastic sectional behavior of steel members obtained from the 2SM-FS with those of the direct integration method using the two-surface plasticity model in stress space (2SM-SS) and experiments. KEYWORDS steel members, sectional behavior, analysis, cyclic loading, two-surface plasticity model INTRODUCTION From the viewpoint of limit state design of steel structures, investigating and clear understanding of the cyclic behavior of steel members and structures are important to prevent their collapse under severe earthquakes. Recently, cyclic behavior of steel members and structures has extensively been studied by taking into consideration material nonlinearity as well as geometrical nonlinearity. As for material nonlinearity, the two-surface plasticity model (2SM-SS) for structural steels (Mamaghani et al. 1995, Shen et al 1995, Mamaghani 1996), among many constitutive models in the stress space, has been developed and applied to the cyclic nonlinear analyses of steel structures. The 2SM-SS has shown the excellent capability to predict the cyclic behavior of steel structures (Mamaghani et al. 1996, Banno et al. 1998). However, the finite element structural analysis with constitutive model in the stress space requires generally much longer computational time than that with the constitutive model in the force space. Therefore, application of constitutive model in the stress space may be limited to the analyses of simple structures.

284 The present paper is concerned with the cychc elastoplastic sectional behavior of steel members subjected to the combined axial force and bending moment. Recently, the two-surface plasticity model in force space (hereafter referred to as 2SM-FS) has been developed by the author and his coworkers (Suzuki et al. 1995, Mizuno et al. 1996, Mamaghani et al. 1997) with extension of concepts used in the 2SM-SS, which takes accurately into account the cyclic elastoplastic behavior of structural steels such as the yield plateau, Bauschinger effect and strain hardening (Mamaghani 1996). In the following sections, first the basic concepts of the 2SM-FS are presented and discussed. Then the detailed procedures for determining of the model parameters are presented and their values for hollow circular, I and box sections are given. Finally the accuracy of the 2SM-FS is discussed and verified by comparing the results of analyses on the cyclic sectional behavior of steel members obtained from 2SM-FS with those of the experiments and direct integration method using 2SM-SS. It has been shown that the 2SM-FS is a promising model and has the advantage of more larger computational speed and requires less computer memory than that of the 2SM-SS. TWO-SURFACE PLASTICITY MODEL IN FORCE SPACE (2SM-FS) To predict elastoplastic sectional behavior of steel members under cyclic loading the 2SM-FS has been developed based on the assumptions that the cross-section remains in plane after deformation, there is no distortion of the cross-section, only normal stress acting on the cross-section and, no local buckling occurs (Mizuno et al. 1996, Mamaghani et al. 1997). The requirements for developing 2SM-FS are based on the experimental evidences that the key behavioral characteristics of the material response under cyclic loading, such as the decrease and disappearance of the yield plateau, the Bauschinger effect and cyclic strain hardening, exhibited at the stress level by the steel members thus propagates to the force (stress-resultant) level provided that no local buckling occurs (Mamaghani et al. 1995, Mamaghani 1996). Therefore, the 2SM-FS has been developed with a simple extension of concepts used in the 2SM-SS. Basic Concepts of the 2SM-FS The 2SM-FS, as in the 2SM-SS, utilizes the bounding surface formulation in force space. The 2SMFS formulation uses two nested curves: an inner loading curve, and an outer bounding curve, as schematically shown in Fig. 1 in two-dimensional normalized force space; axial force n = N/Ny versus bending moment ni = M/My. Ny and My denote the yield axial load and yield bending moment of the cross-section respectively. The inner loading curve represents the locus of loads and moments that causes the initiation of yielding at some point on the cross-section. The outer curve represents the load state at which a limiting stiffness of the cross-section is achieved. As shown in Fig. 1, at the initial unloaded state the loading and bounding curves coincide with the initial yield curve and yield plateau curve respectively.

^1 Yield plateau curve

Initial yield curve

Loading curve (a)

Loading curve

Bounding curve (b)

Figure 1: Loading curve in contact with: (a) yield plateau curve; (b) bounding curve

285

Once the loading point has contacted the loading curve, the response is governed by a number of hardening rules which determine subsequent elastoplastic behavior. As the cross-section is loaded inelastically, both the loading and bounding curves may translate (kinematic hardening), contract or expand (isotropic hardening), to model phenomena such as strength degradation, the Bauschinger effect and cyclic strain hardening. The degree of plasticity at the cross-section is a function of the distance between the two curves. In the following, some important features of the 2SM-FS will be presented and discussed. Plastic M o d u l u s The plastic modulus E^ associated with loading curve is used to prescribe the plastic flow under the assumption of the associated flow rule. In 2SM-FS, the same equation for E^ in 2SM-SS is used. That is, the value of E^ is a function of the distance S between the two loading and bounding curves and is taken as: EP = E^o-^h{6)-^

(1)

Oin-O

where EQ, which is a function of plastic deformation, and h{S) are the current plastic modulus of bounding curve and the shape parameter as in the 2SM-SS respectively. 6 is the distance from loading point P to conjugate point R on the bounding curve and is measured in Euclidean norm, see Fig. 2. 6in is the value of 6 at first contact with the loading curve. As shown in Fig. 2, the conjugate point R on the bounding curve is defined to have the same direction from the center of bounding curve as the direction of the loading point P from the center of the loading curve. Note that the value of 6, measured in the dimensionless force space n versus in, is used in Eqn. 1 after multiplying by the yield stress ay. Effective Plastic Strain Curve Based on the definition of the effective plastic strain surface in plastic strain space for multiaxial 2SM-SS, the effective plastic strain (EPS) curve is defined for the 2SM-FS in the nondimensional plastic strain space; e^/ey versus (jf/cpy, see Fig. 3, as follows:

He''/ey,4>'/4>y)=l^^-V.]

+(^-ri^

- p'= 0.0

(2)

in which (r/e, 77^) and p are the center and radius of the curve respectively. £y and 0^ denote the yield values of the axial strain and curvature of cross-section respectively. The EPS curve which represents a memory of maximum plastic deformation that the cross-section has ever experienced through the loading history, expands and translates if $((5^ + deP)/ey, {(jf + d(j)P)/(t)y) > 0. The evolution of loading and bounding curves in 2SM-FS is related to the size p of EPS curve. Updated EPS curve Bounding curve

\

Loading point

/

Conjugate point

Previous EPS curve Figure 2: Definition of S

Figure 3: Effective plastic strain (EPS) curve

286 Hardening Rule The hardening rule adopted in the 2SM-FS defines evolution of loading and bounding curves in a way to ensure that the two curves be tangential to each other when they contact. The loading curve at the initial unloaded state coincides with initial yield curve FQ, see Fig. 1(a), defined by: Fo(m,n) = |m| + | n | - l = 0

(3)

When the cross-section is loaded it undergoes elastic deformation until the loading point reaches the initial yield curve from where plastic flow starts and subsequent loading curve evolves and begins moving towards the yield plateau curve Fy, see Fig. 4(a), defined by: Fy{m,n)

+ n^'

(4)

1= 0

where ci and C2 are constant values related to the type of cross-section and material; fy is a shape parameters. As shown in Fig. 4(a), the yield plateau curve, which represents the locus of loads causing elastic-perfectly plastic load-deformation behavior, governs the evolution of the loading curve before it ceases to exist as it does in the 2SM-SS. Subsequent loading curve before the yield plateau disappears is given by: /(m,n,a^,an,r)

= Oi

Hl-0,)

-1.0 = 0 (5) rfy where (a^,an) is the center of loading curve; r = K/KQ is the size of loading curve, K, is the radius of loading surface and KQ = ay is the radius of initial yield surface in 2SM-SS. The same expression for K as in 2SM-SS is used just by replacing the radius p of the effective plastic strain surface in 2SM-SS with products of the radius of EPS curve p and yield strain £y; pSy. Similar to the 2SM-SS, the loading curve in the 2SM-FS softens isotropically, as a function of the size of p, to provide experimentally observed decreasing zone of elastic behavior of the steel and its effects at the force level (Mamaghani et al. 1995). In Eqn. 5, 9i whose value changes from 1 at the initial unloaded state to 0 when the loading curve contacts the yield plateau curve, is the evolution tracer of the loading curve before the yield plateau disappears and is defined by: Oi = min (^^^/(^fn)? i^ which S^^ is the value of S measured to the yield plateau curve, and S^^ is the S^^ value of the current loading path at first contact with the loading curve. The sign 'min' indicates that Oi assumes the minimum value through the whole loading history to guarantee transformation of loading curve is irreversible. The loading curve progressively change in shape and assumes the same shape with that of the yield plateau curve when they tangentially contacts at the loading point, as shown in Fig. 1(a). After the yield plateau curve disappears as a function of the size p of EPS curve and the cumulative plastic work as in the 2SM-SS, the loading curve progressively changes in shape and moves towards the bounding curve F^,, see Fig. 4(b), defined by: Fb{m,n,prn,Pn,rb)

-A

nfb I + \

n

1= 0

(6)

where c^ and C4 are constant values related to the type of cross section and material; f^ is a shape parameters; r^ = K,/I^O is the size of bounding curve, R = the radius of bounding surface in 2SM-SS and is defined in a manner similar to K described above. (Pm^Pn) is the coordinates of the center of bounding curve. As shown in Fig. 4(b), the bounding curve governs the evolution of the subsequent loading curve which is expressed by:

287

Yield plateau curve

Loading curve

(a) Before yield plateau disappears

Bounding curve

(b) After yield plateau disappears

Figure 4: Evolution of the loading and bounding curves f{m,n,am,Oin,r)

= 62 + (1-^2)

+ rh

( ^ ) "

-1.0 = 0

(7)

in which O2 whose value changes from 1 when the yield plateau disappears to 0 when the loading curve hits the bounding curve, is another evolution tracer of the loading curve and is defined by: 02 = min (S^yS^^), where 6^^ is the S value measured to the bounding curve; 6^^ is the 6^^ value of the current loading path at first contact with the loading curve. 62 assumes the minimum value through the whole loading history and plays the same role as Oi does before the yield plateau disappears. The loading curve tangentially contacts the bounding curve at the loading point and assumes the same shape with that of the bounding curve as shown in Fig. 1(b). The two curves remain tangent on further loading until unloading occurs. The instantaneous translation of the loading curve associated with the load increment (dm, dn) occurs along PR following the Mroz type of hardening rule given by (Aa^, Aa„) = C^C^'mj^m), in which [Um, I'm) is the unit vector in the direction of PR as shown in Fig. 4. C^ is the step size of translation and is determined through the consistency condition of df = 0. To consider random cycling and identify smaller plastic excursions relative to previous larger excursions the concepts of memory curve and virtual bounding curve are used in the 2SM-FS as they are used in the 2SM-SS. 2SM-FS PARAMETERS The material properties for JIS SS400 equivalent to ASTM A36 and the 2SM-FS parameters related to the 2SM-SS are given in the paper by Shen et al. 1995. In this section, the direct integration method (Minagawa et al. 1988) is used to determine the model parameters related to the strength curves (loading, yield plateau and bounding curves) in 2SM-FS. In this approach, the section analyzed is divided into elemental areas, as shown in Fig. 5 for a hollow box section. The incremental stress-strain relation for each elemental area is described by the uniaxial 2SM-SS. The stress resultants of axial force N and bending moment M are calculated simply by summing the contribution of each elemental area over the cross-sect ion. The axial strain e and curvature 0, with a prescribed ratio as shown in Fig. 6, are increased incrementally and the axial force and bending moment are calculated. From the n - e^ and

288

A£=0

H

(a) Elemental areas

(b) Strain

(c) Strain increment

Figure 5: Subdivision of cross-sect ion and strain distribution for a box section m — 0P curves the values of n^ and lUi corresponding to the yield plateau and bounding curves are determined for a specific ith loading path, cross-section and material, example of which is shown in Fig. 7 for a box section. The results for different loading paths are plotted in the n versus m coordinate system and the values of ci to C4, /y, and /^ are determined by fitting the yield plateau and bounding curves using the least square method, as shown in Fig. 8 for a box section with steel SS400. The values of model parameters are examined for different sectional parameters; ratio of flange area to web area Aj/A^ for box and I sections, and ratio of diameter to thickness D/t for a circular section. The results for a typical example with a box section is shown in Fig. 9 . The model parameters determined for the circular, I and box sections corresponding to steel SS400 are given in Table 1.

Figure 6: Loading paths Yield plateau curve Bounding curve

Figure 7: Definition of initial bounding line ^>. -^

c

o~1.5

0

Figure 8: Definition of the yield plateau and initial bounding curves

_J

I

0.5

1

L_

1.5

2

2.5

0

0.5

1

1.5

(a) Yield plateau curve (b) Bounding curve Figure 9: 2SM-FS parameters for a box section

2

2.5

289 Table 1: 2SM-FS parameters related to strength curves (steel SS400) Parameter Cl C2

fy

1.23 + 1.10 4-

C3 C4.

h

1.29 + 1.08 +

I or box sections 1.0 ^mexpi-^miAf/A^)} Q.2>lexp{-2.29{Af/A^)} 1.0 Qmexp{-^m(Af/A^)} 0.26e.T.7;{-1.92(yl//ylt^)}

circular section 1.0 1.73 1.30 + 0.33ex-p{-0.11(D/0} 1.0 1.67 1.36-1.19 X 10-3(D/0

VERIFICATION OF 2SM-FS The cyclic sectional behavior of steel members are analyzed using the 2SM-FS and the results are compared with those of the direct integration method (DI) using 2SM-SS and experiments (Minagawa et al. 1988). The results for two typical examples will be presented in this section. The first example is a hollow box section subjected to combined proportional axial load and bending moment. The box section has a size oi B = H = 125mm, flange thickness of tj = 8.7mm, web thickness of t^ = 6.1nim. The assumed material is steel SS400. Fig. 10 compares the normalized axial strain e/Sy versus axial load n and curvature 0/(/>y versus bending moment m for the 2SM-FS and direct integration method using 2SM-SS. As shown in this figure, a good correlation between the two analytical models is achieved indicating the accuracy of the 2SM-FS.

-10

DI(2SM-SS) 2SM-FS

Figure 10: Comparison between the 2SM-FS with the direct integration method using 2SM-FS The second example is a H-shaped section of H125 x 125 x 6.5 x 9 with steel SS400 which is tested by Minagawa et al. (1988). Fig. 11 illustrates comparison between the two analytical models and experiments for the load-deflection curves associated with three different loading histories. As shown in this figure, both the 2SM-FS and 2SM-SS provide an excellent prediction of the cyclic elastoplastic behavior for the entire hysteresis curves under random cyclic loading histories, owing to the reasons that they: (a) take accurately into account the Bauschinger effect, which has the effect of softening and reduction in stiffness on the hysteresis curve; (b) correctly treat the yield plateau and cyclic strain hardening of the material. It is worth noting that the 2SM-FS has the advantages of its simplicity and larger calculation speed which is about ten times faster as compared with the 2SM-SS. The cyclic behavior of steel members such as beam-columns and frames subjected to cyclic loading is also predicted by the general purpose finite element program FEAP incorporating the 2SM-FS for sectional elastoplastic behavior through Bernolli-Euler beam element. The results by the 2SMFS compares well with the experimental data and the numerical results from the 2SM-SS. These results will be presented in the conference.

290 M(kN.m) 60 r

M(kN.m)60

-0.08

-0.12

M(kN.m) 60 1

0.16

-0.24

(j)(l/m) Expt. -60 DI(2SM-SS) 2SM-FS Figure 11: Comparison with experiments (7/125 x 125 x 6.5 x 9, SS400)

CONCLUSIONS This paper was concerned with the cyclic elastoplastic sectional behavior of steel members subjected to the combined axial force and bending moment using the 2SM-FS. First the basic concepts of the 2SM-FS were presented and discussed. Then the procedure for determining of the 2SM-FS parameters was presented and the values of the model parameters for circular, I and box sections corresponding with steel SS400 were given. Finally the accuracy of the 2SM-FS was verified by comparing the cyclic elastoplastic sectional behavior of steel members obtained from the 2SM-FS with those of the direct integration method using 2SM-SS and experiments. It was concluded that while both the 2SM-SS and 2SM-FS provide reasonable accuracy, the 2SM-FS has the advantages of simplicity and more larger computational speed than that by the 2SM-SS. That is, the 2SM-FS is a promising model to carry out cyclic inelastic analysis of large scale steel framed structures. REFERENCES Banno, S., Mamaghani, Iraj H.P., Usami, T., Mizuno, E. (1998). Cyclic Elastoplastic Large Deflection Analysis of Thin Steel Plates. J. of Engrg. Mech., ASCE, 124:4, 363-370. Mamaghani, I.H.P., Shen, C , Mizuno, E., Usami, T. (1995). Cyclic Behavior of Structural Steels. I: Experiments. J. Engrg. Mech., ASCE, 121:11, 1158-1164. Mamaghani, I.H.P. (1996). Cyclic Elastoplastic Behavior of Steel Structures : Theory and Experiments. Doctoral Dissertation^ Department of Civil Engineering, Nagoya University, Japan. Mamaghani, I.H.P., Usami, T. and Mizuno, E. (1996). Inelastic Large Deflection Analysis of Structural Steel Members Under Cyclic Loading. Engineering Structures^ UK, 18:9, 659-668. Mamaghani, I.H.P. and Kajikawa Y. (1997). Cyclic Inelastic Sectional Behavior of Steel Members. Proc. of the 52th Annual Meeting, JSCE, Japan, I-A78, 156-157. Minagawa, M. Nishiwaki, T. and Masuda, N.(1988). Prediction of Hysteresis Moment-curvature Relations of Steel Beams. J. of Struct. Engrg., JSCE, 34A, 111-120. Mizuno, E., Mamaghani, I.H.P., Usami, T. (1996). Cyclic Large Displacement Analysis of Steel Structures With Two-surface Model in Force Space. Proc. of Int. Conf. on Advances in Steel Structures^ Pergamon, 1, 183-188. Shen, C , Mamaghani, I. H. P., Mizuno, E. and Usami, T. (1995). Cyclic Behavior of Structural Steels. H: Theory. J. Engrg. Mech., ASCE, 121:11, 1165-1172. Suzuki, T., Mamaghani, I.H.P., Mizuno, E., Usami, T. (1995). Finite Displacement Analysis of Steel Structures with Two-Surface Plasticity Model in Stress Resultant Space. Proc. of the 50th Annual Meeting, JSCE, Japan, 1:A, 110-111.

Session A5 NEW STRUCTURAL PRODUCTS

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

293

RESEARCH ON COLD FORMED COLUMNS AND JOINTS USING IN MIDDLE-HIGH RISE BUILDINGS Y.Chen^

Z.Y. Shen^ Y.Tang^

G.Y.Wang^

^Department of Structural Engineering, Tongji University, Shanghai 200092, CHESFA ^Shanghai Institute of Mechanical & Electrical Engineering, Shanghai 200040, CHINA

ABSTRACT Cold-formed steel rectangular columns with large width-to-thickness ratio are experimentally and numerically researched for the purpose to use this kind of members into middle-high rise building frames in the area where earthquake risk exists but is not a predominant factor. Results of loading tests on cold-formed RHS compressive specimens are briefly introduced. The emphasis in this paper, however, is put on the discussion of the characters of the members against cyclic loads. Inelastic multispring element model is adopted to simulate the behaviors of the columns. The method to calibrate the model parameters by static loading tests and its reasonability is also described. Furthermore, beam-tocolumn joints are tested under cyclic loads and the results are analyzed. The research work indicates that under limited conditions there is the possibility to construct middle-high steel frames by thinwalled cold-formed steel columns.

KEYWORDS Steel frame. Cold-formed member. Rectangular hollow section. Beam to column joint, Hystereis loop, Local buckling. Post buckling deformability. Cyclic loading. Load test. Multi-spring element model

INTRODUCTION Cold-formed steels with large width-to-thickness ratio are chiefly used as main structural members in single story or low rise building frames, or taken as secondary members in roof or wall systems. The slender plates let the moment of inertia and section modulus larger than those of compact sections if the same amount material is consumed. On the other hand, by the post buckling strength, the member can also stand against great loads acting on it. Thus lighter structural system becomes possible. To middle-high rise buildings, for example, buildings having ten or more stories, which are to be constructed on the soft ground, to lessen the weight of upper structures is much concerned by engineers.

294 Though advantages of thin-walled cold-formed steel have great attraction in this case, the engineers face other kinds of problems. One of them is the behavior of the members under cyclic loads, which is unavoidable if a building is just located in the area where earthquake breaks out occasionally. The loading capacities and deformabilities after local buckling having happened during the cyclic loading process have not yet studied thoroughly. In this paper, the authors report their work about this research. The results of tests on cold-formed specimens will be briefly described at first. Numerical model called inelastic multi-spring element is introduced then, and how to calibrate basic parameters by static loading tests is discussed. By using this model, the hystereis characters of the thin-walled cold-formed columns are investigated. Finally, the beam-to-column joint tests under cyclic loads are reported.

LOADING TESTS ON COMPRESSIVE RHS MEMBERS Test Outline All of the test specimens have the same section shape as shovm in Figure 1. Two cold-formed channels with lips are welded together to form a rectangular hollow section (RHS), with the outer size 300x380 mm. The curved comer radiuses of outer surfaces are 5.83, 3.5 or 2.5 times of the plate thickness of 3 mm, 5 mm and 7 mm, respectively. This section is decided according to the column prototype, which has good resistance against to torsional deformation. Another merit is less width-to-thickness ratio at the edge where two lips link each other.

-300-

Figure 1: Test specimen section

Two types of materials, low carbon steel and weatherproof steel, are adopted for test specimens. Table 1 lists main mechanical properties by tension coupon. Q235 is popular material while lOPCu-Re is newly used for building structures for its good corrosion protective natures and relatively higher strength. TABLE 1 MATERIAL PROPERTIES

Steel Type

Q235

lOPCu-Re

Nominal Thickness (mm) Yield Stress (MPa) Strength (MPa) Prolongation (%) 3 297 420 40 5 225 375 43 7 215 392 41 3 362 477 39 330 477 5 38 277 445 7 30

The loading condition is monotonically static one. Three loading cases, stub columns subjected to compressive loads, long columns to axially compressive loads, and to eccentrically compressive loads, are programmed. Table 2 classifies the specimens corresponding to these loading cases. In the specimen code, the first letter S refers to stub column, L to long column, respectively. The second letter C or E means the centrally or eccentrically compressive load. The third one, Q or R, identifies the

295 steel material Q235 or lOPCu-Re. The first Arabic number followed is the nominal thickness of specimen in millimeter. The listed section area and length between end plates are based on measured data. To the long columns, however, considering the pin-supporter at two ends, the curved length is about three meters. This is just equal to a normal height of a story. TABLE 2 SPECIMEN LIST

1 Specimen Area Length Specimen Area Length Specimen Area Length Specimen Area Length Code (mm^) (mm) Code (mm^) (mm) Code (mm2) (mm) Code (mm^) (mm) SC03-2 4382 1056 SCQ3-1 4390 1060 SCR3-2 4231 1052 SCR3-1 4239 1052 SC05-2 7269 1058 SCQ5-1 7098 1054 SCR5-2 6849 1052 SCR5-1 6831 1053 SC07-2 9333 1044 SCQ7-1 9286 1055 SCR7-2 9544 1052 SCR7-1 9575 1052 LCR3-2 4392 2750 LCR3-1 4561 2748 LCQ3-2 4140 2750 LCQ3-1 4149 2750 LCQ5-2 7149 2750 LCR5-2 6713 2752 LCR5-1 6815 2750 LC05-1 7081 2748 LCR7-2 9595 2749 LCR7-1 9628 2750 LCQ7-2 9428 2750 LCQ7-1 9200 2750 LE03-2 4207 2750 LER3-1 4385 2750 LEQ3-1 4359 2750 LEQ5-2 7009 2748 LEQ5-1 7016 2749 LER5-2 6672 2748 LER5-1 6941 2744 LER7-2 9632 2748 LER7-1 9848 2748 LEQ7-2 9400 2750 LEQ7-1 9462 2748 For stub columns, loading and boundary conditions follow CRC stub-column test procedure. In the case of long column specimen, two universal ball bearings are set up on both ends. For axial loading tests, geometrical loading center was carefully fixed. In the eccentrically loading tests, the eccentricity was 100 mm from the strong axes. Test Results Typical curves of loading versus deformation are shown in Figure 2. Figure 2a is record of stub column tests, where the horizontal axis is vertical compressive deformation. Figure 2b is eccentrically compressive loaded column and the horizontal axis is lateral displacement measured at the middle height of the specimen. The graphs of stub columns indicate three stages in the loading process: nearly elastic deformation, inelastic deformation but with loading increasing, and degradation. The origin of the second stage is perhaps due to residual stress, while local bucking occurs with the load increases. When local buckling was obviously observed, the curve reaches its peak value.

2500 r N ( k N ) _ _ T e s t ^ C Q 7 ^ / /

2000

7mm, Analysis"

~-

1500 1000 " // 500

^ ^ 3mm, Analysis

If

r 0 0 .0

TesTsCQpi . 0.2

. 0.4

. 0.6

. Dis.(cm) 0.8 1.0

Figure 2a: Stub column

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2b: Eccentrically loaded column

Table 3 lists the peak loads, Nm, and the relative value, Nm/Ny. Here, nominal yield force, Ny, is the product of measured gross section and the yield stress according to Table 1. The loading capacities of 3 mm thickness stubs are lower than Ny due to early local buckling, whereas to all of 7 mm thickness

296 stubs, the peak loads exceed their yield loads. In the cases of eccentrically loaded columns, the maximum axial loads range from 0.35 to 0.76. This relatively higher axial force should be the one of the reasons that the loading versus deformation curves as shown in Figure 2b degrade sharply after their ultimates. TABLE 3 MAXIMUM LOAD CAPACITY OF TESTED SPECIMENS

Specimen Code SC03-1 SC05-1 SC07-1 LCQ3-1 LCQ5-1 LC07-1 LE03-1 LEQ5-1 LEQ7-1

Nm (kN) 956 1764 2323 686 1510 1980 556 1065 1500

Nm /Ny 0.73 1.09 1.16 0.56 0.95 1.00 0.45 0.67 0.76

Specimen Code SCQ3-2 SCQ5-2 SC07-2 LC03-2 LCQ5-2 LCQ7-2 LE03-2 LEQ5-2 LEQ7-2

Nm (kN) 951 1767 2328 629 1550 2115 432 1085 1543

Nm /Ny 0.73 1.09 1.16 0.51 0.96 1.04 0.35 0.67 0.76

Specimen Code SCR3-1 SCR5-1 SCR7-1 LCR3-1 LCR5-1 LCR7-1 LER3-1 LER5-1 LER7-1

Nm (kN) 923 1970 2910 942 1835 2645 659 1300 1970

Nm /Ny 0.60 0.87 1.10 0.57 0.82 0.99 0.42 0.57 0.72

Specimen Code SCR3-2 SCR5-2 SCR7-2 LCR3-2 LCR5-2 LCR7-2

Nm (kN) 951 1932 2896 900 1815 2550

Nm /Ny 0.62 0.86 1.09 0.57 0.82 0.96

LER5-2 1280 0.58 LER7-2 1887 0.71

ANALYSIS MODEL Analysis Model and Calibration of Parameters Inelastic multi-spring element model for analysis of steel structural members, Ohi(1992), is used. A steel member is divided into elastic element and inelastic elements along its length. In the plastic zone of the member, cross section is substituted by several axial inelastic springs, shear and torsional springs. The advantages of the model have at least the following aspects proved by previous experimental and theoretical research. It can simulate the steel member behavior including the material and section features, such as yielding, hardening, Bauschinger effect, local buckling and so on. By supposing fivepiece linear skeleton curves and calibrating its parameters by monotonically static loading tests on stub columns, the complex hystereis behavior of a member can be simulated in good agreement with its physical prototype, Takanashi (1992), Chen (1996). However, the previous application was concentrated on the compact section members, and the local buckling was considered to happen in inelastic stage. To slender section with large width-to-thickness ratio, different features from the compact section should be reflected in the skeleton curves and parameters.

Flange

Web

Figure 3: Skeleton Curves of Inelastic Spring

Axial Spring

297 For the given rectangular section, two types of axial springs are supposed. One represents the flange, lips and curved comers, locating at or near to the flange; the other is used to replace the straight part of the webs. The skeleton curves are sketched in Figure 3. Nine parameters are necessary to stipulate the skeleton. Parameters, Ki, Kc2,Kc3 and KT2, KT3 are stiffnesses for different stages in both tensile and compressive sides, and PCY,PCU,PTY,PTU define the four turning points. Those are decided by formula (l)to(9). Ki=EAi/Li Kc2=min{ 0.5Ki, KiAie/Ai} Kc3= -0.05Ki, for spring at web location, or = 0.02Ki, for spring at or near flange location KT2=0.5KI KT3=0.02KI

PcY=min{ O.SfyAi, acrAi} Pcu=min{ fyAi, [fu-(fu-fy)(b/t-24)/(be/t-24)]Ai }, for b/t>24 = fuAi, for b/t

f

!-±18

[t=1.2

F

150

|^E,-2l0000N/bm>

:: -1

M

i^ULA

z~

:-:-

--\ *

": • ii

- _:[:

|-p

.71

Jt, (N/mm/tam)

Figure 2: Effect of the connection slip modulus on composite bending stiffness. From Figure 2 it can be seen that the composite bending stiffness (EI^^J could theoretically be up to 78% higher than that of the joist and board without any shear connection (EI^^. However, for a practical shear connection comprising self-drilling screws of 4.2 mm diameter at 150 mm centres, the effective stifftiess of the composite floor is only 38% greater than that of the non-composite section. Line 1 of Figure 2 was determined considering the gross cross section of the cold-formed joist. Line 2 shows the stiffness of the composite section allowing for reductions in the joist capacity due to local buckling (note that the reference £7^„ is still for the gross non-composite section). The convergence of lines 1 and 2 at high levels of connection stiffness is because an increased ability to transfer compressive forces from the top flange of the joist into the board reduces the problems associated with local buckling of the steel section. Non linear shear connection Small-scale pull-out tests have been carried out at the University of Oulu [Leskela (1997)] to determine the load-slip characteristics of self-drilling screws at a steel-plywood interface. A typical non-linear (NLE) response for a 4.2 mm self-drilling screw, a linear elastic (LE) representation of this response, and a schematic of a test specimen, are shown in Figure 3. Finite element modelling (FEM) was used to analyse a member assuming a non-linear load-slip relationship for the shear connectors as shown in Figure 3. A typical load-deflection plot from the FEM output is shown in Figure 4, alongside a comparable result based on linear elastic calculations. It can be concluded from the curves that up to a deflection of span/250, for the example considered, a linear elastic approach is acceptable for design. Design recommendation On the basis of the studies described above, the linear elastic design method described in Annex B of EC5 Design of Timber Structures [ENV 1995 (1993)] is proposed for determining the bending stiffness of steel-board floors. The composite bending stifftiess may be rewritten as a function of the bending stifftiess of each component about its own axis, and the degree of composite interaction (represented by a factor y j :

320

5. '500 i

i

.a

S

—•—LE -O—NLE (Diana)

^ Mkkpan dis|4accineiit (nun)

Figure 3: Connector load-slip characteristic.

^^com Ell EAi e^ y^

bending stifftiess of the composite section bending stiffness of component / about his own axis axial stiffness of component / distance between the centre of gravity of the total cross-section and that of component / composite interaction factor

r.= Up a k

¥\^x^ 4: Load-midspan displacement.

1? a E^Ai XE2A2 p- K E^A + E2A2

connector spacing stifftiess of the shear connection

Further testing and study will be undertaken to validate these design recommendations. Long-term properties and strength criteria will also be formulated in the final design reconmiendations.

INITIAL TEST SERIES In parallel with the theoretical work undertaken at TNO, an initial series of composite floor specimens was tested at British Steel's Welsh Laboratories [Grubb (1999)]. Full details of the test programme are given in Table 1. Parameters studied included: •

joist depth and profile - lipped C, S and others (timber joists were also tested for comparison)

•

steel thickness

•

support detail (masonry or steel) and degree of end fixity (rotational restraint at the joist ends)

•

type of flooring material

•

glueing of the joints in tongue and groove floor boards

•

degree of connection between the joists and boards

321

Testing Arrangement Bare steel joist stif&iesses were determined by strapping two sections back-to-back to produce an Isection that would not distort laterally under the applied loading. The joists were supported on Universal Beam (UB) supports at each end, with a 20 mm diameter bar welded to the top of each UB to ensure a true simple support, and a span of 4.2 m. A central point load was applied using a hydraulic pump and cylinder system, and mid-span deflection was measured. The stiffnesses {EI) of the non-composite joists, derived from measured deflections, were; Series A, 366.6 kNm^; Series B, 208.5 kNm'; Series C, 329.2 kNm'; Series D, 253.7 kNml

Steel edge support

Masonry (simple or built-in)

Plasterboard ceiling

Figure 5: Arrangement for composite floor tests The test arrangement for the composite floor "panels" was as shown schematically in Figure 5. Each panel comprised eight joists at 400 mm centres, with floor boarding attached to the topflangesand a plasterboard ceiling attached to the bottom flanges. End support was provided by masonry blockwork walls for tests A1 to CI (see Table 1), giving a clear span of 4.2 m between the wall faces. Openings in these walls enabled the joists to either sit on the masonry or, by the provision of packs and wedges, to be effectively encased in the wall. The sides of each panel were supported on cold-formed steel fabrications. For test series D, the joist ends were supported on the steel fabrications. Static loads were either uniformly distributed (UDL) over the floor area, line loads or point loads. Distributed loading was applied using a combination of 40 kg sand bags and 10 lb (4.5 kg) dead weights. Line loads were applied using the dead weights. Point loads were applied using a large dead weight normally used for testing cladding. The values of static loading considered were as follows: UDL of 0.5, 1.0 or L5 kN/m^ •

line load of 1.0 kN/m point load of 2.0 kN

Both the UDL and line load tests were used to determine the effective stiffness of each floor system from measured deflections. Deflections were measured using a Dumpy level, modified to give readings to an accuracy of 0.1 mm, at mid, quarter and third span points of each joist. The purpose of the tests with a point load applied centrally on the floor was to assess load sharing between the joists for the various systems.

322 TABLE 1 DETAILS OF TEST PROGRAMME

[xest Joist

Support

|AI

NBSJ^

Simple, on masonry

|A2

Floor

Shear connection

Ceiling

18mm T&G^ glued

Screws^ at 300mm centres

12.5mm PB'

NBSJ

Encastre, in masonry 18mm T&G glued

Screws at 300mm centres

12.5mm PB

|A3

NBSJ

Encastre, in masonry 18mm T&G glued

Screws at 150mm centres

12.5mm PB

|BI

150_L2E

Simple, on masonry

18mm T&G

Screws at 300mm centres

12.5mm PB 12.5mm PB

B2

150_1.2S

Simple, on masonry

18mm T&G glued

Screws at 300mm centres

|B3

150_1.2S

Encastre, in masonry 18mm T&G glued

Screws at 300mm centres

12.5mm PB

|B4

150_1.2S

Encastre, in masonry 18mm T&G glued

Screws at 150mm centres

12.5mm PB

|B5

150_1.2i:

Encastre, in masonry Knauf panelcrete^

Screws at 300mm centres

12.5mm PB

|B6

150_1.2S

Encastre, in masonry 18mm T&G glued

Screws at 150mm centres, glued to joists^

12.5mm PB

|B7

1501.22

Simple, on masonry

18mm T&G glued

Screws at 300mm centres

12.5mm PB resilient bar

|B8

150_1.2S

Simple, on masonry

18mm T&G, 30mm mineral wool, 2xl5mmPB

Screws at 300mm centres

2xl2.5mm PB

CI

200x50 Timber

Encastre, in masonry 18mm T&G

Screws at 300mm centres

12.5mm PB

Dl

150_1.6S

Simple, on studs

18mm T&G

Screws at 300mm centres

12.5mm PB

D2

150_1.6S

Simple, on studs

18mm T&G

Screws at 150mm centres

12.5mm PB

D3

1501.62

Simple, on studs

18mm T&G glued

Screws at 300mm centres

12.5mm PB

D4

150_1.6S

Simple, on studs

18mm T&G glued

Screws at 150mm centres

12.5mm PB

Notes: 1 2 3 4 5 6

|

NBSJ is a new joist being developed by British Steel T&G is tongue and groove chipboard Screws were TFC36 (or TFC38 for test B5) PB is plasterboard Knauf panelcrete is a 16 mm cement particle fibre board Board-to-joist glue was S4 mastic sealant

In addition to the static load tests, the response of the various floors to dynamic loading was also considered. Dynamic loads were appUed in combination with various levels of UDL by dropping a 3 kg sand bag from a height of 1 m using a tripod mounted magnetic release system. Measurements were taken using four accelerometers placed on the floor and connected to a data logger. Test Results The results presented in Table 2, and the discussion that follows, concentrate on the tests with UDL. Full results may be found in Reference [Grubb (1999)]. For each specimen, effective stiffness {EI^^ values were derived from measured deflections under 0.5,1.0 and 1.5 kN/m^, and averaged to minimise the effect of experimental errors. For the joists supported on or in masonry, a span of 4.3 m was used to derive the effective stiffness values, assuming the points of support to be 50 mm behind the wall faces. The following conclusions can be drawn from the results given in Table 2:

323

TABLE 2 TEST RESULTS Test

Al

A2

A3

Bl

B2

B3

B4

B5

B6

B7

B8

CI

Dl

D2

D3

D4

EU (kNm^)

369

482

503

206

227

258

251

266

290

252

305

295

286

294

360

373 1

End restraint A comparison between A2 and Al suggests a 31% improvement in effective stiffness for a composite floor system that is built-in rather than simply supported. A similar comparison between B3 and B2 suggests only 14%. However, consideration of the deflections measured under line loading [Grubb (1999)] revealed that between A2 and Al there was a 26% improvement, and between B3 and B2 there was a 30% improvement. It therefore seems likely that the B series UDL result given above is misleading, and that the beneficial effect of providing a practical level of endfixity(rotational restraint) is between 25 to 30%. Basic composite system With a basic floor 'slab' comprising dry jointed chipboard, fixed with screws at 300 mm centres, test Bl compared with bare steel test B suggests a -1% improvement in stiffness. Although clearly this result is influenced by experimental inaccuracies, it seems reasonable to conclude that any improvement is minimal. A similar comparison between Dl and D suggests a 13% improvement. It is possible that the greater improvement for the D series tests is because the increased steel flange thickness enabled a stiffer shear connection to be achieved. This will be investigated in future tests. Glueing the joints in the floor boards For a floor comprising chipboard with glued tongue and groove joints, fixed with screws at 300 mm centres, B2 compared with B shows a 9% stiffness improvement compared with the bare steel joist. This represents a 10% improvement above the B series composite system with dry joints. A similar comparison for the A series (Al compared with A) suggests only a 1 % improvement for the composite system with glued joints compared with the bare steel joist. One of the reasons why the improvement for Al appears small as a percentage is because of the high stiffness of the bare steel section. These trends were supported by the line load tests. Enhanced shear connection Various comparisons (A3 vs A2, B4 vs B3, D2 vs Dl, D4 vs D3, B6 vs B3) show that neither halving the spacing of the screws, nor glueing the board to the joists, increases the stiffness by more than approximately 5% compared with the basic shear connection of screws at 300 mm centres. The tests under line load indicate more significant gains when the shear connection is improved (up to 15% stiffness enhancement), suggesting a need to take into account the type of loading when assessing composite behaviour. This phenomenon needs further study. 'Improved' floor boarding A comparison between test B5 (dry butted panelcrete boarding), and B3 (chipboard with glued joints), shows only a 3% improvement in stiffness. However, given the practicalities of glueing tongue and groove board joints on site, perhaps the use of panelcrete would be justified.

324 CONCLUSIONS Although the test results are in broad agreement with the theory, suggesting that composite interaction could reasonably lead to a 20 to 30% improvement in stiffness, further testing and analysis is needed to comprehensively validate the proposed design rules. Firstly, it will be necessary to carry out a series of pull-out, or similar, tests to predict the load-slip behaviour of various forms of shear connection (varying screw type and spacing, board type, joisttiiicknessetc) to cover the test specimens discussed above. Further testing of floor systems should include consideration of longer spans, up to, say, 6 m, for which the benefits of composite interaction will be more important. It is also clear that whilst comparisons between the different composite floor arrangements tested so far seem reasonable, it is more difficult to benchmark these against experimental values for the bare steel joists. This is probably due to the different test arrangement for the non-composite specimens. Future testing will need to take this into consideration. ACKNOWLEDGEMENTS The work reported in this paper was undertaken as part of a much larger project entitled "Design tools and new applications of cold-formed steel in buildings", for which funding received from the ECSC is gratefully acknowledged. SCI received additional financial support from the UK Government Department of the Environment, Transport and the Regions. The experimental work reported in this paper was undertaken at British Steel Strip Products Welsh Technology Centre, who contributed both technically and financially. The assistance of John Grubb in the experimental programme is also gratefully acknowledged. REFERENCES ENV 1995-1-1 1993. Design of timber structures. General rules and rules for buildings. CEN, Brussels GRUBB P. J. (1999). Assessment of serviceability performance of light steel floors. The Steel Construction Institute, Ascot, UK. LESKELA M. V. (1997). Load slip properties for steel-wood connectors. Department of Civil Engineering, University of Oulu, Finland.

Poster Session P3 NEW STRUCTURAL PRODUCTS

This Page Intentionally Left Blank

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

327

DESIGN OF GLULAM ARCHED ROOF STRUCTURES WITH STEEL JOINTS Professor Karl Oiger Faculty of Civil Engineering, Tallinn Technical University Ehitajate tee 5, Tallinn 10986, Estonia

ABSTRACT In the paper the design and construction problems of renovation of an old theatre roof are presented. The theatre was designed by Finnish architects A. Lindgren and V. Lonn and erected at the beginning of this century (1910 ... 1913). At the end of the World War II it was bombed (1944) and partly burned dovm and then after 1.5 years of staying in ruins in 1945-50 it was restored (see Figure 1). Now after half a century, in 1995 owners of the theatre wanted to develop and to arrange new halls under the roof without changing the outside shape of the building. Consequently the roof trusses occupying the space under the roof had to be replaced by some other structural arrangement (see Figure 2). There were 3 under-roof rooms altogether with floor surface 1400 m^ where the original roof trusses were replaced with new system of glulam elements: arches, longitudinal glulam beams and concave rafters, which were covered by boarding (cladding) and crosswise roof-laths. In the present paper design and rebuilding problems are presented.

KEYWORDS Glulam structures, steel elements, analyse, design, fabrication, mounting.

1. DESCRIPTION OF THE NEW CARRYING STRUCTURE The carrying system and its shape was in many respects prescribed by the old roof and purposed the new halls. Two under-roof halls will be used for giving concerts and one by ballet dancers for training. Structural and constructional problems were caused by the original shape of the roofs characterized by combination of convex and concave surfaces (see Figure 1 and 2). The primary roof carrying structure was built during rebuilding after World War II.

328

Figure 1: General view of the theatre Estonia

Figure 2: Section of the primary roof structure and the idea of replacing of structure The previous structure was replaced by glulam arches (Figure 2), longitudinal beams (purlins), concave and straight rafters, boarding supported on the rafters and roof-laths for tiles (see scheme on Figure 3, plan of one hall Figure 4 and structure during mounting Figure 5). Main arches have the span of 13.8 (h= 7.75) m, 15.7 (h = 7.3) m and 17.7 (h = 8.25) m, depending on roof width. At the ends of rooms radial arches used to form double-curved roof surface. The cross-section of the glulam arches varies between hi = 750 - 950 mm, bi = 120 mm. The height of the cross-section and curvature of arches changes along the element. All arches everywhere consist of two glulam elements, connected by galvanized bolts. So the real cross-section width of the arch is 240 mm.

329

Figure 3: Scheme of the carrying structure of a part of roof

Figure 4: Plan of the carrying structure of a hall

Figure 5: Carrying system by mounting

330

Spacing of the arches and respectively span of the purUns depended on the possibilities to support arches on the old brick wall and varied between 3410...8000 mm Cross-section of the purlins are h, = 700...800 mm and b, = 140...160 mm. Curved and straight rafters have cross-section of hi x bi = 300 x 100 mm and are placed with spacing of 1 m (Figure 6)

Figure 6: Curved (concave) rafters and main arch

Figure 7: Final roof section

Cross-section of all arches and purlins were determined not only by statical calculations but also by necessity to achieve the needed fire resistance (60 minutes). Rafters are covered with boarding upon what are nailed vertical and horizontal roof-laths to fix underroof ojver and roof tiles. Gyproc plates and vapour seal are placed under rafters, thermal insulation is placed between rafters (see roof section in Figure 7), Steel joints have to cany relatively heavy loads transferred from one glulam element or structure to another. Also the horizontal reaction from rafters resuhing from vertical loading has to be transferred to the walls and to the purlins. Usual joints are so called steel-to-timber dowelled joints, where steel plates are placed into grooves cut m timber elements and connected by dowel-type fasteners. Steel plates and dowels are covered

331 with zinc. Some steel joints elements are shown in Figure 8... 10. Steel plates were made from Fe 360, because usually in normal joints special strength problems by normal joint did not arise.

Figure 8: Welded steel joint element (purlin bracket) for supporting purlins on arches

Figure 9: Welded steel joint element for connecting three purlins in one joint .„*.

1

si

*,..

• • • * '

50j

^

200

J_

--T

ot

^1

L >

Figure 10: Steel joint element for connecting braces between arches The original ceilings of the theatre were not designed to carry heavy hall loads (5 kN/m x 1.5 + new bigger self weight). So we had to design a new floor-carrying structure from steel beams, timber floor beams, and boarding. We had also to solve acoustics problems, because under-roof hall rooms lie on existing theatre or concert halls, and there is need to use both rooms at the same time. Steel beams of

332

the floor (Figure 11) served also as tie-rods for the glulam arches, as arches were supported directly on the ends of steel beams. For steel beams profiles HE A 600 and HE 400 were used.

^^^ Figure 11: Steel beams of the new floor Roof, floor and other structures were designed by EKK Ltd. under guidance of the author of this paper.

2. SOME RESULTS OF THE ANALYSIS AND THE MAIN DESIGN PROBLEMS Analysis of the structure was carried out by FEM. Some problems arose during selecting the structural scheme, because it depended on the shape and measures of the old roof We found out that the best option is to use the system shown in Figure 3...6. At the same time lawer too long part of the arches left without any bracing. Also the theatre wanted the space between arches to be left free. Another unfavourable circumstance is that all load comes to the top area of the arches. The third problem is that concave rafters cause horizontal force on brick wall and purlins and its value depends on the stiffness of the wall. At the same time strength of the old wall is extremely low. Some moment curves and displacements for the main arch with concave rafters and brick wall are shown in Figure 12.

Figure 12: Moment curves and displacements of the main arch and rafter for vertical over the span uniformly distributed load (a) and wind load from the left and snow load on the left half of the span (b)

333

In every case radial forehead arches cause compression forces (about 200 kN) that we have to transfer along first short purlins to the bracing system between arches in longitudinal direction of the hall. Whole system, that consists of main arches, radial arches at one or both ends of the halls, purlins and rafters was also calculated as a spatial system. In case of taking into account the continuous boarding upon rafters, carrying system of the whole roof will act as a spatial ribbed shell system and actual inner forces and displacements of main carrying elements will be noticeably lower than the calculated ones. The main design problems were: Timber structures Fitting the structure to the old original outside shape Steeljoints and elements There were no special problems, because it is rare occasion, when in this case steel joint elements determine joint dimensions, except in case of fire resistance design. Other steel structures Steel profiles HEA 400.. .600 were used for new floor beams and as tie-rods (except one case). In the tight conditions of mounting works we had to assemble the steel tie-rods and steel floor beams on site, putting them inside the old roof trough first slits in the roof Fire resistance of the whole system was a special problem, as 60 min fire resistance was required. For timber structures it was achieved by adequate thickness of glulam elements and for steel joint elements, which were placed into grooves sawn in glulam arches and beams, with corresponding thickness of the covering timber layer. Other structures The original walls had complicated internal structure with cavities, some of them unexpectedly appearing during construction works. Old brick walls have relatively low strength. We dared to use only 0.5 MPa under reinforced concrete support rafts, that determined measures of these rafts. The second big problem was caused by alternating measures of the old brick walls, about 300 mm in plane.

3. FABMCATION AND MOUNTING PROBLEMS Fabrication Glulam structures were prefabricated by an Estonian company "Polva Glulam". Resorcinolformaldehyde glue was used. It was the first experience for the company to produce curved glulam elements of changing curvature and at the same time with changing height of cross-section. Some difficulties were caused by cutting deep grooves for the connections at the ends of elements. Glulam strength class was GL 24. Moisture content in the factory about 10 %.

334 Mounting Protection of the interior of the theatre from rainwater during replacing step by step the old carrying system by the new one caused serious problems as the roof construction works were carried out in various weather conditions. Special measures had to be taken to stabilize the original structure during construction work. Additional difficulties caused very tight working conditions. The period from beginning of the design to the end of mounting works lasted only 5.5 months. Building works were possible only during vacation of the theatre staff

Figure 13: Interior one of the halls

CONCLUSIONS The described roof-carrying system was the first of its kind in Estonia and the experience obtained could be used in other similar structures. The final result is shown in Figure 13. Timber as a lowenergy and ecological material should be used much widely in various buildings, especially in such a country as Estonia, where about 45 % of the surface is covered with woods. We can have good structural solutions by rational using of lightweight complex timber and steel structures.

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

335

FIXED COLUMN BASES IN ASTRON STRUCTURES Andrej

Belica

Dr.Ing., Astron Building Systems, Commercial Intertech S.A. P.O. Box 152, L-9202 Diekirch, Luxembourg

ABSTRACT Astron is one of the leading European producers of the light-weight steel buildings. Basic component of it's modem structural system is the welded, built-up frame, normally having pinned bases. Fixed bases are used in about ten percent of the Astron frames. They belong to the most expensive frame elements due to the big portion of hand work needed for their production. Presented paper shows the running results of the design attitude planned to be used for the standard fixed bases. Annexes J and L of the Eurocode 3, Part LI were the ground for an analysis. Special attention has been focused on eccentric bases used for the buildings with limited structural space.

KEYWORDS Astron, steel structures, building, frame, base, fixed, clamped. Annex J, Annex L, EC3

1. INTRODUCTION Fixed bases in Astron structures have various forms that suit the best to the particular building exploitation - "tailor made solutions". Nevertheless in the development process some of the forms became more accepted due to their manufacturing convenience. Production costs influence a designer to use these more economical - "standard solutions" anytime it is possible. Columns are supposed to have parallel flanges with the same width but their thickness can be different. Figure 1 shows the anchor bolt distribution on the external flange side. Identical bolt distribution is available on the internal flange side. That provides 25 different fixed base shapes called standard with Astron. Base plate width, equal to flange width, is taken into account. All base stiffeners have the same dimensions. An anchor bolt diameter can be 18,24 or 30 mm.

336 Case 1:

Case 2:

V

Case 3:

. Case 4:

V

V

"1

i

1 • 1 •

• •

',

\

T"

V

LI

4T

\

• •

•

-

•

i

I

W 1* *

Case 5:

V

:i A1 ..1 11

11 • •

•

• •

•

• •

\

Figure 1: Anchor bolts distribution - external flange side 2. THEORETICAL BACKGROUND Design principles given in Eurocode 3, Part 1.1 were taken as the base for investigation. Though Eurocode 3 does not offer the complex solution for the fixed base design, rules given in Annex J [1] valid for the tension part of a beam-to-column joint and the ones given in Annex L [2] for a compression zone can meet in one attitude. The equilibrium of internal forces (see Figure 2) is given by: _ (1)

NRd=A,ff-fj-^FtiRd

(2)

Annex J (tension)

external side

\

Annex L (compression)

MRd

i

i

\< .

NRd

column Co

Fti,Rd Ft2,Rd t2,Rd F, rt3jRd

'-^ h

internal side

mtttttttttttttttttttttttti

> .

J CgOfAeff

Figure 2: Equilibrium of internal forces Applied design procedure: • resistance of the tension part Vp^iRd and the concrete bearing strength fj are determined from the known geometry of the steel base and the concrete block • active effective area A^ff for the given normal force N^ is calculated from the equation (1) • center of gravity r,, of the active effective area A^Q is established • moment resistance MR^ of the base loaded by normal force N^ is calculated from the equation (2)

337 2.1 Tension Part of the Base The model of an equivalent T-stub is used to determine the tension resistance of the base. The design tension resistance F^i^Rd was taken as the smallest one from the three base plate failure modes.

(3)

/m

complete yielding of the base plate:

F H , - 4 - M pl,l,Rd

anchor bolt feilure with yielding of the base plate:

F^j = (2Mpi2,Rd+n-2Bt^Rd)/(m+n)

anchor bolt failure:

v^^, = 19- B^^^

and the failure of column web in tension: where:

Mp,,i,Rd = 0.25. Y}.m

• tp • fy / Ym

(4) (5) (6)

pM^beff-t^-fy/rMo

(7)

^^,2M = 0-25 'Y,hs,2 ' tp • fy / ^MO

(8) n = e^in but n < 1.25 • m

(9)

Bt^Rd - tension resistance of anchor bolt; fy - yield stress; ;'MO - "i^terial safety factor The effective length of an equivalent T-stub determined for the particular yield pattern, relating to the 5 cases of bolt distribution (see Figure 1) are summarized in Table 1. Corresponding geometrical dimensions are indicated in Figure 3. The parameter a for stiffened base plates can be taken from the [1] Figure J.27 (J.P.Jaspart favoured us with the computer routine for the purpose of the present investigation). TABLE! EFFECTIVE LENGTH OF AN EQUIVALENT T-STUB Bolt - row considered individually; Bolt row I 1 (case 2,3,5) circular li = 27niist pattern I2 = Tcnist + 2est I I3 = 7an2(n + 2ex noncircular I4 = aim pattern I I5 = a2m2(i) l6 = Cx + aiHist - (2mst + 0,625est)

T" 2

ai = f(A.i, X2) \ ot2 = f(A,i, ^2) A.1 = nist / (nist+est) \ h = m2(i) / (m2(i)+ex) A,2 = m2(n / (nist+est) j A.2 = nist / (m2(i)+ex) niin(li, I2,13,14, Is, h, h) min(l4,15, le, I7)

(case 1,3,4,5) Ig — 27rm

3 (case 4,5) lio — 27im

111 =can

I a = f(Xi, X2) I A.i = m / (m+e) j X2 = m2(2) / (m+e e) I I min(l8,19) I I9

a = f(>.i, ^2) A.1 = m / (m+e) X2 = m2(3) / (m+e) A,2=n: min(li

Bolt - row considered as a part of a group of bolt - rows: 2 + 3 (case 4,5) 1 Bolt row 1 circular li4 — 7im + p I12 = Tim + p pattem 1 noncircular li3 = 0,5p + a m - (2m + 0,625e) li5 = 0,5p + a m - (2m + 0,625e) pattem a = f(A,i, X2) a = f(A-i, X2) A 0,4. Anytime it is possible, bases with symmetrical anchor bolts distribution are advanced. An example where the usage of the asymmetrical base is suitable will be shown in next. Wall girts - cold formed Z profiles - can be in two positions, external or internal. Its choice depends on the building exploitation and on the customer's wish. In the case of internal girts is the distance between the outside column web edge and the inside wall-panel edge 25mm (see Figure 7b). Such solution eliminates the anchor bolts outside the external column side. Considered frame geometry and loading are shown in Figure 7a. Internal forces Md(i) and Nd(i) for five standard load combinations and interaction charts for the three bolt configurations are shown in Figure 7b. It can be seen that the moment resistance of sjmimetrical base I is not sufficient. Symmetrical base II covers only combinations 1, 2, 4 and 5. Application of the asymmetrical base III (symmetrical solutions were given already out) has allowed increase of the moment resistance in that part of the interaction chart, in which the uncovered combination 3 occurred. 10%

Loading: 8m

m»

16 m

^

frame weight; roof dead load: additional dead load: snow load: wind load:

Figure 7a: Geometry of the frame and the loading

0,12 0,15 0,75 0,50

kNW kNW kNW kNW

342

• iext V

intl

i 1 25 mm

\

JH^^

; 1

1 T

i I II •

•1

'II *

* 1

! n

iu^ •IM

• • • • 1

in

II-•

- i ••

web: 600.5 external flange: 180.6 internal flange: 180.7 bolts: M20 base plate thick.: 16mm Figure 7b: Mpe S fame sm^ctore effected the mam frames calculations and led to reduction of steel consumpSon S frames based on the assumption that tenninal structures pro^dde sufficient bracing. As the resuh goo^ load redistribution in cross section for short industrial buildings was achieved AU mam elements were produced from low-alloy steel - equivalent to 18G2A steel An example of a frame structure under construction with an 18 m. span is shown in figure 5.

Figure 5.

347

Polish and Finnish system diifer basicall>^ in the constmction of all joints in cross section of the building. Figure 6 shows comer joints in MET.\LPL AST-PR ACT A (6a) and in PRACTA system (6b). b) SYSTEM PRACTA

a) SYSTEM METALPLAST-PRACTA

Figure 6.

BASIC CALCULATION ASSLIMPTIONS On the basis of e\idence presented in section [3], it was found that the joint flexibility in PRACTA system increases rapidly for the moment close to the maximum calculation moment. The results of tested M-f relation shows a diagram in figure 7 (curx^e 1).

VERSION I 21350/3 II 2E350/2,5

A Ix A Ix

CROSS - SECTION | 2 3 1 33,30 55.80 45,30 24000,0 6650,0 5400,0 28,0 50,50 45,30 24000,0 5800,0 4570,0

Figure. 7

VERSION I 2E350/3 2E350/3 II 21350/2,5 21350/2,5

A Ix ly A Ix

Ll^

3 33,3 5400 5400 28,0 4570,0 4570,0

CROSS - SECTION A B 17,4 10,0 100000,0 3330,0 6200,0 0,01 17,4 10,0 100000,0 3330,0 6200,0 0,01

c 13,6 574,0 28,6 13,6 574,0 28,6

348 As there was no conesponding data for MET.ALPLAST-PRACTA system and all available calculations were made for cross section structures with rigid joints only, new t>pe of joints with calculation results close to rigid joints had been developed. The real M-f characteristic curve of steel joints was proved to differ significantly from the calculation results. For details see reports from a scientific conference [8] and numerous articles, especially papers published by Chen group [2]. The real M-f characteristic curve of joints developed for IV^IETALPLAST-PRACTA system is a subject of experimental and numerical testing. It seems that calculations made for frame systems with rigid comer joints combined with foundation calculations result in an underestimation of cross section displacement.

Figure 8.

Figure 9.

349 To obtain an approximate value of cross section displacement it was necessary to determine joint rotation of two simple calculation models of joints with characteristic close to that of comer joints in frames with a 14 and 18 m span. The M-f relation cur\/es of the two calculation models are shown in figure 7. Figure 8 and 9 present pictures of actual joint solutions constructed for the tests. First results of the tests will be presented at the scientific conference. All calculations of cross section structures were made with regard to remarks discussed in sections 4, 5, 6 and 7 and apart from the M-f relation outlined above were based on the following assumptions: - a test bar called a module was 2,0 m long and was anchored to the column as well as the beam in an analogical way in both systems. This was possible thanks to the use of structures with spandrel beams, purlins as well as columns and spandrel beams in gable walls and diaphragm placed within the cross section frame as well as cut-off walls at the point where sheeting elements are joined with cross section. - calculation value of cross section was obtained by the reduction of the wall thickness of profiles by 0,15 mm. - no allowances were made for waiping of cross section in ends joints of modules. Stability of the system was ensured by cross and vertical bar bracing of the longitudinal structure of the system and use of self-tapping screws for fastening the roof made from METALPLAST sandwich panels and horizontal panels (sheeting).

CONCLUSIONS Structural solutions applied in JS^IETALPL AST-PR ACT A system allowed to achieve low-steel consumption with labour consumption comparable to other light-weight frame structures. Table 1 presents material consumption for example frame structures with a 12, 16 and 18 m span and the basic building length of 44,0 m. TABLE 1 STEEL CONSLIIv/IPTION FOR SELECTED TYPES OF STEEL FRMIE Steel consumption for selected t>pes of steel frames Type of frame Consumption Cross section Sheeting Bracing Total consumption indicator unit kg/m2 14,80 12,94 28,12 0,38 18,0x6,5x44,0 4,33 kg/'m3> 2,28 1,99 0,06 52,60 46,00 100% 1,40 % 12,04 kg/'m2 12,57 25,03 0,42 16,0x6,5x44,0 kg/m3 2,19 2,29 4,56 0,08 48,10 100% 50,20 1,70 % The results presented in table 1 show a relatively low steel consumption per 1 m2 of the \dew of the building. It is lower than in PRACTA and HARD system. Also the proportion of the total steel consumption used for sheeting is different than in traditional frame structures. In light-weight steel frame systems, there is always liiglier steel consumption for sheeting due to reduction of steel consumption for the main cross section. This relation can also be observed in METALPLAST-PRACTA system. In light-weight steel frame systems it is necessary to evaluate the actual joint flexibility to obtain more accurate estimation of displacement and internal forces. In an attempt to cope with this problem authors of this paper carried out tests in which they analysed both comer and alloy joints with structural solutions

350

identical with those developed for MET.ALPLAST-PRACTA system. Evaluation of theflexibilit}^of that type of joints can be veiy helpful in designing thin wall frame structures.

Bibliography: [1]

Brodka J., Ko^owski A.: Sztywnosc i nosnosc w?^6w podatnych, Oficyna wydawnicza Politechniki Rzeszowskiej, Bialystok-Rzeszow, 1996.

[2]

Chen W.F., Kishi N.: Semirigid Steel Beam-to-Column Connections. Date Base and Modelling. Journal of Structural Engineering ASCE, Vol. 115, Nol, Jan, 1989.

[3]

Tarmo Mononen: Practa-leight construction system. 5th International Conference: Modem Building materials structures and techniques - Wilno

[4]

DIN 18800/2 Stahlbauten. Stabilitatsfalle, Kniken von Staben und Stabwerken. November 1990.

[5]

CEN/TC 250, Eurokode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, ENV 1993-1-1.

[6]

EC3-88C4-D3, Eurocode 3, annex A - cold formed tliin - gauge members and sheeting, January-1989

[7]

ECCS - Application of Eurocode 3, Examples to Eurocode 3, 1993, No 71.

[8]

W^Ay podatne w konstrukcjach stalowych. n Konferencja Naukowa, Rzeszovv, 1998r.

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

3 51

DESIGN CONSIDERATIONS FOR LIGHT GAUGE STEEL PROFILES IN BUILDING CONSTRUCTION Haluk Coskun Department of Civil Engineering Design, Eregli Iron and Steel Works Co. 67330 Kdz.Eregli - Turkey

ABSTRACT As the issues like protection of environment, minimisation of energy consumption, sources of material supply, cost minimisation and material strength become the main concerns of today's construction world; steel emerges as an highly qualified material in all aspects. It also gains an additional importance in the countries facing unescaple earthquakes with various damaging effects. Depending upon the reasons above, light gauge steel profiles are increasingly being used in residential buildings as an alternative to wood. These sections proved very well in supplying the structural requirements of building construction. On the other hand, hot rolled steel structures are very common in industrial and commercial sectors in most of the countries. That is why it is aimed to underline the advantages of using light gauge steel members in the complimentary parts of these buildings. A comparison is made between the light gauge steel and hot rolled sections for purlin and girt design and supplied economy is shown in terms of the amount of steel used. Since the material is relatively new for the design community, some of the design concepts are also considered. Regarding the scientific and technological advancement on light gauge steel industry, necessary development steps, required investments and institutions are specified for related countries.

KEYWORDS earthquake, light weight members, steel, purlin, girt, design principles, safety, economy, development

IMPORTANCE OF LIGHT WEIGHT STEEL IN THE EARTHQUAKES Natural disasters are threatening human life by causing structural failures of buildings for many years. Earthquake is very important one of this kind of disasters considering its damages. Turkey is an earthquake country taking place in a highly effected seismic zone. % 95 of population and % 91 of total land area are endangered by this natural phenomena. It is also worth to note that, % 98 of industrial buildings are in the earthquake areas in Turkey. On the other hand, it is well known that human losts are caused by collapsing heavy floors and heavy walls after the earthquakes. This is not the case for a properly designed steel structure. It is generally observed that, only the brick walls come down while the rest of the steel buildings are in good condition. Alternative materials must be considered in place of concrete under the realities of earthquakes in the past years causing heavy burdens for the societies. So, there are enough reasons to consider light gauge steel for building construction due to its flexibility, high strength and light weight properties.

352 LIGHT GAUGE STEEL PROFILES IN PURLIN AND GIRT DESIGN One of the favorable advantages of light gauge steel is the possibility of producing wide range of shapes with different sectional sizes. Since the sections are shaped by cold forming of steel sheets, they are produced more easily comparing with hot rolled profiles. Because of the various shapes and sizes, profile selection is satisfying in regarding the economical use of sections. Designers can easily choose suitable and economical sections matching the required sectional strength. A comparative study for purlin and girt design is given below to explain the relative advantages of light gauge steel members against the hot rolled sections as the material weight and supplied economy are concerned. As they are given in the table 1 and table 2, 1/10 roof slope, 5 and 6 meters double span purlin and girt systems are chosen for comparison. 35 kg / m2 dead load, 75 kg / m2 snow load and 60 kg /m2 wind suction and 60 kg/m2 side wall wind force are considered.(loads are as defined in related Turkish Design Code, TS 498 ) Hot rolled sections are taken as "U" shaped profiles with usual sizes and statical values are calculated accordingly. On the other hand, cold formed profiles are chosen as "C" and "Z"shaped sections with their related sectional data. The profile details are all given in the tables 3, 4 and 5. After the profile selection, the amounts of steel materials used for the unit areas are given in the table 6 and 7. It must be also noted that, the steel grades with higher yield strengths are prefered with the increasing thickness of light gauge sections. The yield strength for hot rolled sections is 2400 kg / cm2 where as it is 3570 kg / cm2 for light gauge steel in this analysis. But the additional cost due to strength increase can be negligible comparing with material weights. Consequently, as it is seen on the tables, it is quite obvious that there is a great economy in using light gauge steel members when they are used in complimentary parts of industrial and commercial buildings constructed with heavy load bearing hot rolled shapes.

TABLE 1 ECONOMICAL PROFILE SELECTION IN PURLIN DESIGN Roof Cover : Trapezoidal Steel Sheet Slope : 1/10 Span 5 meters - 2 Span Continuous Beams 6 meters- 2 Span Continuous Beam Hot Rolled Section Light Gauge Section Hot Rolled Section Light Gauge Section Spacing Profile Profile Weight Profile Weight Profile Weight Weight 10,6 202.Z.15 1,2 m. 172.Z.14 3,60 USP 120 13,3 4,21 USP 100 10,6 202.Z.18 1,4 m. 172.Z.15 3,85 USP 120 13,3 USP 100 5,03 10,6 232.Z.16 1,6 m. 202.Z.15 4,21 USP 120 13,3 USP 100 5,11 232.Z.16 13,3 4,21 USP 140 16 1,8 m. USP 120 202.Z.15 5,11

TABLE 2 ECONOMICAL PROFILE SELECTION IN GIRT DESIGN Side Cover : Corrugated Steel Sheet Span 5 meters - 2 Span Continuous Beams 6 meters 2 Span Continuous Beam Hot Rolled Section Light Gauge Section Hot Rolled Section Light Gauge Section Spacing Profile Weight Profile Weight Weight Profile Weight Profile 172.C.14 10,6 142.C.14 3,16 USP 120 13,3 3,60 1,4 m. USP 100 172.C.14 16,0 3,60 13,3 142.C.14 3,16 USP 140 1,6 m. USP 120 172.C.14 13,3 142.C.14 3,16 USP 140 16,0 3,60 1,8 m. USP 120 172.C.16 13,3 4,11 142.C.15 3,38 USP 140 16,0 2.0 m. USP 120 N.B.: 1- All the weight values are in kg./m. 2- The purlins and girts are assumed to be restrained at each third point of the span with the sag bars in the weak axis. 3- "USP" stands for "U Shaped Profile"

353

TABLE 3 i Sectional Reference il42.C.14 142.C.15 172.C.14 172.C.16

SECTIONAL PROPERTIES Nominal Dimensions Section Properties Weight Area Depth Flange t Ixx lyy Zxx Zxc Ryy Cy mm kg/m cm2 mm mm cm4 cm4 cm3 cm3 cm cm 3,16 4,03 142 64 1,4 133.3 22 18,77 17,31 2,33 1,95 3,38 4,31 142 64 1,5 142,40 23,5 20,05 18,93 2,32 1,95 3,60 4,59 172 69 1,4 218,70 28,80 28,43 22,90 2,49 1,99 4,11 5,24 172 69 1,6 248,70 32,60 28,92 27,31 2,48 1,99

Po 1 N/mm2 321,1 325,1 304,20 315,60

Q 0,65 0,67 0,56 0,63

TABLE 4 SECTIONAL PROPERTIES Section Properties Nominal Dimensions t Section Weight Area Depth Top Bottom Ixx lyy Zxx Ryy Cy Reference kg/m cm2 mm Flange Flange mm cm4 cm4 cm3 cm cm 1,4 213,4 41,5 24,57 2,99 6,01 60 65 1 172.Z.14 3,60 4,59 172 1,5 228,1 44,2 26,25 2,98 6,01 60 65 172.Z.15 3,85 4,91 172 1,5 332,2 44,2 32,59 2,86 6,00 60 65 |202.Z.15 4,21 5,35 202 1,8 395,8 52,2 38,83 2,84 5,99 60 65 202.Z.18 5,03 6,41 202 69 76 232.Z.16 5,11 6,50 232 1,6 532,7 69,0 45,34 3,24 6,94

Cx cm 8,69 8,69 10,19 10,19 11,75

Po N/mm2 304,2 310,2 295,4 312,1 287,7

TABLE 5 Sectional Reference

luspioo USP 120 USP140

SECTIONAL PROPERTIES Dimensional Properties t = rl r2 Weight Area D B W Cy cm cm cm kg/m cm2 cm cm cm 5 0,6 0,85 0,45 1,55 10,6 13,5 10 5,5 0,7 0,9 0,45 1,60 13,3 17,0 12 6 0,7 1 0,5 1,75 16,0 20,4 14

A cm 1,8 1,9 2,15

Ixx cm4 206 364 605

Section Properties Zxx ixx lyy Zyy iyy cm3 cm cm4 cm3 cm 41,2 3,91 29,3 8,49 1,47 60,7 4,62 43,2 11,1 1,59 86,4 5,45 62,7 14,8 1,75

B/2 ,y'

*

=^^

I--

ffi

ff HE

x'L-

m SECTION

142 172

aj

A L 43 13 43 14

U SECTION A B

E F 142-262 21 19 42 44

354

TABLE 6 THE SUPPLIED ECONOMY IN THE PURLIN DESIGN Roof Cover : Trapezoidal Sheet 6 meters - 2 Span Continuous Beams Building Span: 15 m. Hot Rolled Sec. Light Gauge Sec. Num.of Weight for Unit Weight for Unit Building Area Spacing Purlin Building Area kg/m2 kg/m2 4,49 16 1,2 m. 14,19 12,41 14 4,70 1,4 m. 12 10,64 4,09 1,6 m. 3,41 10,67 1 1,8 m. 10

Saved Steel kg/m2 9,70 7,71 6,55 7,26

Roof Slope: 1/10

Building Span: 20 m. Hot Rolled Sec. Light Gauge Sec. Num.of Weight for Unit Weight for Unit Purlin Building Area Building Area kg/m2 kg/m2 20 13,30 4,21 18 11,97 4,53 10,64 16 4,09 3,58 14 11,20

| Saved Steel kg/m2| 9,09 7,44 6,55

7,62 1

TABLE 7 THE SUPLLIED ECONOMY IN THE GIRT DESIGN Side Cover : Corrugated Sheet 6 meters - 2 Span Continuous Beams Building Height :10 m. Building Height 8 m. Hot Rolled Sec. Light Gauge Sec. Hot Rolled Sec. Light Gauge Sec. Num.of Weight for Unit Weight for Unit Saved Num.of Weight for Unit Weight for Unit Saved Steel Steel Girts Wall Area Wall Area Spacing Girts Wall Area Wall Area kg/m2 kg/m2 kg/m2 kg/m2 kg/m2 kg/m2 11,64 8,49 7 3,15 7,76 10,64 2,88 1,4 m. 8 9,30 2,70 6 12 8,68 7 11,20 2,52 1,6 m. 2,70 9,30 8,68 6 7 2,52 12 11,20 1,8 m. 2,57 7,43 10 5 2,47 7,13 6 9,60 2,0 m

1- Purlin Weight / Unit Area = Number of Purlins in Building Span X Unit Weight of Purlins /Building Span 2- Girt Weight / Unit Area = Number of Girts in Building Height X Unit Weight of Girts / Building Height DEVELOPMENT STEPS REQUIRED FOR TECHNICAL ADVANCEMENT OF LIGHT GAUGE STEEL IN THE RELATED COUNTRIES In order to maintain technical development for the use of light gauge steel in the construction industry, some of the required steps can be considered as below for the developing countries: 7- Increase of Steel Production : First of all, as a preferable and alternative material in the building industry, production of steel members must be increased in order to supply an increasing demand in the near future. It became more clear that the structural steel will take a high ranking place for the building industry in the next century. Statistical figures are supporting this comment in the futuristic projections. It is expected to have an increase in steel demand as % 2,9 globaly and % 5,1 in the developing countries in afew years ahead. It is also predicted that, this demand will go up to % 8 in the countries with % 5-6 growing economies. [1] These figures are implying a potential development for the light gauge steel sector as the supplied economy and the other advantages are considered. As a result, steel producers must supply a potential increase in steel demand by organising their production capacities accordingly. 2- Formation and Cooperation of Related Organisations : Technical and scientific research is very important for the improvement and new innovations of light gauge steel concept. Because of the thin walled nature of the material, light gauge steel members need to be tested to determine the required stability criterias. Today's development level is owed to these kind of test studies together with numeric analysis. It must be stated that, a great portion of this research was realised by scientific community with valuable sponsorship of steel producers or their institutional organisations. Contributions of universities, intitutions on steel structures, steel producers and related govermental organisations are highly appriciated in Europe and North America. Similar research organisations must be founded and cooperate in the same way in the other countries as well.

355

Some steel production firms have their own research and development departments. These departments must take the subject into their agenda. 3- Promotional Works: Steel production firms will have lots of benefits in producing and promoting light gauge steel profiles as the market potential and supplied economy are concerned. But as in the case of other building products, some promotional works must be supplied since the material is relatively new for building construction in some countries. Especially, the profiles must be introduced to the construction community with their load bearing properties, economical use and the other structural advantages.

DESIGN CONSIDERATIONS FOR LIGHT GAUGE STEEL MEMBERS Although it is very advantageous to use light gauge steel members for different parts of building constructions, a special design approach is needed for the thinness of the material used unlike the case in hot rolled sections. Some of the design arguments considering the ongoing developments can be summarised as follows: 1 - These sections lack torsional rigidity and prone to twisting during the handling and erection. In the case of channels, the shear center is located some distance away from the back of the web and even under the loading normal to the flanges, there is a tendency for twisting. Consequently, it is essential for the members to be designed with adequate lateral and torsional strength. 2 - Bending, shear, combined bending and shear, web crippling and combined bending and web crippling strength of the sections must be checked in sequence while making flexural design. Torsional and torsional-flexural buckling are important criterias to be considered in compression members. It must be also stated that the determination of elastic buckling stress is very important for the design. It can be very effective to use a numerical method to determine these stresses. [2] 3 - Bending capacity of beams supporting a standing seam roof system under gravity loads like purlins is greater than the bending strength of an unbraced member or may be equal to the bending strength of a fully braced member. It is proposed to determine a bending strength reduction factor "R" for this purpose. [4] But it can be practical for design purpose to assume that properly attached roof covering material provides continuous lateral and torsional restraint to the top flanges of purlins. 4 - Bending strength of "C" and "Z" section beams having the tension flange attached to deck or sheathing with compression flange unbraced (like purlins or girts subjected to wind suction), can also be calculated in design using a reduction factor "R" under special conditions as specified by some sources. [4] 5 - It is up to designer whether to consider the strength increasefi-omcold work of forming the sections for more economical design or omit it as an additional factor of safety. 6 - Light gauge steel member design is difficult with hand calculations comparing with hot rolled sections. In fact, some additional structural theory is also introduced for its design due to the thinness of the material used. These are some of the reasons which may contitute a hesitant approach by the design community. But, as a result of increasing number of material production and sofl:ware firms, engineering design is quite simplified. Most of the producers are supplying their technical documents and design information about their products. Therefore it is possible to use load versus section tables or easy to use software packages. 7 - As it is well known, thermal conductivity of steel is much higher than the other building materials.

356 That is why, cold bridging effects of exterior wall studs must be considered in commertial building design. This effect occurs, when the heat is collected on the inside flanges of the wall stud and transferred to the outside of wall through the metal web. In order to have a good heat insulation, a sequence of principles can be stated as follows: a - An exterior insulation like foam panels must be installed under the exterior finish. b - Wide insulation batts can be used to fill the cavity of studs. c - Studs produced with special thermal considerations for plumbing holes not to cause any heat lost must be used d - A special type of insulation material can be used that can be sprayed in, fills all the cavities and seals any gaps e - An insulation material must be supplied between bottom tracks and foundation. 8 - Form deck profiles can be used successfully for the floor design in commercial and industrial buildings. They can easily span between the floor joists to serve as a formwork for cast in place concrete systems. These are quite favorable since they diminish the need for formwork and supply the tensile strength to concrete. 9 - As the fire ratings of wall covering materials increase everyday, this safety property must be considered in material selection. 10 - Squeaky floors may be a source of complaint if a proper floor design and construction are not fulfilled. Some possible causes of squeaks and vibrations in floors can be specified as follows: a - The movement between the bottom track of the non-bearing interior wall partitions and floor system may cause the squeaks. This generally occurs near the center of the (maximum deflection point) joists. This can be corrected by adding screws between bottom wall track and subfloor where the squeaks occur. b - It can be also a cause of squeaky floors if the joist system is not stiff enough. A proper design will solve the problem. 11 - Spans of about 24 meters are possible for light gauge steel trusses if the spacings are about 0.6 meter on center. But it must be stated that, light gauge profiles can be easily damaged by industrial equipments like cranes and trucks for such a long span length. 12 - Naturaly a protective coating gains a special importance for these thin walled sections. They are untolerable to allow any sectional material lost due to the corrosion which may cause a serious decrease in strength. In general, 3 types of coating methods are defined as to be galvanised, galfan and galvalume.[5] One of these coatings can be used as having the highly protective property of zinc against corrosion. 13 - Horizantal load bracing design must be made against earthquake and wind loads. Cross bracing, lateral bracing with solid bridging or lateral bracing with cold rolled channels can be used for these purpose. 14 - The use of proper type fastening screw is necessary for the full performance of structural members. Externally threaded tapping screws are widely used for this purpose. Self drilling and self piercing types are existing. [3] Screw selection charts of different products can be used in design by considering the given strength values. A suitable rpm tool must be used for different screws sizes.

357

15 - Light gauge steel concept is open to new innovations for different crossectional shapes under the condition that the safe load bearing and structural properties are verified by adequate tests.

CONCLUSIONS 1 - Light gauge sections are gaining popularity in building industry due its various advantages against the other building products. 2 - It can be very economical design when the light gauge steel members are used for some complimentary parts of industrial and commercial buildings such as for purlins, girts, wall studs, floor joists, etc. nearby the heavy load bearing hot rolled sections. 3 - An adequate design method must be followed considering the thin walled nature of the material in accordance with its theory. 4 - Necessary institutions and required cooperation must be realised to develop light gauge steel concept in the related countries.

ACKNOWLEDGEMENTS The author would like to thank Eregli Iron and Steel Works Co. for giving permission for this publication. He would also like to express his gratitudes to "Metsec Building Products Limited, Oldbury, Warley - West Midlands / England" for supplying sectional data for light gauge steel sections.

REFERENCES 1 - Eregli Iron and Steel Works Co. (1996). Steel Market Research, Turkey 2 - Benjamin W. Schafer.(1998) Elastic Buckling Stress and Cold Formed Steel Design, Center For Coldformed Steel Structures, University of Missouri-Rolla 3 - Marge Spencer. (1997), Technical Note On Light Gauge Steel Construction, (565-c) Light Gauge Steel Engineering Association, Nashville, TN, USA 4 - A.I.S.I. (1996). Cold Formed Steel Design Manual 5 - A.I. S.I. (1996). Durability of Cold Formed Steel Framing Members.

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

359

COMPUTER-AIDED DESIGN OF STEEL STRUCTURES IN MATRIX FORMULATION Janusz MURZEWSKI Faculty of Civil Engineering, Politechnika Krakowska 31-155 Krakow, ul.Warszawska 24/L-13, Poland

ABSTRACT A new tool for computer-aided design is presented. It has been called design template. It is different from computer programs. The difference is that design procedures can be easily verified and modified by the designer himself Global loads and influence matrices for unit loads are introduced and new matrices of load effect interaction and combination are defined. Second order theory is taken into account by means of a non-iterative formula and matrix procedures are invented how to discover the most unfavourable extreme cases. Improvements of some unsound clauses of the Eurocodes 1 and 3 are added. KEYWORDS Computer aided design, load combination, load effect interaction, structural analysis, sway frame. INTRODUCTION Three modules create a design template (Murzewski, 1997): • Module #1 - global analysis of a frame or another structural system preceded by load specification, • Module #2 - resistance of cross-sections or members with interaction of internal moments and forces • Module #3 - action effects with consideration of load combination and sway amplification. Every module has 3 parts: A. Line of constants, B. Line of variables, C. Design algorithm. The logical resuh „1" or „0" at the end of the template indicates whether the selected structural element is safe or unsafe. Line D of numerical results may by useflil if errors in data or equations are suspected. Simple templates follow rules of particular design standards. Multiplex templates allow to select an optional method of design at the start. They may be helpful for comparative analyses and calibration of a new method of design. The semi-probabilistic method of partial factors according to European prestandards ECl (Eurocode 1, 1993) and EC3 (Eurocode 3, 1992) is applied in this short paper. Steel columns and beams with bisymmetrical cross-sections are treated in #2 , no more than 5 independent loads are admitted in #3. Application and modification of a template will be easy for anybody who knows the design standards and a mathematical computing program e.g. Mathcad or Mathematica.

360 MODULE # 1 Every independent action Fi is characterised by • global force | Fi | , i = 0, 1, 2, ... which is equal to the sum of all vertical forces for the first load arrangement, but in the case of wind - the sum of horizontal forces, • arrangement of real distributed and concentrated forces with weight factors defined in several variants v = 0, 1,2, ... normed so that the sum of the forces is 100% in the first arrangement, • second-order equivalent forces Hi of the same magnitude as the relative vertical components Pi of the load Fi but horizontal. Load Analysis Densities of structural materials are inserted to the line of constants. So are lengths and other geometrical quantities of structural members which are necessary to determine the dead load. There are also unit values of variable actions taken from standard specifications and modulus of elasticity E necessary for temperature effect evaluation. But the cross-section area A and second moment I of the cross-section are inserted to the line of variables. Their trial values will be improved in an iteration process of calculations. A three-dimensional influence matrix c which transforms applied loads into load effects and the vector of global forces F will be the result of the Module #1 calculations. Elastic or Elastic-Plastic analysis of the structural system is necessary to get the influence matrix c . Elastic Analysis Second order theory may be performed in such a way that normed loads are applied to the perfect frame, without out-of-plumb of columns. First, elastic analysis is performed for real loads; then another elastic analysis is done for equivalent 2-nd order forces. Explicit formulae may be found in design manuals for portal frames and other simple structural systems. They may be copied to the template and they replace formulae lefl; in the computer memory from another design project. The same template may be used in the fiiture and again new formulae will replace the old ones. If a complex structural system cannot be solved by means of simple formulae, a special computer program must be used but the results have to be inserted in matrix form to the environment of design template. Influence Matrix Three-dimensional matrices a and b are components of an influence matrix c. It keeps a complex form because the amplified sway ^ is not known in advance Cijv = aijv+bijv-(t) ,

(1)

aijv - three-dimensional matrix which changes normed loads Fi/|Fi| into load effects, bijv - the same - for the 2-nd order forces Hi. The subscripts mean: i = 0, 1, 2, 3, 4 independent loads, in another notation i = g, q, s, t, w for G = Fo - permanent load, Q = Fi - imposed load, S = F2 - snow, T = F3 - temperature, W = F4 - wind action, respectively;

361 V = 0, 1, 2... - variants and particularly Go - characteristic permanent load, yGi - upper design value, YG2 - lower design value, Fjo - no action of the i-th variable load, Fn , Fi2... - action arrangements j = 0, 1, 2... load effects, in another notation: j = m - for bending moment Ms, j = n - for axial force Ns, j = v - for shear force Vs, j = cj) for column sway and j = 6 - for beam deflection .

777

777

777

777

Figure 1: A frame with imperfection (|)o and the amplified sway ^ MODULE #2 Nominal value of yield strength fy and partial safety factor YM are inserted to the line of constants. There are also free lengths and spacing of stifiFeners necessary for stability verification. Eccentricity Cn and more detailed geometrical characteristics of the cross-section are inserted to the line of variables. So called relative interaction matrices rs = rm for cross-section design and rs = rn for member design will be results of Module #2. Interaction Matrices If bending moment Ms occurs simultaneously with axial force Ns and/or shear force Vs, the ultimate limit state of a cross-section will be reached earlier than in the case of single Ms action, action effect A new concept of equivalent load effect Seff will help to check the ultimate limit state of structural element in the case of interacting moment and forces Seq 0 (compression) or An = min( A, 0,792 Anet fu/fy) if Ns< 0 (tension); Av - shear area equal for Classes 1, 2, 3, approximately Ay « h tw for I sections. It is advisable to differentiate Ay for Class 1, 2 and Class 3, in the same proportion as the section modulus Wc because Av.el _.I-tw/Si/2 _W,,

(6)

w„, 5(9

0

8(91

. n

1/2 2/3 5/6 1 -V

1 5/62/31/2

Figure 2: Piece-wise linearised limit curves for resistance of a cross-section Checking m-v interaction according to point EC3/5.4.7 is useless because it will be always satisfied if only verification EC3/5.4.9 is positive. Secant linearisation is done for v > 1/2 in order to change the standard parabolic curve (2-v - 1)^ into a polygon. The secant linearisation of limit state locus is always conservative. Inaccuracy may be so small as we wish thanks to further fragmentation of the curves. Non-dimensional interaction matrix mje is created by coefficients of linear equations moe-Ms + mie-Ns + mo2-Vs = MR 1 0 0 where the rows and the columns

moi 1 0

0,75 0 0,5

0,75 moi 0,75 0,5

0,45 0 0,9

0,45 moi 0,45 0,9

(6) 0,3 0 1

0,3 moi 0,3 1

j = m, n, V - load effect components for cross-sections, e = mo, mn, mv, nv, vm, vn - interaction ranges at the ultimate limit state;

moi = 0,5 + btf/A but moi ^ 0,75 for Class 1 and 3 and moi= 1 for Class 3 cross-sections.

(7)

363 Vector r of extended cross-section cores is defined and the related matrix rm is derived, 1 Wc/A„ V3-Wc/Av

and

rnije = diag (rj)-mje

(8)

Next, the relative matrix rm is multiplied by the complex influence matrix c and an effective matrix of interaction crm is obtained The effective matrix crm will be multiplied by the global force vector F and it will give the equivalent moment Meq. This will be done in Module #3 . Structural Member Design Member design is different from cross-section design if the axial force is compressive. Particularly, for Ns > 0 , a reduction factor Xm = XLT for beam buckling is determined from EC/5.5.2, Xn = X for column buckling - from EC3/5.5.1 and Xv =ST^/fy is for simple post-critical method of shear buckling according to EC3/5.6.3. Each reduction factor Xj depends on respective slendemess Xj. The so called „non-dimensionar slenderness X ofthe point EC/5.5.1 should be corrected: _

instead of

>^i

X=-

f,

(9)

The reason of correction is that at least equal safety factor should be applied to median resistance of slender columns NR = n^EAJX^ (X -^ oo) as it is for the median plastic resistance of thick columns NR = fmA (X -^ 0). Representative estimate is fm/fy= 1,21 for Fe360 steel (Murzewski, 1989); that is why correction (9) has been proposed. It is introduced already to Polish standard specifications. Since axial force Neq has been defined as the equivalent action effect for members, inverse cores pj [m'^] are defined for member design and buckling factors %] , j = 0, 1,2, are introduced to non-dimensional interaction matrix n . A reduced bending moment kMs is introduced depending on equivalent uniform moment factor PM (EC3/Fig.5.5.3) and a relative interaction matrix rn is derived

Pj^

the rows the columns

A/Wc 1 V3A/Av

1/XLT

->

nje=

0 0

k 1/Xmin 0

0 0

-^

rnje = diag(pj)-nje

(10)

1/Xv

j = m, n, v - the load effect components for members, e = mo, mn,, vo - instability mode interaction ranges.

The interaction curve m-n of column instability is concave therefore secant linearisation would be unsafe. A reduction factor 1 - A has to be evaluated apart for the compression resistance NR. The EC/5.5.4 limit state equation is reformulated in non-dimensional coordinates , m + n=1-A .

(11)

A correction of the EC/5.5.4 equation has been suggested (Murzewski, 1997) and the reduction element A may be evaluated by iterations beginning from a trial value of eccentricity en= MSNR/MRNS ; l^y'Xmin

0,4 • e„

YM-XV

1 + e^

,._...,_,.,, ^^

^, .

,

instead ofthe EC value Ap-,

^'Xr. YM'XV

•mn

(12)

364 Inaccuracy of the revised AJM formula is less than 2%; however, it gives always real results while the mathematically non-homogeneous equation (11) with ARC gives complex numbers both for small and very large values en .

0,5/x 1/x Figure 3: Concave limit curve for a member under eccentric compression MODULE #3 Intended lifetime of the structure tu and initial out-of-plumb angle ^o are inserted to the line of constants A preliminary value of amplified sway (j) is inserted to the line of variables. It shall be determined exactly for the definite load case and load combination which will be known later. Combination Matrix Ferry-Borges and Castanheta (1971) defined a discrete model of load combination where the loads Fi are ordered with respect on their repetition numbers in a reference period tref .There are 2""^ possible combinations. Turkstra (1972) introduced a simplified model where one variable load Fcis dominant and other non-dominant actions are taken in their point-in-time values. Thus n combinations have to be taken into consideration. The Eurocode 1 recommends a similar combination rule where steady values Fi in elementary time periods 6i are taken instead of point in time values. The ECl reference period is tref = 30 years and the combination factors are \\f = 0,7 for imposed loads and \\f = 0,6 for climatic actions (Table 1). Murzewski proved (1996) that the extended Turkstra combination rule delivers lower estimate of combined load effect than the Ferry-Borges and Castanheta's model predicts so the ECl values are unsafe. Anew combination rule has been derived (Murzewski, 1996). It gives safe upper bound estimate of combined action effect. The elements of the new combination matrix are as follows \|/ic = 1 - Ui ln(5O/0i), M/ic = 1 - Ui ln(50/ec) , v|/ic = 1 (if tu=50 years)

if i < c , if i > c , if i = c ,

(13)

where c = 1, 2, ... n - subsequent numbers of the dominant actions, Ui - the Gumbel coefficients of variation for the maximum action in the reference period. The values Ui and elementary periods 0i have been identified so that the same values vj/ic = 0,7 and 0,6 appear. The new matrix v|/ is symmetric (Table 2).

M^ic

Q

s

T W

1 1 0,6 0,6 0,6

TABLE 1 2 3 0,7 0,7 1 0,6 0,6 1 0,6 0.6

4 0,7 0,6 0,6 1

M/jc

Q

s T W

1 1 0,7 0,7 0,7

TABLE 2 2 0,7 1 0,6 0,6

0,7 0,6 1 0,6

0,7 0,6 0,6 1

365 If the intended lifetime tu is different than tref = 5 0 years, the combination factor v|/ii of dominant loads are different than 1, but other combination factors do not change (Murzewski, 1996) V„=l+Uiln(^).

(14)

Extreme Values The 1-st extreme value procedure aims to discover the most unfavourable variant v for every load Fi . More precisely, either maximum or minimum load case for equivalent effect of combined effective load matrix crs is searched; however, not necessarily the extreme effect of each particular load is included because they can have opposite signs, maxcrsie = maxCcrs^^J

and mincrsie = min(crsj^J

V

for i = 0, 1, 2, 3, 4 .

(15)

V

The maxcrsie values usually are non-negative and the mincrsje values are non-positive because zero variants of variable loads can be selected. Matrix product renders the total load effect, maxSce = vj/yFci* maxcrSie

and

minSce = minn/yFci * mincrsie.

(16)

The 2-nd extreme value procedure aims to discover the most unfavourable load effect combination c and interaction range e , MaxS = max(maxS,J ,

MinS = max(maxS,J

c, e

(17)

c, e

and finally we get the absolute load effect Seq = max ( MaxS , |MinS|).

(18)

as a matter of fact it is not necessarily the final result, even if the cross-section is all right, because the sway (j) has to be amplified for the same values v, c , e as the load effect under consideration.. Sway Amplification The Eurocode 3 recommends amplification of the initial imperfection ^o if the frame is classified as a sway frame. Classification whether a frame is sway or non-sway is cumbersome. That is why the computer template will treat every frame as sway fi'ame. Since several years, neither iterations of trial values of sway (j) nor approximate amplification factors (EC3/5.2.6.2) are necessary because explicit solution has been derived (Murzewski, 1992) .

EI-^oSign((|)J + Xa(|)i-vi/YFi

*=

EI-Eb•

(a) Time-Displacement curve

(b) Failure picture of the specimen

Figure 2: Examples of graphics and image data

FUNCTIONS OF THE SYSTEM Because the system is developed on WWW, users need the browser which has Virtual Machine (VM)

381 having Java version over 1.1 (Netscape Communicator 4.5 or Internet Explorer 4.0 is recommended) in order to access the system. The URL of this prototype system is: http://falcon.civil.nagoyau.ac.jp/mdiss/index.html. Some functions and contents of the system will be introduced in the following of this section: Seismic Test Results Figure 3 shows an example to retrieve the results of cyclic loading tests. Once the shape and name of specimen are chosen, the details about the specimen such as dimension can be retrieved. In addition, the available image and graph of the history of the displacement can be visualized in another window such as the U5-2C Load-Displacement as shown in Fig. 3.

wmmm J

Biiefc "fyf

4M)fi

>S£». a;Mat •fc

fo» f*«w*y..,',» »

Choose the shape of specimen Choose the data name

Show the image data and the text data

Show the result and the measure of specimen

Show the Load -Displacement or Time-Displacement curve

Figure 3: Retrieval result of cyclic loading test This database includes a function to draw the graph of the relationship among the major parameters such as the slenderness parameter A, the width-thickness parameter of flange Rf and the maximum displacement //„,,,^. In addition, a function to estimate an approximated equation by means of nonlinear minimum square method is added. The more data will be appended in future, the better information will be able to supply for users. Figs. 4(a) and 4(b) show the cyclic loading test results of unstiffened and stiffened specimens, respectively. The vertical axis in the figure means H^^/Hy and the horizontal axis means the product of R^ and X. This product value in the horizontal axis is found to have a good correlation with the value of H^^/Hy in Suzuki and Usami (1997). The solid curve in the figure represents the mean curve, and the dashed curve represents the lower limit curve that is calculated by the mean value minus a standard deviation.

382 4

4 •

Test Point 1

3

•

>.

M-S Curve!

I

\.\ ^

I

^X^^-l_

£

3

•

Test Point 1 M-S Curve!

*

•

•

TO

E I

*'"--«uM?

0

1 0

0. 35

0.25

0. D5

0.45

(a) Unstiffened specimen

0.15

0.25

0.35

(b) Stiffened specimen

Figure 4: Cyclic loading test results Figures 5 and 6 show the data distributions according to the database and the actual steel pier properties (Nakai et al. 1982), respectively. These data include the slenderness parameter /I, the width-thickness parameter of flange Rf and the plate thickness of flange t. The mean value "m" of the slenderness parameter X from the database as shown in Fig. 5 is larger than the mean of the actual steel pier data. The reason of the difference is that no specimen whose value of X is less than 0.2 has been detected because the specimens whose value of X is less than 0.2 make only slight meaning for examination of buckling. In these two distributions about the values of X, the distributions of are similar while the slenderness parameters are larger than 0.3. On the widththickness parameter Rj, the mean values of both distributions are very near, however the shapes are a little bit different. Comparing these two distributions with respect to flange thickness /, it can be noticed that the tests of about quarter scale model specimen have been performed.

121=0.41 ' B Pseudo-dynamic n Cyclic loading

f

111=0.52

160

m= 5.4

240

•1

120

j

H Pseudo-dynamic E3 Cyclic loading

80 40

L^

n

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.

3

5 7 t (mm)

Figure 5: Distributions of seismic specimen profiles in MDISS

383 m=0.32

100

1 1

80

1I

4)

60 40

m=21

ni=0.49

60

I

I

20

20

60 40

-

20

-

I I II

0 0.2

0.4

0.6

0.2

0.8

1 8 r-.r 0.4 0.6 O.t

1

.1

1

1 >

n

0

10

J

20

30 40 t(mm)

50

Figure 6: Distributions of actual steel piers profiles Ultimate Strength Test Results of Structural Steel Members Figure 7 shows an example of the window of the column database of MDISS using Netscape browser. The horizontal axis of the right hand side figure means the non-dimension buckling parameter of the slenderness parameter A, and the vertical axis represents the non-dimension value Fu/Fy. Here, Fu is the maximum load, and Fy is the measured yield load. Furthermore, the Euler's buckling curve or some other kinds of design curves are also plotted on the graph for the purpose of comparison. The parameter option form in the right side of the window enables to choose the shape of specimens and the non-dimensionalized value of test results. In addition, the system has a function to represent the specimen number whose result point is clicked by mouse on the graph at the bottom right. When the specimen number is inputted in the field of the left side of this graph, the specimen data are represented in detail under the input field. The system enables to compare the test results and some design curves.

^

>^ % ^ ^ j^ijm^^htw

^

^< ^

ili mfi»*{Mimimm^iMtmm&immm,

Input field of the reference number' of data

^ m

//falcon civil "atova^u acj^md

^^II^A ,/2/, 1, = 3 0 0 / ^

^p

= 65/V^

^r

= 150/V^

c

CpG = 286000Q Q = 1.75 +1.05(M, / Afj) + 0.3(A/, I M^^f < 2.3

= 11200

Q = 1.0

The slendemess parameter resulting in the smallest value of/^^ governs.

RpG - 1

/?, = 1.0 - 0.1(1.3 + a, )(0.81 - m) < 1.0

-0.0005«fi-^1.1.0

R^ = 1.0

for non hybrid girders

Tension - flange yield:

Buckling:

Ki-S^R^R,F^

M„ = min(A/„„ A/„2)

Figure 3: Key steps of design - (a) Flexural design requirements

391

No stifTeners required.

-Yes


15 and r/t 15 are usually designed by beam theory. In dependence of geometric parameters calculation using shell theory and a quasi static wind pressure distribution around circumference reveals distinctive differences to the results determined by beam theory. Only on

408

structures with a slendemess ratio h/r < 15 structural analyses considering both geometric and physical non-linearities have been carried out yet, Greiner & Derler (1995). In this geometric range the influence of physical non-linearities is minor. The failure mode is characterized by luff-ward buckling in the upper half of the cylinder. In contrast the shells investigated here (h/r > 15) are determined by the combination of physical an geometrical non-linearities. The analyses presented here are limited to quasi static loading. The wind pressure distribution on the circumference is assumed according to EC 1 Part 2.4 (1994), i.e. pressure on the luff meridian and suction at the flanks as well as in the rear zone. For design reasons a stiffener at the top edge is always arranged. Therefore the investigations are concentrated on those shells. To be able to describe the damage potential of the collapse and to prevent the main failure reason by the design it is necessary not just to determine the collapse load level but also to explain the behaviour of the collapse once the bearing capacity is reached. Non-linear structural analyses are therefore succeeded beyond the first instability point up to the deep post-buckling area.

MERIDIONAL FORCE DISTRIBUTION The fundamental standard EC 3 Part 3.2 (1997) allows to determine the stress resultants using a linear shell theory. The meridional forces calculated this way with the wind pressure distribution according to EC 1 Part 2.4 (1994) deviate considerable to those of the beam theory. Especially at the clamped support essentially higher tension stresses occur at the luff meridian. Fig. 1 shows the distribution of meridional forces along the height at three chosen meridians for a unstiffened shell and a shell ring stiffened at the top with a height h=50m, a slendemess ratio h/r = 30 and a wall-thinness ratio r/t = 123 (t = wall-thickness) at normalized load factor NLF = 1.0, which characterizes the elastic limit load of the beam theory. In the transition range between shell and beam it is useful to mark geometries by the deviation of the meridional forces at the clamped support calculated by shell theory and beam theory. The exaggeration factor of the meridional force of the unstiffened shell at the clamped support at the luff meridian compared to the same force of the beam theory is called as a^ according to Peil & Nolle (1988).

\

o lee meridian A flank meridian 0 luff meridian s=0 s=1 beam theory

Fig. 1 Meridional force distribution along the cylinder height of the unstiffened shell (s=0) an the shell stiffened at the top (s=l) at normalized load factor NLF=1.0, linear shell theory; h/r=30, r/t=123, a,=1.80

/ 4

o lee meridian A flank meridian 0 luff meridian a,= 1.4;r/t=61 a,= 1.8;r/t=123

a, = 2.2:r/t=188

Fig. 2 Meridional force distribution along the cylinder height at normalized load factor NLF=1.0 for varying a^; lin. shell theory, h/r=30, ring stiffened top

409 At the clamped support the meridional forces of the unstiffened shell and the shell stiffened at upper edge are nearly identical. In Fig. 1 a/amounts to 1.8 for the example shell. Remarkably, different shells with the same exaggeration factor a^ also have the same distribution of the meridional forces along the cross section as well as along the height by calculation with the linear shell theory. Therefore the exaggeration factor a^ is well suitable to characterize the mechanical behaviour of different geometries. The squatter the shells and the thinner the walls the more the shells deviate form the conditions of the beam theory, i.e. the lager is a^. The reason for these differences to the stress conditions of the beam theory is the ovalising of the cross section due to the cos2(p-part of the wind pressure on the circumference. If the cross section ovalising is constrained, e.g. at the clamp or due to a ring stiffener, additional meridional forces occur, which decay only gradually along the height of the shell. Therefore the higher a^ the more the ring stiffener at the top causes a significant increase of the meridional forces in the upper section of the shell (Fig. 2). The maximum meridional compression forces are decisive for the stability behaviour. Their maximum value is reached for all geometries at the clamped support cross section, however not always at the lee meridian but moving to the flanks for a^ > 1.6. The meridional force distribution along the height reveals considerable compression forces in the upper section of the shell which deviates from the conditions of the beam theory (compare Fig. 2). For a^ > 1.6 the lee-ward compression maximum moves from the clamped support cross section up to 60% of the shell height for high a^. Even at the luff-ward meridian - the so called tension side - compression forces occur at the upper cylinder half for the top ring stiffened shell. The meridional compression force conditions point on three areas where buckling may occur: the base area, the rear part up to 60% of the shell height and the luff-ward meridian in the upper shell section.

FAILURE MODES General Non-linear analyses reveal three different instability modes in the range of h/r>15 and r/t.

71

D, ^r. (Di+DsjttV

y

L

^

4

71

„,iV

4 , ^ , ^n^Tc' Tt J \-

.

IT,' 2^iL

where Xj = — the location of the within-the-span ties in a third of the span for a given o case. (Fig. 3)

420 W. mm in 2

,•*•'

p,

n

u-

4]

^ \ 3 .••••'""

\ \ \

10

-150

0.2 K kN/cm

0.4

Figure 3: Dependence of deflection on loads: 1,2 - experimental and theoretical curve for a panel; 3,4 - experimental and theoretical curve for a supporting inner sheet between the points of its joining to the basic corrugated metal sheet. Using (3), we calculate N^^from the conditiond3/dN = 0 . But in this case it can result in awkward and inconvenient calculations. It is an easier way to calculate by relation graph 3(N)(Fig.4). 9/9mjax

1.5 1.0

1 — 1

f

/

05 0 -0.5

2

/ /—

-1.0 ^1.5 -2.0

-25 01 K kN/cm

02

Figure 4: Calculations of a critical force in loosing stability by a supporting sheet in the whole constructon: 1 - character of energy barrier; 2 - theoretical value of the critical force; 3 - experimental value of the critical force. We must emphasize the fact that thorough researches of the laminated compressed-bent structures, including those which are deviated from the given paper [5], have shown, that this kind of the local loss of stability by the elements of the whole constructions are a characteristic feature. They actually become prevailing as it was considered up till now.

421 STATIC TESTS Axial Displacement of Joints The static tests of the panels intend to particularize the details of their stress deformation and stress limits. This part of report refers only to the tests of the wall panel supported with an additional sheet material on the interior side. The behaviour of the construction is characterized by special features due to its constituent structure and compound stresses. The first step of the experimental tests was to determine the bearing capacity and axial displacement of discrete ties of the basic layer and the supporting one because the stress deformation are connected with them. The analysis of the tested samples has shown that the main reason for displacement in jonts was crumpling of the metal of the steel profile at the ties. The calculation of the joints made according Building Codes 11-23-81* "Steel Constructions" has shown that the bearing capacity of the joints for the self-tapping screw must be Nj^ =1.358 kN (group I), for the bolt N^ =1.631 kN (group IV). The breaking load obtained during the tests was P=7.125 kN for the samples of group I, P=5.056 kN for the samples of group IV. This exceeds the design load correspondingly in 5 and 3 times. Longtudinal-and-Cross Bending The next stage was testing the panels for determination of their stress deformation and stress limits. The report presents the basic outcomes of the tests of panels in compression with bending. This kind of stress largely corresponds to the conditions in which the load-bearing walls work and also is possible in coverings. As the tests have shown, the static work of the composite constructions of this kind in longtudinal-and-cross bending has its individual specifics. The samples have a real span (L = 3.0 m). Taking into account the uniaxial load, their width was limited by two waves of the steel profile H57-750-0.8. The supporting layer was 4 mm waterproof building plywood. The eccentric of the longtudinal load was 8 cm. The stress limit of the panel at the design load has corresponded to the data obtained by calculations of the panel being a composite construction. The forces between the basic layer and the auxiliary one have been distributed in proportion to their stiffness. The stress limit of the construction in accordance with the calculations has been defined by loosing the total stability of the panel at loads of 1.59 kN per cm. But the tests have shown that loosing of the bearing the capacity can occur at less loads as a result of local buckling of the corrugated steel sheet edges. We should take into account the fact that the stress limit of the panel can be defined by loosing the local stability of the supporting sheet as a result of transient buckling. The value of the critical force in the supporting layer out of 4 mm plywood in transient buckling is about 0.2 kN per cm. in particular. But it satisfies the requisite bearing capacity of the walls for the buildings of the worked out system in most cases. (Fig. 3) The relative deflection of the panel at a normal longitudinal force of 0.1...0.15 kN per cm. is 1/350... 1/900 of the span (Fig. 4) and that meets the requirments of the exhisting norms of Building Codes 11-23-81*, ect. The displacement and the stability of the supporting sheet is satisfactorily described by the worked out methods of calculation to an accuracy of 15...30%.

422 CONCLUSION The walls of Monopanel constructions can be designed for different types of buildings (mainly for resedental and public ones). In cottages which are wide spread in Russia, architects intended to present traditional design for living in the Northen and the Southern parts of the country and for the Western way of living. Thus, the cottages have up to 4 levels and in some cases there is a basement floor (the most expensive and the most labour-consuming variant). There can be a garage, a cellar, a sauna, showers, a water-closet, storerooms in the basement floor, on the ground floor there can be a hall, a kitchen, a sitting-room with windows at difTirent levels, bedrooms; on the first floor there can be a garret, a study and a balcony. The Western type of cottages is a two-story building. On the ground floor there is a kitchen with a dining area, a dining-rom, a sitting-room with a large fire-place and a library. On the first floor there are bedrooms (up to 5). One of them is the host's bedroom with a separate bathroom. The sitting room is of two story height. In additions, there are wide terraces at the level of the ground floor. One terrace is located near entrance, the other one is located in the direction of a garden. We should ephasize, that the given space-planning solutions are only advisible. Monopanel system can meet any custumer's requirements as well as architect's. We should notice that one of the trends in Russia, where Monopanel system is utilized, is atticks of restored or erected buildings made of traditional buildings materials. The above-stated information allows to consider that a new effective system of buildings with light weight bearing and protecting constructions on the basis of thin steel sheets has been worked out. It is based upon thorough researches. The feasibility study assessment, the practical experience of production and the application have shown that the submitted solution is more economical than constructions out of traditional building materials (eg. masonry) in a view of heat losses, material consumtion and labour input. REFERENCES 1. Chistyakov A.M., Samokhina I.A., Ilienko E.A. (12*^-15*^ September 1995). Sandwich panels with rigid phenolic foam cores of low combustibility. Third in Internetional Conference on Sandwich Construction 1,Southhampton, UK, p. 71-79. 2. Rzhanizin A.R. (\9%6).Combined rods and plats, Moscow, p. 261. 3. Tampion F.F. (1988). Metal panels strukches, Leningrad, - p . 14, 165. 4. Umanskiy A.A. (1961). Building mechanics of aircraft, Moscow, p. 84-87. 5. Rass F.V. (1996). The "leap-like" loss of stability of the outer layer of a non-zentrally compressed triple-layer rod. Mechanics of composite materials, Riga, p. 525-530.

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

423

LOCAL BUCKLING OF HOT-ROLLED AND FABRICATED SECTIONS FILLED WITH CONCRETE B. Uy^ and H.D. Wright^ ^School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, AUSTRALIA ^Department of Civil Engineering, The University of Strathclyde, Glasgow, G40NG, UK

ABSTRACT The use of hot-rolled and welded composite sections has increased over the last few decades. These include standard hot-rolled universal and hollow sections together with fabricated sections which are filled with concrete to provide composite action which increases both their stiffness and strength. These types of composite sections have been used mainly for columns and beams as part of overall steel and composite framed structures. The presence of the concrete will restrain local budding of the steel section and two important benefits that can be achieved will be highUghted in this paper. Firstly, the abihty to restrain local budding may allow larger plate slendemess limits to be adopted which can have significant benefits for the cost of fabricated structural members. Secondly, since most hot-rolled sections are designed as compact sections, the effect of the presence of concrete is to allow a greater ductility of the member and a significant increase in the post-local buckling strength of the steel section. This has an important effect on the design of composite frames using advanced analysis. Both of these issues have been considered in this paper by the use of an inelastic finite strip analysis, together with an in depth treatment of residual stress distributions. The finite strip analysis has been undertaken for clamped steel plates to model the restraint offered by the concrete. The analysis is undertaken for a variety of boundary conditions and residual stress distributions in order to allow it's application to a wide selection of possible structural configurations.

KEYWORDS buildings, composite construction, finite strip method, local buckling, steel construction, welded sections

INTRODUCTION Composite structural members have been used in buildings and bridges for well over a century. However, their increased usage recently, particularly in multi-storey building construction has

424

galvanised a plethora of research into identifying the true behaviour of these members. The issue of local and post-local buckling of both hot-rolled and welded sections made composite with concrete will be considered in this paper. In particular, the effects of residual stresses and assumed boundary conditions will be considered. The improved understanding of this phenomenon will have far reaching benefits which include the economical design of structural steelwork coupled with an improved understanding of their behaviour which will be of benefit to the design of these members using advanced structural analysis concepts.

LOCAL BUCKLING OF CONCRETE FILLED STEEL SECTIONS The beneficial effect of the restraint of concrete provided to the local budding of steel sections was first identified by Matsui (1985). In this study he determined the post-local buckling strength of rectangular hollow steel sections filled with concrete to be 50% greater than that for the equivalent hollow sections. An energy method was used by Wright (1993) to determine the local buckling behaviour of steel plates restrained by a rigid medium, which was appUcable for concrete filled steel sections for a large variety of boundary conditions. A finite strip method was developed by Uy and Bradford (1994) to consider the local buckling of cold formed steel plates in profiled composite beams. This analysis was appUcable to plates undergoing non-uniform compression, however it ignored the effect of residual stresses caused through the bending process. Uy et al (1998) augmented this model to include the effects of residual stresses and increased yield stresses. Uy (1998) modified the finite strip method for fabricated welded box sections, to incorporate the effects of residual stresses associated with the welding process. This study showed the effects of residual stresses to be quite significant to the elastic local buckling behaviour, however it was found to be less significant for the results of inelastic local buckUng associated with stocky steel plates. The increase in local buckling strength due to the change in local buckling mode is illustrated for both hollow & filled sections in Figure 1.

(a) Local buckling half-^»ravelength

(a) Local buckling balf-wavelengtb

(i) Hollow Column (ii) Concrete Filled Column Figure 1: Local buckling of box sections, (Uy, 1998)

FINITE STRIP METHOD A semi-analytical finite strip method originally developed by Cheung (1976) was modified to consider elastic and inelastic local buckling of plates with clamped loaded edges by Uy and Bradford (1994). This finite strip method satisfies zero slope and out-of-plane displacement at the ends of the strip by using a sine squared function and thus adequately models the restraint offered by the concrete. This section outUnes the augmentation of this model to incorporate the presence of residual stresses for both the hot-rolled and welded cases. The method has been fully outlined by Uy (1998) and only pertinent changes required for incorporating residual stresses will be oudined herein.

425 Residual Stresses - Flange Outstands Hot-Rolled Sections Hot-rolled sections develop residual stresses due to the different rates of cooling after rolling of various regions of the cross-section. Generally the flange-web junction offers restraint to cooling and tensile residual stresses are developed in the vicinity of this region. In order for axial equilibrium of the sections to be maintained, compressive residual strains then develop in the unsupported flange regions of the section. These residual stress distributions are illustrated in Figure 2 (a). Fabricated Sections Fabricated steel sections which are formed into I sections are welded at the flange web junction using longitudinal fillet welding. This approach also renders residual stresses upon cooUng due to the different thicknesses throughout the section. The residual stress distributions developed in a fabricated I section are shown in Figure 2 (b).

aa

aav

aa

acy

(a) Hot-rolled I-section (b) Fabricated I-section Figure 2: Residual stress distributions Residual Stresses - Box Sections Hot-Rolled Sections Box sections formed by hot rolling, also develop residual stresses upon cooling due to the non-uniform thicknesses around the entire section. Generally restraint to cooUng occurs at the comers of these sections and tensile residual stresses will therefore develop. In order for axial equilibrium to be achieved compressive residual stresses are thus developed at the unsupported regions of the component plates of these sections. This form of stress distribution is illustrated in Figure 3 (a). Fabricated Sections This study will consider those columns fabricated with four component plates, where welding takes place at the vertices of the box. Due to the cooling at the weld region, residual tensile stresses will develop. These tensile stresses must be compensated with compressive stresses in the unsupported regions. A typical idealised stress distribution is illustrated in Figure 3 (b).

(a) Hot-rolled box section (b) Fabricated box section Figure 3: Residual stress distributions of hot rolled box section

426 PARAMETRIC STUDY The parametric study considered the cross-section types of a flange outstand and a box section when concrete infill is included. In this study a single plate was used and the boundary conditions were varied to reflect both simply supported and fixed conditions. For hot-rolled sections, the plates of both the webs and flanges are generally compact and thus a fixed support at the junction is generally a good approximation. However, for welded sections where the component plates may be slender, the restraint offered by these junctions may be more appropriately considered as simply supported. Yield and Plastic Limits The local buckling analysis determines both the local budding stress and strain for the sections investigated. The yield slendemess limits can be determined using the elastic local buckling analysis approach, to ascertain the yield hmits of the sections in question, the stresses at which local buckling took place were monitored from the analyses and the point at which the local buckhng stress was equivalent to the yield stress, was termed the yield slendemess limit. However, in order to determine the plastic plate slendemess limits, one must consider the local buckUng strains of the sections. A plastic slendemess limit, can be defined as a slendemess limit which will allow a plastic hinge to develop. For a composite section a plastic hinge will develop if concrete crushing is allowed to occur in a compression zone. Thus if the steel remains in an unbuckled state up to and beyond this strain, the plate may be considered compact. For normal strength concrete the value of this limiting strain is generally taken to be about 3000 |ie. The analysis undertaken herein assumes that the plastic slendemess Umit is defined when the local buckling strain is twice that of the yield strain of 1500 |Lie for 300 MPa yield steel. The parameters used in the analysis include:

a=^;

p=^.

cy„

y=2^,

5=i^

t

where: b=unsupported plate width; t=plate thickness; 8oi=local buckling strain; ey=yield strain; aoi=local buckling stress; ar=level of compressive residual stress; ay=yield stress. Flange Outstands The analysis for the flange outstand where the section is assumed to have the residual stress characteristics of a hot-rolled section was undertaken for both simple and fixed supports. The results of these analyses are illustrated in Figure 4. The inclusion of residual stresses has a marked effect on the elastic local buckhng stress, however it has a neghgible effect on the local buckhng stress in the inelastic range. Furthermore, for this type of section the level of compressive residual stress has very Httle impact on the local buckhng stress. The level of compressive residual stress increases as the heat affected zone increases in width. As the heat affected zone represents one of tensile residual stress, an increased compressive residual stress, will improve the restraint offered at the flange-web junction. Thus there is very little difference observed in the local buckhng stress for a variation in residual stress. A yield slendemess hmit of 15 and 25 were obtained for the simply supported and fixed boundary conditions respectively.

427 1.25

i

1

ot=0.0 0.75 T

-•-ot=0.3 -»-0=0.5

1 1

0.5 0.25

-*-ot=l.o| 1

0 0

10

20

30

40

50

60

70

P 2 1.5 • K>

0=0.0

0.5

•

^ 10

20

-^0=0.5

s^

00

-•-0=0.3

s>

1 •

30

40

50

-*-a;=l.o|

-—~> 60

70

P

(a) Simply supported junction (b) Fixed junction Figure 4: Non-dimensional buckling stress and strain versus slendemess limit for hot-rolled section The analysis for a flange outstand where the section is assumed to have the residual stress characteristics of a welded section was undertaken for both simple and fixed supports. The results of these analyses are illustrated in Figure 5. The effect of the level of residual stress has a very significant impact on the elastic local buckhng stress. Now, since the level of tensile residual stress in the flangeweb junction is at yield an increase in the heat affected zone causes an increase in the magnitude of compressive residual stress. For this case, this significandy reduces the elastic local buckling stress. A yield slendemess limit of 13 and 22 are obtained for each of these boundary conditions. 1.25 1

I \

0.75 0.5 0.25 0

0=0.0 -•-o^O.l

\

1 0

-•-a?=0.2

10

20

^^^^^;:r:^~— 30

40

50

60

-00 10

—

__—xr^^^^ 1250

1500

1750

2000

Traverse length (mm)

1 -\ S

6 -

"^ ,

—O— columns

.o and ^1 given in Rasmussen & Rondal (1997a), the values of a, p, XQ and Xi shown in Table 1 were obtained (upon making minor rounding adjustments). The column curves derived from these values are compared with the a-, b- and c-curves of the ECCS Recommendations (ECCS, 1978) in Fig. 1. As shown in the figure, the curves are in excellent agreement for A > 0.3. The discrepancy observed for A,-..

1.0

*-, *"'*

0.9 0.8

fo.2(N

50

250

150

350

1

450

Figure 4: Values of the correction factor r| The correction coefficient r| has been fitted in such a way that the use of the plastic hinge method provides the same result, in terms of ultimate load bearing capacity, as the F.E.M. simulation approach (Figure 3). The factor Ti has been found to be mainly a function of the hardening features of the alloy, customarily related to the conventional yield stress fo.2 (Mazzolani, 1995), and can be considered independent of the structural scheme, as shown by Mandara & Mazzolani (1995). The following expression is provided for r| in EC9: Tl =

1 a+bfo.2

2 (fo.2 in N/mm)

(6)

463 De Matteis, Mandara & Mazzolani (1999) have observed that a suitable criterion for the evaluation of T| could be also to assume that the concentrated plasticity model has the same strain energy at collapse as the actual structure. This seems to be more appropriate from the physical point of view for the prediction of the ultimate structural response, in particular when strongly hardening alloys are involved. Such an assumption leads in average to decreasing values of r| compared to EC9 as far as the material strain hardening increases. According to the r|-criterion, the ductility demand can be defined in a conventional way, starting from the available material ductility. The conventional ultimate curvature x^ = 5%^= %5 or 10%e = %io (see Figure 3) can be evaluated on the basis of the nominal ductility properties of the alloy under consideration as a function of the curvature at the elastic limit Xe- For this purpose, the structural alloys are divided into two groups, depending on whether the above mentioned conventional curvature limits are reached or not. Thus, alloys are classified as brittle when the ultimate tensile deformation is sufficient to develop a bending curvature %u equal to Xs- Similarly, they are defined as ductile when this limit is equal at least to %io. The curve of r| values is shown in Figure 4, as a function of the elastic limit stress^.2, of the geometrical shape factor oco, and of the curvature limits %5 and Xio. As values for the ultimate deformation, it is assumed 4% < 8u ^ 8% for brittle alloys and 8u > 8% for ductile alloys.

PREDICTION OF ROTATION CAPACITY For Class 1 sections, as well as for Class 2 and 3 sections when no local buckling occurs, EC9 Annex G supplies some provisions for the modellisation of the section behaviour as well as for the evaluation of the rotation capacity. The elastic and post elastic behaviour of the cross-section may be modelled by means of the moment-curvature relationship, written in the Ramberg-Osgood form (Mazzolani & Piluso, 1995):

%0.2

Mo.2

|^Mo.2j

where M0.2 and X0.2 are the conventional elastic limit values corresponding to the attainment of the proof stress fo.2, m and k are numerical parameters which can be expressed through the following formulas: ^^log[(lO-aiQ)/(5-a5)] \og((XxQla^) '

^^^S-a^ ^lO-aio ma5 majo

^^^

as and aio being the generalised shape factors corresponding to curvature values equal to 5 and 10 times the elastic curvature, respectively. The stable part of the rotation capacity R is defined as the ratio of the plastic rotation at the collapse limit state 0p = 0p - 0e to the elastic limit rotation 0e (Figure 5): ^=^ Oe

= ^^Z^ = ^ - , ©e ©e

(9)

where 0u = is the maximum plastic rotation corresponding to the ultimate curvature %u- The following approximate formula is provided for the evaluation oiR: R = aMj(l + 2ka;jl7Ym + l ) - l

(10)

where m and k have been defined before. The code provides approximate relationships for the evaluation of as and aio according to the material hardening properties and to the geometrical shape factor ao.

464 M/Mo.2

^

Class 1 sections

a M,j •

-

-

\ ^ ^ C l a s s 2, 3, 4 sections

/

Ri

/

j = 2,3,4

' ^' 1

(Ou/Be).

(Ou/ee)^

e/Be

Figure 5: Definition of rotation capacity CONCLUSIVE REMARKS The Eurocode 9 represents the most recent among the european Eurocodes and has been developed during the last years by the activity of the CEN TC 250/SC9 Committee (Chairman Prof. P.M. Mazzolani), giving rise to die most advanced and comprehensive european code in the field of aluminium alloy structures presently available. It is now entering into the conversion phase, leading to the possibility to collect comments and remarks from the member Countries. At the same time, a great amount of research work is presently in progress all over Europe in order to supplement the codification activity with a suitable background literature. Since the plastic design of aluminium structures has been widely desdt with in the code and represents indeed one of its most innovative aspects, the validation need is particularly felt with regard to this subject. In this perspective, the authors hope that this paper contributes to illustrate the main features of the Eurocode provisions in this field and, at the same time, opens a gate to improve its operational aspects. REFERENCES De Matteis, G., Mandara, A. and Mazzolani, P.M. (1999). Interpretative Models for Aluminium Alloy Connections, Proc. of the 4* ICSAS Conference, Helsinki, 20 - 23 June. Mandara A and Mazzolani P.M. (1995). Behavioural Aspects and Ductility Demand of Aluminium Alloy Structures, Proc. of the 3"* ICSAS Conference, Istanbul, 24 - 26 May. Mandara A. (1995). Limit Analysis of-Structures made of Round-House Material, XV CTA Conference, Riva del Garda, October. Mazzolani P.M. (1995). Aluminium Alloy Structures, E & PN SPON, London. Mazzolani, P.M. and Piluso, V. (1995). Prediction of the Rotation Capacity of Aluminium Alloy Beams, Proc. of the 3"^ ICSAS Conference, Istanbul 24 - 26 May. Mazzolani, P.M., Paella, C, Piluso and V., Rizzano G. (1996). Experimental Analysis of Aluminium Alloy SHS-Members subjected to local Buckling under Uniform Compression, Proc. of the 5* Intemational Colloquium on Structural Stability, SSRC Brazilian Session, Rio de Janeiro, August 5-7. Mazzolani, P.M. (1998a): Bemessungsgrundlagen fur Aluminiumkonstruktionen (Design Principles for Aluminium Structures), Stahlbau 67, Sonderheft Aluminium. Mazzolani, P.M. (1998b): Design of Aluminium Structures according to EC9, Proc. of the Nordic Steel Construction Conference, Bergen, Norway, 14-16 September.

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

465

On the Design of New Tram Vehicles Based on the Alusuisse Hybrid Structural System Alois Starlinger and Simon Leutenegger Alusuisse Road & Rail Ltd., Buckhauserstrasse 11, CH-8048 Zuerich, Switzerland

ABSTRACT In close cooperation with the leading suppliers of modem mass transit vehicles a hybrid design concept has been developed for body shells that uses longitudinal hollow aluminum extrusions combined with foam core sandwich panels of aluminum face layers. In addition to the reduction of mass, the hybrid design helps to save production costs. These savings are primarily based on the simplified assembling technique via the new Alusuisse Alugrip comer bolt system and via viscoelastic adhesives. Furthermore, the insulation inherent to foam core sandwich shells and the outstanding impact behavior offer additional benefits to the final customer. Based on the example of a new low floor tram vehicle the essential features of the hybrid stmctural concept are described. In order to improve the fatigue behavior of the vehicles, a special joining technique linking orthogonal aluminum extrusions without any welding operations has been developed (i.e., the Alugrip system). The joining of the sandwich composite elements to the aluminum extrusions is a further challenging design step in the development of a hybrid stmcture. In order to meet all the stmctural criteria complex finite element analyses have been performed. The stmctural integrity of the body shell has been proven in statical testing and in operation.

KEYWORDS Hybrid Design, Composites, Comer Bolt System, Viscoelastic Adhesives, Tram Stmctures, Aluminium Extmsions, Finite Element Analyses, Production Cost Reduction.

466 INTRODUCTION In close cooperation with the leading suppliers of modem mass transit vehicles a hybrid design concept has been developed for body shells that uses longitudinal hollow aluminium extrusions combined with foam core sandwich panels of aluminium face layers. The combination of large scale aluminium extrusions and sandwich composites provides high flexural rigidity combined with low weight. The design of the joints between the aluminium and composites components is one of the most challenging steps when developing hybrid structures. For that reason, state of the art joining techniques like the Alusuisse Alugrip corner bolt system and viscoelastic bonding have been developed. The substitution of traditional design components like locally stiffened metal components (e.g., orthotropic shells with welded ribs) by structural sandwich composites panels has been saving weight as well as manufacturing and assembling costs. These savings are primarily based on the modular design concept and on the simplified assembling technique via the new Alusuisse Alugrip corner bolt system and via viscoelastic adhesives. Based on the modular concept the assembling procedure of the railway structure is simplified. Underframe and side walls may be assembled first. Then, interior components are transported and fixed easily (e.g., seats can be transported through the still open roof structure in the case of railway coaches). In the very last assembling step, the sandwich components are implemented. The application of viscoelastic adhesives allows for an additional insulation between the metal structure and the sandwich components. The sandwich structure itself has a sound dampening behavior towards vibrations. Furthermore, the insulation inherent to foam core sandwich shells and the outstanding impact behaviour offer additional benefits to the final customer. Nevertheless, composites design requires a more sophisticated design procedure to cope with local failure mechanisms, creeping effects, aging effects, temperature dependence, etc. For that reason, complex nonlinear structural analysis models have to be developed even in the early steps of the development process. The finite element method has been proven as one of the most efficient methods to meet that challenge. In order to reduce the time to market all the development process has to be speeded up, especially when engineering composites components. In the following sections the typical development steps required by the new hybrid design concept are outlined. Based on the example of a new low floor tram vehicle the essential features of the hybrid structural concept are described.

ALUSUISSE HYBRID DESIGN CONCEPT The Alusuisse hybrid design concept uses longitudinal hollow aluminium extrusions combined with foam core sandwich panels of aluminium and GRP face layers (see figure 1). Large scale extrusions of aluminium alloys of high corrosion resistance are available up to lengths of 30 metres and more. These extrusions positioned in the longitudinal direction of the hybrid structure provide the major contribution to the overall compressive as well as the bending stiffness. Usually the floor solebar is dimensioned as a large scale extrusion. If the moments of bending inertia are limited due to the low floor design, the roof cantrail section has to provide the stiffness required. In this case the sidewall pillars and their joints to the longitudinal extrusions have to be stiff enough to transfer the forces between the roof and the floor module. For that reason a special joining technique linking orthogonal aluminium extrusions without any welding operations has been developed: the Alugrip comer bolt system. This type of joint is characterized by high static strength as well as by outstanding fatigue performance. Since in a low

467 floor vehicle the center of gravity is relatively high, considerably stiffness and strength of those joints is required to be able to withstand the high loads resulting from lateral acceleration. The floor module has to sustain the passenger loading and to transfer the forces to the longitudinal extrusions. Shallow aluminium extrusions and sandwich panels can be optimized with repect to weight to fulfill those criteria. Especially shaped C-channels in the longitudinal extrusions offer easy clamping possibilities of seat structures. In regions of high longitudinal forces like coupler forces welded submodules can be applied to transfer the loads to the large scale extrusions. Since due to the low floor design heavy have to placed on the top of the hybrid structure the roof panel has to be designed stable enough to withstand those loads. Beyond that, the roof panel considerably contributes to the torsional stiffness of the hybrid vehicle by linking the roof cantrails. Sandwich panels are best suited to meet those demands. The sandwich panels are usually linked by viscoelastic adhesives to the roof cantrails. The dimensioning of these joints represents the most challenging step when designing hybrid structures. The sidewall panels contribute to the overall shear stiffness of the hybrid structure. Due to the large openings required by wide doors the shear stiffness may be considerably reduced. The application of glued side windows can help to compensate that lack in shear stiffness, but usually the width of the sidewall pillars is increased. In addition to the fixation of the seat structures in the floor panels those loads are often directly introduced into the sidewU structure. In that case the sidewall extrusions have to be stiff enough to prevent local bending and buckling. Again, C-channels running in the longitudinal direction of the structure offer easy clamping possibilities. The front cab structures of state-of-the-art light rail vehicles are usually designed in a complex threedimensional shape. Due to the high production costs that complex shape can hardly be manufactured in metal design. For tht reason the front cab modules are preferably designed as GRP-composites sandwich shells. These composites front cab modules are fixed by viscoelastic bonding to the metal structure.

Composites Components Composites components applied in hybrid strcutures usually consist of sandwich shells. Roof and floor panels of dimensions up to 14 meters in length and up to 2.5 meters in width are manufactured as sandwich panels with aluminium face layers and with structural foam core materials. The thickness of the face layers covers a range of 0.8 mm up to 2.0 mm. The aluminium sheets are bonded to the core layer by applying PU adhesives. The front cab modules consist of sandwich shells with GRP face layers from 1 to 6 mm. The complex three-dimensional shapes of the front modules are manufactured with the RTM method. Structural foam materials like AIREX R82 based on PEI (Polyetherimide) and AIREX C70 based on PVC combine excellent stiffness, low density, excellent fatigue performance, high impact strength and high thermal and climate sustainability. PEI materials are proven for their excellent fire rating which even meets stringent underground metro criteria. Beyond that, PEI material can be easily recycled. The sandwich panels provide smooth surfaces. Thus, further surface finishing operations are not necessary any more. The sandwich composites show excellent fatigue and crash behavior. Since

468 sandwich panels are best suited to sustain surface loading like wind pressure, single loads can only be tolerated when the sandwich structure is locally reinforced by stiffener extrusions and inserts. The edge stiffeners should be designed with special Z-shaped flanges to provide sufficient bonding area for the viscoelastic adhesives. Since the composites components can be manufactured in closer tolerances than the metal components of the sidewall, the thickness of the viscoelastic adhesices can be chosen large enough to overcome any geometrical mismatch. Due to the visocelastic bonding the assembly process is simplified and shortened. In the case of local damage the sandwich panels can be easily repaired.

Alugrip Corner Bolt System The Alugrip comer bolt system has been developed as a special joining technique to link orthogonal aluminium extrusions without applying any welding operations (see figure 2). This system is based on friction. The tightening of bolts pushes a bracket against two comer elements which themselves are pressed against the interior surface of extmsion flanges. Due to the relatively high friction forces the comer bolt system is fixed. The stiffness and the stength of the Alugrip system can be adjusted by the pretensioning. Since the torque moment can be applied exactly, the pretension forces can be reproduced readily. The bolts of the Alugrip system are secured by adhesive capsules. The assembly of the Alugrip system can be performed in a quick and easy way, even by unskilled workers provided a certified torque moment can be provided. The stiffness and strength of the Alugrip comer bolt system has been determined in a longterm series of experimental testing. The system is characterized by high static strength as well as by outstanding fatigue performance.

Joining by Viscoelastic Adhesives Since the stmctural approval of the hybrid design concept cmcially depends on the joints between sandwich components and the aluminum stmcture, the application of new joining techniques has become the major challenge in the design development. Viscoelastic PU adhesives have been favored, because they allow for a smooth continous transfer of the interface forces into the sandwich structure. In contrast to rivets and bolts where local force peaks are induced, adhesive layers provide a sufficient interface length in order to reduce the local stress level considerably. Thus, the load transfer into the sandwich structure is readily established without any requirement of additional edge reinforcements of the sandwich shell. Since viscoelastic adhesives require relatively thick adhesive layers, any tolerance mismatch inherent to the linking of large scale modular stmctures can be compensated easily. The thickness of the adhesive layers can be adjusted in a wide range from 5 to 15 mm. Furthermore, viscoelastic adhesives provide outstanding dampening characteristics. Local vibration modes can be restricted. Due to the good insulation features of the PU adhesives any heat flow between the components to be joined is eliminated (i.e., thermal uncoupling). Any different thermal elongation rates between different materials can be buffered in the adhesive layer. Since the stifftiess properties of viscoelastic adhesives are rather low in comparison to the properties of high strength adhesives used in the aerospace industry, the cross section of the adhesive layer as well as the length of the bond have to be optimized to reach the joint stiffness required.

469 Since the mechanical properties of viscoelastic adhesives are strongly nonlinear, the stiffness parameters depend on the temperature, on the loading velocity, on the geometry of the cross section of the adhesive layer, and on the type of loading. For those reasons reliable material data have to be determined for the structural analyses. In cooperation with the leading suppliers of viscoelastic adhesives like SIKA Industry the stiffness and strength parameters have been determined in a long term test program. Thus, the structural behavior of the adhesive layer can be evaluated w^ith respect to static as well as fatigue loading conditions. Special attention is required to check for creeping effects induced by longterm loading, especially at elevated operating temperatures.

FINITE ELEMENT ANALYSIS OF HYBRID STRUCTURES In order to meet all the structural criteria complex finite element analyses have to be performed when developing hyrid structural components. Since lightweight structures are often inclined to undergo large deformations, geometric nonlinear analyses and global instability considerations may be required for the determination of the design limits. Due to the reduced shear stiffness inherent to sandwich structures (as a result of the low density core), shear effects and local failure phenomena have to be incorporated as additional design criteria (see Rammerstorfer et.al. (1994)). In several postprocessing steps the stress tensors in the integration points have to be checked on the following local failure modes: - short wavelength buckling of the face layers - shear buckling of the core - intracell bucking of the face layers (only with honeycomb cores) - cell wall buckling of the cell walls (only with honeycomb cores). In order to determine the local failure limits local bending efffects and multiaxial stress states have tobe taken into account, since the critical buckling stress is considerably influenced by those conditions. For that reason, a method originally developed by Stamm and Witte (1974), extended by Starlinger (1991), is recommended to determine the local failure limits. Since the finite element models of railway vehicles in hybrid design easily lead to complex mesh sizes (i.e., 900000 degrees of freedom and more), special modelling techniques have been derived to reduce the numerical efforts. Joints like adhesive layers and rivets are represented by sets of translational springs. In order to take into account the stiffness contribution of the adhesive layers a special spring model has been developed to: Three translational springs represent a substitute model for the adhesive continuum (see figure 3): spring stiffness for tension/compression: spring stiffness for shear:

kx = ( E . l . b ) / h ky = kz = (G . 1. b) / h

(1) (2)

where E represents the Young's modulus of the adhesive layer, G represents the shear modulus of the adhesive layer, b represents the width of the adhesive layer, h represents the thickness of the adhesive layer, and 1 represents the length of the finite element adjacent to the adhesive layer. In dependence on the loading nonlinear material parameters are defined for the material parameters of the adhesive.

470 CASE STUDY: COMBING LOW FLOOR TRAM In the design of a low floor tram light weight design and low production costs are the decisive key factors. Due to the reduced clearance between the floor module and the rail surface, there is no room available to place heavy aggregates below the floor. For that reason, all the heavy equipment units like the inverter, the air conditioning unit, the braking resistor, etc., have to be placed on the roof As a consequence the center of gravity is higher than in traditional tram vehicles. Due to the elevated position of the center of gravity the joints between the sidewall and the floor module are highly loaded. Only high strength joints like the Alugrip comer bolt system can sustain load levels of that magnitude. In cooperation with DUEWAG Duesseldorf, a member of the SIEMENS group, the low floor tram COMBING has been developed (see figure 4). The base version structure consists of a welded floor module, a sandwich roof panel, sidewall panels with glued windows, and a 3-dimensionally shaped front cab module (see figure 4). Due to the modular concept a large variety of design options can be realized without requiring an additional structural approval of the new layout. The sidewall pillars are linked with the Alugrip comer bolt system to the floor and to the roof cantrail. The sandwich roof panel is glued with viscoelastic adhesives to the roof cantrail. All the heavy aggregates are positioned on the top of the composites roof panel. Due to the foam core the loads had to be introduced by additional profiles to smoothen any local peaks. The welded floor structure helps to transfer the coupler loads to the large scale longitudinal floor extmsions. The stmctural analysis of this tram vehicle proved the stmctural integrity of the new design for the load history specified. The stiffness of the stmcture has been sufficient to fulfill all deformation criteria. In order to evaluate the stmctural integrity of the Alugrip comer bolt system special postprocessing routines were developed that took into account the results of a longterm fatigue test program. Based on the hybrid design concept a prototype and several preseries vehicles have been built. The static strength and stiffness were verified in a stringent stmctural test program. Since the successful completion of this test program the tram vehicles have been operated under service conditions without detecting any major fatigue problems. Due to the successful introduction of this design concept a whole family of COMBING trams is now in development. The good stmctural behavior as well as the low production costs have been proven as key features of the Alusuisse hybrid stmctural system.

References Stamm K. and Witte H. (1974). Sandwichkonstmktionen - Berechnung, Fertigung, Ausftihmng, Springer-Verlag, Berlin, Germany. Starlinger A. (1991). Development of Efficient Sandwich Shell Elements for the Analysis of Sandwich Stmctures Accounting for Large Deformations and Global as well as Local Instabilities. VDI Verlag,Reihe 18, Nr. 93, Diisseldorf, Germany. Rammerstorfer F.G., Starlinger A., Dominger K. (1994). Combined Micro- and Macromechanical Considerations of Layered Composite Shells. International Joumal for Numerical Methods in Engineering, Vol. 37, pp. 2609-2629.

471 Figures

Figure 1: Alusuisse Hybrid Structural System

Figure 2: Alugrip Comer Bolt System

472

Composite Shell Element Adhesive Layer FE Node

Figure 3: Finite Element Modelling of Adhesive Layers

Figure 4: Combino Tram Structure

Session A7 ALUMINIUM AND STAINLESS STEEL STRUCTURES

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

475

THE STRUCTURAL USE OF ALUMINIUM: DESIGN AND APPLICATION Federico M. Mazzolani Professor of Structural Engineering Department of Structural Analysis and Design University of Naples "Federico 11", Italy

ABSTRACT The increasing use of aluminium alloys in the structural engineering field poses several problems to the designer. This paper want to give a contribution in this direction by treating different aspects. First of all, the reason of the choice of these materials in the existing applications covering a w^ide range of structural typologies. The second aspect is related to the codification at international level. Particular emphasis is given to the incoming Eurocode 9, w^hose main innovative features are specifically outlined.

KEYWORDS Aluminium structures , Design, Codification, Eurocode 9.

INTRODUCTION Aluminium alloys w^ere initially used for applications v^here there was virtually no substitutive material: it was the case of the aeronautical industry. Afterwards the use of aluminium alloys rapidly spread into many fields both structural and non structural (windowfi*ames,door furniture, claddings, industrial chemistry, armaments). Since many years, these materials are successfully used in transportation, such as the rail industry (subway coaches, sleeping cars, ), the auto industry (containers for trucks, motorcars, moving cranes, ) and the shipping industry (civil and military hydrofoils, motorboats, sailboats, ). A parallel trend for aluminium alloys consists on their use in the so-called civil engineering structures, CIDA (1972), where these material can be considered as new and they have also to complete with steel, the most widely used metallic material in this field. In the early fifties, when the first building structures made of aluminium alloy appeared in Europe, a big limitation undermined this kind of application: the inadequacy or quite complete absence of recommendations, making the structural design difficult for consulting engineers and controlling Bodies.

476 Nowadays, this gap is going to be completely filled up at European level, starting from the first edition of the ECCS Recommendations issued in 1978, ECCS Committee T2 (1978) and Mazzolani (1980, 1981), and going on at the present time with the preparation of the Eurocode EC9, CEN-TC 250/SC9 (1998) and Mazzolani (1998 a). What probably is still actmg in negative sense is the lack of information about the potential of these materials in structural applications, being their peculiar advantages very seldom considered by structural engineers, who are much more familiar with steel structures, despite the publication of "ad hoc" volumes on the design of aluminium alloy structures, Mazzolani (1995 a). For this reason, a continuous comparison between the two metallic materials, aluminium and steel, is necessary in order to emphasise the specific characteristics and the advantages, as well as sometimes the disadvantages, of aluminium alloys as structural material. It can lead to identify the design criteria which must be followed in order to make the use of aluminium alloys actually competitive with steel in the range of structures.

DEVELOPMENT OF STRUCTURAL APPLICATIONS The success of aluminium alloys as constructional material and the possibility of a competition with steel are based on some prerequisites which are connected to the physical properties and the technological features, Mazzolani (1995 b, 1998 c). Summing-up, the following statements can be assessed: a. Aluminium alloys represent a wide family of constructional materials, whose mechanical properties cover the range offered by the commun mild steels. b. Corrosion resistance normally makes it unnecessary to protect aluminium structures, even in aggressive environments. c. The lightness of the material gives advantages in weight reduction, but it is partially offset by the necessity to reduce deformability, which gives a high susceptibility to instability. d. The material itself is not prone to brittle fracture, but particular attention should be paid to those problems in which high ductility is required. e. The extrusion fabrication process allows to produce individually tailored shapes to be designed (Figure 1). f As connection solution, it is possible to have either bolting, riveting and welding, without any difficulties involved.

Figure 1: Shapes of extruded profiles

477

After these preliminary remarks, it is possible to state that aluminium alloys can be economical, and therefore competitive, in those applications in which full advantage is taken of their above prerequisites. In particular: A. Lightness makes it possible to: simplify the erection phases; transport fiilly prefabricated components; reduce the loads transmitted to foundations; economise energy either during erection and / or in service; reduce the physical labour. B. Corrosion resistance makes it possible to: reduce the maintenance expenses; provide good performance in corrosive environments. C. Functionally of structural shapes, due to the extrusion process, makes it possible to: improve the geometrical properties of the cross-section by designing a shape which simultaneously gives the minimum weight and the highest structural efficiency; obtain stiffened shapes without using built-up sections, thus avoiding welding or bolting; simplify connecting system among different component, thus improving joint details; combine different fiinctions of the structural component, thus achieving a more economical and rational profile. The best fit from the application side can be obtained in some typical cases, which are characterised in getting profit at least of one of the main basic properties: lightness, corrosion resistance and functionally. The main cases of structural applications belong to the following groups: 1. Long-span roof system in which live loads are small compared with dead loads. They include reticular schemes of plane and space structures (Figure 2).

1

W^

%ia«W(i\#

Figure 2: The roof of the Tribune of the football Stadium in Guayaquil, Equador. 2. Structures located in inaccessible places far from the fabrication shop, so the transport economy and ease of erection are of extreme importance (Figure 3). It is the case of prefabricated elements such as electrical transmission towers, stair cases, provisional bridges, which can be carried by helicopter completely assembled. 3. Structures situated in corrosive or humid environments. They cover many types, such as swimming pool roofs, river bridges, hydraulic structures and offshore superstructures.

478

Figure 3: «. A bridge is entirely transported by helicopter. h. A stair-case is erected by crane 4. Structures having moving parts, so that lightness means economy of power under service. They are mainly moving bridges, both for pedestrian and motorcars, as well as the ones rotating on circular pools m the sewage plants. 5. Structures for special purposes, for which maintenance operations are particularly difficult and must be limited, as in case of masts, lighting towers, sign motorway portals, and so on. In the field of large-span one story buildings, the portal frame scheme has been used since the early fifties for industrial buildings which particularly exploited the fiinctionality of extruded shapes. Some applications of plane schemes with large span (50 to 70 m) are made in the field of aircraft hangars, warehouses, airport building, sport-halls. The field of space structures gave rise to a wide number of systems, which particularly utilise the prerequisites of the material technology (Figure 4).

Figure 4: The Sport-hall of Quito, Equador.

479 The most wide applications in the field of large roofing using reticular space structures have been done in South America (Figure 5). Many applications of geodetic domes have been erected in U.S.A. for multipurpose activities with diameters over 100 m (Figure 6). All the main typologies used in steel in the field of bridges have been also experienced in aluminium. It is interesting to observe how the bridge deck can be made of special extruded parts, which are shaped as stiffened plates able to be transversally joined without fasteners. A very convenient application is the one of foot-bridges, which have been successfully built with variable span from 20 to 60 m. In case of motor-bridges, the American experience to use concrete-aluminium composite system has been transferred also in Europe, pointing out a series of new problems which deserved a more wide investigation, Mazzolani & Mandara (1997). A range of great interest is the one of the moving bridges due to the energy consuming reduction in service. They have been built both for motor cars and for pedestrians.

Figure 5: The reticular space structure (60x60) Interamerican Center of San Paolo, Brazil.

Figure 6: The "Spruce Goose" is the world's largest clear-span aluminium dome 415 feet in diameter.

480 A very attractive activity has been developed in the field of suspended bridges. Startmg from some first suspended foot-bridge, this scheme has been successfully revalued m some cases of refurbishment of old suspended bridges of the 19* Century, made of masonry piers and of a combination of iron plus timber as decks. They have been upgraded by using a new aluminium alloy deck (Figure 7). This allowed to obtain the following advantages:

Figure 7: The Groslee bridge with 175 m span; the deck is a composite aluminium-concrete structure. -

due to the lightness of the new deck, the service load can be increased without reinforcing the existing structures (piers and sometimes cables); - due to the corrosion resistance, neither any painting protection has been used nor further maintenance. By means of this system, three bridges have been retrofitted in France, Mazzolani & Mele (1997). Many towers for electrical transmission lines have been erected in Europe. In particular, the delta scheme with three cables allowed the aluminium solution to be more competitive than steel, due to the special shape of the extruded profiles (L, T, Y), stiffened by lips. Lighting towers are frequently used in the urban habitats, specially in France. Two special towers for parabolic antennas have been recently erected in Naples (Figure 8), Mazzolani (1989). Many portal frames for traffic signs on highways have been built in France, Italy, Switzerland and Netherlands. It is quite obvious that these fields are particularly attractive due to the corrosion resistance. As hydraulic applications, mention can be made to: - water pipelines - water reservoirs - sewage plants In the last field, the rotating crane bridges of the cylindrical pools are very often made in aluminium alloys. There are example in Austria, Germany, Great Britain, Italy.

481

Figure 8: The "Information Tower" and the "ENEL Tower" for parabolic antennas, both in Naples. The first application in Italy consisted in 8 bridges of the Turin sewage plant (Figure 9), Mazzolani (1985). In addition to all these possibilities, it seems important to underline that the offshore applications can be considered the main future trend for aluminium alloys. In fact, they offer to this industry enormous benefits under form of cost savings, ease of fabrication and proven performance in hostile environments.

Figure 9: The rotating crane bridge of the Turin sewage plant.

482

Stair towers, mezzanine flooring, access platforms, walkways, gangways, bridges, towers and cable ladder systems can all be constructed in pre-fabricated units for simple assembly offshore or at the fabrication yard (Figure 10).

Figure 10: The superstructures of an offshore platform Mobility and ease of installation are maintained even for larger structural elements, such as link bridges and telescopic bridges. Helidecks have been made by using aluminium alloy since the early seventies, so they have now a fiilly tried experience. They offer large weight reduction, meeting the highest safety standards and providing up to 12% cost saving. Complete crew quarters and utilities modules, from large purpose-built modules to flexible prefabricated units, have been recently developed. The modules may be used singly or assembled in group to form multy-story complexes, linked by central transverse corridors and stair towers (Figure 11).

Figure 11: Five stories crew quarter on an offshore platform

483 The innovative contents of Eurocode 9 Part 1.1 "General Rules" is basically given by the introduction, for the first time in a structural aluminium code, of the analysis of the inelastic behaviour starting from the cross-section up to the structure as a whole. The classification of cross-section has been done on the basis of experimental results, which come from an "ad hoc" research project supported by the main representatives of the European Aluminium Industry, which provided the material for specimens. The output has been the assessment of behavioural classes based on the b/t slendemess ratio, according to an approach qualitatively similar to the one used for steel, but with different extension of behavioural ranges, which have been based on the experimental evidence, Mazzolani et al. (1996 b), and confirmed by numerical simulation, Mazzolani et al. (1997). For members of class 4 (slender sections), the check of local buckling effect is done by means of a new calculation method which is based on the effective thickness concept. Three new buckling curves for slender sections has been assessed considering both heat-treated and work-hardened alloys, together with welded and non-welded shapes, Landolfo & Mazzolani (1995, 1997). The problem of the evaluation of internal actions has been faced by considering several models for the material constitutive law from the simplest to the most sophisticated, which give rise to different degrees of approximation. The global analysis of structural systems in inelastic range (plastic, strain hardening) has been based on a simple method which is similar to the well known method of plastic hinge, but considers the typical parameters of aluminium alloys, like absence of yielding plateau, continuous strain-hardening behaviour, limited ductility of some alloys, Mandara & Mazzolani (1995). The importance of ductility on local and global behaviour of aluminium structures has been emphasised, due to the sometime poor values of ultimate elongation, and a new "ad hoc" method for the evaluation of rotation capacity for members in bending has been set up, Mazzolani & Piluso (1995). For the behaviour of connections, a new classification system has been proposed according to strength, stifftiess and ductility, Mazzolani et al. (1996 a). This approach is now under numerical check, De Matteisetal. (1998). Fire Design is a transversal subject for all Eurocodes dealing with structural materials. For Aluminium Structures it has been codified for the first time according to the general rules which assess the fire resistance on the bases of the three criteria: Resistance (R), Insulation (I) and Integrity (E). As it is well known, aluminium alloys are generally less resistant to high temperatures than steel and reinforced concrete. Nevertheless, by introducing rational risk assessment methods, the analysis of a fire scenario may in some cases result in a more beneficial time-temperature relationship and thus make aluminium more competitive and the thermal properties of aluminium alloys may have a beneficial effect on the temperature development in the structural component, Forsen (1995). The knowledge on the fatigue behaviour of aluminium joints has been consolidated during the last 30 years. In 1992 the ECCS Recommendations on Fatigue Design of Aluminium Alloy Structures have been published, Kosteas (1992), representing a fiindamental bases for the development of Eurocode 9. It was decided to characterise Part 2 of EC9 in general way, giving general rules applicable to all kind of structures under fatigue loading conditions with respect to the limit state of fatigue induced fracture. It has been done contrary to steel, for which Part 2 is dealing with bridges only. Three design methods has been introduced: - Safe life design - Damage tolerant design - Design by testing Five basic groups of detail categories have been considered: - non-welded details in wrought and cast alloys; - welded details on surface of loaded members; - welded details at end connections; - mechanically fastened joints; - adhesively bonded joints.

484 Even if all these kinds of application do not strictly belong to the civil engineering range in the classical sense, it can be noticed that the boundaries of the Building Industry are more and more becoming wilder and less traditional; it is sure that the Aluminium Industry v^ill take a good profit from this new scenario.

DEVELOPMENT OF INTERNATIONAL CODIFICATION Owing to the increasing use of aluminium alloys in construction, several countries have published specifications for the design of aluminium structures. It is due to the efforts of the EGGS Committee for Aluminium Structures and of its working groups that the first edition of the European Recommendations for Aluminium Alloy Structures became available in 1978. These Recommendations represent the first international attempt to unify computational methods for the design of aluminium alloy constructions in civil engineering and in other applications, by using a semiprobabilistic limit state methodology. Immediately after during the eighties the UK (BS 8118), Italian (UNI 8634), Swedish (SVR), French (DTU), German (DIN 4113) and Austrian (ON) specifications have been published or revised. Since 1970 the EGGS Committee on Aluminium Alloy Structures has carried out extensive studies and research, in order to investigate the mechanical properties of materials, their imperfections and their influence on the instability of members. On the basis of these data, for the first time, the aluminium alloy members have been characterized as "industrial bars", in accordance with the current trends of the safety principles in metallic structures, Mazzolani (1995a, c). Among the research programs in this fields, undertaken with the cooperation and support of several European countries, buckling tests on extruded and welded built-up members were carried out at the University of Liege, in cooperation with the University of Naples and the Experimental Institute for Light Metals of Novara, Italy. The use of "ad hoc" simulation methods which allow all the geometrical and mechanical properties, together with their imperfections to be taken into account, has led to satisfactory results in the study of the instability phenomena of columns and beam-columns. The analysis of these experimental and numerical results demonstrated the major differences between the behaviour of steel and aluminium. In particular the buckling curves, valid for extruded and welded bars with different cross-sections and different alloys, have been defined and they have been used in many national and international Codes, including ISO and Eurocode. In the last decade the research reached satisfactory levels also in other fields, such as the local buckling of thin plates and its interaction with the global behaviour of the bar, the instability of two-dimensional elements (plates, stiffened plates, web panels) and the post-buckling problems of cylindrical shells. The time being is characterized by the activity in progress for the preparation of the Eurocode for Aluminium Alloy Structures (EC9), within the Committee CEN-TC 250/SC9.

THE MAIN FEATURES OF EC9 The unavoidable complexity of a code on Aluminium Structures is essentially due to both the nature of the material itself (much more "critical" and less known than steel), which involves the solution of difficult problems and demands careful analysis. In this case the need for the code to be educational as well as informative and not only normative has been particularly determinant, Mazzolani (1998 a). The present edition of the Eurocode 9, GEN-TG250/SC9 (1998), is based on the most recent results which has been achieved in the field of aluminium alloy structures, without ignoring the previous activities developed within EGGS, ECCS-Committee T2 (1978), and in the revision of outstanding codes, like BS 8118, Bulson (1992).

485 The use of finite elements and the guidance on assessment by fracture mechanism have been suggested for stress analysis. The importance of quality control on welding has been particularly emphasised.

REFERENCES Bulson, P.S. (1992). The New British Design Code for Aluminium BS 8118, Proceedings of the 5^^ International Conference on Aluminium Weldments, fNALCO, Munich. CEN-TC250/SC9 (1998). Eurocode n. 9: Design of Aluminium Structures, (pr ENV 1999-1.1; 1.2; 2). CIDA. (1972) Structures in Aluminium. Aluminium. - Verlag, Dusseldorf De Matteis G., Mandara A. and Mazzolani F. M. (1998). Numerical Analysis for T-Stub Aluminium Joints. Proceedings of the f^^ International Conference Engineering Computational Technology, Edinburgh, Scotland. ECCS - Committee T2 (1978). European Recommendations for Aluminium Alloy Structures. Forsen N. E. (1995). Fire Resistance, Chapter 10 in Mazzolani F. M. Aluminium Alloy Structures (second edition), E & FN SPON, an imprint of Chapman & Hall. Kosteas D. (1992). European Recommendation for Fatigue Design of Aluminium Structures, Proceeding of the 5^^ International Conference on Aluminium Weldments, INALCO, Munich. Landolfo R. and Mazzolani F. M. (1995). Different approaches in the design of slender aluminium alloy sections. Proceedings oflCSAS '95, Istanbul Landolfo R. and Mazzolani F. M. (1997). The Background of EC9 design curves for slender sections. Volume in honour of Prof J. Lindner. Mandara A. and Mazzolani F. M. (1995). Behavioural aspects and ductility demand of aluminium alloy structures. Proceedings oflCSAS '95. Mazzolani F. M. (1980). The bases of The European Recommendations for design of aluminium alloy structures. Alluminio n.2. Mazzolani F. M. (1981). European Recommendations for Aluminium Alloy Structures and their comparison with National Standards, Proceedings of the /^ Int. Light Metal Congress, Vienna. Mazzolani F. M. (1985). A new aluminium crane bridge for sewage treatment plants. Proceedings of the 3rd International Conference on Aluminium Weldments. Munich. Mazzolani F. M. (1989). Torre in lega di alluminio per antenne paraboliche (Tower in aluminium alloy for parabolic antennas), Alluminio e Leghe, n. 2.

486 Mazzolani F. M. (1995a). Aluminium Alloy Structures (second edition), E & FN SPON, an imprint of Chapman & Hall, London Mazzolani F. M. (1995b). Globaal overzicht van constructieve aluminium toepassingen in Europa. Aluminium in Beweging, Utrecht. Mazzolani F. M. (1995c). Stability problems of aluminium alloy members: the ECCS methodology, in Structural Stability and Design (edited by S. Kitipomchai^ G.J. Hancock & M. A. Bradford), Balkema, Rotterdam. Mazzolani F. M. and Piluso V. (1995). Prediction of rotation capacity of aluminium alloy beams. Proceedings oflCSAS '95. Mazzolani F. M., De Matteis G. and Mandara A. (1996 a). Classification system for aluminium alloy connections, lABSE Colloquium. Mazzolani, F. M., Faella C , Piluso V. and Rizzano G. (1996 b) Experimental analysis of aluminium alloy SHS-members subjected to local buckling under uniform compression. Proceedings of the 5^^ Int. Colloquium on Structural Stability, SSRC, Brazilian Session, Rio de Janeiro. Mazzolani F. M. and Mandara A. (1997). Plastic Design of Aluminium - Concrete Composite Sections: a Simplified Methods. Proceedings of the International Conference on composite Construction - Conventional and Innovative. Innsbruck, Austria. Mazzolani F. M. and Mele E. (1997) Use of Aluminium Alloys in Retrofitting Ancient Suspension Bridges. Proceedings of the International Conference on composite Construction - Conventional and Innovative. Innsbruck, Austria. Mazzolani F. M. Piluso V. and Rizzano G. V. (1997). Numerical simulation of aluminium stocky hollow members under uniform compression. Proceedings of the 5^^ International Colloquium on Stability and Ductility of Steel Structures, SDSS '97, Nagoya Mazzolani F. M. (1998a). Design of Aluminium Structures according to EC9. Proceedings of the Nordic Steel Construction Conference 98, Bergen, Norway. Mazzolani F. M. (1998b). Bemessungsgrundlagen fur Aluminiumkonstruktionen (Design Principles for Aluminium Structures). Stahlbau Spezial: Aluminium in der Praxis (Aluminium in Practice), Ernst &Sohn. Mazzolani F. M. (1998c). New developments in the design of alummium structures. Proceedings of the S'^^ National Conference on Steel Structures. Thessaloniki. Greece.

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

487

ALUMINIUM BUILDING AND CIVIL ENGINEERING STRUCTURES

^ Department of Structural Design, Eindhoven University of Technology, Eindhoven, The Netherlands ^ Division of Structural Engineering, TNO Building and Construction Research, Delft, The Netherlands

ABSTRACT The design of aluminium structures in building and civil engineering applications is different from designing in traditional materials. This difference is based on the physical and mechanical properties of the material and the freedom of cross-sectional shape provided by the extrusion manufacturing process. This freedom of shape is a key to obtain optimal sections and therefore optimal structures in aluminium. However, structural design is not purely a matter of mechanics; production, maintenance, erection and human preferences may be the decisive factors during the design process. As a case, this paper discusses a bridge design for an international contest in The Netherlands. The design process of the bridge will be discussed, and in addition its influence on the prospects for aluminium structures in The Netherlands.

KEYWORDS Aluminium, structures, design, bridges, cross-section, shape, extrusion

INTRODUCTION In The Netherlands, there is increasing interest in the use of aluminium for structural applications. This is observed not only in an increase in projects, but also in the variety of application fields, the reasons for applying aluminium, and its application in distinctively different designs. The favourable strength to dead weight ratio is the driving force for structural application of aluminium in the transport and aeroplane sector. Combined with the excellent corrosion resistance of the 5000- and 6000- alloys, it prevents the necessity to apply protective layers and resulted in the application of offshore helidecks, crew cabins, silos, tanks and bridges. Nowadays the increase in aluminium applications in structures is due to a combination of aspects like: low dead weight, use of extrusions, durability and low maintenance costs. Additionally aluminium is favourable in special

488 circumstances like: extremely low temperatures, attack of chemicals (machinery) and drink-water purification applications (hygiene). The dead weight of the structure is the main load in large suspensions (roof structures, bridges). Application of aluminium reduces the dead weight of the structure and therefore results in a vast material reduction. A decrease in dead weight may be used to increase the allowable live load, a key aspect in bridge renovation, simplifies assembly and erection, and reduces transport costs. Finally it offers additional cost reduction in moveable structures like bridges and roofs (stadiums) aluminium due to reduced operating costs and smaller operating machinery. Though suitable for structural applications for decades, designers have been reluctant to apply aluminium structures. However, change is unavoidable. The actual building of aluminium structures proved aluminium to be a viable alternative, which leads to acquaintance with the material, to insight in the involved risk and costs, and therefore in trust to build new aluminium structures. Alumimum Material characteristics Density p [kg/m^J 2700 Modulus of elasticity [N/mm^] 70000 Shear modulus [N/mmj| 27000 Poisson's modulus v 030 2310"^ Linear elongation coefficient ai [K*^]

Steel 7800 210000 81000 0.30 23-10"^

Figure 1: Material characteristics

STRUCTURAL ALUMINIUM The design of aluminium structures depends highly on the specific properties of aluminium alloys and the extrusion manufacturing process. Taking optimal advantage of these properties is a key to obtain optimal designs. Alloy Properties Pure aluminium is not suitable for structural applications because of the low values of its mechanical characteristics. However, many alloys are available with a large variety of excellent mechanical and physical qualities. The appropriate alloy depends on the specific application. Generally the advantages of aluminium alloys are: • low density, of approximately one third of steel, see Figure 1; • good strength and toughness properties, also at very low temperatures; • large variety of possible cross-sectional shapes of profiles and connection elements; • good workability; • high corrosion resistance due to a tough oxide-layer; • excellent to recycle without a decrease in quality; Some disadvantages are: • relatively low modulus of elasticity, approximately one third of steel; • low melting point (± 650 °C) and low strength at high temperatures; • more susceptible to fatigue than a comparable steel structure;

489 In most cases the disadvantages of the material can be met by changing the cross-sectional shape. For example, increased susceptibility to local buckling due to the modulus of elasticity can be prevented by extruding stiffeners on plates, while a change of cross-sectional shape can reduce peak stresses to decreases its susceptibility to fatigue. Extrusion The main manufacturing process for sections used in aluminium structures is extrusion. This in addition to more traditional procedures like: rolling, folding, and bending of sheets and sections. The extrusion process consists of putting billets of heated aluminium (± 400 °C) into a container and subsequently pressing these billets through a die by a piston. The shape of the die is therefore decisive for the cross-sectional shape of the profile. This process yields simple open but also very complicated and often closed profiles, see Figure 2.

Figure 2: Complex shapes of extruded aluminium profiles / Maximum dimensions The limitations of the extrusion press and the restrictions of the material characteristics of the applied alloy restrict the engineer in the design of the cross-sectional shape. The capacity of the extrusion press is decisive for the maximal dimensions of cross-sections. Currently the largest diameter for extruded profiles is approximately 500 to 600 millimetres. However, assemblage of sub-elements can yield even larger and more complex profiles.

STRUCTURAL ALUMINIUM DESIGNS Though designing in aluminium resembles designing in steel, essential differences exist. These differences occur in various steps of the design process and are inherent to structural design and the aluminium and extrusion properties. The arbitrarily chosen diagram of Figure 3 describes five different design steps to obtain a product. Though described as a linear process, the actual design process will be cyclic. This means a constant repetition and improvement of previous steps. Know or establish expected environment

Establisli performance requirements

Establish concepts

Figure 3: Design process

Evaluate and optimize concepts

490 In general the field of building and civil engineering is characterised by unique designs. For example, a bridge design will be used only for one specific crossing or, at most, for a limited number. Therefore, only limited funds (and time) are available for the design phase when compared to other engineering fields like aerospace and mechanical engineering. With limited time and funds available for calculation and optimisation, structural designs tend to be traditional, repetitive and conservative, using simplified rules from design recommendations. Only in specific situations is the actual mechanical behaviour considered more thoroughly. These situations occur when, for instance: design rules are inconclusive, loading conditions are severe, the mechanical behaviour is unsure, or architectural demands are high. Aluminium structures exhibit the same characteristics as general structural designs. However, there are some major differences when comparing steel and aluminium structures. First of all, more experience is available with building in steel than there is with building in aluminium. Consequendy, examples and design recommendations for specific applications are less available. Secondly, the freedom of the cross-sectional shape of aluminium extrusions is unparalleled by steel. Thus enabling different shapes to result in more economical designs, but increasing the design costs during the initial phases of the design process. Design criteria 1. Cross-s«c|

web

Y:

2. WeM$ 3. Exttttsioul

^

Concepts 6. Local 1>itq 7. W^Miag S, A&semMyl

I

A

Sum

Design criteria 2

3

4

5

2

I

2

0

2

2

U

B

flanges

C

n A B C

m A B

c

t

^

t^

it.

+^

Figure 4: Design steps As described in a previous section, aluminium material characteristics differ from those of steel and result in different design aspects, like increased buckling or fatigue susceptibility. Though the state of the art Eurocode 9 represents an important upgrade of the design rules, those of steel are still more extensive, more widely applicable, and accepted in more application fields. To increase the complexity, the possibilities of extrusion allow for a wide variety in cross-sectional shapes. Though this allows for improved designs, optimisation of the cross-sectional shape and the addition of functions, it does increase the complexity of the design. Traditionally, design rules are based on standard shapes like I, O, or U, which have been optimised and summarised in handbooks. In addition design rules for the structural behaviour could be tested for those simple profiles. However, complex extruded shapes can not be caught as simple into handbooks or design rules. Thus more design effort is necessary to develop aluminium structures. Evaluation of the design process for an aluminium overhang structure resulted in Figure 4, see Mennink et al (1998).

491 Starting points of cost minimisation and compliance with the building regulations led to a set of eight criteria (e.g. costs, ease of extrusion, erection and maintenance). Concepts were developed which had the possibility to satisfy the requirements and were evaluated and optimised until a final design was obtained. From the figure can be seen that the final shape is much more complex than it would be in a comparable steel structure. As a result, the design, calculation and optimisation of aluminium extruded profiles is far more timeconsuming and difficult than it is for traditional steel structures. However, the resulting structure can result in a more optimal solution, gaining advantage in erection and assembly and reducing weight or improving the instability behaviour. While the design process is cyclic and very complex, the quality of the design is highly dependent on the input, imagination and experience of the designer.

ALUMINIUM BRIDGES During 1997 and 1998 an international contest was held for the building of 58 pedestrian and traffic bridges in the planned residential area of "Leidschenveen", near The Hague, The Netherlands. Within the tradition of Dutch city planning the area is criss-crossed with canals. To accommodate traffic circulation, a bridge system had to be designed for 15 different bridge types, consisting of 45 combined pedestrian/cycling and 13 traffic bridges. Each type has its own width and lane configuration (traffic, cycling and pedestrian lanes) and a different angle (up to 20 degrees) to cross the canals. The design contest consisted of two steps. In the first step a good hundred (international) parties presented an initial design. A jury narrowed the group down in the second step to a selection of five parties, which were to develop a detailed design. One of the selected five was an aluminium alternative initially designed by Jan Brouwer Associates (architect) and TNO (engineering). The paper focuses on the initial and detailed designs, and describes its influence on the future of aluminium structures in The Netherlands. Initial Design The design concept of the aluminium alternative was to develop a lens-shaped bridge built up out of longitudinal beams of extruded elements, see Figure 5. The lens shape accommodates both architectural and constructive demands, providing a slender "wing" to cross the water, which adapts itself to the change in bending moment of the bridge. In addition the concept of longitudinal beams provides a modular system that can be used for any required bridge width. The cross-section of the longitudinal beams makes optimal use of the extrusion possibilities of aluminium. The beam is built up out of two identical extrusions separated by a flat plate. The triangular shape of the extrusions provides a torsion-stiff profile, is able to withstand local point loads, and prevents local buckling. The extruded shapes are designed to accommodate the connections, which highly simplifies the assembly of the bridge elements. The bridge height is varied by means of a variation of the height of the web-plates; thus optimising the use of material. Detailed Design When selected into the next round of the design contest, additional (aluminium) partners were attracted to actually build the bridge: Hydro Aluminium (Norway) and Bayards Aluminium Constructions (The Netherlands). Within this combination, TNO and Hydro Aluminium made the design, while Bayards and Hydro performed production. Though the concept of the bridge remained the same the detailed design differed highly from the initial design, see Figure 5.

492 Bridge deck sections in transverse direction replace the triangular top flanges of the beams (longitudinal direction) of the initial design. According to experimental and numerical work, this kind of profile is well suited to withstand concentrated wheel-loads. By rotating the deck sections from the longitudinal to the transverse direction, their stiffness is used to reduce the number of longitudinal beams.

Figure 5: Initial design, detailed design, and extrusions Two types of deck sections are developed. The first section is used for the pedestrian bridges, while the second section is used for the traffic and combined traffic/pedestrian bridges. The span of the sections (web distance) depends on the design width of the bridge and the type of lane (traffic or pedestrian). In addition flat bottom plates and solid extrusions replace the triangular bottom flanges. The replacement simplifies the production of the beams, while the solids are easier to curve.

Production / Durability Shop production of the aluminium bridges provides an optimal construction climate, allowing for robotic welding and a continuous production process. The bridges are transported by boat from the construction yard to the site, which allows for a bridge width of up to six meters. Therefore, most bridges can be build and transported bodily. Only the largest traffic bridges (width 11.7 meters) are build up out of two pieces. Therefore, combined with the low dead weight of the aluminium bridges, the transport- and erection costs remain limited. The aluminium is appHed unprotected; no coating is needed because of the excellent corrosion resistance of the alloy. However, for aesthetic reasons only, the handrails will be anodised. While no coating is applied, the bridges lack maintenance and therefore result in an environmental conscious design.

Design and Caiculation The design and calculation of the bridges is an interactive process leading to an optimised detailed design. Calculations by hand were performed to estimate dimensions, numerical calculations (with the TNO finite element program DIANA) were performed to establish the influence of shear lag and finally the design was checked according to the codes. The bridges are checked using the design rules of the state of the art Eurocode 9, while applying the load characteristics of the Dutch bridge code NEN 6788 (1995). According to this code the pedestrian and traffic bridges are respectively classified as Class 300 and Class 450, in which the classes account

493 for the loads due to vehicles of 300 kN and 450 kN respectively. In case of the pedestrian bridges this means an accidental ambulance or police car. Major design aspects were the spreading of wheel loads, shear lag of the bottom plate, optimisation of the cross-sectional shape of the extruded profiles, and production and aesthetics of the bridges. The bridges (even the pedestrian) had to be designed for relatively large wheel loads. The design was highly improved by applying a deck-system, which is designed specifically for this phenomenon and is used in e.g. offshore helidecks. 1FEMGV

TNO

1 Model: BRUiS40 LC1: Load easel Element EL.SXX.G SXX Surface: 2 Max/Min on model set Max = .55E8 Mln = -.175E9

H G F E D C B A

'

^

F

** - ' ^ ^ ^ 1 ^ - • ^ - ^ "

n

f

K

H.

"

F"

--^"-^ °

^- c

P ^

40.0 34.3 28.6 22.9 17.1 11.4 5.71 0.00

" P

c p

D

c

P

n

n

Figure 8: Shear lag of the bottom plate Optimisation of the bridge dimensions (minimising weight) led to thin bottom plates with thicknesses of respectively eight and ten millimetres for the class 300 and class 450 bridges. Combined with a curved bottom plate, shear lag provided a major problem, see Figure 8. Shear lag is the effect that only part of a stressed plate is active. If not properly taken into account it results in a reduced stiffness of the bridge and stresses higher than accounted for. Though design rules are available for steel bridges, there are none for aluminium. In addition, production (too little stiffness of the plates during construction) and aesthetics (thin plates tend to hang through) were problematic. Therefore was chosen to increase the plate thickness of the bottom plate, which solved these problems however at the cost of an increase in material. Evaluation Looking back, the aluminium design did not win the contest. Because, in the opinion of the jury, the internal shape of the detailed design is not as "exciting" and more traditional than the initial design was. However, the aluminium design did receive wide publicity and has incited several Dutch cities to build aluminium bridges. Good reasons existed to change the design. First of all, neither time nor funds were available to fully develop the initial design, where experience was available with the bridge deck profile of the detailed design. Secondly, the initial design was unable to cross the channels at an angle. And finally, even the pedestrian bridges had to be designed for large localised wheel loads. The resulting detailed design is much more flexible than the initial design, especially for those bridges not perpendicular to the water. Additionally it is a modular system, the optimal number of longitudinal

494 beams is chosen dependent on the actual dimensions (widths) of the bridges. Only two deck profiles had to be developed for respectively the Class 300 and Class 450 bridges. The detailed design is competitive with traditional materials like steel and concrete, especially when the life-cycle costs are taken into account. Though no cost calculations have been performed on the initial design, it was estimated that the price would have been increased because of increased engineering costs, more material used, and a more difficult fabrication. The initial design seamed, to the jury, more flexible and innovative than the detailed design, but in fact it was less. In addition, the interior of the bridge would have been invisible when built. Summarised, the "human factor" was a key issue in the judgement of the bridges. Other Dutch cities have different opinions, which will result in the building of comparable aluminium bridges in the near future.

CONCLUSIONS As is shown by the designs for the Leidschenveen international bridge contest, aluminium provides an economic and viable alternative for traditional materials like steel and concrete. However, an optimal design is not the only and governing factor for the actual application of aluminium. During the bridge contest there were three key aspects. First of all, the novelty of applying aluminium was a decisive reason to choose the aluminium alternative into the next round. Secondly, the jury found the interior of the bridge of the global design more innovative and "exciting" than the detailed design. Therefore aesthetics was decisive, even though the interior will never be seen again after construction. And finally, of course, it is a money issue. Five designs were developed that emphasized on aesthetics rather than costs. While the offered prices can thus not be compared directly, in the end they were. The initial design costs for 58 bridges were approximately 10 million euros for all five designs. Therefore the perspectives for aluminium structures are excellent. Recent designs have provided substantial attention to the possibility of using them, confidence in their capabilities, and therefore willingness for their use in building and civil engineering applications.

REFERENCES Nederlands Normalisatie Instituut (1995). NEN 6788 Design of Steel Bridges; Basic requirements and design rules, Delft, The Netherlands (in Dutch) Mennink J., Soetens F., Snijder H.H., Hove van Mw. B.W.E.M., and Straalen van U.J. (1998). Design of aluminium cross-sections with complex shapes. Proceedings of the f^ International Conference on Joints in Aluminium, Cambridge, UK Soetens F. and Mennink J. (1998). Design of aluminium cross-sections with complex shapes. Proceedings of the Nordic Steel Construction Conference 98, September 14*-16*, Bergen, Norway CEN (1998). ENV 1999-1-1 Eurocode 9: Design of Aluminium Structures, Brussels, Belgium

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

495

Design of mechanical fasteners for thin walled Aluminium-structures Karlfriedrich Pick Hoogovens Aluminium Bausysteme GmbH Koblenz, Germany

ABSTRACT Normally the design loads are evaluated from test results by statistical means. These results represent the ultimate load bearing capacity because they describe the practice as close as possible. In most cases the values are written in more or less official tables. Nevertheless in some cases it will be necessary to use theoretical calculations to obtain the load bearing capacity i.e. in order to perform a preliminary design or if tabled values or test results are not available. On the basis of former investigations and some additional research projects new formulas have been developed and approved by some hundred test results in order to be able to calculate the load bearing capacity of the fasteners in dependence of their failure mode. It can be shown that a safe design is possible, taking into account also the material of the supporting members as Aluminium, steel or timber as well as other influences like additional bending stresses or flexible substructures. As not only the design is responsible for the safety but also the working up, some examples of an insufficient performance can demonstrate the importance of a proper workmanship. In a modified form the formulae have become part of the relevant chapter of Eurocode 9 .

KEYWORDS Mechanical fasteners, shear, tilting and pull-out, tension, pull-through, self tapping screws, self drilling screws, blind rivets, thin walled structures, supports from Aluminium, steel and timber

INTRODUCTION Mechanical fasteners for the connection of thin walled elements from Aluminium are either self tapping or self drilling screws or blind rivets, which are commonly used to fasten trapezoidal or similar sheets to the substructure or to connect the sheets together (Figure 1). Additional to their load bearing function these fasteners are normally used at the outside of a building where they have to withstand directly atmospherical influences. Therefore they must be resistant against corrosion, carry repeated loads as wind, keep the fastening tight over a couple of years and - as they are applied to fasten wide spread flat elements - it must be able to install them from one side only.

496

^

Self tapping screw

Self drilling screw

Blind rivets

Figure 1: Mechanical fasteners for thin walled Aluminium-structures These functions need special performance, above all concerning the requirement of self sealing of the fastening which will be fulfilled by special washers of metal-elastomer type where the elastomeric is vulcanised to the metal part of the washers. Since the washers are stiff enough they also improve the load bearing capacity of the fastening. It has to be distinguished between round washers which shall be used in any case and special formed washers (following the trapezoidal or sinusoidal shape of the sheeting) which can improve the load bearing capacity and the sealing if they are stiff enough.

DESIGN OF FASTENERS In order to define the load bearing capacity of a fastening system it is essential to know its behaviour under loading and the failure mode. Principally there are two kinds of loading shear forces (rectangular to the axis of the fastener) caused by temperature movement, dead loads in walls, diaphragm actions etc. tension forces (in line with the axis of the fastener) caused by wind-suction, hanging loads or constrained loads by temperature etc. For Aluminium-structures used as wall-cladding and roof-decking elements mainly the tension forces are the more important ones. Failure modes of fastening loaded in shear or tension Due to comprehensive research work has been found out that there has to be distinguished in principle three different failure modes for both types of loading, depending on the material strength of the fasteners or the sheets and their dimensional ratios as well as the dimensions of the washers (Figures 2 and 3).

497 Shear of fastener .^S,\^>.TT>S^'>NS'>N^'^^'^'^^^'^'^

Tilting and pull out of fastener

fnnnmy^mmi Tearing of sheet (sheeting)

Tearing of sheet by yielding (substructure)

Figure 2: Failure under shear loading

Tension failure of fastener

t

^

t

x///y//////^/y^yy'77^7y////////////7Z\

I

9

I

Figure 3: Failure under tension forces

498 An important influence on the failure mode and the load bearing capacity which shall not be neglected is the carefulness in the fabrication-process and the installation of the fasteners. Design of screws Normally the design loads are derived from test resuhs by statistical means. These results represent the ultimate load bearing capacity because they describe the practice as close as possible. In most cases the values are written in more or less official tables. Nevertheless it will be necessary in some cases to use theoretical calculations to obtain the load bearing capacity i.e. in order to perform a preliminary design or in the absence of tabled values or test results. On the basis of former investigations namely in Sweden and some research projects in Germany the following formulas have been developed. They are approved by some hundred tests results. Nevertheless it is important to notice that their validity is limited, and strictly speaking comprises only that range of experience which is verified by tests. In some cases these formulas look similar to those for steel or timber constructions, as they have also been used as a basis. Screws loaded in shear The structure (sheet) near the head of the screw is made of Aluminium and ti is its nominal thickness. The substructure consists of Aluminium or steel and its nominal thickness is tn if tii = t,

: FQ=l,6.Rn,-Vt,^.dG

but

:

< 1,6-RnT

ti -do

if t „ > 2 , 5 - t , : FQ = l , 6 . R n , -

ti • do

if ti < tn < 2,5 • ti FQ ti, tn do Rm

: linear interpolation can be performed

shear force nominal thicknesses of sheet resp. substructure diameter of thread > 5,5 mm the smaller one of the minimum tensile strength of either sheet or substructure

The formulas are valid for thread forming screws from steel or stainless steel with A- or B-thread. Values of tensile strength Rm > 260 N/mm^ shall not be taken into account. Ifti>tii take ti = tn The drilling holes have to be performed according to the recommendations of the fabricators. The supporting member consists of timber FQ < 1,6 • Rm • ti • do F Q = 5,31 s d s = 42,5 • ds^

for4 • d s < s < 8 for 8 • ds < s

if the sheet fails ds

if the substructure fails

499 The minor value is valid.

[N] [mm] do [mm] Rm [N/mm^

FQ ti

S [mm] ds [mm]

dK [mm]

shear force nominal thickness of sheet diameter of thread (5,5 mm < dc < 8 mm) minor tensile strength of sheet, not more than 260 N/mm^ to be taken into account penetration depth into timber support diameter of the part of a timber screw without thread; take ds = 0,5 (do + dK) with dK as the inner diameter of the screw if the shear plane is in the part of the thread nominal diameter of screw

The formulas are valid for thread forming screws or timber screws from steel, stainless steel or Aluminium with A-thread in a substructure made of conifers of a special quality. These formulas have been adapted from the standard for timber structures. It has been found out that thread forming screws which normally are utilised for thin walled metal structures have a higher load bearing capacity than real timber screws. Therefore the values written above are on the safe side. The shear force of the screw can be calculated according to FQ = 0 , 4 . A K

[kN]

where AK [mm^] is the net tensile stress area of the screw. The formula is valid for screws made of steel or stainless steel. Screws loaded in tension The tensile force of a screw can be calculated according to FT

= 0,6 • AK [kN]

where AK [mm^] is the net tensile stress area of the screw made of steel or stainless steel. Pull out of screw The supporting member is made of steel or Aluminium FT = Rm-Vt„^-dG The supporting member is made of timber F T = 6 -SG-do = 72 • do^

FT

for for

4 • dG<SG< 12 • do 12 • dG ^ SG

FT [N] tensile force Rm [N/mm^] ultimate tensile strength of the supporting member tii [mm] nominal thickness of > 0,75 mm (steel) the supporting member > 0,9 mm (Aluminium)

500

do

[mm]

SG [mm]

diameter of thread 6,25 mm < do < 6,5 mm (steel / Aluminium) 5,5 mm < do < 8 mm (timber) penetration depth ofthread into timber support

The formulas are valid for self tapping screws made of steel, stainless steel or Aluminium (only for timber) with A- or B-thread (only for steel or Aluminium). Thicknesses of supporting members of more than 5 mm (steel) respectively 6 mm (Aluminium) as well as tensile strength values of more than 400 N/mm^ (steel) or 250 N/mm^ (Aluminium) shall not be taken into account. Timber shall be of a special quality (i.e. conifer with a strength of more than 155N/mm2). The diameter of the drilling hole shall be in accordance with the manufacturer's recommendations. Pull through Fp = ttL •ttE• aM • 6,5 • ti • Rm • V dD/22 Fp [N] ti [mm] < 1,5 mm Rm [N/mm^] do [mm] >14cm

tensile force nominal thickness of sheet minor tensile strength of sheet diameter of washer

The formula is valid for thread forming screws made of steel, stainless steel and Aluminium. The metal part of the washer must be at least 1 mm thick; if it is made of Aluminium aM = 0,8. The width of the upper (adjacent) flange of the sheet shall not be wider than 200 mm. If the height of the profiled sheet is less than 25 mm the values for Fp have to be reduced by 30%. In the case of fastening the sheets in their lower flanges additional tensile stresses round the screw holes will occur at intermediate supports of multiple span system if the sheets are loaded by uplift loads such as wind suction. These stresses can influence the pull through tensile capacity. It has been found out by special investigations that it is necessary to take these stresses into account for sheets with tensile strength values of Rm > 215 N/mm^. As these stresses at the bottom of the flanges round the screw holes are not known in most cases it has been tried to find an expression for their influence by known parameters. Therefore a reducing factor ttL has been introduced depending on the span L representing the bending moment and by this the additional tensile stresses: Up to a span of 1,5 m there is no reduction (at = 1). At a span of more than 4,5 m the value of a t = 0,5. Between both spans a linear interpolation is possible (aL = 1,25 - L [m] / 6). A further reduction of the load bearing capacity can be caused by the way of performing the connection (i.e. out of the middle of the flange, or light gauge elements as supporting members, or two screws near together). The factor an represents these cases, values can be taken out of the following figure 4. If there are two or more of these cases applicable, only that one with the minor value has to be taken into account and no superposition is necessary.

501 lower flange

upper flange

bu[mml

fastening

aE

^

^ 1,0

I

I

I

bu150:0,7

y^ 0,7

^ •

0,9

^

0,9

1,0

I 1 1 !• ' 0,9

0,9

Figure 4: Reduction factor ae for different types of connections Blind rivets Similar formulas has been developed for blind rivets since the failure mechanism is comparable to that of screws. They can be found, as well as the formulas for screws, in a modified form in part 1.3 of Eurocode9.

SAFETY CONCEPT Wind pressure on a building is not a static but a repeated loading which consists of wind gusts coming out of different directions with different intensity. To make them easier to handle in practice in most load-regulations these repeated loads are transformed by statistical means into static loads, but the origin must be kept in mind. Especially the fasteners at the outside of the building have to withstand these loads because they carry flat constructions of big formats which are directly exposed to the wind. Therefore a certain safety against these repeated loads must be required, and a value of 1,3 is estimated by the building authority as sufficient enough. Former comparative investigations have shown that in the case of failure by pull out or break through under tension forces, or in all cases of shear loaded fasteners, a safety of 1,3 against repeated loading is covered by a factor of 2,0 against static load - the static failure load divided by 2 will always be smaller than the repeated failure load divided by 1,3. If the connection fails by pulling the head of the fastener through the sheet, the failure under repeated load is covered by a (static) safety factor of 3,0. This connection is more sensitive against dynamic loading because of the high stresses round the drilling hole in the sheeting. The comparison to failure under static loading has been investigated because static tests are much more easier and cheaper to carry out than dynamic tests which on one hand represent better the real behaviour of the connection but on the other hand require a rather high amount of testing procedure. This knowledge has to be taken into account in the respective formulas with the result of a uniform safety factor of 2,0. If the concept of partial coefficients is applied, the loading must be multiplied with YF =1,5 and the resistance side must be divided by YM=1,33. Normally the load bearing capacity of connections in thin walled Aluminium-structures influenced by pull through failure is comparatively small. It can be improved remarkably by the application of a special reinforced washer with proper bending stiffness which causes a larger pressure area by the help of which the impact load under the head of the fastener can be distributed to a wider area in the sheet.

502 APPLICATION The clear design of the connection is only one of the important aspects. Another is the installation of the fasteners. Experience shows that there are more mistakes possible than in design procedure, mainly through (see some selected examples in figure 5) wrong combinations of materials which will cause galvanic corrosion (as screws or blindrivets from carbon-steel or containing copper) utilising tools or parts of them which are not fitting to the fastener (the fastener, the tool and the connected parts have to be seen as one system!) setting the fasteners too strong or too soft (which causes leakage and corrosion damage in the substructure) connecting together too thick or too thin components by therefore too short or too long fasteners (which causes leakage and failure).

Drilling hole too wide (reduced load bearing capacity)

Drilling hole too narrow (jamming and twisting off of the screw, shear off of the thread)

Setting the fastener too soft (washer not close enough to surface, connection leaky)

Setting the fastener too tight (washer deformed or destroyed, connection leaky, reduced capacity)

Screw too short (not enough thread screwed in. Drilling flute too short (no penetration of the reduced load bearing capacity) substructure, damage of the thread) Figure 5: Examples for defective installations of screws (selection)

CONCLUSION The application of fastening systems with self tapping or self drilling screws or blind rivets is only safe enough in combining a clear design with a careful working up. The existing products represent a very high quality in performance, the methods of design by test or calculation are available and approved to be on the safe side. It is only necessary to utilise them and combine them with an adequate careftil execution in practice. The urgency to fulfil these requirements is given by the fact that these small connections are directly exposed to the atmosphere where they have to withstand partly very high forces on very huge buildings.

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

503

NUMERICAL MODELLING OF THE BEHAVIOUR OF STAINLESS STEEL MEMBERS IN TESTS G. Sedlacek and H. Stangenberg Institute of Steel Construction, RWTH Aachen 52074 Aachen, Germany

ABSTRACT ENV 1993-1.4 Supplementary rules for stainless steels may be used as a complementary code to the other parts of Eurocode 3 to design structural elements made of stainless steel. The rules specified in ENV 1993-1.4 adopt the system of cross-sectional classification according to b/t- ratios of compressed elements specified for normal structural steels to decide on plastic or elastic analysis or plastic, full elastic or effective resistances though the stress-strain curves for austenitic steels do not exhibit a distinct yielding plateau on the level Rpo,2 as ferritic steels do. Also the rules for column buckling, lateral torsional buckling and plate buckling that all are related to a distinct plastic or elastic or effective material behaviour have been applied for stainless steels due to the fact that experimental evidence was not available to justify more favourable rules. In the ECSC project Development of the use of stainless steel in construction tests have been carried out to investigate the specific behaviour of stainless steel members at the above mentioned limit states. The numerical studies presented in this paper aim at a numerical simulation of the behaviour of the test specimens under the test conditions in order to understand the phenomena and to calibrate the numerical models to the tests. In a second step parameter studies are performed with such calibrated models to develop sufficient data for preparing reliable engineering models for the design. KEYWORDS Stainless steel, bending moment resistance; classification of cross sections; strain hardening; lateral torsional buckling of beams

504 INTRODUCTION In order to improve the design rules in ENV 1993-1 A, Supplementary rules for stainless steels an ECSC project - development of the use of stainless steel in construction - is being carried out with the following partners:

-

The Steel Construction Institute, UK Avesta Sheffield AB Research Foundation and Lulea University of Technology, Sweden Outokumpu Polarit Oy and VTT Building Technology, Finland Ugine S A and CTICM, France Studiengesellschaftfur Stahlanwendung and RWTH Aachen, Germany CSM, Italy

An objective of this project is to find out, in what way the specific stress-strain curve of austenitic steels, that does not exhibit a distinct yielding plateau on the level Rpo,2, influences the ultimate resistances of cross sections with elements in compression and the resistances to buckling and lateral torsional buckling. For these limit state phenomena the rules for cross-sectional classifications and ultimate resistances as given in ENV 1993 for ferritic steels had been adopted in ENV 1993-1.4 because no experimental evidence was available to support more favourable rules for stainless steels. The procedure applied in the project is stepwise. In the first step tests have been carried out to study the phenomena. In the second step numerical simulations of these tests have been performed in order to understand the behaviour observed and to calibrate the numerical model so that it gives reliable results. In the third step parameter studies have been undertaken to investigate the influence of the various parameters and to develop sufficient data for preparing engineering models suitable for improving the code. This paper mainly presents the results of numerical simulations performed for tests carried out by VTT Building Technology. As the project is only expected to be finished by the end of 1999, works are still going on and conclusions will be expected later.

INPUT DATA AND SIMULATION MODEL The test reports with all documentations of the planning and execution of the tests including all necessary data for dimensions, imperfections and material properties as well as all data reported during the tests were provided by VTT. The Finite Element calculations were performed with MARC7.2- Software using the interactive Preand Postprocessor MENTAT32. 8-node thick shell elements including transverse shear effects with 6 geometric degrees of freedom per node were used. The material behaviour was modelled by the true stress strain curve using: ein

=ln(l+eTest)

CTtrue = CTjest ( 1 + Ejest )

(1) (2)

The material was steel grade 1.4301 for which fig.l shows the true stress-strain curve for given samples taken from a given source material. This curve was modelled by an initial elastic part representing the initial modulus of elasticity and a subsequent polygonic approach with 15 segments. The plasticity was specified by the von Mises-yield-surface.

505

350" 300" ^250-

S10A

E ^

200-

~~S10B

i 150-

— S10C

z

0)

S6A

10050-

0

0.5

1

Strain [%]

Figure 1:

True stress-strain behaviour from tested samples of the material (steel grade 1.4301)

The Finite Element models were applied to beams tested in bending and lateral torsional buckling under 4-point loading and to compression loaded in columns. The test specimen for bending was laterally restraint at the points of load introduction to prevent lateral displacements and torsion, whereas the test specimen for the lateral torsional buckling resistance had a larger slendemess and could perform free movements between the supports.

Figure 2:

Schematic presentation of bending resistance tests of I-beams and adequate Finite Element model

Residual stresses were taken into account for columns of small and medium slendemess, whereas for beams they were neglected after sensitivity studies showed that their influence was small. The Finite Element meshes were determined after studies with various types and sizes of elements and mesh configurations in the areas of load introduction.

506

Figure 3:

Schematic test set-up for beams tested in lateral-torsional buckling

0,7 f, 0.7 f , p ^ E^S o.7f,nn

N]0.7f,

k 0,7 f. V D,7f,|D WMm%.

0,7 f.

Figure 4:

Distribution of residual stresses

C O M P A R I S O N OF TEST R E S U L T S W I T H C A L C U L A T I O N S In the following figures 5, 6, 7 representative deformation shapes of the test specimen as calculated are shown together with load-displacement curves from tests and calculations. Tables 1, 2 and 3 give a comparison between the ultimate resistances reached in the tests and calculations that reveals a good complyance between tests and numerical results of the FE-models.

507 Bending resistance of I sections TABLE 1 BENDING RESISTANCE IN TESTS AND NUMERICAL SIMULATIONS 1-160x80-60

1-160x160-60

1-320x160-60

Maximum test load Ftest[kN]

411

687

835

Maximum FE load FpElkNl

413

673

833

1,005

0,980

0,997

specimen

FFE /Ftest

1-160x160-BO (measurement point 2) 700 •

-

600-

^::::^=^^^*=*^*'*~^^-—

500-

zj «

400-

0)

u o u.

300 •

200-

1

100-

—Test

1 0

-*~ Numerical Analysis 5

10

15

20

25

30

Displacement [mm]

Figure 5:

Deformed shape and vertical displacement of specimen I160x160-BO at mid-span

Lateral-torsional buckling of beams TABLE 2 LATERAL TORSIONAL BUCKLING RESISTANCE IN TESTS AND NUMERICAL SIMULATIONS specimen

1-160x80-61

1-160x80-62

1-160x160-61

Maximum test load Ftest[kN]

366

251

578

Maximum FE load FpElkN]

350

238

582

0,956

0,948

1,007

FpE /Ftest

508 1-160x160-61 (measurement point 4) 600

•

•

•

r——..____

500 400

z 0

300 •

0 IL 200

Test

100

" ^ Numerical Analysis -20

0

20

40

60

80

Displacement [mm]

Figure 6:

Deformed shape and horizontal displacement of top flange of specimen I-160x 160-B1

Buckling of columns about the weak axis TABLES FLEXURAL BUCKLING RESISTANCE IN TEST AND NUMERICAL SIMULATION

specimen Maximum test load FtestCkN] Maximum FE load FFE [I >^^. y•^ _ I _ jij,

I I

I I

I I

I I

I I

I I

I I

I r

I 1

I I

I T

I 1

I 1

I T

100

200

300

400

500

600

700

"-"^ I ^ - A j

^^. n ''•'•t*'^"

800

± I

900

Temperature [C] •

RHS 30x30x3 Test result

RHS 40x40x4 Design resistance|

RHS 30x30x3 Design resistance

x

RHS 40x40x4 Test result

Figure 4: The relative values of fire resistance of eccentric loaded columns. The relative values of test values are determined by dividing the test value by design resistance at normal temperature Standard test In addition to the fire tests described above, one test for an unprotected column according to the standard time-temperature curve ISO 834 (1975) was performed. The material of the column was EN 1.4571 with a load of 60 kN, which is equivalent to a load level of 0.42. The column collapsed after 34 minutes' standard fire. Figure 5 shows the measured temperatures of the furnace and the column in a standard fire test. 1000 900 1

800 700

- \

500 300 200 100

1

\

"

1

600 400

1

_^,^-T-^^?^^^^^^ ^"'"^ .^^^-^ j ^ ^ 1 T/7

\ Jr

1/ /

w

if

/

1

\— \

-f

\ 1

1

1

1

;

!

;

:

1

I

1

1

\

\

\

\

\

10

15

20

25

30

1 35

40

Time (min) - ISO 834-

• Average of furnace temperatures |

Figure 5: Fire test according to ISO 834 for column RHS 40x40x4. Material EN 1.4571

530 CONCLUSIONS The concentric and eccentric compression tests were performed for hollow sections ( X = 1.11... 1.52) cold-formed from austenitic stainless steel of type EN 1.4301 and EN 1.4571. Fire resistance tests on columns were performed at VTT Building Technology at the Laboratory of Fire Technology. Based on the comparison of calculated and experimental results, the same formulae may be used to determine the ultimate buckling load under fire action as at normal temperature, only the mechanical properties (modulus of elasticity and yield strength) are reduced at elevated temperatures. The temperature may be assumed to be uniformly distributed throughout the cross-section and along the column. The simple calculation method is valid when the modified slenderness X is below 1.5 at normal temperature. The possibilities of using austenitic stainless steels in load-bearing structures without fire protection seem quite realistic, when the parametric or local fire is adapted or the fire resistance time is 30 minutes or less according to the ISO 834 standard fire-temperature curve. The class requirement of 30 minutes might cause overestimation in normal temperature design. Depending on the slenderness and cross-section dimensions, after 30 minutes' standard fire the load level for material EN 1.4301 0.25.. .0.35 and for material EN 1.4571 can be over 0.40. REFERENCES Ala-Outinen, T. 1996. Fire resistance of austenitic stainless steels Polarit 725 (EN 1.4301) and Polarit 761 (EN 1.4571). Espoo: Technical Research Centre of Finland. 33 p. + app. 30 p. (VTT Research Notes 1760). Ala-Outinen, T. & Oksanen, T. 1997. Stainless steel compression members exposed to fire. Espoo: Technical Research Centre of Finland. 41 p. + app. 30 p. (VTT Research Notes 1864). ENV 1993-1-2. 1995. Eurocode 3: Design of steel structures. Part 1.2: Structural fire design. Brussels: European Committee for Standardization (CEN). 64 p. ENV 1993-1-4. 1996. Eurocode 3: Design of steel structures. Part 1.4: General rules. Supplementary rules for stainless steels. Brussels: European Committee for Standardization (CEN). 55 p. ISO 834. 1975. Fire resistance tests. Element of building construction. Switzerland: International Organization of standardization. 16 p. Outinen, J. & Makelainen, P. 1997. Mechanical properties of austenitic stainless steel Polarit 725 (EN 1.4301) at elevated temperatures. Espoo: Helsinki University of Technology, Steel Structures, Report 1.20 p. Talja, A. & Salmi, P. 1995. Design of stainless steel RHS beams, columns and beam-columns. Espoo: Technical Research Centre of Finland. 51 p. + app. 37 p. (VTT Research Notes 1619).

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

531

TESTS ON COLD-FORMED AND WELDED STAINLESS STEEL MEMBERS Asko Talja VTT Building Technology, P.O. Box 18071, FIN-02044 VTT, Finland

ABSTRACT The design of stainless steel members is frequently based on the rules published for carbon steel, although stainless steel exhibits fundamentally different material stress-strain behaviour. An extensive test series including tests on sheetings, Z-sections restrained by sheeting, welded I-sections and circular hollows sections was carried out. The tests comprised bending tests, web crippling tests, concentric and eccentric axial compression tests, stub column tests and material tests. This paper presents the content of the test series and the methodology of the testing procedures used in the experimental investigations. INTRODUCTION The tests form part of the ECSC-sponsored research project, "Development of the use of stainless steel in construction". The main objective of the structural tests was to provide test data on the effects of the rounded stress-strain curve on the design of stainless steel members. The rounded stress-strain curve affects plastic resistance, buckling of plates and members, and deflections. Material work-hardening during the roll-forming increases the strength, but just like the welding process, it also affects the residual stresses. The test series has been performed for • the development of design expressions for Eurocode 3 Part 1.4 (1996) for stainless steel members, and • calibration of the numerical methods for subsequent studies on section shapes and geometries not yet tested. The shapes of profiles manufactured for the tests are shown Figure 1. This paper describes the test programme and details of the experimental investigation. The results with comparisons of the predicted capacities based on design expressions and finite element analyses will be presented by other partners of the project (e.g. Burgan et al. 1998). To enable direct comparison with the results from finite element analyses and other theoretical solutions, it was highly desirable that the fixings were fully fixed or free and that the true dimensions and material properties were measured. SHEETINGS Three different stainless steel trapezoidal sheeting profiles were tested (Figure 1). The web and flanges of the first sheeting, RAN-45, were without stiffeners and the nominal height of the profile was 44 mm. The second sheeting, RAN-70, of nominal height 66 mm had one stiffener in both flanges but

532

the web was not stiffened. The third sheeting, RAN-113, of nominal height 113 mm had one stiffener in both flanges and two stiffeners in the web. All the sheetings were roll-formed from material grade EN 1.4301 (AISI 304) of 0.6 mm nominal thickness. Single-span tests for both gravity and uplift load and three internal support tests with different spans were carried out for all sheetings (Table 1). In addition, one double-span test was performed for the RAN-70 sheeting.

Figure 1: Cross-sections of the members manufactured for the experiments TABLE 1 SPANS OF THE TESTED MEMBERS

Bending resistance tests (A is pressure load, B is uplift load) Span (m) RAN-45A 2.3 RAN-45B 2.3 RAN-70A 3.1 RAN-70B 3.1 RAN-113A 3.8 RAN-113B 3.8

Double span test

Web crippling tests

RAN-45A RAN-45A RAN-45A RAN-70A RAN-70A RAN-70A

Span(m) 0.28 0.40 0.70 0.80 1.2 2.1

RAN-113A RAN-113A RAN-113A

Span (m) 1.0 1.9 3.7

0

RAN-70

Span (m) 3.1

20

40

Displacement (mm)

Figure 2: Double-span test ofRANlOA: Final deformations at the support and measured displacements at the mid-span and at the support

533

The test set-up and single-span, double-span and internal support testing were carried out according to the testing procedures given in ENV 1993-1-3 (1996) for profiled sheets. An example of the deformations and measured deflections is shown in Figure 2. Z-SECTIONS RESTRAINED BY SHEETING One stainless steel lipped Z-profile of height 175 mm used in structural assembly was tested. The assembly comprised two Z-beams braced with sheeting. The span of the assembly was 3.6 m. The sheeting profile was RAN-45. The Z-profile of 1.5-mm nominal thickness was press-braked and the sheeting of 0.6-mm nominal thickness was roll-formed from material grade 1.4301 (AISI304). Tests were performed for bending due to pressure and uplift load, eccentric compression due to force at the restrained and free flange, and concentric compression alone and together with pressure and uplift load. Altogether 7 tests with different loading conditions were performed (Table 2). The dimensions of the stainless steel members and test assembly are equivalent to those of carbon steel (Kolari and Talja 1994). TABLE 2 SCHEMATIC VIEW OF DIFFERENT LOAD CASES FORZ-SECTIONS

llllUIJIIllUl

\/-v /-\_/~w\.< Ul tion zone. The number of the cycle corresponO ding to the development of the first cracking 100 was dependent on the displacement amplitude of the cyclic test being as much greater as 50 smaller is the displacement amplitude. By in^ , \ , 0 creasing the number of cycles these cracks 0 20 40 60 80 progressively propagated towards the flange DISPLACEMENT (mm) edges up to the complete fracture of one flange which produced the complete loss of the load Figure 2: Monothonic tests carrying capacity. This behaviour gave rise to .

\

.

\

.

s

542 HEA160-C1

HEA160-C2

(amplitude = 6.0 mm)

(amplitude = 9.01

HEA160-C3 (amplitude =

P"-

cTdn Id cnmplrt^rrachii^ =

CTClKl cnnpkornic nrr=233

^^^^

v^^^^^^^g^ ^^^^y\ ,

§ .9

•^ _, -

cycles A«|>4cl«